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ABSTRACT BOOK

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E July 5th - Sunday

9:00-10:15 Plenary Talk 1: Halil Mete Soner, Title: Stochastic target problems, Engineering Auditorium ENGZ50, Chair: Fikri Karaesmen

10:15-10:45 Coffee Break

Room ENG B05 ENG B11 ENG B15 ENG B30

Simulation Control and Learning Stochastic systems Stochastic

Applications

10:45-12:25

Barış Tan - Analytical Methods for Simulation and

Optimization of Dynamic Systems

Gideon Weiss - Parallel Skilled Based Service

Under FCFS and Infinite Bipartite

Matching

Yichuan Ding - Heavy-Traffic

Approximation for Service Systems

Turgay Ayer - Stochastic Modeling

in Healthcare

Behnaz Hosseini Marko Boon Rob Wang Gordon Pang

Yijie Peng Jean Mairesse Jing Dong Nermin Elif Kurt

Andrea Matta Seva Shneer Amy Ward Turgay Ayer

Gideon Weiss Yichuan Ding Mohsen Bayati

12:25-14:00 Lunch

14:00-15:40

Hamed Jalali - Simulation

Esa Hyytia - Control, Scheduling and Optimization of Queueing Systems

Nelly Litvak - Random Graphs: New Models and

Methods

Ali Devin Sezer - Stochastic

Processes and Their Applications

Hamed Jalali Dieter Claeys Lasse Leskelä Stephan Johannes Ankirchner

Ebru Angun Sofia Villar Mariana Olvera-Cravioto Nabil Kazi-Tani

Christelle Vergé Samuli Aalto Pim van der Hoorn Tuan Phung-Duc

Onur Bakır Ianire Taboada Sergey Foss Ali Devin Sezer


16:10-17:50

Jose Blanchet - Perfect Simulation

of Stochastic Networks and

Queues

John Birge - Learning

Mor Harchol-Balter -

Queues with Redundant Jobs

Nilay Argon - Stochastic Modeling

for Healthcare Operations

Jing Dong Matthew Stern Mor Harchol-Balter Ivo Adan

Xinyun Chen Bora Keskin Kristen Gardner Jori Selen

Karthyek Murthy Georgios Fellouris Emina Soljanin Mark Lewis

Enlu Zhou Andrew Li Baris Ata Nilay Tanik Argon

18:00 Welcome Reception (Yapi Kredi Plaza Koc University) and APS Business Meeting (18:15 room CAS Z48)

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EJuly 5th - Sunday

9:00-10:15 Plenary Talk 1: Halil Mete Soner, Title: Stochastic target problems, Engineering Auditorium ENGZ50, Chair: Fikri Karaesmen


Room ENG B16 ENG B29 ENG B18 ENG B21Stochastic

Applications Stochastic Processes Stochastic Control Stochastic networks

10:45-12:25

Mehmood Khan - Maintenance and

Other Service Applications

Isaac Sonin - Stochastic Processes, Optimal Stopping and

Related Problems

Flora Spieksma - Theory of Markov Decision Processes

Kemal Gursoy Doncho Donchev Michael Katehakis

Andrei Sleptchenko Michael Grabchak Pelin G., Canbolat

Mehmood Khan Alexander Slastnikov Bora Keskin

Abhishek Abhishek Isaac Sonin Herman Blok

12:25-14:00 Lunch

14:00-15:40

Ali Eshragh - Stochastic

Applications in Flow Lines Services and Healthcare

Tutorial Speaker: Antonis Economou

Title: Strategic customers in

queueing systems: Bridging observable and unobservable

models, Chair: Barış Ata

Sophie Weiss

Rene Bekker

Raïsa CarmenAli Eshragh


16:10-17:50

Ethem Canakoglu - Risk Management,

Finance and Energy

Alex Belloni - Probability and Optimization

Sandeep Juneja - Stochastic Multi-armed Bandits

Amy Ward - Fork-Join Networks

and Large-Scale Markov Chains

Bikramjit Das David A. Goldberg Nahum Shimkin Erhun Ozkan

Hiroshi Toyoizumi Yehua Wei Rahul Jain Yuan Zhong

Soumyadip Ghosh Sasa Pekec Assaf Zeevi Guodong Pang

Ethem Çanakoğlu Alessandro Arlotto Sandeep Juneja Siddhartha Banerjee

18:00 Welcome Reception (Yapi Kredi Plaza Koc University) and APS Business Meeting (18:15 room CAS Z48)

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E July 6th - Monday

9:00-10:15Plenary Talk 2: Kavita Ramanan, Title: Infinite-dimensional scaling limits of stochastic

networks, Engineering Auditorium ENG Z50, Chair: Mine Çağlar



Stochastic Processes

Finance and Revenue

Management

Stochastic Applications

Stochastic Networks

10:45-12:25

Ad Ridder - Markov Chain Analysis and Applications

Dan Iancu - Revenue

Management and Financial

Considerations

Serhan Ziya - Stochastic Models

in Healthcare

John Hasenbein - Stochastic Network

Approximations

Bernd Heidergott Do Young Yoon Carri Chan Gordon Pang

Haralambie Leahu Yonatan Gur Sarang Deo Itai Gurvich

Jia-Ping Huang Santiago Balseiro Navid Izady Jim Dai

Ad Ridder Dan Iancu Serhan Ziya John Hasenbein

12:25-14:00 Lunch

14:00-15:40

Tahir Hanalioğlu - Renewal-Reward

Processes and Their Applications

Alexandra Chronopoulou -

Stochastic Systems in Finance

Qiong Wang - Inventory, Assembly

and Production Management

Neil Walton / Bert Zwart-

Proportional Resource Allocation

and Scheduling

Basak GeverMathieu

RosenbaumDavid Goldberg Bernardo D’Auria

Ozlem Ardic Mustafa Pinar Erkut Sonmez Philippe Robert

Aslı Bektas KamislikAlexandra

ChronopoulouQiong Wang Jiheng Zhang

Brendan Patch Naveed ChehraziMelda Ormeci

MatogluYuan Zhong


16:10-17:50

Arka Ghosh -Stochastic Systems

Süleyman Özekici - Risk, Price and

Inventory Models

Onno Boxma - Performance Analysis

of Communication Systems

Olvera-Cravioto -Applications and

Algorithms for Random Graphs

Itai Gurvich Caner Canyakmaz Murtuza Ali Abidini Uwe Muehlich

Subhamay Saha Hans Frenk Petra Vis Nelly Litvak

Amy Ward Süleyman Özekici Thomas Meyfroyt David Gamarnik

Asaf Cohen Odysseas Kanavetas Joris Walraevens Ninguyan Chen

18:00 Conference Dinner (Bosphorus Boat Tour)

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EJuly 6th - Monday

9:00-10:15Plenary Talk 2: Kavita Ramanan, Title: Infinite-dimensional scaling limits of stochastic

networks, Engineering Auditorium ENG Z50, Chair: Mine Çağlar



Stochastic SystemsQueues and

ControlMDP


10:45-12:25

Peter Taylor/Sophie Hautphenne -

Matrix Analytic Methods in

Stochastic Models

Moshe Haviv - Decentralized

Priority Selection

Nur Sunar - Stochastic

Control and Optimal Stopping

Applications

Stella Kapodistria- Scheduling: Novel

Techniques and Applications

Michel Mandjes Liron Ravner Nur Sunar Flora Spieksma

Azam Asanjarani Binyamin Oz Canan Ulu Laurens Smit

Peter Taylor Moshe Haviv Semih Sezer Yu Zhang

Sophie Hautphenne Olga Boudali Hayriye Ayhan Stella Kapodistria

12:25-14:00 Lunch

14:00-15:40

Wen Sun - Stochastic Networks

Tutorial Speaker: Assaf Zeevi

Title: Online learning

with moving targets, Chair: Zeynep Akşin

Mohammadreza Aghajani

Rahul Jain

Danielle Tibi

Wen Sun


16:10-17:50

Jim Dai - Asymptotic Analysis

for Stochastic Systems

Bora Keskin - Stochastic Control

Applications / Dynamic Learning

Hayriye Ayhan / Spyros Reveliotis -

Control

Antonio Gomez-Corral -Stochastic Models in Biology

Krishnamurthy Iyer Hamid Nazerzadeh Mark Lewis Laleh Tafakori

Subhonmesh Bose Bangrui Chen Lerzan Örmeci Philippe Robert

Siddhartha Banerjee Eli Gutin Spyros Reveliotis Antonio Gomez-Corral

Jamol Pender Bora Keskin Hayriye Ayhan Tristan Stark

18:00 Conference Dinner (Bosphorus Boat Tour)

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E July 7th - Tuesday

9:00-10:15Plenary Talk 3: John Tsitsiklis, Title: An analysis of sparse, limited flexibility,

service architectures, Engineering Auditorium ENG Z50, Chair: Apostolos Burnetas



Stochastic

networks and processes

Service Applications



10:45-12:25

Yunan Liu - Network

of Queues: Approximations

and Control

Onno Boxma - Service

Applications

Mohsen Bayati - Stochastic and

Statistical Learning Models

Antonio Gomez-Corral -Stochastic

Models in Epidemics

A. Korhan Aras Alex Kuiper David Gamarnik Sophie Hautphenne

Junfei Huang Patrick Metzler Gah-Yi Vahn Martin Lopez Garcia

Shuangchi He Evin Uzun Jacobson Mohsen BayatiNancy Hernandez-

Ceron

Guodong Pang Onno Boxma Murat Erdogdu Piero Manfredi

12:25-14:00 Lunch

14:00-15:40

Alessandro Zocca -Resource Sharing in Communication

Networks

Stella Kapodistria - State-of-the-art Applications of Service Systems

Philippe Chevalier -Stochastic Models

for Operations Management

Jing Dong - Asymptotics

in Queues and Matching Graphs

Neil Walton Ivo Adan Tanja Mlinar Xiaowei Zhang

Fabio Cecchi Jori Selen Nico Vandaele Pascal Moyal

Bart Post Rick Boere Alejandro Lamas Peter Kovacs

Alessandro Zocca Marko Boon Philippe Chevalier Jing Dong


16:10-17:50

Andreas Lopker - Stochastic Processes

Philippe Afeche - Strategic Behavior

in Queues

Tolga Tezcan - Application

of Stochastic Processes

Mustafa Hayri Tongarlak - Sustainable Operations

Management

Michael Chek Hin Choi Xiaoshan Peng Burak Büke Damla Usar

Peter Straka John Yao Kenan Arifoğlu Özge İşleğen

Geoffrey Decrouez Nahum Shimkin Tolga TezcanMustafa Hayri

Tongarlak

Andreas Lopker Philipp Afeche Sasha Stolyar Nicola Secomandi

18:00 Free Night

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EJuly 7th - Tuesday

9:00-10:15Plenary Talk 3: John Tsitsiklis, Title: An analysis of sparse, limited flexibility,

service architectures, Engineering Auditorium ENG Z50, Chair: Apostolos Burnetas



Stochastic Control New Directions in Applied Probability Simulation Queues – Limit

Theorems

10:45-12:25

Alan Scheller-Wolff -

Markov Decision Processes

David Goldberg - New Directions in

Applied Probability

Ebru Angun - Stochastic

Optimization

Bruno Gaujal - When Limits Help

in Large Scale Stochastic Systems

Opher Baron Yashodhan Kanoria Guzin Bayraksan Jonatha Anselmi

Elvin CobanSiddhartha Banerjee

Ehsan Mehdad Ana Busic

Sherwin Doroudi Neil Walton Raghu PasupathyPanayotis

Mertikopoulos

Emre Nadar Jamol Pender Saul Toscano Benny Van Houdt

12:25-14:00 Lunch

14:00-15:40

Sheng Qiang - Price and Portfolio

Optimization

Tutorial Speaker: Dan Adelman

Title: An Overview of

Approximate Dynamic

Programming, Chair: Pelin G.

Canbolat

Ioannis Baltas

Efe Çötelioğlu

Sheng Qiang

Andrew Saweljew


16:10-17:50

Bora Çekyay - Sochastic Control

and Network Problems

Alessandro Arlotto -

New Directions in Applied Probability

Shane Henderson - Simulation

Optimization

Itai Gurvich - Limit

Theorems and Approximations

Ningyuan Chen Alessandro Arlotto Enlu Zhou David Goldberg

Bora Çekyay Yash Kanoria L. Jeff Hong Junfei Huang

Jaron Sanders Tauhid Zaman Raghu Pasupathy Yunan Liu

David Lipshutz Kuang Xu Susan R. Hunter Pascal Moyal

18:00 Free Night

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July 8th - Wednesday


Service

ApplicationsQueues


Stochastic Models

9:00-10:40

Douglas Down - Queues with

Computer Applications

Barış Balcıoğlu - New Perspectives

in Queueing Models

Özge Büyükdağlı -Inventory and

Production

Peter Taylor - Markov Modulated Stochastic Models

Esa HyytiaApostolos Burnetas

Zeynep Turgay Guy Latouche

Maialen Larranaga Peter Jacko Rim Essifi Eleonora Deiana

Jacob Leshno Raik Stolletz Sinem Özkan Matthieu Simon

Douglas Down Barış Balcıoğlu Özge Büyükdağlı Sarah Dendievel


11:00-12:40

Jiheng Zhang- Applications of Approximate

Queueing Analysis

Antonis Economou

-Optimization and Strategic Behavior

in Queueing

Balaji Prabhakar -Algorithms and Big

Data Methods in some Real World

Systems

Michel Mandjes - Mixed Poisson Models, and

Related Queueing Systems

Junfei Huang Athanasia Manou Shiva Theja Britt Mathijsen

Rouba Ibrahim Opher Baron Chinmoy Mandayam Mariska Heemskerk

Zhenghua Long Sherwin Doroudi Balaji Prabhakar Marijn Jansen

Bert Zwart Yoav Kerner Damon Wischik Michel Mandjes

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EJuly 8th - Wednesday


Stochastic Networks

Healthcare Applications

Queues Performance

Stochastic Processes

9:00-10:40

Guodong Pang - Asymptotics in Stochastic

Networks

Burhaneddin Sandikci -Stochastic

Models in Healthcare

Harsha Honnappa - Queues

Performance

Masayiko Miyazawa - Queues and Stochastic Processes

Sergey Foss Peter van de Ven Bart Kamphorst Dwi Ertiningsih

Arka Ghosh Mucahit Cevik Ioannis Dimitriou Masakiyo Miyazawa

Yoni Nazarathy Tolga Tezcan Justus Arne Schwarz Nurgul Okur Bekar

Justin DeanSaloumeh

SadeghzadehHarsha Honnappa Farida Kachapova

10:40-11:00

11:00-12:40

Marko Boon - Single-Server

Networks: Applications, Analysis and Asymptotics

Tutorial Speaker: Michael Katehakis

Title: Multi-armed bandit problems, models

and algorithms, Chair: Lerzan

Örmeci

Stella Kapodistria

Jan-Pieter Dorsman

Erik Winands

Maria Remerova

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RS Halil Mete Soner, ETH Zurich, Switzerland

Halil Mete Soner is a professor of mathematics at the Swiss Federal Institute of Technology in Zurich (Eidgenossische Technische Hochschule Zurich). He holds a Senior Chair at the Swiss Finance Institute. His research is concentrated on nonlinear analysis in partial differential equations, stochastic processes and mathematical finance. He co-authored a book, with Wendell Fleming, on viscosity solutions and stochastic control; Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, 1993 (second edition in 2005).

John N. Tsitsiklis, Massachusetts Institute of Technology, US

John N. Tsitsiklis is a Clarence J Lebel Professor of Electrical Engineering, with the Department of Electrical Engineering and Computer Science (EECS) at MIT, affiliated with the Laboratory for Information and Decision Systems (LIDS) and the Operations Research Center (ORC). He is currently serving as the Chair of the Council of the Harokopio University in Greece. His research interests are in the fields of systems, optimization, control, and operations research. He is a coauthor of Parallel and Distributed Computation: Numerical Methods (1989, with D. Bertsekas), Neuro-Dynamic Programming (1996, with D. Bertsekas), Introduction to Linear Optimization (1997, with D. Bertsimas), and Introduction to Probability (1st ed. 2002, 2nd. ed. 2008, with D. Bertsekas).

Kavita Ramanan, Brown University, US

Kavita Ramanan is a professor at the Division of Applied Mathematics at Brown University. She is a recipient of the Erlang Prize of the INFORMS Applied Probability Society and a fellow of the IMS (Institute for Mathematics and Statistics). Her research lies in the area of probability theory, stochastic processes and their applications, including stochastic analysis, large deviations, Gibbs measures, measure-valued processes and applications to stochastic networks.

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PLENARIES AND TUTORIALS

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Lk 1 July 5th - Sunday 9:00-10:15

Room: ENG Z50

Plenary Talk 1

STOCHASTIC TARGET PROBLEMS

Speaker: Halil Mete Soner

Chair: Fikri Karaesmen

In a stochastic target problem, the controller tries to steer a stochastic state process into a prescribed target set with certainty. The state is assumed to follow stochastic dynamics while the target is deterministic and this miss-match renders the problem difficult and without any correlations between the noise processes it is not possible to achieve this goal. However, when there are such degeneracies and/or correlations of the noise process, one exploits them to determine the initial positions from which this goal is feasible. These problems appear naturally in several applications in quantitative finance providing robust hedging strategies. As a con-venient solution technique, we use the geometric dynamic programming principle that we will describe in this talk. Then, this characterization of the reachability sets will be discussed in several examples.

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TUTO

RIA

L 1July 5th - Sunday 14:00-15:40

Room: ENG B29

Tutorial 1

STRATEGIC CUSTOMERS IN QUEUEING SYSTEMS: BRIDGING OBSERVABLE AND UNOBSERVABLE MODELS

Speaker: Antonis Economou

Chair: Barış Ata

The economic analysis of service systems with strategic customers is a fast-growing area in the queueing lite-rature, that complements the earlier studies that concerned the performance evaluation, the design and the dynamic control of systems with passive (i.e., non-deciding) customers.

In such studies, a certain reward-cost structure is imposed on a queueing system that quantifies the custo-mers’ desire for service and their dislike for waiting. The customers are allowed to make decisions as to whet-her to join or balk, to stay or renege, to buy priority or not etc. Then, the collective behavior of the customers is analyzed as a game among the potential customers and the basic problem is to determine the correspon-ding symmetric strategy equilibrium profiles.

In the literature, there are basically two kinds of models: Observable and unobservable. In the observable models, the customers make their decisions after observing the state of the service system, whereas in the unobservable models the customers decide, relying only on their knowledge of the system parameters, wit-hout observing it.

In the present talk, we will focus on models that lie in between these two extreme information cases. After a very brief introduction in the area of strategic queueing, we will present several ideas that bridge the obser-vable and the unobservable models, such as partially observable models, models with a mixture of observing and unobserving customers and models with delayed observations.

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July 6th - Monday 09:00-10:15 Room: ENG Z50

Plenary Talk 2

INFINITE-DIMENSIONAL SCALING LIMITS OF STOCHASTIC NETWORKS

Speaker: Kavita Ramanan

Chair: Mine Çağlar

Many stochastic networks are too complex to be amenable to an exact analysis. An established framework is instead to obtain tractable approximations that provide qualitative insight into the dynamics and whose accu-racy can be rigorously justified in a suitable (asymptotic) regime of network parameters via limit theorems for suitably scaled state processes. It turns out that in many cases, to establish scaling limits it is fruitful to use an infinite-dimensional Markovian representation of the state dynamics. We illustrate this in the context of two different classes of models: randomized load balancing models in the presence of general service times, and single-server networks with scheduling policies (such as Earliest-Deadline-First or Shortest-Remaining-Pro-cessing-Time) that employ a continuous parameter to prioritize the service of different jobs. Although the nature of the infinite-dimensional representations is rather different for the two classes of models, we show that they both lead to tractable scaling limits that can be used to identify interesting (and sometimes count-er-intuitive) qualitative phenomena. This is based on various joint works with Mohammadreza Aghajani, Rami Atar, Anup Biswas and Haya Kaspi.

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TUTO

RIA

L 2July 6th - Monday 14:00-15:40

Room: ENG B29

Tutorial 2

ONLINE LEARNING WITH MOVING TARGETS

Speaker: Assaf Zeevi

Chair: Zeynep Akşin

The term ‘online learning’ in the title is used in reference to sequential decision making problems in which structural primitives such as the objective function/costs/utilities/underlying distributions/etc. are only par-tially known to the decision maker a priori, and need to be estimated over the course of play. The term ‘mov-ing targets’ reflects an additional characteristic, common in many applications, where these model primitives are in addition changing over the relevant problem horizon.

This talk will survey a few representative problems of this variety with the aim of elucidating basic tradeoffs that pertain, in particular, illustrating some interactions between exploration (information collection), ex-ploitation (optimizing actions), memory (information depreciation), and approximation (computational trac-tability considerations).

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July 7th - Tuesday 09:00-10:15 Room: ENG Z50

Plenary Talk 3

AN ANALYSIS OF SPARSE, LIMITED FLEXIBILITY, SERVICE ARCHITECTURES

Speaker: John N. Tsitsiklis

Chair: Apostolos Burnetas

It is well known that resource pooling (or, equivalently, the use of flexible resources that can serve multiple types of requests) significantly improves the performance of service systems. On the other hand, complete resource pooling often results in higher infrastructure (communication and coordination) costs. This leads us to explore the benefits that can be derived by a limited amount of resource pooling, and the question of whether a limited amount of pooled resources can deliver most of the benefits of complete resource pooling.

We consider a service system with independent job streams and servers, where each server can only serve a relatively small number, d, of job streams. We wish to design a service architecture (an assignment of d streams to each server) so that the system has as large a capacity region as possible, and a scheduling policy under which queueing delays become vanishingly small as the system size, n, increases. After reviewing the pros and cons of a simple, modular architecture, we show that our objective can be accomplished by combin-ing an expander graph architecture and a batching policy. This improves upon earlier results that involved a random graph and whose guarantees held only under high probability probabilistic.

(Joint work with Kuang Xu.)

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RIA

L 3July 7th - Tuesday 14:00-15:40

Room: ENG B29

Tutorial 3

AN OVERVIEW OF APPROXIMATE DYNAMIC PROGRAMMING

Speaker: Dan Adelman

Chair: Pelin G. Canbolat

While many real-world problems can be formulated as Markov Decision Processes, they have historically found limited use in practice because of Bellman’s curse of dimensionality. However, in the last decade or so, a group of academicians have been steadily pushing the area of approximate dynamic programming. A wide range of practical applications have now been explored, many with great success, in areas ranging from sup-ply chain and distribution management, dynamic pricing and revenue management, marketing/advertising, and others. In this talk, we will give an overview of basic approaches to approximate dynamic programming. After briefly discussing simulation-based methods and exact methods, we will then focus for the bulk of the talk on math programming approaches. Finally, we will conclude with a summary of some future research directions.

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RIA

L 4 July 8th - Wednesday 11:00-12:40

Room: ENG B29

Tutorial 4

MULTI-ARMED BANDIT PROBLEMS, MODELS AND ALGORITHM

Speaker: Michael Katehakis

Chair: Lerzan Örmeci

In its basic form a MAB problem involves sampling sequentially from a finite number of populations or `bandits', where each population is specified by a sequence of random variables representing the reward received when the time population is sampled. This model has played a central role in the analysis of diverse systems, such as models of reinforcement learning and Markovian Decision Problems. After a survey of the methods of Thomson Sampling, Robbins sparse sequences, Gittins indices, Lai Robbins (1985), and Burnetas and Katehakis (1996) maximal convergence rate results, we discuss recent work on multi‐objective, and risk adverse optimization models. The talk will end with a presentation of recent work on: a) the MAB problem with time commitments, b) an investigation of the short term losses due to ignorance of system parameters versus long term optimality, and c) the construction of policies that are asymptotically optimal in the sense of achieving the Katehakis Burnetas (1996) lower bound on the logarithmic rate of increase of the regret for the cases of i) Normal populations with unknown means and unknown variances , and ii) Uniform with unknown supports .

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July 5th - Sunday 10:45-12:25Room No: ENG B05

Simulation - Barış Tan - Analytical Methods for Simulation and Optimization of Dynamic Systems

OPTIMIZATION MODELS FOR GENERATING SAMPLE PATHS OF FLUID FLOW MODELS WITH A FINITE BUFFER

Behnaz Hosseini, Barış Tan

We present a mathematical programming representation for simulation of the dynamics of a two-state fluid flow system with an intermediate finite buffer. Each stage is represented with a discrete state space-continuous time semi-Markov process with the given transition time distributions between the states and a set of flow rates associated with each discrete state.

Although a methodology exists to analyze Markovian continuous flow systems with a finite buffer, discrete event simulation is the only way to analyze these systems when the transition times are generally distributed.A mathematical programming representation is used to generate simulated sample realizations of the system. Our representation is based on determining only the critical time instances of the sample path due to state transitions, flow dynamics and changing flow rates. Using the critical time instances leads to significant computational improvement compared to using a discrete-event simulation that approximates a continuous flow with a discrete flow.

We show that a simulated sample realization of this system can be obtained by developing a mixed-integer linear programming formulation.

The optimal production flow rate control problem can be analyzed by considering a special case of this model where the downstream station represents a constant demand source. We show that the optimal value of the threshold policy for this problem can be obtained by solving a quadratic integer program.

By analyzing a range of systems, we show that the mathematical programming representation is a viable method for performance evaluation and optimization of continuous-flow systems with a finite buffer.

PROBABILITY OF CORRECT SELECTION: MORE MAY NOT BE BETTER!

Yijie Peng

We present a simple counterexample where the probability of correct selection decreases with additional sampling under certain allocation schemes. We then characterize the general setting where this phenomenon may occur, which highlights the importance of an appropriate allocation scheme. Simulation experiments illustrate our findings.

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SAMPLE AVERAGE APPROXIMATION USING MATHEMATICAL PROGRAMMING MODELS FOR INTEGRATED SIMULATION-OPTIMIZATION OF QUEUING SYSTEMS

A. Matta, A. Alfieri, G. Pedrielli

Optimization of discrete event systems conventionally uses simulation as a black-box oracle to estimate performance at design points suggested by a separate optimization algorithm. Isolating the simulation model in this way fails to exploit an important advantage of simulation: simulation codes are white-boxes, at least to their creators. In many situations it is possible to fully integrate the simulation model and the optimization algorithm. In this work, we use mathematical programming methodology to model both discrete event system dynamics as well as to optimize some system variables in a queueing system. The framework adopted is that of Sample Average Approximation (SAA). The use of mathematical programming makes SAA be applied differently. Pros and cons of using mathematical programming models and solution algorithms in the SAA framework will be analyzed in terms of assumptions needed by the algorithm as well as its convergence properties.


Control and Learning - Gideon Weiss - Parallel Skilled Based Service under FCFS and Infinite Bipartite Matching

A DESIGN HEURISTIC FOR SKILL BASED PARALLEL SERVICE SYSTEMS

Marko Boon, Ivo Adan, Gideon Weiss

We study a queueing model of parallel servers of types � � �� serving customers of types � � �� under the policy FCFS‐ALIS. Customers arrive in stationary streams, join the queue and then abandon or get served. Service is skill based, which is described by a compatibility graph �, where arc �� indicates that server type �� can serve customer type ��. Service times depend on both server and customer type. This generic model is motivated by novel modes of service seen in areas as diverse as manufacturing, call centers, health care, data server farms and online retailers. The design in terms of workforce, skills and service level decisions is an extremely challenging problem. In this paper we propose a heuristic design algorithm to determine, for given data and desired service mode, which can be quality or efficiency driven or both, the required workforce levels to meet target levels of service quality and work division. The algorithm is validated through detailed simulation of three representative examples, indicating that it is remarkably robust and effective.

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NEW RESULTS ON THE STABILITY OF THE MATCHING MODEL

Jean Mairesse, Ivo Adan, Ana Busic, Gideon Weiss

The bipartite matching model and the matching model are two closely related models that have been introduced recently. The matching model is the simplest one to describe: colored items arrive one by one, in an i.i.d. way, in a buffer and depart from it by pairs of “matched” items. There are a finite number of colors and the allowed matchings depend on the colors only. When several matchings are possible, the indeterminacy is resolved in a first-in-first-out way. The stability conditions for the model have been established and a product form result has been proved for the bipartite version. Here we complete the picture by proving a Loynes type result based on a non-elementary monotonicity property.

LOCAL STABILITY OF MULTI-DIMENSIONAL MARKOV CHAINS

Seva Schneer, Ivo Adan, Sergey Foss and Gideon Weiss

We consider multi-dimensional Markov chains where some of the components are known to grow infinitely. Even though the entire process in this case cannot be stable, it is possible for the other components to have a limiting distribution. We start by presenting a number of classical processes for which this phenomenon may occur and proceed to present conditions sufficient for such “local stability” to hold.

FLUID MODELS AND STABILITY OF SKILL BASED PARALLEL SERVERS QUEUES UNDER FCFS

Gideon Weiss, Hanqin Zhang

We consider skill-based parallel-server queues udder first-come-first-served and assign-longest-idle-server policy. Under general assumptions on customer arrival and service times, the fluid limits of the system dynamics are established. The fluid limits are then used to characterize system stability by the fluid-model stability approach. Moreover, an important system performance measure, matching rate, is also obtained for some special cases.

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23


Stochastic systems - Yichuan Ding - Heavy-Traffic Approximation for Service Systems

CONVERGENCE TO EQUILIBRIUM FOR BROWNIAN QUEUEING MODELS

Rob J. Wang, Peter W. Glynn

We discuss, in some detail, the rate of convergence to equilibrium for one-dimensional reflected Brownian motion (RBM) with negative drift. This model is widely used as an approximation to heavily loaded queues. As part of this discussion, we consider the spectral representation for RBM, and the related eigenstructure of the infinitesimal generator of RBM. We also discuss the implications of these results for performance analysis of queues, as well as the planning of steady-state queueing simulations. Finally, we briefly contrast the theory with that associated with the Ornstein-Uhlenbeck process, which acts as an approximation to many-server queues (e.g. call centers).

THE IMPACT OF DELAY ANNOUNCEMENTS ON HOSPITAL NETWORK COORDINATION

Jing Dong

We investigate the impact of delay announcements on the coordination within hospital networks using a combination of empirical observations and numerical experiments. We show that patients take delay information into account when choosing emergency service providers and that such information can help increase coordination in the network, leading to improvements in performance of the network, as measured by Emergency Department wait times. Our results also indicate that the level of coordination that can be achieved is limited by the patients’ sensitivity to waiting, the load of the system, the heterogeneity among hospitals, and, most importantly, the method hospital use to estimate delays. We show that delay estimators that are based on historical average may cause oscillation in the system and lead to higher average waiting times when patients are sensitive to delay.

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24

COMPENSATION AND STAFFING FOR STRATEGIC EMPLOYEES: INCENTIVIZING SPEED-QUALITY TRADE-OFFS IN LARGE SERVICE SYSTEMS

Amy Ward, Dongyuan Zhan

Most common queueing models used for service system design assume the servers work at fixed (possibly heterogeneous) rates. However, real-life service systems are staffed by people, and people may change their service speed in response to their compensation incentives. The delicacy is that the resulting employee service rate affects the staffing, but also the staffing affects the resulting employee service rate. Our objective in this paper is to find a joint staffing and compensation policy that induces optimal service system performance.

We do this under the assumption that there is a trade-off between service speed and quality, and employees are paid based on both. Each employee selfishly chooses his or her service speed in order to maximize the expected payment. For large service systems, we describe the symmetric equilibrium service speed and show the conditions under which a critically-loaded, quality-driven, efficiency-driven, or mixed regime emerges under a first-best policy.

A MULTI-QUEUE SERVICE SYSTEM WITH CUSTOMER CHOICE

Yichuan Ding, Mahesh Nagarajan, George Zhang

This paper introduces a multiclass heavy-traffic queuing game for modeling a stochastic service system. In such a system, service providers of heterogeneous quality are distributed at different locations and heterogeneous customers arrive randomly at the system. Based on his or her own valuation of the service, the quality differentiation of the service providers, and the real-time queue length information, an arriving customer chooses one of these service providers to obtain the service. Many practical systems fit this type of model such as the system of matching kidney donors (resources) with patients waiting for kidney transplants (customers), a multi-hospital system in which a patient chooses an emergency department (ED) to visit based on the waiting times that are disclosed online, or customers choosing international land border-crossings in a region based on the characteristics of service providers and congestion information. We prove the existence of the unique equilibrium customer queue-length vector for the multiple locations of service providers. In particular, when the reneging rate is relatively small, the system can be approximated by a multi-dimensional Ornstein-Uhlenbeck process centered at that equilibrium. Our model provides a useful tool for predicting the performance, evaluating the efficiency, and optimizing the parameters of the multi-queue service systems with customer choice.

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25


Stochastic Applications - Turgay Ayer - Stochastic Modeling in Healthcare

SCORE-BASED ANTICIPATIVE TRANSFER REQUESTS IN INTENSIVE CARE UNITS

Gordon Pang, Mehmet Yasin Ulukus, Andrew J. Schaefer, Gilles Clermont

The efficient operation and management of ICUs are critical to providing high quality of care while managing costs. In this study, we show that stochastic and dynamic understanding of patient physiology can significantly improve ICU operations and predictions. In that regard, we first construct a new Transfer Score to estimate readmission and death probabilities upon transfer to a lower level of care unit. Our score makes better predictions compared to the ones in the literature. We build a stochastic model of patient length of stay (LOS) based on patient physiology, the transfer score process as well as transfer delay dynamics. This is the first attempt to provide an explicit LOS model, which will enhance the understanding of LOS dynamics and improve predictions. We then show that a score-based anticipative transfer request policy combined with effective use of clinical markers can significantly decrease transfer delays without increasing the capacity. We present a Markov Decision Process (MDP) model for the transfer request problem and solve it via approximations.

A STYLIZED BED ALLOCATION MODEL WITH A TWO CLASS LOSS SYSTEM

Nermin Elif Kurt, E. Lerzan Örmeci

Hospital managers apply certain policies when developing daily schedules to admit patients to the hospital wards. Severeness of the patient, expected length of stay of the patient, expected number of incoming patients during the day or cost of the hospitalization are some factors that the manager considers during the decision making process. In this study, we develop a simplified model to understand the effects of admission, early discharge and overflow policies on the expected cost of the system. The model assumes that two hospital wards (main ward and overflow ward) serve two types of patients (severe and mild). The recovery time of a severe patient is represented by a two stage hypoexponential distribution whereas that of a mild patient follows an exponential distribution. The main ward serves both patient types, but the overflow ward can only accommodate mild patients. A mild patient may be discharged before his/her complete recovery upon an arrival of a severe patient. We show that early discharge decision is not optimal unless all beds are occupied in the main ward. Furthermore, we establish the conditions which ensure that a patient type is “preferred” or “strongly preferred,” where patients of a preferred type are always accepted when there is at least one free bed and patients of a strongly preferred type are accepted even when the ward is full by early discharge decision. Finally, we verify our findings with a numerical analysis whose data is taken from a private hospital in Turkey.

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26

OPTIMAL LIVER CANCER SURVEILLANCE IN PATIENTS WITH HEPATITIS C

Turgay Ayer

Hepatocellular carcinoma (HCC) is the most common type of liver cancer and the fastest growing cause of cancer-related death in the United States (US). Most HCC cases are attributed to hepatitis C virus infection, which has now become endemic in the US, affecting almost 4 million people. Although surveillance in hepatitis C-infected patients can detect HCC and improve survival, the optimal surveillance policy remains unknown. In this study, we systematically analyze cost-effectiveness of surveillance policies from a societal perspective using mathematical modeling. Our modeling framework allows us to capture that disease progression depends on both unobservable and observable states, but surveillance decisions should only depend on observable states. We characterize several structural properties of the model, which reveals several interesting managerial insights. We carefully calibrate our model based on related clinical trials and observational studies. Our numerical analysis shows two main results with important policy implications. First, we find that, unlike the one-size-fits-all type policies as in the current guidelines, the optimal surveillance interval should be stratified based on patients’ liver disease stage. Second, we find that expanding surveillance to non-cirrhotic patients with advanced fibrosis improves the cost-effectiveness of HCC surveillance.

FORECASTING EMERGENCY DEPARTMENT WAIT TIMES

Mohsen Bayati

We propose a Combined Method (combining fluid model estimators and statistical learning) to forecast the wait time for low-acuity patients in an Emergency Department. Using historical data from four different hospitals, we show that the Combined Method is more accurate than stand-alone fluid model estimators and statistical learning, and also more accurate than the rolling averages that hospitals currently use to forecast the ED wait time. We show empirical validation of our findings using historical data and post-implementation data for a hospital in California.

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July 5th - Sunday 10:45-12:25Room No: ENG B16 Stochastic Applications - Mehmood Khan - Maintenance

ON MAXIMIZING THE AVAILABILITY OF SYSTEMS COMPOSED OF REPAIRABLE COMPONENTS

Kemal Gürsoy, Michael Katehakis

In this talk we revisit the problem of optimal dynamic allocation of a repairman to failed components of a reliability system.

It is assumed that the components’ lifetimes and repair times are geometrically distributed with parameters that may depend on the individual component. It is assumed that repaired components restart as good as new, and pre-emptions of repair are allowed.

We discuss problems of computing repair allocation policies to sets of failed components that maximize the system availability for the cases in which components are statistically independent and for the case in which there is a dependence of the failure probabilities on the working set of components.

INTEGRATED ENGINEERS AND PARTS PLANNING

Andrei Sleptchenko, Ahmad Al Hanbali, Henk Zijm

In this research, we analyze the integrated optimization problem of spare parts supply and workforce allocation in a maintenance system. We assume multiple failure types that arrive randomly to the repair system with exponentially distributed inter-failure times. For each failure, a service engineer with a necessary replacement for the failed component needs to be sent. If an engineer or a replacement part is not available, the failure will be outsourced and is considered as lost for our model. The objective it to minimize the total cost that consists of the inventory costs for the spare parts, costs for the pool of engineers and costs of the lost orders (or outsourcing costs). We developed a joint optimization model for the number of engineers and the stock levels that combines Markov Process based model with Mixed Integer Programming formulation. In addition to it, we developed a simple and efficient approximation method for evaluation of the system costs and optimization heuristics that can be used in cases with many failures types. The developed approximation and the optimization heuristic are time efficient and show very good results (<2% cost difference) compared to results produced by the exact MIP optimization. This allows us to use the proposed heuristic for analysis and optimization of larger systems (with more failure types) as demonstrated in a real-life case study with more than 50 failure types.

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28

AN INTEGRATED DETERMINATION OF PRODUCTION QUANTITY, PROCESS PARAMETERS AND MAINTENANCE INTERVALS

Mehmood Khan, Matloub Hussain and Abdul-Rahim Ahmad

Operations managers would always be keen to be able to decide about product design, process plan and maintenance schedules all at the same time for their key-workstations. Enlightening this line of research, this paper attempts to simultaneously determine an optimal production and maintenance schedule along with quality parameters of a machine that undergoes probabilistic deterioration while in process. This would be the very first model that integrates maintenance/restoration activities to the economic-production-quantity-quality literature. The total costs of this unique model will be the sum of setup cost, cost of quality loss, inspection-restoration cost and holding cost. Taguchi quality loss function would be utilized to assess the loss during production. Decision variables include the initial setting (process mean and tolerance), production quantity and restoration intervals. The contribution of this model is the integration of activities that control production, process as well as maintenance. The proposed model would be illustrated with numerical examples.

STATIONARY ANALYSIS OF A MULTI-TYPE QUEUE WITH DEPENDENT SERVICE DURATIONS

Abhishek Abhishek

We study a queueing system with multiple service types and correlated service times having general distribution functions. Specifically, we assume that there are � different service types, each having their own service time distribution ��, � � 1,2, . . , �, which may be chosen arbitrary. The service type of any job is dictated by the service type of the previous job and the service duration of the previous job. This correlation is encoded in the conditional joint distribution �� which is the probability that the service duration of the nth job does not exceed �, � � �, and that the �� 1�st job is of type �, given that the �th job is of type �. We investigate this queueing system in equilibrium and find the joint distribution of an arbitrary customer's service duration and service type by determining the partial generating functions. The solution technique entails the derivation of a linear system of equations for the partial generating functions and then determining a set of � unknown coefficients by employing Rouche's to determine the number of zeros of a determinant inside the unit disc.

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Stochastic Processes - Isaac Sonin - Stochastic Processes, Optimal Stopping and Related Problems

EXIT LEVELS AND PROBABILITIES OF BESSEL DIFFUSIONS

Doncho Donchev

We discuss two problems, which concern the exit probabilities and densities of Bessel diffusions in the case of two-sided boundaries. We introduce the concept of exit levels, and find large families of boundaries for which the exit probabilities admit a closed representation. Next, applying the theory of conditional processes, we show that the problem concerning the exit densities can be reduced to two simpler problems - a two-sided exit probability problem for the original process and a one-sided boundary problem for a suitable conditional process.

APPLICATIONS OF THE STOCHASTIC KNAPSACK TO INTERNET AD PLACEMENT

Michael Grabchak

One of the most important questions for internet companies is how to choose which ad to display in order to maximize their revenue. In this talk we discuss the situation when the company gets paid only if they receive a certain number of clicks by a prespecified time. We formulate this as a stochastic knapsack problem and give several strategies.

OPTIMAL STOPPING IN ONE-DIMENSIONAL DIFFUSION: APPLICATION TO REAL OPTIONS

Alexander Slastnikov

The paper deals with optimal stopping problems that arise in real options theory. We describe a variational approach to solving optimal stopping problems for diffusion processes, as an alternative to the traditional approach based on the solution of the free-boundary problem. We study smooth pasting conditions from a variational point of view, and give some examples when the solution to the free-boundary problem is not the solution to the optimal stopping problem. Special attention is paid to threshold strategies that allow reducing optimal stopping problem to more simple one-parametric optimization. Necessary and sufficient conditions for threshold structure of optimal stopping time are derived. We apply these results to both investment timing and optimal abandon models.

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30

INSERTION - A NEW OPERATION IN MARKOV CHAINS

Isaac Sonin


Stochastic Control - Flora Spieksma- Theory of Markov Decision Processes

ON THE STRUCTURE OF AVERAGE REWARD OPTIMAL POLICIES FOR MULTI-CHAIN MDPs

Michael Katehakis

We explore certain properties of the average cost optimality equations of the Markovian decision processes (MDP) model that provide further Insight into the structure of optimal policies in terms of the transient and closed classes of states they induce.Second, we give a new direct and intuitive proof that a policy defined by equalizing actions of pertinent optimality equations is optimal. Although our method applies to more general cases we discuss the case of finite state space and action sets, so that we can employ finite matrices in the analysis in order to avoid the basic features of the theory being obscured by additional formalism.

STATIONARY OPTIMAL POLICIES IN RISK-AVERSE MARKOV POPULATION DECISION CHAINS

Pelin G. Canbolat

This talk discusses the existence and the characterization of stationary optimal policies in infinite-horizon Markov population decision chains with risk-averse exponential utility. For each of three special cases, viz. (i) positive, (ii) negative, and (iii) contracting systems, sufficient conditions that guarantee the convergence of successive approximations to the minimum expected disutility and the existence of a stationary optimal policy are obtained. Under these conditions, the performance of policy improvement and geometric programming techniques and their limitations are explored.

A Markov chain (MC) observed only when it is outside of a subset is again a MC with a well‐known transition matrix . This matrix can be obtained in a few iterations, each requiring operations, when the states from are “eliminated” one at a time. We modify these iterations to allow for a state previously eliminated to be “reinserted” into the state space in one iteration. This modification sheds a new light on the relationship between an initial and censored MC, and introduces a new operation – “insertion” into the theory of MCs. An algorithm using this operation was applied for finding an optimal strategy and the value function for a Markov Decision Process.

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31

OPTIMAL DYNAMIC PRICING WITH DEMAND MODEL UNCERTAINTY: A SQUARED-COEFFICIENT-OF-VARIATION RULE

FOR LEARNING AND EARNING

Bora N. Keskin

We consider a price-setting firm that sells a product over a continuous time horizon. The firm is uncertain about the sensitivity of demand to price adjustments, and continuously updates its prior belief on an unobservable sensitivity parameter by observing the demand responses to prices. The firm’s objective is to minimize the infinite-horizon discounted loss, relative to a clairvoyant that knows the unobservable sensitivity parameter. Using partial differential equations theory, we characterize the optimal pricing policy, and then derive a formula for the optimal learning premium that projects the value of learning onto prices. We compare and contrast the optimal pricing policy with the myopic pricing policy, and quantify the cost of myopically neglecting to charge a learning premium in prices. We show that the optimal learning premium for a firm that looks far into the future is the squared coefficient of variation (SCV) in the firm’s posterior belief. Based on this principle, we design a simple variant of the myopic policy, namely the SCV rule, and prove that this policy is long-run average optimal.

A ROADMAP TO STRUCTURAL PROPERTIES FOR UNBOUNDED RATE MDPs

Herman Blok

We study structures of optimal policies of MDPs under the expected average cost criterion. However, unbounded rate MDPs do not admit uniformisation, hence structural properties cannot be derived directly via value iteration. To obtain properties for these kinds of MDPs a number of intermediate steps need to be taken. We will present a roadmap to obtain structural properties for the value function as well as for the optimal policy.

We will illustrate this with two problems: the infinite server farm problem [Adan et al. 2015] and the -competing queues problem with abandonments.

The structure is as follows. Since the value iteration algorithm needs extra validation for convergence in the average cost criterion, we obtain the results via the vanishing discount approach. So we consider the MDP under the total discounted cost criterion. On this we apply a rate truncation to allow uniformisation. For these bounded rate MDPs we use value iteration to show structural properties of the value function as well as the optimal policy. Then a limit argument ensures that these properties are carried over to the unbounded MDP. Finally letting the discount factor go to zero requires a continuous time Tauberian theorem stating that the Abel and Cesàro limit are equal. This gives the desired results for the average cost criterion.

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Simulation - Hamed Jalali- Simulation

KRIGING-BASED SIMULATION OPTIMIZATION WITH HETEROGENEOUS NOISE: A COMPARISON

Hamed Jalali, Inneke Van Nieuwenhuyse, Victor Picheny

Simulation-optimization refers to optimization of the performance of simulated systems. Among the different techniques, kriging-based methods (also referred to as Bayesian optimization) are widely applicable and are tailored to time-consuming simulation (where each replication takes a long time to execute). While most existing kriging-based methods focus on deterministic or noiseless simulation (e.g., the famous Efficient Global Optimization or EGO algorithm), recently, there has been a growing interest to adapt these methods for stochastic or noisy simulation. In spite of promising results, most of these adaptations are initially designed for homogeneous noise while heterogeneous noise (i.e., the noise is stronger in some regions of the objective function than others) is much more prevalent in real-life applications. Therefore, the behavior of these methods with heterogeneous noise is an open and critical question.In this paper we evaluate the performance of five state-of-the art algorithms, derived from the original EGO strategy, for simulation optimization with heterogeneous noise. To this end, we apply these algorithms to minimize three well-known benchmark test functions, perturbed with heterogeneous Gaussian noise. Although in real-life problems the noise can follow a large variety of structures, we focus here on two somewhat “extreme” but realistic cases: 1) favorable scenario: the noise variance decreases as the objective value decreases; therefore we have the minimum noise at the global minimum, and 2) unfavorable scenario: the noise variance increases as the objective value decreases; therefore we have the maximum noise at the global minimum.

RISK-ADJUSTED JOINT OPTIMIZATION OF BASE-STOCK LEVELS AND COMPONENT ALLOCATION IN A HYBRID SYSTEM

Ebru Angun

This talk considers a multi-component, multi-product, periodic-review hybrid (re)assemble-to-order system that uses an independent base-stock policy for the inventory replenishment of the components. At the beginning of each period, end-of-lease cores are returned. Because the quality of cores is random, they are tested, graded, and sorted into four pre-specified quality levels. Then, the random, jointly and continuously distributed demands for the products are realized. In the problem, partial fulfillment is not allowed. Furthermore, the system quotes a predetermined response time window for each product, and it penalizes the system if the demand is not satisfied within its time window. This problem is formulated through a risk-adjusted two-stage stochastic programming model, where the first-stage decisions are the base-stock levels for all components, and the second-stage decisions are the allocations of components to different products. The risk adjustment is achieved through the conditional value-at-risk or mean-upper semideviations risk measures of order p. After some simple manupulations, the resulting problem can be solved through a standard method such as the L-shaped method. The resulting problem is analyzed numerically under different distributions, for different stages of products in their life cycle, etc.

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33

AN ISLAND PARTICLE ALGORITHM FOR RARE EVENT ESTIMATION

Christelle Vergé, Jérôme Morio, Jimmy Olsson, Pierre Del Moral, Eric Moulines

Estimating rare event probability with accuracy is of great interest for safety and reliability applications. Nevertheless, it is not just evaluating a risk or a probability but estimating the law of random phenomena that leads to critic events. Some parameters of the model or density parameters of input random variables in the system may be fixed by a user for simplicity reasons. From a risk analysis point of view, it is interesting to determine the impact of such tuning of parameters on the realization of a critic event. In the present work, we design a new algorithm which belongs to the island particle models including a parameter, referred to by us as interacting Particle Markov Chain Monte Carlo (iPMCMC). It is a sequential algorithm that samples from the law of parameters of a system conditionally to a rare event, and allows to determine the parameters that, in case of bad estimation, tend to increase the rare event probability value. It also gives an estimate of the rare event probability maximum taking into account the likelihood of the parameters. We prove the convergence of iPMCMC seeing it as a classical sequential Monte Carlo algorithm on an extended state space, and we establish asymptotic properties analyzing it through parameterized Feynman-Kac framework. We illustrate the convergence of iPMCMC on an analytic case, and we apply it to estimate the fallout zone of a launch vehicle booster.

VALUE OF INTELLIGENCE ON TERRORISTS’ TARGET CHOICE

Niyazi Onur Bakır, Buğra Ersü

This study presents an analytical approach that is intended to determine the value of intelligence in a ho-meland security problem with multiple targets under different scenarios. The aim of this study is to make a comparative analysis of the defender strategies and to evaluate intelligence sources available to the defen-der. There are two potential targets for which the defender allocates security resources to counter attacks from two independent terrorist groups. The defender bears the costs of maintaining security, receiving the additional intelligence and suffers losses in case of a successful attack. The defender can make a resource allocation decision either in the light of intelligence or without additional intelligence. We assume that intel-ligence will be precise in the sense of perfect information. Analysis partial intelligence is an extension that is

planned for future work.

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Control and Learning - Esa Hyytia - Control, Scheduling and Optimization of Queueing Systems

A POLLING MODEL FOR AN ORDER-PICKING WORKSTATION

Dieter Claeys, Ianire Taboada, Jose Oscar Fajardo, Fidel Liberal

We investigate the performance of parts-to-picker warehouses with remotely located order-picking workstations. More specifically, we deduce mathematical expressions for the order flow times, which are defined as the time between release and completion of a customer order in the warehouse. To this end, we model an order-picking workstation as a polling system so that order flow times correspond to cycle times in the polling system. The order-picking policy of each order picker handling one order at a time to avoid picking errors is captured by a specific service discipline, which, to the best of our knowledge, has not been studied before. As this service discipline does not satisfy the so-called branching property, we establish stochastic bounds for the order flow times. These bounds are shown to be adequate and aid in setting targets for the throughput of the storage area, which, in turn, is the input rate of the workstations. As research on parts-to-picker warehouses has traditionally focused on optimizing operations in the storage area, we believe our results are complementary to those established in literature.

NOVEL BANDIT-BASED SOLUTIONS FOR PRACTICAL STOCHASTIC SCHEDULING PROBLEMS

Sofia Villar

Over the past 50 years bandit-based solutions, and particularly the concept of index policy introduced by Gittins and Jones, have been fruitfully deployed to address a wide variety of stochastic scheduling problems arising in practice. Deriving such solutions poses various research challenges, but offers significant computational and performance advantages. In this talk I will illustrate this point by presenting recent results from two Bayesian bandit models of the optimal allocation of patients in a clinical trial and the scheduling of sensors in a network to detect smart targets, in which either the derivation of an index policy or the practical implementation of existing index policies poses complex research questions. Part of this talk is based on recent joint work with Jack Bowden and James Wason.

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OPTIMAL TRADE-OFF BETWEEN SIZE-BASED AND OPPORTUNISTIC SCHEDULING: THE WHITTLE INDEX APPROACH

Samuli Aalto, Pasi Lassila, Prajwal Osti

We consider the optimal opportunistic scheduling problem for data traffic in a wireless cell with time-varying channels. The scheduler itself operates in a very fast timescale of milliseconds, but the objective function is related to minimizing the holding costs in a much longer timescale, at the so-called flow level. The Whittle index approach is a powerful tool in this context since it renders the flow-level optimization problem tractable, even with heterogeneous users. Until now, this approach has been applied to the opportunistic scheduling problem to generate non-anticipating index policies that may depend on the amount of attained service but do not utilize the exact size information. We derive a size-based (i.e., anticipating) index policy by applying the Whittle index approach in a novel way. By a numerical study based on simulations, we demonstrate that the resulting size-aware index policy systematically improves performance when compared to the state-of-the-art opportunistic schedulers. As an important side result, we show that the opportunistic scheduling problem is indexable when the flow sizes follow the Pascal distribution, and we derive the corresponding Whittle index, which generalizes earlier results.

SCHEDULING TRAFFIC FLOWS IN EMERGING WIRELESS NETWORKS

Ianire Taboada, Jose Oscar Fajardo, Fidel Liberal

This work deals with the problem of scheduling traffic flows in emerging wireless networks. In actual mobile implementations the channel state feedback from the users to the radio schedulers may be deployed based on periodic reports in order to avoid uplink signaling overhead. In this work, we study how to design schedulers under partially observable channels. For that aim, we employ the Whittle index approach using a Partially Observable Markov Decision Process. We further analyze the performance of the derived scheduling solution, both in single-class and multi-class scenarios. Real-world channel quality traces are employed in experiments. Finally, we conclude that the proposed scheduling index rule outperforms well-known channel-aware disciplines.

This work has been partially funded by the Spanish Ministerio de Economía y Competitividad (MINECO) under grant TEC2013-46766-R: QoEverage - “QoE-aware optimization mechanisms for next generation networks and services” and the COST Action IC 1304.

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Stochastic Systems - Nelly Litvak - Random Graphs: New Models and Methods

DIRECTED RANDOM INTERSECTION GRAPHS

Lasse Leskelä, Mikko Kuronen

Unlike most classical random graph models studied in the literature, many real-world social and data networks are directed, and neighboring nodes are often statistically correlated. In this talk I will discuss a mathematical model of a large directed graph where stochastic dependence between neighboring nodes is built into the model by randomly assigning each node two sets of attributes. An ordered node pair is connected if one node demands an attribute that the other node supplies. In this talk I will describe conditions on the model parameters for the existence of a giant strongly connected component, and results describing the expected size of the forward and backward components of a typical node in the network.

WEIGHTED BRANCHING PROCESSES IN INFORMATION RANKING ALGORITHMS AND PARALLEL QUEUES WITH SYNCHRONIZATION

Mariana Olvera-Cravioto

We present in this talk two problems from the BigData era that lead to the analysis of max-plus branching stochastic fixed-point equations. The first problem is related to the analysis of information ranking algorithms on complex networks, e.g., Google’s PageRank, where we show that the rank of a randomly chosen node in a directed configuration model converges in distribution, as the size of the graph grows, to the endogenous solution to a linear branching stochastic fixed-point equation. The second problem analyzes a queueing network with many servers where arriving jobs are split into several pieces to be processed in parallel, with the constraint that all pieces must begin their service in a synchronized fashion. We show that the waiting time of jobs before they start their service converges in the Kantorovich-Rubinstein distance, as the number of servers grows, to the endogenous solution to a maximum branching stochastic fixed-point equation (i.e., high order Lindley equation). The endogenous solutions to these and other branching stochastic fixed-point equations can be constructed on a weighted branching process, and their tail behavior can be explicitly characterized, providing interesting insights into both problems.

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37

DISTANCES IN DIRECTED RANDOM GRAPHS WITH FINITE VARIANCE

Pim van der Hoorn

The distance between two arbitrary nodes in random undirected graphs with finite degree variance, generated by the configuration model, has been shown to scale logarithmically in the size of graph. We extend this analysis to the setting of directed random graphs. We keep the framework as general as possible by only assuming mild conditions on the degrees, including finite variance, while, for instance, still allowing for arbitrary dependence between the in- and out-degree of nodes.

Moving to the directed setting generates subtle difficulties when applying standard random graph techniques, such as coupling the graph with a branching tree, since we need to control both in- and out-degrees simultaneously. Using a coupling with a so-called thorny branching tree instead, we show that the distance between to randomly chosen nodes in the directed configuration model also scale logarithmically in the size of the graph. Furthermore, similar to the undirected case, we analyze the fluctuations around these sample means.

INFINITE BIN MODELS AND PREFERENCE ATTACHMENT SCHEMES

Sergey Foss

We will introduce a number of infinite bin models in discrete and continuous time, and discuss conditions for existence of regenerative structure and related problems.Links with preference attachment schemes will be discussed too.

The talk is partly based on joint papers with T. Konstantopoulos (MPRF, 2003), D. Denisov and T. Konstantopoulos (AnnAP, 2012), S. Zachary (AdvAP, 2013), J. Martin and P. Schmidt (AnnAP, 2014).


Stochastic Applications - Ali Devin Sezer - Stochastic Processes and Their Applications

ON OPTIMAL STOPPING WITH EXPECTATION CONSTRAINTS

Stephan Ankirchner, Maike Klein, Thomas Kruse

The talk is about optimal stopping with the constraint that the expectation of any stopping time has to be bounded by a given constant. We show that by introducing a new state variable one can derive a dynamic programming principle. This allows to characterize the value function as the solution of a PDE and to obtain a verification theorem. Finally we compare our approach with alternative solution methods and discuss some examples.

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OPTIMAL BOUNDS FOR RISK MEASURES AND INSURANCE APPLICATIONS

Nabil Kazi-Tani

The theory of convex risk measures has attracted a great deal of attention among applied probabilists and insurance specialists. In this talk, we revisit some of the applications in reinsurance. In particular, we give optimal bounds for law invariant convex risk measures, using simple step function approximation techniques. We show how this can be used for pricing issues, regulation and model uncertainty problems.

EXACT ANALYSIS OF MULTISERVER QUEUES WITH SETUP TIME

Tuan Phung-Duc

Recently, queues with setup time are extensively studied because they have applications in power-saving data centers. In data centers, idle servers are turned off in order to save energy and are turned on again when the workload increases. The off-servers need some setup time to be active so as to serve waiting customers. In this study, we present exact solutions for multiserver queues with setup time based on a generating function approach and a matrix analytic method. We prove a conditional decomposition showing that the queue length under the condition that all the servers are busy is decomposed into the sum of two independent random variables which have a clear physical interpretation. In particular, the first variable is the conditional queue length of the corresponding model without setup time and the second variable is the number of customers standing before a tagged waiting customer under the condition that one the server is setting up and the rest of servers are busy.

EXIT PROBABILITIES AND BALAYAGE OF CONSTRAINED RANDOM WALKS

Ali Devin Sezer

Let X be the constrained random walk on �� representing the queue lengths of a stable Jackson network and x be its initial position. Let ��be the first time the sum of the components of X equals n. �� is a key performance measure for the queueing system represented by X; stability implies �� → 0exponentially. Currently the only analytic method for approximating �� is large deviations analysis, which gives the exponential decay rate of ��. Finer results are available via rare event simulation. We develop a new method to approximate �� and related expectations. The method has two steps: 1) with an affine transformation, move the origin onto the exit boundary of ��; take limits to remove some of the constraints on the dynamics; this yields a limit unstable constrained walk Y, 2) Construct a basis of harmonic functions of Y and use them to apply the superposition principle. The basis functions are linear combinations of log‐linear functions and come from solutions of harmonic systems, which are graphs whose vertices represent points on the “characteristic surface” of Y; the edges between the vertices represent conjugacy relations between the points, the loops represent membership in “the boundary characteristic surfaces.” Using our method we derive simple and almost exact formulas for �� for d‐tandem queues, similar to the product form formulas for the stationary distribution of X. The same method allows us to approximate the Balayage operator mapping f to � → ��1�� for a range of stable constrained walks in 2‐dimensions.

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39


Stochastic Applications - Ali Eshragh - Stochastic Applications in Flow Lines Services and Healthcare

OPTIMIZATION OF BUFFER ALLOCATIONS IN STOCHASTIC FLOW LINES WITH LIMITED SUPPLY

Sophie Weiss, Raik Stolletz

Sample-based optimization of buffer allocations allows for flexible modeling of real-world features that occur in flow lines. We introduce a limited supply of the line, which follows an (s, q) -order policy. Furthermore, we develop a search algorithm to determine the optimal buffer capacities, which is based on the equal protection criterion. To further improve the performance of the algorithm, we generate lower bounds for each individual buffer. The numerical study evaluates the impact of the order policy on the optimal buffer allocation as well as the performance of the search algorithm compared to existing approaches.

ACHIEVING GOOD SERVICE LEVELS IN QUEUES: CONTROLLING THE TAIL

Rene Bekker, Ger Koole

In many service systems, the service level is defined in terms of the probability that the waiting time exceeds some threshold (i.e., in terms of tail probabilities). A prominent and well known example is the 80/20 rule in call centers. The systems studied in the literature are typically controlled based on the number of customers present or the workload, although this is not the primary objective. In this talk, we examine ways to study queues where the control is based on accumulated waiting time, e.g. where a secondary server may be activated. Moreover, we study a model variant where the secondary server cannot be directly activated due to the service of other types of jobs (also known as call blending). Finally, if time permits, we fluctuations in the service level that manifests on a considerably larger time scale than expected.

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40

BOARDING PATIENTS IN THE EMERGENCY DEPARTMENT: A QUEUEING PERSPECTIVE

Raïsa Carmen, Inneke Van Nieuwenhuyse, Benny Van Houdt

Boarding patients are patients that require admission to the hospital after being treated in the emergency department (ED) but are stranded in the ED because of a lack of beds in the hospital wards. Boarding patients occupy valuable resources and prevent newly arriving patients from starting treatment. This “inpatient boarding” phenomenon is considered to be a big problem in many EDs all over the world and has been associated with increase increased ambulance diversions, worse patient outcomes, frustration among medical staff, higher patient length of stay, loss of revenue, and higher mortality rates.

Our aim is to use queueing models to gain intuition about how boarding patients affect the performance of the ED and thus optimal capacity planning (both staff and beds are considered). As the queueing models remain relatively basic, the results and intuitions are expected to be generally applicable in many settings.

FISHER INFORMATION, STOCHASTIC PROCESSES ANDGENERATING FUNCTIONS

Ali Eshragh

In this talk, we deliver our theoretical and numerical results on the Fisher Information for the birth rate of a partially-observable simple birth process involving n observations. Our goal is to estimate the rate of growth, lambda, of a population governed by a simple birth process. We may choose n time points at which to count the number of individuals present, but due to detection difficulties, or constraints on resources, we are able only to observe each individual independently with fixed probability p. We discuss the optimal times at which to make our n observations in order to maximize the Fisher Information for the birth rate lambda. Finding an analytical form of the Fisher Information in general appears intractable. Nonetheless, we find a very good approximation for the Fisher Information by exploiting the probabilistic properties of the underlying stochastic process. Both numerical and theoretical results strongly support the latter approximation and confirm its high level of accuracy. However, this approximation is limited to the number of observations. Eventually, we utilised the concept of generating functions to calculate the Fisher Information efficiently.

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41


Simulation - Jose Blanchet - Perfect Simulation of Stochastic Networks and Queues

PERFECT SAMPLING OF GI/GI/c QUEUES

Jing Dong

We introduce the first class of perfect sampling algorithms for the steady state distribution of multi-server queues with general interarrival time and service time distributions. Our algorithm is built on the classical dominated coupling from the past protocol. In particular, we use a coupled multi-server vacation system as the upper bound process and developed an algorithm to simulate the vacation system backwards in time from stationarity at time zero. The algorithm has finite expected termination time with mild assumptions on the interarrival time and service time distributions.

EXACT GRADIENT SIMULATION FOR STOCHASTIC FLUID NETWORKS IN STEADY STATE

Xinyun Chen

We develop a new simulation algorithm that generates unbiased gradient estimators for the steady-state workload of a stochastic fluid network, with respect to the throughput rate of each server. Our algorithm is based on the perfect sampling algorithm developed in Blanchet and Chen (2014), and the infinitesimal perturbation analysis (IPA) method. We illustrate the performance of our algorithm with two multidimensional examples, including its formal application in the case of multidimensional reflected Brownian motion.

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42

TAIL ASYMPTOTICS FOR DELAY IN HALF-LOADED GI/G/2 QUEUES WITH HEAVY-TAILED JOB SIZES

Karthyek Murthy, Jose Blanchet

We consider two-server queues processing regularly varying job sizes and discuss asymptotic bounds for the tail distribution of steady-state waiting time in two server queues. In particular, we consider the half-loaded regime where each server processes incoming jobs at a rate equal to the rate of their arrivals and identify several interesting phenomena.

When the incoming jobs have finite variance, there are basically two types of effects that dominate the tail asymptotics. While the quantitative distinction between these two manifests itself only in the slowly varying components, the two effects arise from qualitatively very different phenomena (arrival of one extremely big job (or) two big jobs). Then there is a phase transition that occurs when the incoming jobs have infinite variance. In that case, only one of these effects dominates the tail asymptotics, the one involving arrival of one extremely big job.

SAMPLING FROM MODEL MANIFOLDS WITHOUT DERIVATIVE INFORMATION

Enlu Zhou, Chang-han Rhee, Peng Qiu

We introduce an algorithm that generates samples from a specified distribution on a manifold, based only on the ability to evaluate the mapping defined by the parametrization of the manifold. In particular, we do not assume the ability to evaluate the derivatives of the mapping nor the answer to membership inquiry on the manifold. Our approach is useful when the manifold is analytically intractable and highly nonlinear-for example, in studying complex regulatory networks in systems biology where the mapping is typically defined by the solution of a system of ordinary differential equations. We will present convergence properties of our algorithm and some illustrative numerical examples.

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43


Control and Learning - John Birge - Learning

LEARNING TO COMPETE AGAINST DYNAMIC PRICING STRATEGIES

Matthew Stern

Our research examines the impact of competition on the design and implementation of dynamic pricing strategies. Observing posted prices in each period, as well as their own private demand realizations, firms compete for revenues while learning the parameters of their underlying demand curves. As the market matures and firms learn from the experience of selling their products, each firm becomes more proficient at setting prices; however, the resulting increased level of competition may cause profits to decline for the industry as a whole. By minimizing a notion of regret relative to the single-period Nash equilibrium, we examine pricing strategies that balance the tradeoffs between actively learning the demand parameters, pricing to optimize current period rewards and favorably influencing the rival firms’ future prices.

WHEN DOES PASSIVE LEARNING WORK? DYNAMIC LEARNING WITH CERTAINTY-EQUIVALENCE CONTROL AND LEAST SQUARES ESTIMATION

Bora Keskin, Assaf Zeevi

We consider a dynamic control and estimation problem where a system manager sequentially chooses controls and makes observations on a response variable that depends on chosen controls and an unknown sensitivity parameter. After every observation, the system manager estimates the unknown parameter and uses a myopic certainty-equivalence policy to determine subsequent controls based on estimates. We show that the estimates under such a myopic learning policy can converge with positive probability to a confounding estimate that is not necessarily equal to the true value of the unknown parameter, resulting in poor asymptotic accuracy of estimates. We also derive conditions under which this myopic learning policy can achieve better accuracy performance.

EFFICIENT METHODS FOR QUICKEST DETECTION OF EPIDEMICS

Georgios Fellouris, Sriram Somanchi

The problem of epidemic detection can be naturally formulated as a multistream sequential change-detection problem. Indeed, a change, the onset of the epidemic, occurs at some unknown time and modifies the dynamics of various stochastic systems (streams) that are monitored in real time. The goal is to detect the change, as well as the subset of affected streams, as soon as possible, so that any action can be taken in time, while avoiding false alarms and the disruption that they cause. In this talk, we will apply various quickest detection techniques to the problem of epidemic detection. The proposed methods will be illustrated on simulated data, which may agree or disagree with our modeling assumptions, as well as on real data. Finally, we will discuss differences and similarities with existing techniques in the epidemic detection literature.

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44

COST EFFICIENT CLOUD COMPUTING

Andrew Li

The Cloud is now part of the typical consumer’s lexicon, and with it, come a variety of practical resource allocation problems. We address a key problem faced by enterprise users: completion of batch computational workloads under a deadline at minimal cost. For example, online retailers frequently update personalized product recommendations based on new transactions. This problem is made more challenging by the introduction of market-driven prices, which are volatile and notoriously difficult to model as stochastic processes. We formulate this as a dynamic optimization problem and present an algorithm that completes batch jobs on time and at low cost. Our algorithm does not require stochastic price models as input, but instead makes use of past prices in a simple, scalable way that requires no tuning. We demonstrate that its performance is robust to price behavior by showing a performance guarantee under a common adversarial model. In addition, the algorithm enjoys performance guarantees for a variety of stochastic price processes. In experiments using price traces, our algorithm closely matches the clairvoyant optimal policy; experiments on a real-world, open-source implementation show similar performance. Along the way, we introduce a generalization of the classical Secretary Problem. We include historical information in the standard problem and are able to quantify the value of this history. In particular, we give bounds on the performance of any online algorithm, show that our algorithm achieves these bounds, and demonstrate that the long-standing best achievable performance ratio of 1/e can be improved with history by our algorithm.


Stochastic Systems - Mor Harchol-Balter - Queues with Redundant Jobs

COUNTERINTUITIVE RESULTS IN SYSTEMS WITH REDUNDANT JOBS

Mor Harchol-Balter, Kristen Gardner, Samuel Zbarsky, Sherwin Doroudi, Esa Hyytia, Alan Scheller-Wolf

Using the analysis developed in the previous talk, we investigate several specific redundancy structures in small (two or three server) systems. We derive simple forms for the distribution of response time of both the redundant classes and non-redundant classes, and we quantify the “gain” to redundant classes and “pain” to non-redundant classes caused by redundancy. We find some surprising results. First, in many cases the response time of the redundant class follows a simple Exponential distribution and that of the non-redundant class follows a Generalized Hyperexponential. Second, once a class is fully redundant, it becomes “immune” to any pain caused by other classes becoming redundant. We also compare redundancy with other approaches for reducing latency, such as optimal probabilistic splitting of a class among servers (Opt-Split) and Join-the-Shortest-Queue (JSQ) routing of a class. We find that redundancy outperforms JSQ and Opt-Split with respect to overall response time, making it an attractive solution. We also study performance in scaled systems, and find that many of our counterintuitive results continue to hold as the number of servers increases.

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45

QUEUES WITH REDUNDANCY: EXACT ANALYSIS

Kristen Gardner, Samuel Zbarsky, Sherwin Doroudi, Mor Harchol-Balter, Esa Hyytia, Alan Scheller-Wolf

Recent computer systems research has proposed using redundant requests to reduce latency. The idea is to run a single request on multiple servers and only wait for the first completion (discarding all remaining instances of the request). However no exact analysis of systems with redundancy exists. Here, we present the first exact analysis of systems with redundancy. We allow for any number of classes of redundant requests, any number of classes of non-redundant requests, any degree of redundancy, and any number of heterogeneous servers. We determine an appropriate state space to represent the system and describe the system evolution as a Markov chain. In all cases we derive the limiting distribution on the state of the system.

QUEUES FOR DATA ACCESS FROM CODED DISTRIBUTED STORAGE

Emina Soljanin

Users of cloud systems demand that their data be reliably stored and quickly accessible. Cloud providers today strive to meet these demands through over-provisioning: keeping processors ready to go at all times and replicating data over multiple servers. Codes for distributed storage introduce redundancy in order to both reliably maintain the stored data and efficiently maintain the guaranteed reliability of the storage network under node failures. For example, a file consisting of chunks can be encoded and the coded chunks distributed across multiple disks so that reading only a subset of the disks is sufficient to reconstruct the entire file. Or a file can be encoded so that each of its chunks can be recovered by reading one of several small disjoint groups of coded chunks. In distributed storage practice, coding based solutions are already being used instead of replication, as a more efficient way to provide storage reliability. But code design also determines the ways in which data can be retrieved from coded storage, and thus how available it is under given data request patterns and the request scheduling policies. This talk will address data retrieval quality of service in distributed storage implementing different coding schemes when data retrieval processes are modeled by fork-join like queues determined by the coding scheme in use.

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46

ORGANJET: OVERCOMING GEOGRAPHICAL DISPARITIES IN ACCESS TO DECEASED DONOR KIDNEYS IN THE UNITED STATES

Barış Ata

There are significant disparities in waiting times and access to transplant across different geographical areas in US. We propose an operational solution that offers affordable jet services (OrganJet) to patients on the transplant waiting list, allowing them to multiple list in different, and possibly very distant, donation service areas (DSA) of their choosing. First, using a fluid approximation we formulate the patients’ problem of choosing a location to multiple list as a selfish routing game in which each patient tries to minimize his “congestion cost”, i.e. maximize his life expectancy. Through a combination of numerical, simulation and analytical results, we show that multiple listing can lead to a significant improvement in geographic equity. In the special case when sufficiently many patients can multiple list, the geographic inequity disappears. Moreover, the supply of deceased-donor organs increases under multiple listing, leading to more transplants and saved lives.

We also consider a diffusion approximation and study the resulting multiple listing game. The equilibrium outcome under the diffusion approximation is a second-order perturbation of that under the selfish routing formulation.


Stochastic Applications - Nilay Tanık Argon - Stochastic Modeling for Healthcare Operations

JOINT QUEUE LENGTH DISTRIBUTION IN EXPONENTIAL PRIORITY SYSTEMS

Ivo Adan

The exact steady-state analysis of multi-dimensional Markov processes is often intractable, and if tractable, the analysis is in most cases restricted to dimension two. In this talk we focus on the analysis of an exponential multi-class priority system, which can be modeled by a multi-dimensional structured Markov process. Based on matrix-analytic techniques and probabilistic arguments, we develop a recursive method to exactly calculate the joint queue length distribution.

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47

THE SNOWBALL EFFECT OF CUSTOMER SLOWDOWN IN CRITICAL MANY-SERVER SYSTEMS

Jori Selen

Customer slowdown describes the phenomenon that a customer’s service requirement increases with experienced delay. In healthcare settings, there is substantial empirical evidence for slowdown, particularly when a patient’s delay exceeds a certain threshold. For such threshold slowdown situations, we design and analyze a many-server system that leads to a two-dimensional Markov process. Analysis of this system leads to insights into the potentially detrimental effects of slowdown, especially in heavy-traffic conditions. We quantify the consequences of underprovisioning due to neglecting slowdown, demonstrate the presence of a subtle bistable system behavior, and discuss in detail the snowball effect: A delayed customer has an increased service requirement, causing longer delays for other customers, who in turn due to slowdown might require longer service times.

EMERGENCY DEPARTMENT PATIENT FLOW WITH DETERIORATION

Mark Lewis, Carri W. Chan, Douglas G. Down

We consider the problem of patient flow in an emergency department. Patient’s health deteriorates over time while a single medical service provider must decide where to allocate its efforts between patients in different health states. Since the rate of deterioration depends on the number of patients in each class the decision-making scenario is confounded.

DYNAMIC CASUALTY DISTRIBUTION IN THE AFTERMATH OF A DISASTER

Nilay Tanık Argon

In the aftermath of a disaster, emergency responders must transport a large number of casualties to medical facilities, using limited transportation resources (such as ambulances). Decisions about where to send the casualties are typically made in an ad hoc manner by responders on the scene. Using a dynamic programming model, we develop heuristic policies that use limited information such as mean travel times and congestion levels, to determine (a) how to allocate ambulances to casualty locations and (b) which medical facility should be the destination for those ambulances. We consider two types of transportation resources: those that are dedicated to a specific casualty location but can be dynamically routed to different medical facilities, and those that can be dynamically reallocated both to different casualty locations and medical facilities. In a simulation study, we incorporate casualty survival rates and service times for different types of traumatic injuries and show that the proposed heuristics can provide substantial improvement in the expected number of survivors, even when the decision maker has only limited up-to-date information about the system state.

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48


Stochastic Applications - Ethem Çanakoğlu - Risk Management, Finance and Energy

CONDITIONAL EXPECTED SHORTFALL UNDER HIDDEN REGULAR VARIATION

Bikramjit Das, Vicky Fasen

PARTICLE SURVIVAL MODEL, RECORD VALUE PROCESS AND THEIR APPLICATION TO LIMIT ORDER BOOKS

Hiroshi Toyoizumi

The particle survival model, which was originally proposed to analyze the dynamics of species’ coexistence, has surprisingly been found to be related to a non-homogeneous Poisson process. It is also well known that successive record values of i.i.d. sequences have the spatial distribution of such processes. In this research, we show that the particle survival model and the record value process are indeed equivalent. Further, we study their application to determining the optimal strategy for placing selling orders on stock exchange limit order books. Our approach considers the limit orders as particles, and assumes that the other traders have zero intelligence.

The phenomenon of asymptotic independence of extreme events in a portfolio of different risks is well‐known. Many models used in practice exhibit this phenomenon. For example, if the risks risk related to a portfolio of two stocks is given by �� , then it is highly unlikely for both � and � to take high values together. In the context of systemic risk we can think of � as the risk of a stock we are interested in and � as that of the entire market.

We study ��|� � ��which is the expected shortfall of one risk given than the other risk is high under assumptions of multivariate and hidden regular variation. We present an extrapolation method to estimate��|� � ��, provide consistency results and exhibit its applicability in some data examples. This work extends studies done under the regime of asymptotic dependence.

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49

MODELING AND OPTIMIZATION OF TRANSMISSION OPERATIONS UNDER RENEWABLES UNCERTAINTY

Soumyadip Ghosh, Dzung Phan

Real-time operation of a transmission grid must handle increasing uncertainty as the proportion of renewables in the generation portfolio rises. This necessitates probabilistic modelling of the impact of renewables uncertainty over the near-future state of the grid. We present three key steps in the stochastic modelling and optimization of transmission control decisions. We start with the question of efficiently estimating the probability of the occurrence of a congestion event, which is defined here as the event when power flow in a transmission line exceeds critical thermal limits, in a multi-period formulation. Robustness of transmission decisions istypically analysed by performing a contingency analysis (CA), wherein the effects of the decisions are analysed over not just the current state of the grid but also over a set of scenarios of slightly changed network topologies with certain links or nodes offline. Our second step performs a stochastic CA to efficiently estimate the probability of a congestion event in the multi-period formulation given that the network can be in its current state or it may switch to one of the contingency scenarios. Finally, we present a simulation-based method to solve a multi-period stochastic optimization formulation to choose the best mitigation decisions to minimize the chances of experiencing a congestion event. The decision set includes changes in the network topology (transmission switching), or to the load (via shedding, demand-response) or to the generation (via generator off-loading, increased output from fast-acting peakers).

RISK MANAGEMENT APPLICATIONS IN ELECTRICITY MARKETS

Ethem Çanakoğlu

After deregulation in electricity markets contract planning has become one of the major risk management tools for either electricity generation or distribution companies. Participants of the electricity market are exposed to market risk due to the special characteristic of the electricity prices. Since electricity cannot be stored efficiently, spikes are common in the prices resulting as extremely high or negative levels in the price which results as highly volatile structure compared to other commodities or assets in financial markets. High levels of volatility encourage companies to use derivative contracts for risk management. In this paper we analyse the dynamics of spot prices for electricity market with respect to fuel and carbon price dynamics and compare the forecasting power of different models such as econometric time series models and stochastic models with regime switching characteristics. Using different models we price the derivative contracts for electricity and fuel contracts. Then we suggest and solve alternative stochastic portfolio optimization problems with different risk metrics. As a result we compare the expected payoff and risk structure of an electricity company with different risk management strategies.

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50


Stochastic Processes - Alex Belloni - Probability and Optimization

BEATING THE CURSE OF DIMENSIONALITY IN INVENTORY PROBLEMS WITH LEAD TIMES

David A. Goldberg, Linwei Xin

Many classical inventory models become notoriously challenging to optimize in the presence of positive lead times, since the state-space blows up and dynamic programming techniques become intractable. This includes, for example, lost sales models with positive lead times, and dual-sourcing models with positive lead time gap between the two suppliers. In this talk, we will present a new algorithmic approach to such problems, which shows that as the lead time grows large, simple policies become asymptotically optimal. These results are quite surprising, as this setting had remained an open algorithmic challenge for over forty years. In particular, we will show that a simple constant-order policy is asymptotically optimal for lost sales models with large lead times, and provide explicit bounds on the optimality gap which demonstrate good performance even for small-to-moderate lead times. We will also show that the so-called Tailored-Base Surge heuristic for dual-sourcing problems is asymptotically optimal as the lead time gap between the two sources grows large. In both cases, our results provide a new algorithmic approach to these problems, as well as a solid theoretical foundation for the good performance of these algorithms observed numerically by previous researchers. Our approach combines ideas from the theory of random walks and queues, convex analysis, and inventory control.

MANAGING MULTI-PERIOD PRODUCTION SYSTEMS WITH LIMITED PROCESS FLEXIBILITY

Yehua Wei, Cong Shi, Yuan Zhong

In this paper, we study a multi-period production system with multiple products and facilities in the make-to-order environment. The system has limited process flexibility, i.e., each facility has flexibility to produce multiple but not all of the products. To manage the production facilities over multiple periods, we propose a computationally tractable max-weight policy. We demonstrate that the max-weight policy can greatly outperform a naïve greedy policy in both theory and simulation. Moreover, by applying a classical result from the queuing literature, we show that sparse process flexibility structures under the proposed max-weight policy are asymptotically optimal in a heavy traffic regime.

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51

STRATEGIC INVESTMENT IN RISKY PROSPECTS WITH SYNERGIES

Sasa Pekec

Continuous time investment decisions of multiple agents in risky prospects are modeled in a multi-armed bandit setting in which arms (risky prospects) could yield a one-time payoff (success). Each arm type is unknown and is either good or bad. The payoff of a good arm realizes according to a Poisson process with the rate parameter proportional to the current investment amount. A bad arm never produces a payoff. All investments are publicly observed and thus allow for joint learning and for forming common beliefs about arm types.

The payoffs exhibit complementarities: a payoff in one arm (success of a risky prospect) increases the payoff amounts to be obtained from future successes in other arms, i.e., there are synergetic effects out of multiple successes. We show that an index policy need not be optimal due to these payoff synergies. However, optimal investment strategies do have a threshold policy structure, with threshold values depending on information/beliefs about all arms. We also show that, unlike the setting without complementarities, the optimal individual and aggregate investment with cooperating agents is different from that of competing self-interested agents.

FINITE HORIZON MARKOV DECISION PROBLEMS AND A CENTRAL LIMIT THEOREM FOR TOTAL REWARD

Alessandro Arlotto

We prove a central limit theorem for a class of additive processes that arise naturally in the theory of finite horizon Markov decision problems. The main theorem generalizes a classic result of Dobrushin (1956) for temporally non-homogeneous Markov chains, and the principal innovation is that in the present framework the summands are permitted to depend on the current state as well as a bounded number of future states of the chain. We show through several examples that this added flexibility gives one a direct path to asymptotic normality of the optimal total reward of finite horizon Markov decision problems. The same examples also explain why such results are not easily obtained by classical means.

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52


Stochastic Control - Sandeep Juneja - Stochastic Multi-armed Bandits

TIGHT BOUNDS FOR THE INFINITELY MANY-ARMED BANDIT PROBLEM WITH DETERMINISTIC ARMS

Nahum Shimkin, Yahel David

We consider a version of the Multi-Armed Bandit problem which involves a large pool of a-priory identical arms (or objects). Each arm is associated with a deterministic value, which is revealed once that arm is chosen. At each time instant the agent may choose a new arm (with unknown value), or a previously-chosen one whose value is known. The goal is to minimize the cumulative regret relative to the fixed choice of the best arm in the pool. We present a lower bound on the regret, whose time-dependence depends on the functional form of the tail of the distribution of values in the pool, and devise learning algorithms that achieve this bound up to sub-logarithmic terms. We consider several variants of the basic problem, involving an unknown time horizon (anytime algorithm), and a non-retainable arms model where chosen arms are lost if not re-used immediately. Time permitting, we will describe a multi-pool version of this problem.

LEARNING TO MATCH: DECENTRALIZED LEARNING IN MULTIPLAYER MULTI-ARMED BANDITS

Rahul Jain

We consider a setting where there are N arms and M(<N) players. As the players are matched to arms, each of them gets a random reward from an unknown distribution. If two players choose the same arm, there is a “collision” and neither gets anything. The question is: Is there a decentralized learning algorithm (with sublunar regret) that players can choose that asymptotically leads to the optimal player-arm matching? And furthermore, is there a cost of decentralization. We first present a UCB-like decentralized learning algorithm that does not require a dedicated communication channel for coordination and achieves near-log-squared regret. This is not regret-optimal but reveals a structure of the decentralized learning algorithm that can be used for online learning in bandit models. We then present two algorithms dPEGE and dPEGE-beta that combine some of these ideas with ideas from `Thomson sampling’ that achieves near-log expected regret which is order optimal.

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53

CHASING BANDITS

Assaf Zeevi

In a stochastic multi-armed bandit (MAB) problem a gambler needs to choose at each round of play one of K arms, each characterized by an unknown reward distribution. Reward realizations are only observed when an arm is selected, and the Gambler’s objective is to maximize his cumulative expected earnings over some given horizon of plays T. To do this, the gambler needs to acquire information about arms (exploration) while simultaneously optimizing immediate rewards (exploitation). Bandit problems have been studied extensively since their inception. The bulk of this literature is concerned with settings where the reward distributions are stationary, namely, the statistical properties of the arms do not change over time. In this talk we will consider a formulation that relaxes this restriction, highlight some of the key theoretical findings that characterize this setting, and, time permitting, discuss an emerging application domain in the space of online services that motivates our work.

MULTI ARMED BANDIT SAMPLING IN NESTED PORTFOLIORISK MEASUREMENT

Sandeep Juneja, Ankush Agarwal

We consider estimating the probability that portfolio loss exceeds a large threshold within a time horizon when the portfolio comprises of diverse financial derivative securities. The value of the portfolio at any state-time is a conditional expectation that needs simulation estimation. We develop a multi-armed bandit based sampling method to determine whether at any time loss exceeds a specified threshold. For this we also develop computation lower bounds on such estimations and show that the proposed method matches them up to the first order. We identify the decay rate of the mean square error of the proposed estimator and show that it improves upon the decay rate of a popular naive estimator.

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54


Stochastic Networks - Amy Ward - Fork-Join Networks and Large-Scale Markov Chains

CONTROL OF A FORK-JOIN PROCESSING NETWORK WITH TWO CUSTOMER CLASSES AND ONE SHARED RESOURCE

Erhun Özkan

We consider a fork-join processing network in which two job classes arrive to separate servers, then each forks (or splits) into two tasks that require additional processing, and finally the tasks must be joined (or processed simultaneously) before departing the network. There are two “preliminary” servers that perform the first processing step, three “secondary” servers that process the tasks, and two “final” servers that do the end processing. Two of the secondary servers are devoted to each job class and one of the secondary servers is flexible, and must choose which job class to prioritize. We study this network when the flexible secondary server is heavily loaded, so that the scheduling policy has a non-trivial effect on performance. The classic scheduling rule suggests giving static priority to the more expensive job class. This policy performs well when the devoted secondary servers are lightly loaded. However, it performs poorly when the devoted secondary servers are heavily loaded. This is because there is starvation at the final server, due to the synchronization constraint. To correct this, we propose a dynamic state-dependent policy that matches the queue-length of the more expensive jobs at the flexible secondary server with that at the devoted secondary server. The result is to have a negligible number of more expensive jobs at the final server, while still departing the maximum number of less expensive jobs. We prove that our proposed policy is asymptotically optimal in heavy traffic, and use simulation to demonstrate its superior pre-limit performance.

ROBUST SCHEDULING IN A FLEXIBLE FORK-JOIN NETWORK

Yuan Zhong, Ramtin Pedarsani, Jean Walrand

We consider a general flexible fork-join processing network, in which jobs are modeled as directed acyclic graphs with nodes representing tasks, and edges representing precedence constraints among tasks. Both servers and tasks are flexible in the sense that each task can be processed by several servers, which in turn can serve multiple task types. The system model is motivated by the problem of efficient scheduling of both sequential and parallel tasks in a flexible processing environment, which arises in application areas such as cloud computing, and manufacturing.

A major challenge in designing efficient scheduling policies is the lack of reliable estimates of system parameters such as arrival and/or service rates. We call a policy robust if it does not depend on system parameters such as arrival and service rates. We propose a robust scheduling policy for the flexible fork-join network model, and prove that it is rate stable when service rates can be written as products of a task-dependent quantity and a server-dependent quantity. We also provide simulation studies to demonstrate the performance and limitations of the proposed policy.

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55

HEAVY-TRAFFIC LIMITS FOR A FORK-JOIN NETWORK IN THE HALFIN-WHITT REGIME

Guodong Pang, Hongyuan Lu

We study a fundamental fork-join network model with a single class of jobs that will fork into a fixed number of parallel tasks upon arrival, and then join after service completion. Each parallel task is processed at a multi-server station under the FCFS and non-idling service discipline. Service times of the parallel tasks of each job can be correlated. Upon service completion, each parallel task will join a buffer associated with its service station and wait for synchronization. Each job can be synchronized only after all of its tasks are completed, called non-exchangeable synchronization. The goal is to understand the waiting buffer dynamics for synchronization as well as the service dynamics. We show FLLN and FCLT for the number of tasks in each waiting buffer for synchronization jointly with the number of tasks in each parallel service station and the number of synchronized jobs in the Halfin-Whitt regime. All the limiting processes in the FCLT are functionals of two independent processes: the arrival limit process and the generalized multiparameter Kiefer process driven by the service vectors for the parallel tasks of each job.

FAST ESTIMATION OF TRANSITION PROBABILITIES IN LARGE MARKOV CHAINS

Siddhartha Banerjee, Peter Lofgren, Ashish Goel, C. Seshadri

July 6th - Monday 10:45-12:25Room No: ENG B05

Stochastic Processes - Ad Ridder - Markov Chain Analysis and Applications

AN EFFICIENT ALGORITHM FOR COMPUTING THE ERGODIC PROJECTOR OF MARKOV MULTI-CHAINS

Bernd Heidergott

This talk presents an alternative for the power method for approximately computing the ergodic projector of a Markov chain. This new method, called series expansion approximation technique (SEA), is based on series expansions for Markov chains. SEA applies to Markov multi-chains consisting of multiple ergodic classes and possible transient states, and provides a controllable approximation for Markov multi-chain ergodic projectors. SEA allows for the analysis of large-scale network, and we will discuss applications to the Google PageRank. Numerical experiments are included to illustrate the performance of SEA.

I'll present a new bi‐directional algorithm for estimating multi‐step transition probabilities in large state‐space Markov chains, which combines linear algebraic and MCMC techniques. Remarkably, we

show that our algorithm can estimate probabilities of size � in time � � �√��‐ orderwise less than the

time taken by any MCMC procedure. An important application for this procedure is for computing PageRank in large graphs ‐ we show that our algorithm is faster than the state‐of‐the‐art by several orders of magnitude.

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56

SIMULATION-BASED DETECTION OF STABILITY REGIONS FOR MARKOVIAN QUEUE PROCESSES

Haralambie Leahu

Multi-class queueing networks (McQN) extend the concepts of Jackson and/or Kelly networks in that customers visiting multiple times a certain station (node) may experience different (probabilistic) routing and/or service-time. For such networks, stability is a crucial property as it guarantees that the network will be able in the long run to cope with the incoming workload, allowing decision-makers to predict the long-term behavior of the system. Unlike Jackson networks, a sub-critical McQN is not necessarily stable; in fact, sub-criticality (of all nodes) is a necessary but not sufficient condition for stability. We model the dynamics of McQN as a parameter dependent Markov process, where the parameter of interest is the vector of (external) arrival rates and establish a simple characterization of the positive (Harris) recurrence of the underlying process, which allows one to represent the stability region as the support of some limiting functional of the process. Eventually, we design a simulation-based search scheme to approximate the stability set associated with a McQN and illustrate our method in some specific cases where stability conditions are not available in closed form.

WISDOM OF CROWDS IN SOCIAL NETWORKS WITH STRUCTURAL VARIABILITY: FROM CONSENSUS TO TRUTH

Jia-Ping Huang, Bernd Heidergott, Ines Lindner

Wisdom of crowds proposed by Golub and Jackson (2010) is a characterization of growing networks where individual beliefs of people therein converge to the truth in the double limit of time and network size. It extends the classic model of DeGroot (1974) about reaching a consensus with a fixed size network. In this paper we examine the validity of wisdom of crowds when the network structure is variable over time. We observe that when the network is fluctuating between two alternatives, the application of Markov chain theory to analyzing consensus and wisdom of crowds in the literature is no longer appropriate. Numerical examples are provided to demonstrate our observation.

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57

COUNTING VERTEX COVERS IN GENERAL GRAPHS

Ad Ridder

In this paper we consider estimating the number of vertex covers in general graphs by simulating a specific Markov chain on the graph. The transition distribution associated with the zero-variance estimator is not implementable. We will consider two proposal distributions as approximations to the optimal. Both proposals are based on randomization techniques. The first randomization is the classic probability model of random graphs, and in fact, the resulting algorithm shows polynomial complexity (FPRAS) for random graphs. The second randomization introduces a probabilistic relaxation technique that uses Dynamic Programming. The numerical experiments show excellent practical performance in comparison with existing methods.


Finance and Revenue Management - Dan Iancu - Revenue Management and Financial Considerations

OPTIMAL MONITORING POLICY FOR ASSET BASED LOANS

Do Young Yoon, Dan Iancu, Nikolaos Trichakis

We consider a borrowing firm that finances its operations by collateralizing its working assets, e.g., inventory. To mitigate the risk due to the assets’ uncertain valuation, the lender has a monitoring option granting him the right to enforce immediate loan repayment by liquidating the borrower’s assets. Although monitoring is considered as a critical feature of asset-based lending (ABL) in practice, questions about optimal monitoring have not been studied yet. We discuss the choice of optimal monitoring time and optimal liquidation rule that maximize the lender’s return. We first characterize the specific conditions under which monitoring is effective, and characterize their dependence on market parameters. When monitoring is needed, we fully characterize the lender’s optimal liquidation policy, and show that it can have a counterintuitive, non-threshold structure. We derive bounds on the optimal monitoring time, and leverage them to devise simple heuristics, which perform well in numerical studies.

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58

NON-STATIONARY STOCHASTIC OPTIMIZATION

Yonatan Gur

We consider a non-stationary variant of a sequential stochastic optimization problem, in which the underlying cost functions may change along the horizon. We propose a measure, termed variation budget, that controls the extent of said change, and study how restrictions on this budget impact achievable performance. We identify sharp conditions under which it is possible to achieve long-run-average optimality and more refined performance measures such as rate optimality that fully characterize the complexity of such problems. In doing so, we also establish a strong connection between two rather disparate strands of literature: adversarial online convex optimization; and the more traditional stochastic approximation paradigm (couched in a non-stationary setting). This connection is the key to deriving well performing policies in the latter, by leveraging structure of optimal policies in the former. Finally, tight bounds on the minimax regret allow us to quantify the “price of non-stationarity,” which mathematically captures the added complexity embedded in a temporally changing environment versus a stationary one.

DYNAMIC MECHANISM DESIGN WITH BUDGET CONSTRAINED BUYERS UNDER NON-COMMITMENT

Santiago Balseiro

We study the stochastic dynamic mechanism design problem of a firm repeatedly selling items to budget-constrained buyers when the seller has no commitment power. We argue that this problem is generally intractable. Thus motivated, we introduce a fluid model that allows for a tractable characterization of the optimal dynamic mechanism. We leverage our characterization to provide insights into the dynamic structure of the optimal mechanism, and conclude by showing that the proposed fluid mechanism is a good approximation for the original stochastic model as markets become large.

DYNAMIC MANAGEMENT OF LOYALTY PROGRAMS

Dan Iancu, So Yeon Chun, Nikolaos Trichakis

We study the problem of dynamically managing a loyalty program. While originally viewed as marketing efforts, in the last two decades loyalty programs have grown substantially in size and scope, to the extent that they now often significantly interact with other firm functions, including operations, accounting and finance. We develop a dynamic programming model to study the problem of optimally setting prices and point requirements. The model captures all the aforementioned interactions, such as the effect of loyalty programs on sales revenues, rewards redemptions, servicing costs, and earnings. The structure of the optimal policy allows us to provide concrete managerial insights and prescriptive recommendations for the joint management of a loyalty program and pricing.

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Stochastic Applications - Serhan Ziya - Stochastic Models in Healthcare

QUEUES WITH TIME-VARYING ARRIVALS AND INSPECTIONS WITH APPLICATIONS TO HOSPITAL DISCHARGE POLICIES

Carri Chan, Jing Dong and Linda Green

In order for a patient to be discharged from a hospital unit, a physician must first perform a physical examination and review the pertinent medical information to determine that the patient is stable enough to be transferred to a lower level of care or be discharged home. Requiring an inspection of a patient’s ̀ readiness for discharge’ introduces an interesting dynamic where patients may occupy a bed longer than medically necessary. Motivated by this phenomenon, we introduce a queueing system with time-varying arrival rates in which servers who have completed service cannot be released until an inspection occurs. We examine how such a dynamic impacts common system measures such as stability, expected number of customers in the system, probability of waiting and expected waiting time. Leveraging insights from an infinite-server model, we’re able to optimize the timing of inspections and find via theoretical and numerical analysis that 1) optimizing a single inspection time could lead to significant improvements in system performance when the amplitude of the arrival rate function is large, 2) the amount of time between subsequent inspections should be uniform throughout a day, and 3) the marginal improvements of adding additional inspection times is decreasing.

MANAGING VIRTUAL VISITS FOR CHRONIC CONDITIONS

Sarang Deo, Armağan Bayram, Seyed Iravani, Karen Smilowitz

Virtual visits (e.g. via phone, e-mail, web based application) between patients and healthcare providers can assist in managing chronic conditions by providing low cost condition monitoring, symptom treatment and education. This can potentially decrease the number of hospitalizations and emergency room visits. Motivated by these potential benefits of virtual visits, we develop capacity allocation models that decide which patients to schedule given limited availability of both office and virtual visit slots. Our model aims to maximize aggregate health benefits across a cohort of patients enrolled in a chronic disease management program. We model this problem using a dynamic programming framework over a finite horizon, and perform analytical and numerical analyses to identify policies for scheduling patients for different medical interventions

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60

A DISCRETE-TIME MODELLING FRAMEWORK FOR SUPPLY AND DEMAND PLANNING IN OUTPATIENT CLINICS

Navid Izady

In this talk, I will present a discrete-time modelling framework for appointment supply and demand planning in outpatient clinics. The model enables us to capture the complex features observed in a wide range of clinics, including arbitrary appointment request distribution, arbitrary clinic cancellation distribution, and delay-dependent no-show and rescheduling behaviour. Using real data, we show the application of the model for appointment capacity planning in specialty care clinics, patient panel size planning in primary care clinics, and designing slot publication and dynamic visit policies in clinics where the appointment booking process is supported by an online booking system like the Choose and Book system implemented in the UK. We further demonstrate how the model can be used in conjunction with a density forecasting model in situations where there is significant trend or seasonality in the requests arrival process.

ADMISSION AND DISCHARGE DECISIONS IN INTENSIVE CARE UNITS

Serhan Ziya

We consider a stylized, discrete-time model for an Intensive Care Unit (ICU) in which patients’ health conditions change over time with Markovian probabilities. At any point in time, each patient is in one of two possible health stages, one representing a more serious and the other representing a less serious condition in regards to eventual survival. Arriving patients also present in one of two health stages. The ICU has limited bed availability and therefore when a patient arrives at a full ICU, a decision needs to be made as to which patients should be kept in the ICU and which ones should be transferred to general care. Our objective is to make that decision so that the long-run average rate with which patients survive is maximized. We first identify mathematical conditions under which the ICU is always more preferable than general care. Then, under these conditions, which one can assume to hold in practice, we give an almost complete characterization of the optimal patient admission/discharge policy. We find that the optimal policy, in general, depends on the composition of the patients currently in the ICU but our numerical study suggests that even simple policies that do not take such dependence into account, perform quite well.

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Stochastic Networks - John Hasenbein - Stochastic Network Approximations

ERGODIC CONTROL OF MULTI-CLASS M/M/N+M QUEUES IN THE HALFIN-WHITT REGIME

Gordon Pang, Ari Arapostathis, Anup Biswas

We consider a dynamic scheduling problem for a multi-class queueing network with a large pool of statistically identical servers. The arrival processes are Poisson, and service times and patience times are assumed to be exponentially distributed and class dependent. The optimization criterion is the expected long time average (ergodic) of a general (non-linear) running cost function of the queue lengths. We consider this control problem in the Halfin-Whitt (QED) regime. The optimal solution of this control problem can be approximated by that of the corresponding ergodic diffusion control problem in the limit. We introduce a broad class of ergodic control problems for controlled diffusions, which includes a large class of queueing models in the diffusion approximation, and establish a complete characterization of optimality via the study of the associated HJB equation. We also prove the asymptotic convergence of the values for the multi-class queueing control problem to the value of the associated ergodic diffusion control problem. The proof relies on an approximation method by spatial truncation for the ergodic control of diffusion processes, where the Markov policies follow a fixed priority policy outside a fixed compact set.

COLLABORATION AND MULTITASKING IN NETWORKS: CAPACITY AND QUEUE CONTROL IN NESTED ARCHITECTURES

Itai Gurvich, Jan Van Mieghem

We study networks where some activities require the simultaneous processing by multiple types of multitasking indivisible resources. The capacity of such networks is generally smaller than the bottleneck capacity. We first prove that both capacities are equal in networks with a nested collaboration architecture. We then proceed to study how this capacity is achieved through, and affected by, dynamic queue control.

STEIN’S METHOD FOR STEADY-STATE DIFFUSION APPROXIMATIONS

Jim Dai, Anton Braverman

We consider many-server queues with phase-type service time distributions and customer abandonment. We prove the rate of convergence for approximating the stationary distribution of the normalized system size process by that of a piecewise Ornstein-Uhlenbeck (OU) process. For the proof, we develop a modular framework that is based on Stein’s method. The framework has three components: Poisson equation, generator coupling, and state space collapse. The framework, with further refinement, is likely applicable to steady-state diffusion approximations for other stochastic systems.

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62

FURTHER REFLECTIONS ON RBM IN THE OCTANT

John Hasenbein, Ziyu Liang

This work is an extension of earlier work on optimal paths in large deviations for rotationally symmetric RBM in the octant. In particular, for the Harrison-Reiman case, we show that only gradual paths can be optimal. Furthermore, we show that, unlike the general case, spiral optimal paths are not optimal in this case. This conforms with the intuition that approximations of single-class queueing networks are better-behaved than those for multiclass networks.


Stochastic Systems - Peter Taylor/Sophie Hautphenne - Matrix Analytic Methodsin Stochastic Models

THE RUNNING MAXIMUM OF A LEVEL-DEPENDENT QUASI-BIRTH-DEATH PROCESS

Michel Mandjes

In this talk I explain how to determine the distribution of the running maximum of the level in a level-dependent quasi-birth-death process (QBD). By considering this running maximum at an exponentially distributed ‘killing epoch’ , I devise a technique to accomplish this, relying on elementary arguments only; importantly, it yields the distribution of the running maximum jointly with the level and phase at the killing epoch. I also point out how our procedure can be adapted to facilitate the computation of the distribution of the running maximum at a deterministic (rather than an exponential) epoch; this is done relying on the concept of `Erlangization’.

PARAMETER ESTIMATION FOR THE MARKOVIAN TRANSITION COUNTING PROCESS, AN ALTERNATIVE TO MMPP

Azam Asanjarani, Yoni Nazarathy, Sophie Hautphenne

We revisit the popular Markov Modulated Poisson Process (MMPP) and introduce an alternative that we call Markovian Transition Counting Process (MTCP). The latter is simply a point process counting the number of transitions of a finite continuous time Markov chain. We show that MTCP exhibits many of the qualitative, quantitative, analytical and modelling properties of MMPP, but slightly superior in terms of parameter estimation properties. That is, both processes are very useful for modelling traffic streams in various application areas, but as we claim and illustrate, our MTCP alternative is more sensible when actually estimating the parameters of a system.

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63

MARKOV-MODULATED LOSS QUEUES

Peter Taylor

THE CAPACITY VALUE FUNCTION OF MAP/PH/1/C QUEUES

Sophie Hautphenne

In an MAP/PH/1/C queue, customers are lost when they arrive to find C customers already present. Assuming that each arriving customer brings a certain amount of revenue, we are interested in calculating the expected amount of revenue that the queue will lose over a finite time horizon [0,t]. This requires to solve a finite system of matrix difference equations for the expected lost revenue function that corresponds to a transient version of the Poisson equation. We describe different methods to solve this system of equations.


Queues and Control - Moshe Haviv - Decentralized Priority Selection

ACCUMULATING PRIORITY QUEUES WITH STRATEGIC CUSTOMERS

Liron Ravner

In an accumulating priority M/G/1 queue each customer is assigned a positive priority coefficient. This assignment can be either class dependent or a choice made by the customers themselves. The actual (accumulated) priority of a waiting customer is a linear function of his time since arrival whose slope coincides with his priority parameter. The service regime is such that upon service completion the next customer to be admitted is the one who has accumulated the most priority. We study a non-cooperative game where customers can purchase their own priorities. For the case of homogeneous customers with respect to their waiting cost parameter, we explicitly compute the unique pure strategy Nash equilibrium. We further show that this model can display both avoid the crowd and follow the crowd behaviour, for different levels of bidding. For a game with heterogeneous customer types that differ in their waiting costs, we characterize the Nash equilibrium as the solution to a set of polynomial equations and suggest using an iterated best response dynamics in order to compute it. We further consider a non-atomic version of this game where each class of customers can coordinate their bids. We construct a unique pure strategy Nash equilibrium for the resulting game.

We consider a Markov modulated Erlang loss queue with ��servers in which the arrival and service rates depend on the state of an underlying Markov chain ��. We derive a differential equation satisfied by the vector generating function �� ≡ ∑ �� , where �� is the vector of stationary probabilities of states in which there are � customers present. The vector of constants �� can be determined as a solution of a linear equation, obtained by repeated differentiation.

We shall go on to discuss various consequences of the analysis.

We consider a Markov modulated Erlang loss queue with ��servers in which the arrival and service rates depend on the state of an underlying Markov chain ��. We derive a differential equation satisfied by the vector generating function �� ≡ ∑ �� , where �� is the vector of stationary probabilities of states in which there are � customers present. The vector of constants �� can be determined as a solution of a linear equation, obtained by repeated differentiation.

We shall go on to discuss various consequences of the analysis.

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64

SELF-REGULATION OF AN UNOBSERVABLE M/M/1 QUEUE

Binyamin Oz, Moshe Haviv

We consider an unobservable M/M/1 queue where customers are homogeneous with respect to their reward (due to service completion) and with respect to their cost per unit of time of waiting. Left to themselves, it is well known that in equilibrium they will join the queue at a rate that is higher than it is socially optimal. Hence, regulation schemes, under which the resulting equilibrium joining rate coincides with the socially optimal one, should be considered. In this talk we suggest a classification of regulation schemes, based on a few desired properties, and use it to classify schemes from existing literature. To the best of our knowledge, there is no existing scheme that possesses all properties, and in this talk we suggest such one. This novel scheme is based on assigning random priority to each customer, prior to the decision whether or not to join. We also introduce variations of this regulation scheme as well as additional schemes based on randomization.

REGULATING AN OBSERVABLE QUEUE

Moshe Haviv, Binyamin Oz

It is well-known that customers join a queue at a rate which is higher than is socially optimal. In particular, if they observe the queue length prior to deciding whether or not to join, they will use a higher threshold than would have been prescribed by a central planner. The question than is how to change the rules so as selfish customers would still behave in the socially optimal ways. The talk, after surveying a few existing mechanisms, will present some new ones. All of the new mechanisms will be based on deviating from the usually assumed First Come First Serve regime and they do not involve any money transfer. The model of M/M/1 and homogeneous customers will be assumed throughout.

OPTIMAL ADMISSION POLICIES IN BATCH SERVICE SYSTEMS

Olga Boudali, Lerzan Örmeci

We study the admission control problem in a Markovian queue, where customers are served in batches of size K. We assume that each customer brings a reward of R units, and accumulates a waiting cost of C units for every time unit he remains in the system. The administrator of the system is the only decision maker and may admit a customer upon arrival or reject him. In particular, we determine the structure of the optimal admission policy. Moreover, we give several conclusion and interpretations regarding the optimal batch size. Next, we consider a system which receives different types of customers, in the sense that each customer brings a different reward to the system according to his type. We study the effect of these different types of customers in the optimal admission policy and give interpretations.

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MDP - Nur Sunar - Stochastic Control and Optimal Stopping Applications

DYNAMIC PRODUCT DEVELOPMENT AND OPTIMAL LAUNCHFOR A CUSTOMER NETWORK

Nur Sunar

The development and launch of products with local network externalities require a deep understanding of social or commercial relationships among customers because optimal advertising and pricing strategies are dependent on the structure of the social or business relations among customers. Motivated by this fact, we consider a firm that dynamically develops a product to be launched in a market, which is represented by a network of customers, and sets advertising and pricing strategies at the product launch. Using a continuous time Brownian model, we analyze implications of the structure of the customer network for optimal product development and launch strategies of the firm. We introduce new network centrality measures that identify optimal advertising, development and launch strategies of the firm.

RISK AVERSION, INFORMATION ACQUISITION ANDTECHNOLOGY ADOPTION

Canan Ulu

Uncertainty about a technology’s benefits complicates adoption decisions. Should the consumers or firms adopt immediately or wait and gather more information? How does risk aversion affect adoption decisions? Does it encourage decision makers to wait longer?In this paper, we study the impact of risk aversion and uncertainty on technology’s benefits on adoption and information gathering decisions. We develop a discrete time dynamic programming model where a risk averse decision maker can adopt the technology, reject it or wait and gather additional information. We study conditions under which optimal policy has a threshold structure in the decision maker’s beliefs about the benefits of the technology.

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66

DYNAMIC BIDDING FOR MULTIPLE KEYWORDS IN SEARCH-BASED ADVERTISING

Semih Sezer, Savaş Dayanık

We consider an advertiser bidding for multiple keywords on a search engine. Keyword inquiries arrive according to a Poisson process, and at each arrival time the advertiser bids for an ad position on the result page. In this problem, the objective of the advertiser is to find a dynamic bidding policy maximizing its expected revenue over a finite horizon subject to a budget constraint. In this talk, we will discuss the solution of this problem, and we will illustrate some numerical examples.

OPTIMIZING THE INTERACTION BETWEEN ATTENDING PHYSICIANS AND RESIDENTS

Hayriye Ayhan, Sigrun Andradottir

The training of physicians in the United States and in many countries in the world takes place not only in classrooms but also in hospitals where residents practice under the supervision of experienced attending physicians. Because attending physicians work with several residents, examine patients on their own, and have other responsibilities, it is important to understand when an attending physician should switch from his own responsibilities to have a conference with his residents. In this talk, we consider a model where there are always patients waiting to be seen, a patient is first examined by a resident, and then the resident and the attending physician have a conference to form a plan on patient care. Under various settings, we determine the optimal dynamic assignment of the attending physician in order to maximize the long-run average revenue.


Stochastic Applications - Stella Kapodistria - Scheduling: Novel Techniques and Applications

EXPLOSIVENESS PROPERTIES OF COUNTABLE STATE PARAMETRISED MARKOV PROCESSES

Flora Spieksma

Conditions on non-explosiveness of a Markov process is a key ingredient in the study of optimal policies in a Markov decision process with unbounded jump rates as a function of state. It appears to be a crucial condition on a transformed Markov decision process to guarantee continuity properties of the expected cost as a function of the policy. Since a policy can be viewed as a parameter, one can study these problems also in the framework of a parametrised Markov process.

Different conditions guaranteeing suitable non-explosiveness properties are used in the literature. We will discuss necessary and sufficient conditions for non-explosiveness as well as the relation between various conditions encountered in the literature.

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67

A COMPARATIVE ANALYSIS OF SUCCESSIVE LUMPING AND LATTICE PATH COUNTING ALGORITHMS

Laurens Smit

In this talk a successive lumping (SL) based algorithm and a lattice path counting based algorithm (LPCA) are compared both in terms of applicability requirements and numerical complexity. Both type of algorithms make use of rate matrices, and are part of Matrix Analytic Models. To make the comparison between the procedures, we analyse their performance for certain classical queueing and inventory models. We will show that the structure required to use any of the two algorithms is very similar.The main findings presented are: i) When both methods are applicable the SL based algorithms outperforms the LPCA one in speed. We will go in detail how the matrices arising in SL algorithms can be inverted fast. ii) There are classes of problems (e.g., models with (level) non-homogenous rates or with finite state spaces) for which the SL methodology is applicable and for which the LPCA cannot be used. There are also some structures for which LPCA can compute rate matrices, while the SL algorithm cannot. iii) The SL based algorithms always provide a method to compute the steady state distribution, besides constructing the rate matrix.

APPOINTMENT SCHEDULING WITH PATIENT PREFERENCES

Yu Zhang, Vidyadhar Kulkarni

We consider an appointment system where the patients have preferences about the appointment days. A patient may be scheduled on one of the days that are acceptable to her, or be denied appointment. The patient may or may not show up at the appointed time. The net cost is a convex function of the actual number of patients served on a given day. We study the optimal scheduling policy that minimizes the total expected discounted cost over infinite horizon, or the long-run average cost. We present structural properties of the optimal appointment policy, and study heuristic policies. We recommend an index policy that is easy to implement and performs particularly well in comparison to the other heuristic policies and is close to the optimal policy.

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68

SCHEDULING PREVENTIVE MAINTENANCE ON A WIND TURBINE BASED ON QUANTITATIVE DATA

Stella Kapodistria

This presentation focuses on the close relationship between statistical process control and preventive maintenance for a wind turbine. In particular, inspired by a physical model for the power output of the wind turbine, we design a statistical process monitoring procedure. The process is monitored with a control chart with the purpose of quickly detecting shifts to inferior operational states of the wind turbine due to the occurrence of unobservable assignable causes. At the same time, the information collected from the monitoring process is used to determine the overall operational state of the wind turbine, a.k.a. the degradation process of the asset. This degradation process moves in a continuous manner between two extremes (perfect condition and failure) with random measurement errors. Although, these three models, the physical model, the statistical model and the stochastic model, are obviously related, they have been typically treated in the literature independently. In this talk, we highlight the underlying connections between the three models and present a general mathematical model that can be used for the optimal identification of what constitutes sufficient evidence of imminent failure, so as to perform preventive maintenance, taking into account a maintenance cost structure.


Stochastic Processes - Tahir Hanalioğlu - Renewal-Reward Processes and Their Applications

LIMIT DISTRIBUTION FOR RENEWAL-REWARD PROCESS WITH REFLECTING BARRIER

Başak Gever, T. Khaniyev, Z. Hanalioglu

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ASYMPTOTIC RESULTS FOR RENEWAL ‐ REWARD PROCESS WITH GENERAL INTERFERENCE OF CHANCE

Özlem Ardç

In this study, a renewal – reward process with a discrete interference of chance and delay �� is investigated. Under some assumptions, three – term asymptotic expansion for the �� order moment of the ergodic distribution of the process �� is obtained, when� � �. Additionally, kurtosis coefficient, skewness coefficient and coefficient of variation of the process �� are computed. The main result of the study can be given with the following theorem. Theorem. Let the following conditions be satisfied:

i) Random variables ��, �� and �� are independent;�� ii) �� iii) �� iv) �� is non‐arithmetic random variable, v) �� , � � �,2,�, . ..

Then, the following three‐term asymptotic expansion for the �� order moment of the ergodic distribution of the process �� can be written, as � � �:

��

��

�� , where

��

�� ,

�� 2 � � �� .

Here, �� , ��

��, � � 2,�,and � � ��

�� is delay coefficient.

Özlem Ardıç, T. Khaniyev, R. Aliyev

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70

MOMENTS OF AN INVENTORY MODEL OF TYPE (s,S) WITH HEAVY TAILED AND INFINITELY VARYING DEMANDS

Aslı Bektaş Kamışlık

A CORRECTION TERM FOR THE ASYMPTOTIC COVARIANCE OF RENEWAL-REWARD PROCESSES WITH MULTIVARIATE REWARDS

Brendan Patch, Yoni Nazarathy, Thomas Taimre

We consider a sequence of independent and identically distributed random vectors with possibly dependent coordinate elements, where the first coordinate has non-negative support. The first coordinate represents the time between occurrences of events, which we call renewals, and the other coordinates represent rewards, that accumulate at the time of the associated renewal. We provide a second order approximation for the covariance matrix of cumulative rewards in terms of the moments and cross moments of the coordinate random variables. Our expression becomes exact as time goes to infinity and extends a classic result of Brown and Solomon for the variance of an individual reward coordinate. We will illustrate this result in the context of loss networks.

In this study, we consider a renewal reward process that expresses an inventory model of type with heavy tailed Pareto distributed demand and uniform distributed interference of chance.

The most important difference of this study from the other studies in the literature is, we assumed the random variables which represent the amount of demands have Pareto

distribution with . It is known that Pareto

distribution is a member of regularly varying subclass of heavy tailed distributions with infinite variance under mentioned conditions. Hence, in order to obtain the renewal function generated by the heavy tailed Pareto distributed random variables, we used a special asymptotic expansion. We show that this process is ergodic under some weak conditions. Moreover, we obtained asymptotic expressions for the order moments of ergodic distribution of the process , as

. Finally, we tested the accuracy of the approximation formulas by using Monte Carlo simulation methods.

Aslı Bektaş Kamışlık, T. Kesemen, T. Khaniyev, Z. Küçük

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Finance and Revenue Management - Alexandra Chronopoulou - Stochastic Systems in Finance

VOLATILITY IS ROUGH

Mathieu Rosenbaum, Jim Gatheral and Thibault Jaisson

Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that log-volatility behaves essentially as a fractional Brownian motion with Hurst H exponent of order 0.1, at any reasonable time scale. This leads us to adopt the fractional stochastic volatility (FSV) model of Comte and Renault. We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, H<1/2 . We demonstrate that our RFSV model is remarkably consistent with financial time series data; one application is that it enables us to obtain improved forecasts of realized volatility. Furthermore, we find that although volatility is not long memory in the RFSV model, classical statistical procedures aiming at detecting volatility persistence tend to conclude the presence of long memory in data generated from it. This sheds light on why long memory of volatility has been widely accepted as a stylized fact. Finally, we provide a quantitative market microstructure-based foundation for our findings, relating the roughness of volatility to high frequency trading and order splitting.

THE ROBUST MERTON PROBLEM OF AN AMBIGUITY AVERSE INVESTOR

Mustafa Pınar, Sara Biagini

We derive a closed form portfolio optimization rule for an investor who is diffident about mean return and volatility estimates, and has a CRRA utility. The novelty is that confidence is here represented using ellipsoidal uncertainty sets for the drift, given a volatility realization. This specification affords a simple and concise analysis, as the optimal portfolio allocation policy is shaped by a rescaled market Sharpe ratio, computed under the worst case volatility. The result is based on a max-min Hamilton-Jacobi-Bellman-Isaacs PDE, which extends the classical Merton problem and reverts to it for an ambiguity-neutral investor.

SIMULATION-BASED FILTERING FOR INFERENCE IN STOCHASTIC VOLATILITY MODELS

Alexandra Chronopoulou

We consider stochastic volatility models, in which the volatility is modeled by a non Markovian process, both in discrete and continuous time. More specifically, we consider the cases of rough and long-range dependent volatility and we employ a sequential Monte Carlo method in order to estimate the parameters of the model. Specifically, we suggest a Sequential Importance Sampling with Resampling (SISR) algorithm for the approximation of the volatility filter. In addition, we estimate the underlying model parameters on-line, by augmenting the unobserved state of the system. We discuss asymptotic properties of the proposed estimators and we apply our method to S&P 500 data.

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72

DYNAMIC CREDIT-COLLECTIONS OPTIMIZATION

Naveed Chehrazi, Peter Glynn, Thomas Weber

Based on a dynamic model of the stochastic repayment behavior exhibited by delinquent credit-card accounts as a self-exciting point process, a bank can control the arrival intensity of repayments using costly account-treatment actions. A semi-analytic solution to the corresponding stochastic optimal control problem is obtained using a recursive approach. For an affine cost of treatment interventions, the optimal policy in the two-dimensional (intensity, balance)-space is described by the frontier of the connected (but nonconvex) action region. The optimal policy significantly reduces a bank’s loss given default and concentrates the collection effort onto the best possible interventions at the best possible times, so as to minimize the sum of the expected discounted outstanding balance and the discounted cost of the collection effort, thus maximizing the net value of any given delinquent credit-card account.


Stochastic Applications - Qiong Wang - Inventory, Assembly and Production Management

ASYMPTOTIC OPTIMALITY OF TAILORED BASE-SURGE POLICIES IN DUAL-SOURCING INVENTORY SYSTEMS

David Goldberg, Linwei Xin

Dual-sourcing inventory systems, in which one supplier is faster (i.e. express) and more costly, while the other is slower (i.e. regular) and cheaper, arise in many real-world supply chains. However, these systems are notoriously difficult to optimize due to the complex structure of the optimal solution and curse of dimensionality.

Recently, Tailored Base-Surge (TBS) policies have been proposed as a heuristic for the dual-sourcing problem, in which a constant order is placed at the regular source in each period, while the order placed at the express source follows a simple order-up-to rule. Numerical experiments by several authors have suggested that such policies perform well as the lead time difference between the two sources grows large, which is exactly the setting in which the curse of dimensionality leads to the problem becoming intractable. However, providing a theoretical foundation for this phenomenon has remained a major open problem.

We provide such a theoretical foundation by proving that a simple TBS policy is indeed asymptotically optimal as the lead time of the regular source grows large, with the lead time of the express source held fixed. Our main proof technique combines a steady-state approach, novel convexity and lower-bounding arguments, a certain interchange of limits result, and ideas from the theory of random walks and queues, significantly extending the methodology and applicability of a novel framework for analyzing inventory models with large lead times recently introduced by Goldberg et al.

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73

AN ANALYTICAL THROUGHPUT APPROXIMATION FOR CLOSED FORK/JOIN NETWORKS

Erkut Sönmez

Queueing networks featuring fork/join stations are natural models for a variety of computer and manufacturing systems, including parallel computer networks, fabrication/assembly systems, supply chains and material control strategies. Unfortunately, an exact solution for a Markovian fork/join network can only be obtained by analyzing the underlying Markov chain using numerical methods and these methods are computationally feasible only for networks with small population sizes and numbers of service stations. In this paper we present a new, simple and accurate analytical approximation method to estimate the throughput of a closed queueing network that features a single fork/join station receiving inputs from general subnetworks. Our technique first estimates arrival processes from the input subnetworks. It then uses the estimated arrival processes, combined with aggregation, to derive a closed form approximate expression for the network throughput (and other performance metrics) by analyzing a simplified Markov chain. An extensive numerical study reveals that our proposed approximation is highly accurate, especially for large network sizes. Specifically, the accuracy of our approximation improves with (i) increasing population size, (ii) degrading network balance, and (iii) increasing number of stations when network is unbalanced. Furthermore, proposed approximation is in general superior to existing techniques in terms of accuracy, and provides significant computational advantage compared to simulation and existing approximation techniques, the latter of which may even fail to provide a solution.

ON THE USE OF BASE STOCK POLICIES IN ASSEMBLE-TO-ORDER INVENTORY SYSTEMS WITH NON-IDENTICAL LEAD TIMES

Qiong Wang

In Assemble-to-Order (ATO) inventory systems with different replenishment lead times, real-time shortage of components with longer lead times affects the usage of other components that are currently being ordered. This important connection is ignored when base stock policies are used to make replenishment decisions. Nevertheless, going beyond base stock policies in systems with multiple products is known to be difficult. So it is interesting to identify situations in which the loss from using a base stock policy is limited or even negligible.

We consider ATO inventory systems with a general bill of material and K >1 different component lead times. We examine the gap between the inventory cost under a base stock policy from a lower bound, obtained by solving a (K+1)- stage stochastic program (SP), which was established in previous work. We develop our base stock policy by collapsing the (K+1)-stage SP into a two-stage SP and using the first-stage optimal solution of the latter to set base stock levels. We consider an asymptotic region in which the lead times of all components become increasingly long, while the difference between the longest and shortest lead times grow at a slower rate than that of the lead times themselves. We show that in this region, our base stock policy, when used together with a previously developed allocation policy, is asymptotically optimal on the diffusion scale, i.e., the percentage difference between the resulting inventory cost and its lower bound converges to zero.

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74

MANAGING CAPACITY BY DRIFT CONTROL

Melda Örmeci Matoglu, John Vande Vate

We model the problem of managing capacity in a build-to-order environment as a Brownian drift control problem and seek a policy that minimizes the long-term average cost. We assume the controller can, at some cost, shift the processing rate among a finite set of alternatives by, for example, adding or removing staff, increasing or reducing the number of shifts or opening or closing production lines. We introduce a practical restriction on this problem, called the S-restricted Brownian control problem, and show how to model it via a structured linear program. We demonstrate that an optimal solution to the S-restricted problem can be found among a special class of policies called deterministic non-overlapping control band policies. These results exploit apparently new relationships between complementary dual solutions and relative value functions that allow us to obtain a lower bound on the average cost of any non-anticipating policy for the problem even without the S-restriction. Under mild assumptions on the cost parameters, we show that our linear programming approach is asymptotically optimal for the unrestricted Brownian control problem.


Stochastic Networks - Neil Walton / Bert Zwart- Proportional Resource Allocation and Scheduling

PROPORTIONAL SWITCHING IN FIFO NETWORKS

Bernardo D’Auria

We consider a family of discrete time multihop switched queueing networks where each packet moves along a fixed route. In this setting, BackPressure is the canonical choice of scheduling policy; this policy has the virtues of possessing a maximal stability region and not requiring explicit knowledge of traffic arrival rates. BackPressure has certain structural weaknesses because implementation requires information about each route, and queueing delays can grow super-linearly with route length. For large networks, where packets over many routes are processed by a queue, or where packets over a route are processed by many queues, these limitations can be prohibitive. We introduce a scheduling policy for FIFO networks, the Proportional Scheduler, which is based on the proportional fairness criterion. We show that, like BackPressure, the Proportional Scheduler has a maximal stability region and does not require explicit knowledge of traffic arrival rates. The Proportional Scheduler has the advantage that information about the network’s route structure is not required for scheduling, which substantially improves the policy’s performance for large networks. For instance, packets can be routed with only next-hop information and new nodes can be added to the network with only knowledge of the scheduling constraints.

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75

RESOURCE SHARING WITH LOGARITHMIC WEIGHTS

Philippe Robert

The properties of a class of resource allocation algorithms for communication networks are investigated: if a node of this network has requests to transmit, then it receives a fraction of the capacity proportional to log(1+x), the logarithm of its current load. A detailed fluid scaling analysis of such a network with two nodes is presented. It is shown that the interaction of several time scales plays an important role in the evolution of such a system; in particular its coordinates may live on very different time and space scales. As a consequence, the associated stochastic processes turn out to have unusual scaling behaviors. A heavy traffic limit theorem for the invariant distribution is also proved. Finally, we present a generalization to the resource sharing algorithm for which the function is replaced by an increasing function. Generalizations of these results to a star topology are presented.

INSENSITIVITY OF PROPORTIONAL FAIRNESS IN CRITICALLY LOADED BANDWIDTH SHARING NETWORKS

Jiheng Zhang

Proportional fairness is a popular service allocation mechanism to describe and analyze the performance of data networks at flow level. Recently, several authors have shown that the invariant distribution of such networks admits a product form distribution under critical loading. Assuming exponential job size distributions, they leave the case of general job size distributions as an open question. In this paper we show that product form in heavy traffic still holds for general distributions, thus settling the conjecture. More importantly, we establish insensitivity of proportional fairness in heavy traffic.

QUALITATIVE PROPERTIES OF -FAIR POLICIES IN BANDWIDTH-SHARING NETWORKS

Yuan Zhong, Devavrat Shah, John Tsitsiklis

We consider a flow-level model of a network operating under an α-fair bandwidth sharing policy (with α>0). This is a probabilistic model that captures the long-term aspects of bandwidth sharing between users or flows in a communication network. In this talk, we establish a variety of results on the performance properties of the alpha-fair policies, including both transient properties as well as the steady-state distribution of the model. These results include tail bounds on the size of a maximal excursion during a finite time interval and exponential tail bounds under the steady-state distribution. An important aspect of our analyses is the application of various Lyapunov drift conditions.

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76


Stochastic Systems - Wen Sun - Stochastic Networks

RANDOMIZED LOAD BALANCING NETWORKS WITH GENERAL SERVICE DISTRIBUTIONS

Mohammadreza Aghajani, Kavita Ramanan

Dynamic load balancing models are of importance in a variety of applications in computer science and communication networks. A randomized load balancing model that has received a lot of attention is the so-called supermarket model, in which jobs arriving to a parallel server network choose two queues at random and join the shorter of the two. This model and its variants have been analyzed extensively under the assumption of exponential service distributions. However, in many applications, service distributions are typically not exponentially distributed. We introduce a general framework for studying this model in the case of general service distributions with finite mean when a first-come-first serve policy is used within each queue. We use an interacting measure-valued process representation of the state to obtain hydrodynamic or fluid limits. Our framework allows for the study of both the transient behavior of the system and its steady state performance. In particular, we obtain a tractable description of the evolution of the queue length process in terms of a partial integro-differential equation. We use this equation to gain insight into the behavior of the system, and deduce some non-intuitive results on the effect of the properties of the service distribution on system performance.

PERFORMANCE ANALYSIS OF A NETWORK OF TRANSITORY OF QUEUES

Rahul Jain, Harsha Honnappa

Consider a freeway network, such as that covers Los Angeles County in the United States. Traffic arriving at the entire network displays obvious ‘diurnal’ behavior with arrival rates peaking at rush hours. Furthermore, the traffic patterns repeat everyday. The non-stationary and periodic traffic behavior implies that standard performance analysis results on generalized Jackson networks in heavy-traffic do not pertain. In this work, we focus on developing diffusion approximations to the sojourn time in a network of single server queues, over a finite time horizon (or single ‘day’), when traffic is time inhomogeneous. In particular, we adopt a ‘population acceleration’ regime where the number of arrivals to the network is scaled to infinity and service rates are accelerated by the population size - a regime that is particularly relevant to transportation networks. We point out several difficulties in obtaining the diffusion approximations, and show how network topology can be leveraged to simplify the computations.

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77

EQUIVALENCE OF ENSEMBLES AND APPROXIMATION FOR LARGE BIKE-SHARING SYSTEMS

Danielle Tibi, Christine Fricker

A bike-sharing system can be modeled by a closed Jackson network. In this description, bikes constitute the fixed set of “customers,” moving across the network, while bike-riders play the role of servers. Nodes are of two types: finite capacity stations, which operate as one-server nodes, and infinite capacity routes, with infinite-server processing mechanism. The general network is heterogeneous, as regards the routing matrix and the different stations capacities and service parameters. The effect of saturation at some station can be modeled by blocking and rerouting, which well matches the behavior of users at saturated end-of-ride stations. This dynamics preserves the product form (on non-product state space) of the equilibrium distribution of the nodes occupation process - well-known for standard, infinite capacity, Jackson networks. Using a Local Limit Theorem for independent, non-identically distributed random variables, one can prove asymptotic independence of the nodes at stationarity as the network gets large, under mild assumptions. In the statistical mechanics terminology, the equivalence of ensembles - canonical and grand canonical - is proved. The limiting distributions of the different queues are explicit, geometric as concerns the stations. This grand canonical approximation can be used for adjusting the total number of bikes in a way to get an acceptable trade-off between the two types of service failure - namely finding no bike for rent, or no place for parking.

THE TIME SCALES FOR A STOCHASTIC NETWORK WITH FAILURES

Wen Sun, Mathieu Feuillet, Philippe Robert

A transient Markov process with d+1 nodes and an absorbing state is studied to investigate the qualitative behavior of a large scale storage network of non-reliable file servers. Each file has at most d copies and each copy of a file is lost at some rate. The network can duplicate files with at least 1 copy and less than d copies. In this talk it is assumed that the duplication capacity of the system is allocated to the files with the least number of copies. A file with 0 copies is lost for good. When the size of system, i.e. the number of servers, goes to infinity, it is shown that there is a critical value for the mean number of files per server such that below this quantity, the system is stable, it stays away from absorbing state - all files lost. In this case, a quasi-stationary state is reached where most of files have the maximum number of copies. Above this value, the network loses quickly a significant fraction of files until some equilibrium is reached. By a convenient scaling of the time scale, the evolution of the network towards the absorbing state can be described via a stochastic averaging principle. It is shown that the fluctuations conveniently renormalized converge to an Ornstein-Uhlenbeck process.

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July 6th - Monday 16:10-17:50

Stochastic Processes - Arka Ghosh - Stochastic Systems

DECENTRALIZED CONTROLS IN MULTICLASS NETWORKS: BETWEEN DIFFUSION APPROXIMATIONS AND ROBUST OPTIMIZATION

Itai Gurvich, Chaithanya Bandi

The Lu-Kumar network is an instance of a multi-class network in which decentralized priority rules lead to instability of the network even when the resource load is less than a 100 percent. In our work, we study conditions on the network topology that guarantee with such load the stability of the network under any decentralized priority rule. Our approach blends results from diffusion approximations and tools from robust optimization to generate both computational tools and structural insights.

AN ε-NASH EQUILIBRIUM FOR STRATEGIC CUSTOMERS IN HEAVY TRAFFIC

Subhamay Saha, Rami Atar

We consider a multiclass queue with many servers and strategic customers. Upon arrival, customers observe the queue length of the corresponding class and decide whether to join or leave. The payoff for a customer is given by a function of the waiting time, in case he joins, and a fixed cost for not receiving service, in case he leaves. Due to the randomness of the waiting time, this formulates the payoff as a random function of the customer’s decision, for which the notion of an equilibrium is only meaningful in an asymptotic framework. The asymptotics we consider are the Halfin-Whitt heavy traffic regime, where the number of servers grows without bound, and so does the number of customers involved (on any fixed finite time interval).The servers apply a scheduling policy of which the customers are unaware. We identify a strategy that, if adopted by all customers, gives rise to an ε-Nash equilibrium, with probability approaching 1 in the scaling limit.

STEADY-STATE DISTRIBUTION CONVERGENCE FOR GI/ GI/1+GI QUEUES IN HEAVY TRAFFIC

Amy Ward, Chihoon Lee

We establish the validity of the heavy traffic steady-state approximation for a single server queue, operating under the FIFO service discipline, in which each customer abandons the system if his waiting time exceeds his generally-distributed patience time. This follows from early results of Kingman when the loading factor approaches one from below, but has not been shown in more generality. We prove the convergence of the steady-state distributions of the offered waiting time process and their moments both under the assumption that the hazard rate of the abandonment distribution is scaled and that it is not scaled. As a consequence, we establish the limit behavior of the steady-state abandonment probability and mean queue-length.

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79

A MULTICLASS QUEUEING MODEL IN THE MODERATE-DEVIATION HEAVY-TRAFFIC REGIME

Asaf Cohen, Rami Atar

We consider a multiclass single-server queueing control problem in the moderate-deviation heavy-traffic regime with a discounted risk-sensitive cost. In the scaling limit, an optimal control problem associated with the model is shown to be governed by a differential (deterministic) game that can be explicitly solved and that admits an optimal stationary feedback policy. We also present a stationary asymptotic optimal policy that satisfies a state space collapse property.


Finance and Revenue Management - Süleyman Özekici - Risk, Price and Inventory Models

STATIC AND DYNAMIC HEDGING POLICIES FOR AN INVENTORY SYSTEM WITH STOCHASTIC PRICES

Caner Canyakmaz, Süleyman Özekici, Fikri Karaesmen

We consider the joint inventory management and financial hedging problem of a risk-averse firm that sells a commodity-based product. We assume that the customer arrival process and sales prices are governed by the stochastic market price process of the commodity which is correlated with the price processes of financial indices in the market. At predefined trading times in the selling season, the firm has the opportunity to invest in a financial portfolio to manage its exposure to price and demand uncertainties. We employ a two-phase procedure such that an inventory decision is made at first and then the corresponding portfolio which minimizes the variance of the final cash flow from both operational and hedging decisions is found. In the static hedging case, a one-time portfolio involving the decisions at all trading points and securities is constructed at the beginning of the sales season. We give a complete characterization of the optimal base-stock level and the optimal portfolio. In the dynamic hedging case, however, the decision maker observes current inventory, wealth and price levels to change his portfolio dynamically throughout the sales season. In this case, we assume that the financial securities are fairly priced and follow martingale price processes. We characterize the optimal trading policy and base-stock level by using dynamic programming.

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80

ON THE END OF LIFE PROBLEM IN INVENTORY CONTROL

Hans Frenk, Semih Onur Sezer, Sonya Javadi Khatab

We consider an inventory problem of controlling the inventory of spare parts in the final phase of the service life cycle. The final phase starts when the production of a product is terminated and it continues until the last service contract or warranty period expires. Placing final orders for service parts at the end of the production cycle of a product is considered to be a popular tactic to satisfy demand during this period and to mitigate the effect of part obsolescence at the end of the service life cycle. Previous research focuses on repairing defective products by replacing the defective parts with properly functioning spare ones. However, for most of the inventory problems with the product in a no production phase there is typically a price erosion for the new type of product presently in production while repair cost for a defective product of a previous generation stays steady over time. As a consequence, there might be a point in time at which the unit price of a new generation product drops below the repair costs. If so, it is more cost effective to adopt an alternative policy to meet service demands toward the end of the final phase, such as offering customers the new product of a similar type or a discount on a next generation product. This study examines the cost trade-offs of implementing alternative policies for the repair policy using martingale theory techniques.

MEAN-VARIANCE NEWSVENDOR MODEL WITH RANDOM SUPPLY AND FINANCIAL HEDGING

Süleyman Özekici, Müge Tekin

We follow a mean-variance (MV) approach to the newsvendor model. Unlike the risk-neutral newsvendor that is mostly adopted in the literature, the MV newsvendor considers the risks in demand as well as supply. We further consider the case where the randomness in demand and supply is correlated with the financial markets. The MV newsvendor hedges demand and supply risks by investing in a portfolio composed of various financial instruments. The problem therefore includes both the determination of the optimal ordering policy and the selection of the optimal portfolio. Our aim is to maximize the hedged MV objective function. We provide explicit characterizations on the structure of the optimal policy. We also present numerical examples to illustrate the effects of risk-aversion on the optimal order quantity and the effects of financial hedging on risk reduction.

ORDERING POLICIES FOR TWO PRODUCTS WITH DEMAND SUBSTITUTION

Odysseas Kanavetas, Apostolos Burnetas

We consider the problem of ordering for two products with stochastic demand and partial demand substitution. Successive demands arrive at random times, thus the order of arrivals affects the ending inventory and/or shortages. In previous work we have established the supermodularity of the expected cost function, which facilitates the computation of the optimal ordering policy. Here we consider extensions to the case of unknown parameters and suggest adaptive ordering policies.

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81


Stochastic Applications - Onno Boxma - Performance Analysis of Communication Systems

MODELLING AND ANALYSIS OF OPTICAL SWITCHES

Murtuza Ali Abidini

Optical Networks today are a major part of every communication network. From intercontinental transmission lines to wireless routers, optical networks not only form the backbones of such large systems but also find application in device level circuits. There is ongoing research to include them in integrated circuits as well. Among the many advantages that optical networks offer, very high speeds and bandwidth are two of the most important reasons why they could conquer the communication arena. However, due to certain properties of optical carriers, data has to be converted into electrical data at several epochs and reconverted. This acts as bottle neck for many important properties, especially the speed. To overcome this effect, new, completely optical switches are being devised. Unlike typical switches, these don’t have a memory storage buffer, thereby the usual queuing models for performance analysis do not apply. To address this issue, we propose a basic model. We put forward a vacation-type queuing model, and a single-server multi-queue polling model, with the special feature of retrials. Just before the server arrives at a station there is a glue period. Customers (both new arrivals and retrials) arriving at the station during this glue period will be served during the visit of the server. Customers arriving in any other period leave immediately and retry after an exponentially distributed time. We then analyze the several performance measures, like queue length of such a model, and try to optimize them.

WAITING TIME DISTRIBUTIONS FOR POLLING SYSTEMS IN HEAVY TRAFFIC

Petra Vis, R. Bekker, R.D. van der Mei

In a polling model there is one server that serves multiple queues in some prespecified order. We consider a polling model with cyclic server routing and exhaustive service. This means that the server serves the queue until it is empty and then switches to the next queue. In the vast majority of papers that appeared on polling models, it is assumed that at each of the individual queues, customers are served on a First-Come-First-Served (FCFS) basis. In this talk we will study other local service policies like Last-Come-First-Served (LCFS), Random Order of Service (ROS), Processor Sharing (PS) and Shortest-Job-First (SJF). For each of these local policies we derive the scaled waiting time distribution in heavy traffic. For FCFS the scaled waiting time distribution is known to be a uniform times gamma distribution. For other service disciplines, the gamma distribution remains unaffected but we obtain a galaxy of results as alternative for the uniform distribution. In particular, we get uniforms with a point mass in zero, trapezoidal distributions and mixtures of these. Compared to the gated service discipline, we notice that the waiting time results in heavy traffic are now more involved.

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82

ON THE SCALABILITY AND MESSAGE COUNT OF COUNTER-BASED BROADCASTING SCHEMES

Thomas Meyfroyt, Sem Borst, Onno Boxma, Dee Denteneer

As the use of wireless sensor networks increases, the need for efficient and reliable broadcasting algorithms grows. Ideally, a broadcasting algorithm should have the ability to quickly disseminate data, while keeping the number of transmissions low. We analyze the popular Trickle algorithm, which has been proposed as a suitable communication protocol for code maintenance and propagation in wireless sensor networks. Trickle relies on a counter-based suppression mechanism to keep the number of redundant transmissions low. We show that the broadcasting process of a network using Trickle can be modeled by a Markov chain and that this chain falls under a class of Markov chains, closely related to residual lifetime distributions. It is then shown that this class of Markov chains admits a stationary distribution of a special form. These results are used to analyze the Trickle algorithm, its scalability and its message count. Our results prove conjectures made in the literature concerning the effect of a listen-only period. Besides providing a mathematical analysis of the algorithm, we propose a generalized version of Trickle, with an additional parameter defining the length of a listen-only period.

PARTITIONING TRAFFIC CLASSES IN PRIORITY CLASSES: DELAY ANALYSIS OF A MULTICLASS PRIORITY QUEUE

Joris Walraevens

We analyze the delay experienced in a discrete-time priority queue with a so-called train arrival process. An infinite user population is considered. Each user occasionally sends packets in the form of trains: a variable number of fixed-length packets are generated and these packets arrive to the queue at the rate of one packet per slot. Previous studies assumed two traffic classes, with one class getting priority over the other. We extend these studies to cope with a general number M of traffic classes that can be partitioned in an arbitrary number N of priority classes. The lengths of the trains are traffic-class-dependent and generally distributed. To cope with the resulting general model, a Markovian state vector of infinite size is introduced. By using probability generating functions, moments and tail probabilities of the steady-state packet delays of all traffic classes are calculated. The impact of the model parameters are studied through some numerical examples, where we pay special attention to the partitioning of traffic classes in priority classes.

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Stochastic Networks - Mariana Olvera-Cravioto - Applications and Algorithms for Random Graphs

QUANTITATIVE STUDY ON THE INFLUENCE OF GEOMETRICAL AND TOPOLOGICAL RANDOMNESS IN PLANAR ELASTIC NETWORKS

Uwe Muehlich

Cellular materials like foams, bone scaffolds, etc. are commonly interpreted as networks. Here, planar elastic central force networks are exposed to an overall strain and the networks’ response is measured in terms of strain energy density. Topology is varied randomly under the condition that the number of nodes and edges in the network are fixed. The corresponding sampling is performed by means of an M×M Ising-model with fixed magnetization. Subsequently, topology and geometry are varied simultaneously. The major trend in the overall response is determined by topological properties. However, a detailed quantitative study reveals that the network’s response is also affected by an interplay between topological and purely geometrical characteristics. Furthermore, severe clustering effects can be observed. Therefore the problem of defining appropriate measures accounting for the particular type of clustering observed here is addressed.

DEGREE-DEGREE CORRELATIONS IN SCALE-FREE DIRECTED RANDOM GRAPHS

Nelly Litvak, Pim van der Hoorn

Correlation between degrees of neighboring nodes plays an important role in a wide range of processes in real-life networks, such as information diffusion and spreading of infections. We study the behavior of degree-degree dependencies in directed random graphs with heavy-tailed degree distributions. Statistical analysis of such dependencies requires null models, i.e. models that generate uncorrelated scale-free networks. Most models to date however show structural negative dependencies, generated by finite size effects. We analyze these negative degree-degree dependencies, using rank based correlation measures, in the directed Erased Configuration Model. We prove that this model is asymptotically uncorrelated and obtain the scaling for its degree-degree correlations as a function of the power law exponents of in- and out-degrees. We show that this scaling undergoes an interesting phase transition, that is, different scaling applies in different regions of the exponents.

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84

LARGE SUBMATRIX DETECTION IN GAUSSIAN RANDOM MATRICES

David Gamarnik

EFFICIENT SIMULATION FOR WEIGHTED BRANCHING TREES

Ninguyan Chen

Motivated by recent results for the analysis of information ranking algorithms on complex networks and parallel queueing networks with synchronization, we propose an efficient algorithm for simulating the endogenous solution to max-plus branching stochastic fixed-point equations. The aforementioned solutions can be constructed on a weighted branching process, but closed-form expressions for their distributions are in general unavailable. Moreover, unlike a Galton-Watson process, their Laplace transform cannot be directly inverted. Hence, the availability of efficient simulation techniques is extremely important for numerically approximating the distributions and moments of these solutions. Naive Monte Carlo techniques, however, are extremely inefficient in this context due to the geometric growth of a weighted branching process. We describe in this talk an algorithm based on iterative bootstrap whose complexity grows linearly (as opposed to exponentially) in the number of generations of the weighted branching process being simulated. We show the consistency of a wide class of estimators based on this technique.

Iterative Search Procedure is a computationally efficient approach for finding large average submatrices of a real‐valued matrix in the exploratory analysis of high dimensional data. It alternately updates rows and columns until no further improvement can be obtained in terms of the average of the submatrix. However, there is no theoretical understanding of whether this procedure converges to a global maximum.

In this paper, we present first theoretical analysis of the performance of the Iterative Search Procedure for finding large average submatrix in Gaussian random matrices. We show that the Iterative Search Procedure proceeds steps with high probability (w.h.p.), and converges to a local maximum submatrix w.h.p. More specifically, while the average of the global

maximum submatrix is known to be , the average of the submatrix the

Iterative Search Procedure converges to is w.h.p., thus leading to a constant factor gap.

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Stochastic Systems - Jim Dai - Asymptotic Analysis for Stochastic Systems

WHEN FIXED PRICE MEETS PRIORITY AUCTIONS: SERVICE SYSTEMS WITH DUAL MODES

Krishnamurthy Iyer, Jiayang Gao and Hüseyin Topaloğlu

Many service systems offer multiple modes of service, where each mode differs in the price as well as the priority of service. We consider a service system where service is offered via two modes. In the first mode, customers obtain service at a fixed price, and the service discipline is first-in-first-out (FIFO). In the second mode, referred to as the bid-based priority queue, customers submit a bid to obtain service, get serviced in the descending order of their bids, and make payments equal to their bids. We assume the customers are heterogeneous, with different waiting costs, and choose the mode of service strategically on arrival to the system. Under a mild technical assumption, we establish the existence and uniqueness of a symmetric equilibrium, and characterize the structure of the equilibrium strategy. In particular, we show that the equilibrium strategy has a simple threshold structure, where customers with either high or low waiting cost choose to obtain service from the bid-based priority queue, whereas those with moderate waiting cost choose to obtain service from the FIFO queue. We compare the social welfare and the revenue generated in equilibrium under different allocation of service capacity between the two modes, and show that the social welfare is maximized when all the capacity is allocated to the bid-based priority queue. Finally, we investigate the equilibrium of the service system in the limiting regime of large arrival rates and service capacity.

LOCATIONAL MARGINAL VALUE OF STORAGE CAPACITY

Subhonmesh Bose

In the context of power system operations, we consider the system operator’s problem of optimizing the expected cost of dispatch over a finite time horizon, given access to spatially distributed energy storage resources to balance a stochastic net demand process evolving over a transmission-constrained power network. The expected benefit of storage capacity under optimal dispatch policy is shown to be concave and non-decreasing in the energy storage capacities. Naturally, the greatest marginal value of storage is derived at small installed capacities. Our main result characterizes this marginal value in terms of the maximum revenue one can derive through a causal arbitrage against the nodal electricity prices. We also derive an upper bound for the marginal value in terms of the revenue through an arbitrage on the said prices under perfect foresight. This bound is shown to be tight when the cost of dispatch is spatially uniform and the network topology is acyclic. Furthermore, we explore cases when the marginal value approaches the upper bound in the continuous time limit. The analysis provides computationally tractable tools to empirically calculate the locational marginal value of storage from time series data of net demand. Though our formulation is motivated by power systems, the result applies more broadly to networked inventory control problems.

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86

THE VALUE OF DYNAMIC PRICING IN RIDE-SHARE PLATFORMS

Siddhartha Banerjee, Ramesh Johari, Carlos Riquelme

Ride-sharing platforms like Lyft and Uber are one of the fastest growing online marketplaces. Much of their success is ascribed to their ability to do fine-grained fast-timescale dynamic pricing - where prices can react to instantaneous system state, and across very small geographic areas. To explore the value of such fast-timescale dynamic pricing, we develop a model for ride-share platforms, which combines a queueing model for the dynamics of the platform’s operations with strategic models of both passenger and driver behavior. Using this, under an appropriate large-market scaling, we study various aspects of this system - the value of dynamic pricing versus static pricing; the robustness of these policies with respect to imperfect knowledge of system parameters; the effect of heterogeneous ride-request rates and traffic between different locations.

ASYMPTOTIC COUPLING OF STANDARD AND TICKET QUEUES IN HEAVY TRAFFIC

Jamol Pender

Upon arrival to a ticket queue, a customer is offered a slip of paper with a number on it -- indicating the order of arrival to the system -- and is told the number of the customer currently in service. The arriving customer then chooses whether to take the slip or balk, a decision based on the perceived queue length and associated waiting time. Even after taking a ticket, a customer may abandon the queue, an event that will be unobservable until the abandoning customer would have begun service. In contrast, a standard queue has a physical waiting area so that abandonment is apparent immediately when it takes place and balking is based on the actual queue length at the time of arrival.We prove heavy traffic limit theorems for the generalized ticket and standard queueing processes, discovering that the processes converge together to the same limit, a regulated Ornstein-Uhlenbeck (ROU) process. One conclusion is that for a highly utilized service system with a relatively patient customer population, the ticket and standard queue performances are asymptotically indistinguishable on the scale typically uncovered under heavy traffic approaches.


Queues and Control - Bora Keskin - Stochastic Control Applications / Dynamic Learning

DYNAMIC RESERVE PRICES FOR REPEATED AUCTIONS: LEARNING FROM BIDS

Hamid Nazerzadeh

A large fraction of online advertisements are sold via repeated second price auctions. In these auctions, the reserve price is the main tool for the auctioneer to boost revenues. In this work, we investigate the following question: Can changing the reserve prices based on the previous bids improve the revenue of the auction, taking into account the long-term incentives and strategic behavior of the bidders? We show that if the distribution of the valuations is known and satisfies the standard regularity assumptions, then the optimal mechanism has a constant reserve. However, when there is uncertainty in the distribution of the valuations and competition among the bidders, previous bids can be used to learn the distribution of the valuations and to update the reserve prices. We present a simple approximately incentive-compatible and optimal dynamic reserve mechanism that can significantly improve the revenue over the best static reserve in such settings.

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87

UPPER BOUNDS FOR THE CORRELATED BAYESIAN INFORMATION FILTERING PROBLEM

Bangrui Chen, Peter Frazier

Information filtering systems automatically distinguish relevant from irrelevant items (emails, news articles, intelligence information) in large information streams. They typically use a machine learning classifier, trained on past items. However, when filtering for new users, or when item contents or user interests change, sufficient training data may not be available. In this situation, it may be beneficial to actively explore user interests by forwarding items to learn about their relevance, but too much exploration degrades performance. This is the so-called exploration vs. exploitation tradeoff. We present a Bayesian sequential decision-making formulation of this problem, where user interests are described by a linear model, similar in spirit to a Bayesian linear bandit. Although this problem’s solution is characterized by a stochastic dynamic program, this stochastic dynamic program cannot be solved in practice because its complexity scales exponentially with the dimension of the linear model. We construct a computationally tractable upper bound, which scales linearly with the dimension of linear model, and which allows us to bound the optimality gap for heuristic policies. We also prove that our upper bound is asymptotically tight under conditions on the covariance of the prior and the limiting item feature vectors, and show that a heuristic policy based on this upper bound is asymptotically optimal under the same conditions. Numerical experiments suggest that our heuristic policy is at least as good as an upper confidence bound algorithm.

APPROXIMATE GITTINS INDICES FOR MULTI-ARMED BANDITS

Eli Gutin, Vivek Farias

There has recently been considerable interest in applying Bayesian techniques to the frequentist multi-armed bandit problem. A number of algorithms have been designed by assuming a prior on each arm’s reward, including Bayes-UCB, Information-Directed Sampling (IDS) and Thompson sampling. As well as being intuitively appealing, these approaches are competitive in the non-Bayesian setting against frequentist UCB algorithms. To explore the Bayesian framework more, we propose a simple and efficient algorithm which works by approximating the solution to the multi-stage dynamic programming problem used in Gittin’s index theorem. Simulation results shows our method achieve a lower expected Bayesian regret than both IDS and Thompson sampling. We also show that the algorithm attains the Lai and Robbins’ regret bound in the case of Bernoulli rewards.

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88

DYNAMIC PRICING AND PRODUCT DIFFERENTIATION WITH COST UNCERTAINTY AND LEARNING

Bora Keskin, John Birge

Motivated by applications in the health insurance industry, we consider a seller who designs and sells a set of vertically differentiated products to a population of quality-sensitive customers. The seller’s business environment entails an uncertainty about production costs. We characterize the seller’s optimal price-quality schedule in the cases of: (a) static cost uncertainty, and (b) dynamic learning about cost uncertainty through noisy observations on an underlying cost curve. We prove that the seller’s optimal quality allocations in (a) and (b) stand in stark contrast: While a seller facing static cost uncertainty degrades the quality in its product offering, a dynamically learning seller improves the quality of its products to accelerate information accumulation. In the case of dynamic learning, we prove that the seller exercises the most extreme experimentation on less quality-sensitive customers. We also extend our results to the cases of commonly used regulations in health insurance industry, and show how such regulations moderate the interplay between uncertainty and learning.


MDP - Hayriye Ayhan / Spyros Reveliotis - Control

PLANNING HORIZONS AND DISCOUNT RATE SENSITIVITY

Mark Lewis, Anand Paul

A turnpike integer is the smallest finite horizon for which an optimal infinite horizon decision is the optimal initial decision. An important practical question considered in the literature is how to bound the turnpike integer using only the problem inputs. We consider turnpike integers as a function of the discount factor. We show that the turnpike integer, while finite for a fixed discount factor, can approach infinity at specific discount factors. We completely characterize this set of discount factors and find necessary and sufficient conditions for a set of turnpike integers to be unbounded. This finding provides a cautionary tale for practitioners using point estimates of the discount factor to manage the length of rolling horizon procedures. Using a new uniform convergence theorem for finite state and action Markov decision processes (uniform in discount rate) we also obtain additional structural results in turnpike planning horizon theory.

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89

OPTIMAL DISCHARGE AND OVERFLOW POLICIES WITH A STEADY FLOW OF ELECTIVE SURGICAL PROCEDURES

Lerzan Örmeci, Hessam Bavafa, Sergei Savin

We consider a hospital department that can perform several types of elective surgical procedures, where each procedure is differentiated by its revenue and its probability distribution for hospital length of stay. The hospital policy is to schedule the same daily portfolio of surgical procedures. The department is allocated a fixed number of beds. Given that the patients’ length-of-stays are random, it is possible that the number of patients who need to stay overnight can be larger than the number of beds allocated to the department. In such situations, the hospital offers two types of solutions, either having some of the patients overflow to another department or early discharging some of the patients. In this study, we analyze the structure of optimal discharge and overflow policies to maximize the total expected profit.

PERFORMANCE OPTIMIZATION OF STOCHASTIC NETWORKS WITH BLOCKING AND DEADLOCKING EFFECTS

Spyros Reveliotis, Ran Li

This presentation will consider the long-run performance optimization of stochastic networks with finite resources and a resource allocation protocol that can lead to blocking and deadlocking effects. The dynamics of the considered networks are represented in the modeling framework of the Generalized Stochastic Petri Nets (GSPNs) and this representation allows for the synthesis and the representation of the necessary deadlock avoidance policy as a concise subnet superimposed on the original net. In the resultant net, timed transitions model activities corresponding to actual processing operations, while untimed transitions model decisions relating to the resource allocation function. Hence, the scheduling problem of maximizing the long-run performance of the considered GSPN with respect to some reward function defined on the net markings reduces to the problem of arbitrating the conflicts that will arise among the untimed transitions that are activated at the various net markings. In the talk we shall show how such a representation of the considered scheduling problem can lead to the specification of pertinent policy spaces that admit a parsimonious representation in terms of the involved parameters, and how the resultant scheduling problems can be solved effectively through stochastic approximation algorithms. We shall also show that the considered policy spaces correspond to certain aggregations of the underlying state space, and that additional performance enhancement can be attained through controlled partial disaggregation. The entire set of results will be concretized and exemplified through application to capacitated re-entrant lines, i.e., re-entrant lines with finite buffers at the line stations.

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90

OPTIMAL ADMISSION CONTROL FOR TANDEM MARKOVIAN LOSS SYSTEMS

Daniel Silva, Hayriye Ayhan

We study a system of two single-server-queues in tandem. Each queue has an arbitrary finite buffer, arrivals come only to the first station, following a Poisson process, and service times at each station are exponentially distributed. A gatekeeper must decide whether to admit or reject each incoming customer upon arrival. Losses occur when the second station is full at the time of a service completion at the first station. Costs are incurred either when a customer is rejected at the first station or a loss occurs at the second station. The objective is to determine the optimal admission control policy that minimizes the long-run average cost. We identify states for which the optimal action is always to admit incoming arrivals. We define a Prudent Policy which rejects arrivals in all states, except those; and a Greedy Policy which admits in all states where it is possible. We provide necessary and sufficient conditions for the Prudent policy to be optimal and a closed-form expression for the long-run average cost. We also provide sufficient conditions for the Greedy policy to be optimal, as well as a matrix-analytic solution for the long-run average costs. We also propose heuristic policies and use numerical experiments to show that they provide near optimal performance.


Stochastic Applications - Antonio Gomez-Corral - Stochastic Models in Biology

A MODEL FOR CELL PROLIFERATION IN A DEVELOPING ORGANISM

Laleh Tafakori, Peter Taylor

In mathematical biology, there is much interest in building continuum models by scaling discrete agent-based models, governed by local stochastic rules.We shall discuss a particular example: a model for the process that leads to Hirschprung’s disease, a potentially-fatal condition in which the enteric nervous system of a new-born child does not extend all the way through the intestine and colon, as well as various other stochastic models for foetal tissue growth.We start with a simple discrete-state Markov chain model proposed by Hywood in 2012 for the location of the neural crest cells that make up the enteric nervous system, and consider a variety of limiting regimes that lead to partial differential equation models that describe the dynamics of crest cell density as the whole gut grows. The initial discrete-state model has properties that are reminiscent of the totally asymmetric exclusion process with a variable-size lattice. When a neural crest cell proliferates, the size of the whole lattice increases by one and a cell from the underlying domain is left behind the proliferating cell. Since tissue growth is a significant cellular transport mechanism during embryonic growth, it has an indispensable role in the derivation of the correct partial differential equation description.

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91

STOCHASTIC MODELS OF PROTEIN PRODUCTION WITH FEEDBACK

Philippe Robert, Renaud Dessalles, Vincent Fromion

Protein production is the fundamental process by which the genetic information of a biologic cell is synthesized into a functional product, the proteins. For prokaryotic cells (like bacteria), this is a highly stochastic process and results from the realization of a very large number of elementary stochastic processes of different nature. This talk investigates the impact of feedback mechanisms to regulate protein production in bacteria, in particular to reduce the variance of the numbers of different types of proteins produced. For a given protein, the control investigated consists in binding one of the proteins to the promoter of its own gene, preventing the transcription of the gene, and therefore the production of the associated protein. A stochastic model of this particular feedback is introduced. It is analyzed and compared with an analogous stochastic model but without feedback. Results, both theoretical and computational, tend to show the limited impact of the feedback in reducing the variance of protein production. However, in case of radical change in the environment, when the production of a protein has to dramatically change, we show that the model with feedback adapts more rapidly than its counterpart without feedback.

HETEROGENEOUS CONTACTS IN STOCHASTIC SIS EPIDEMIC MODELS

Antonio Gomez-Corral

In this talk, a stochastic model for the spread of an SIS epidemic among a population consisting of N individuals, each having heterogeneous infectiousness and/or susceptibility, is considered and its behavior is analyzed under the practically relevant situation when N is small. The model is formulated as a finite time-homogeneous continuous-time Markov chain X. Based on an appropriate labeling of states, we first construct its infinitesimal rate matrix by using an iterative argument, and we then present an algorithmic procedure for computing steady-state measures, such as the number of infected individuals, the length of an outbreak, the maximum number of infectives, and the number of infections suffered by a marked individual during an outbreak. The time till the epidemic extinction is characterized as a phase-type random variable when there is no external source of infection, and its Laplace-Stieltjes transform and moments are derived in terms of a forward elimination backward substitution solution. The inverse iteration method is applied to the quasi-stationary distribution of X, which provides a good approximation of the process X at a certain time, conditional on non-extinction, after a suitable waiting time. The basic reproduction number R0 is defined here as a random variable, rather than an expected value.

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92

A PURITY-DEPENDENT MARKOV MODEL FOR THE EVOLUTION OF MICROSATELLITES

Tristan Stark, Malgorzata O, Barbara Holland, Bennet McComish, David Lambert

Microsatellites are repetitive regions of DNA where a short motif is repeated many times. Mutations in the number of repeat units occur frequently compared to point mutations and thus provide a useful source of genetic variation for studying recent events. It is thought from empirical studies that the rate of length changing mutations due to slipped-strand mispairing can depend on the purity of the repeat units, i.e. how well they each match the motif. However, most studies that use microsatellite data are based on models that only track the number of repeat units. We introduce a Markov model on a two-dimensional state-space that tracks both the length and purity of microsatellites. This model extends existing models by including point mutation and allowing it to affect the rate of slipped-strand mispairing. We find that models in which impurity reduces the rate of slipped-strand mispairing provide a significantly improved fit to microsatellites derived from whole-genome sequence data.

July 7th - Tuesday 10:45-12:25Room No: ENG B05

Stochastic Networks and Processes - Yunan Liu - Network of Queues: Approximations and Control

A MANY-SERVER HEAVY-TRAFFIC LIMIT FOR THE OVERLOADED Gt/GI/n+GI QUEUE

A. Korhan Aras

We establish a many-server heavy-traffic functional central limit theorem (FCLT) for key performance processes such as potential waiting time, number of abandonment and queue length for the Gt/GI/n+GI queue in the overloaded regime. We obtain a stochastic differential equation (SDE) driven by a Gaussian process in the limit for the scaled waiting time process. The Gaussian limit and Gaussian integral appear in the limit of the departure process which is not a Brownian motion when the service distribution is not exponential. When the arrival process is stationary, a special case of our model can help approximate the steady state distribution for the Gt/GI/n+GI queue.

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93

REFINED MODELS FOR EFFICIENCY-DRIVEN QUEUES, WITH APPLICATIONS TO DELAY ANNOUNCEMENTS AND STAFFING

Junfei Huang, Avishai Mandelbaum, Jiheng Zhang and Hanqin Zhang

Motivated by delay announcement in call centers, we study many-server queues in the ED+QED regime. We propose a refined approximation, allowing the patience time distribution to be rather general. This helps one accommodate customer behavior in response to delay announcements. Our refined approximation not only accurately captures the impact of announcements on system performance, but it also provides insights on call center management-specifically via the optimal solution to the problem of (jointly) staffing servers and making announcements to customers.

OPTIMAL SCHEDULING FOR SYMMETRIC OPEN SHOP NETWORKS

Shuangchi He, Gideon Weiss, Hanqin Zhang

In an open shop network, each customer needs to go through all stations once, but the order of visiting each station is irrelevant. Can this flexibility in service order give us an edge in reducing customer waiting times? In this paper, we consider an open shop network consisting of two stations. The network is said to be symmetric when the service time distributions depend on the service order, and to be asymmetric when the service time distributions depend on the service stations. For the symmetric network, we find routing and sequencing policies that are asymptotically optimal when the open shop network is operated in heavy traffic. We prove that under the obtained scheduling policies, customer waiting times in a symmetric open shop network are asymptotically close to the waiting times in a GI/GI/2 queue with the same traffic intensity. We will also discuss optimal scheduling policies for asymmetric open shop networks.

G/G/N QUEUES WITH SERVICE INTERRUPTIONS

Guodong Pang, Hongyuan Lu, Yuang Zhou

We consider G/G/N queues with renewal alternating service interruptions. The arrival process is general and the service times forms a stationary and weakly dependent sequence satisfying some strong mixing condition. The system experiences up and down alternating periods. Both the arrival and service processes operate normally in the up periods. In the down periods, arrivals continue entering the system, but all servers break down and the amount of service a customer has received will be conserved and resumed when the next up period starts. We assume that the up times are of the same order as the service times but the down times are asymptotically negligible compared with the service times. In the QD and QED regimes, we prove FLLNs and FCLTs for the total count processes and the two-parameter queueing processes tracking elapsed or residual times. The limit processes in the FCLTs are characterized via stochastic integral equations with jumps, and the convergence requires Skorohod M1 topology in the spaces D([0,T],R) and D([0,T],D([0,T],R).

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94


Service Applications - Onno Boxma - Service Applications

EFFICIENT COMPUTATION OF OPTIMAL APPOINTMENT SCHEDULES IN HEALTHCARE

Alex Kuiper, Michel Mandjes

Appointment scheduling plays a prevalent role in healthcare. In practice, a good appointment scheduling minimizes the time lost by both medical personnel and patients. The main challenge is to deal with the uncertainty of patients’ service times, that is, deriving an algorithm that achieves optimal schedules in a wide variety of healthcare settings against low computational cost. This requires a lean and flexible algorithm to capture specific healthcare situations, such as to be able to deal with no-shows, limited waiting room capacity (e.g., MRI, CAT-scan, or surgeries) and different loss functions.In the literature a broad variety of methods are suggested. These include simulation studies, which are often very case-specific, or optimization routines, which in general require restrictive assumptions to overcome the intrinsic complexity of the optimization problem to be solved. We bridge the gap by combining a phase-type fit approach with a lag-order approximation method allowing fast evaluation of optimal schedules in a wide range of settings. We thoroughly assess our results in some practical healthcare situations.

PRECISE CALCULATION OF PROBABILITIES OF THE UEFA ROUND OF 16 DRAW

Patrick Metzler

Computer programs are presented to calculate the probabilities of the results of the UEFA champions league round of 16 draw. 8 champions and 8 runners-up have to be paired. Additional conditions apply. In many journals tables can be found showing how likely it is that champion A will play against runner-up B. Most of the tables are obviously wrong. The better tables employ Monte Carlo simulations. With these only confidence intervals can be determined instead of precise point values. This paper calculates the exact values by scanning the complete probability tree. In view of the true values, pros and cons of the official draw scheme are discussed. Alternative schemes are considered. Statistical entropy as a measure of the overall surprise included in a draw is introduced and used to evaluate draw schemes.

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95

PRIORITY SCHEDULING IN SERVICE SYSTEMS WITH ABANDONMENT AND PREEMPTION

Evin Uzun Jacobson, Nilay Tanık Argon

We study a resource allocation problem among competing customers who may differ in their tolerance for wait. If a customer waits longer than his/her tolerance for wait (which we call the “lifetime”), then he/she leaves the system without receiving any service. On the other hand, if a customer enters service, a random reward is earned. The decision maker knows the type of the customer, which determines the lifetime, service time, and reward distributions for that customer. The objective is to obtain dynamic scheduling priority policies that maximize the total (or average) reward collected.We formulate the problem as a priority assignment problem for a queueing system with multiple types of impatient customers under preemptive service discipline. We study the problem under two main scenarios: (1) the case with a fixed number of customers to be cleared (no future arrivals), (2) the case with customer arrivals. In either case, the objective is to maximize the reward (either total or long-run average). In the analysis, we use sample path methods and stochastic dynamic programming to characterize structures of “good” scheduling policies.

A BLOOD BANK MODEL WITH PERISHABLE BLOOD AND DEMAND IMPATIENCE

Onno Boxma, Shaul Bar-Lev, Britt Mathijsen, David Perry

We consider queueing systems with parallel queues under a Join the Shortest Queue (JSQ) policy in the Halfin‐Whitt heavy traffic regime. Because queues with at least two customers form only when all queues have at least one customer and we expect the number of waiting customers to be of the order , we restrict our attention to a truncated system that rejects arrivals creating queues longer than two. We provide simulation results supporting this intuition. We use the martingale method to prove that a scaled process counting the number of idle servers and queues of length 2 weakly converges to a two‐dimensional reflected Ornstein‐Uhlenbeck process. This limiting system is comparable to that of the traditional Halfin‐Whitt model, but there are key differences in the queueing behavior of the JSQ model. In particular, it is possible for the system to have both idle servers and waiting customers at the same time.

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96


Stochastic Applications - Mohsen Bayati - Stochastic and Statistical Learning Models

JOIN THE SHORTEST QUEUE WITH MANY SERVERS. THE HEAVY TRAFFIC ASYMPTOTICS

David Gamarnik

We consider queueing systems with parallel queues under a Join the Shortest Queue (JSQ) policy in the Halfin-Whitt heavy traffic regime. Because queues with at least two customers form only when all queues have at least one customer and we expect the number of waiting customers to be of the order , we restrict our attention to a truncated system that rejects arrivals creating queues longer than two. We provide simulation results supporting this intuition. We use the martingale method to prove that a scaled process counting the number of idle servers and queues of length 2 weakly converges to a two-dimensional reflected Ornstein-Uhlenbeck process. This limiting system is comparable to that of the traditional Halfin-Whitt model, but there are key differences in the queueing behavior of the JSQ model. In particular, it is possible for the system to have both idle servers and waiting customers at the same time.

THE BIG DATA NEWSVENDOR: PRACTICAL INSIGHTS FROM MACHINE LEARNING

Gah-Yi Vahn

We investigate the newsvendor problem when one has n observations of p features related to the demand as well as historical demand data. Both low-dimensional (p/n = o(1)) and high-dimensional (p/n = O(1)) data are considered. We propose two approaches to finding the optimal order quantity in this new setting - that of Machine Learning (ML) and Kernel Optimization (KO). We show how the feature-based model and solution approaches can be extended naturally to other realistic, “Big Data” situations, such as when one has data on prices, sales, competition, bidding and marketing; when data is censored; when ordering for multiple, similar items or when ordering for a new product with limited data. We show that both solution approaches yield decisions that are algorithmically stable, and derive tight bounds on their performance. We apply the feature-based algorithms for nurse staffing problem in a hospital emergency room and find that (i) the best KO and ML algorithms beat the best practice benchmark by 23% and 24% respectively in out-of-sample cost with statistical significance at the 5% level, and (ii) the best KO algorithm is faster than the best ML algorithm by three orders of magnitude and the best practice benchmark by two orders of magnitude.

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97

A LOW-COST METHOD FOR MULTIPLE DISEASE PREDICTION

Mohsen Bayati

Recently, in response to the rising costs of healthcare services, employers recognize one of the key ways to manage their company’s costs is to incent their workforce to lead a healthier lifestyle. Employers of all sizes have embraced wellness-based incentives to help control costs while maximizing their positive impact on health for both the organization and employees. In particular, these so called “wellness programs” are aimed at reducing the incidence of chronic illnesses such as cardiovascular disease, cancer, diabetes, and obesity. The majority of these wellness programs include an annual health risk assessment (HRA) to detect individuals with the highest risk of developing chronic disease followed by risk reduction interventions targeted at those high risk individuals. However, the overall efficacy of these wellness programs relies on prediction accuracy of the HRA that increases with the amount of data points used by the HRA per employee. On the other hand, capturing more data points per employee increases the cost of HRA itself. In this paper we propose a solution, to address this trade-off, based on multi-task learning and group dimensionality reduction from statistical learning, empirically validate it using data from two different electronic medical records systems, and support its performance theoretically via a stylized model.

NEWTON-STEIN METHOD: SECOND ORDER METHOD FOR GLMs VIA STEIN’S LEMMA

Murat Erdoğdu

We consider the problem of efficiently computing the maximum likelihood estimator in Generalized Linear Models (GLMs) when the number of observations is much larger than the number of coefficients n>>p>>1. In this regime, optimization algorithms can immensely benefit from approximate second order information. We propose an alternative way of constructing the curvature information by formulating it as an estimation problem and applying a Stein-type lemma, which allows further improvements through sub-sampling and eigenvalue thresholding. Our algorithm enjoys fast convergence rates, resembling that of second order methods, with modest per-iteration cost. We provide its convergence analysis for the general case where the rows of the design matrix are samples from a sub-Gaussian distribution. We show that the convergence has two phases, a quadratic phase followed by a linear phase. Finally, we empirically demonstrate that our algorithm achieves the highest accuracy for any fixed amount of time compared to a wide variety of fast algorithms on several datasets.

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98


Stochastic Applications - Antonio Gomez-Corral - Stochastic Models in Epidemics

SENSITIVITY ANALYSIS OF A BRANCHING PROCESS EVOLVING ON A NETWORK WITH APPLICATION IN EPIDEMIOLOGY

Sophie Hautphenne

We perform an analytical sensitivity analysis for a model of a continuous-time branching process evolving on a fixed network. This allows us to determine the relative importance of the model parameters to the growth of the population on the network. We then apply our results to the early stages of an influenza-like epidemic spreading among a set of cities connected by air routes in the United States. We also consider vaccination and analyze the sensitivity of the total size of the epidemic with respect to the fraction of vaccinated people. Our analysis shows that the epidemic growth is more sensitive with respect to transmission rates within cities than travel rates between cities. More generally, we highlight the fact that branching processes offer a powerful stochastic modeling tool with analytical formulas for sensitivity which are easy to use in practice.

ANALYZING STOCHASTIC DESCRIPTORS IN AN SIR EPIDEMIC MODEL WITH HETEROGENEOUS CONTACTS IN SMALL NETWORKS

Martin Lopez GarciaOur aim in this talk is to show how to analyse the SIR epidemic model in an exact way when the population under study is formed by a small highly heterogeneous group of � individuals. Our approach, which amounts to the analysis of the exact 3�‐state continuous‐time Markov chain (CTMC), makes special focus on algorithmic issues, and requires a creative order for the states within the state space � � �� of the CTMC. We propose two different orders, Orders ��and �, which show different advantages and disadvantages from the computational point of view, as well as depending on their applicability in different variants of the SIR epidemic model. Finally, we illustrate our approach by studying the spread of the nosocomial pathogen Methicillin‐resistant Staphylococcus Aureus among the patients within an intensive care unit (ICU). The interest here is in analysing the effectiveness of different control strategies which intrinsically incorporate heterogeneities among the patients within the ICU.

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99

A DISCRETE STOCHASTIC METAPOPULATION MODEL WITH ARBITRARILY DISTRIBUTED INFECTIOUS PERIOD

Nancy Hernandez-Ceron

VARICELLA AND HERPES ZOSTER: EXOGENOUS BOOSTING, PROGRESSIVE IMMUNITY AND THE DILEMMA OF MASS IMMUNIZATION

Piero Manfredi, Gianpaolo Scalia Tomba, Giorgio Guzzetta, Stefano Merler, Piero Poletti, Emanuele del Fava

Varicella and herpes zoster (HZ) are different diseases caused by the varicella-zoster virus (VZV). After varicella infection, the virus remains dormant in the host and can reactivate into HZ possibly due to waning cell-mediated immunity (CMI). Hope-Simpson (1965) formulated the “exogenous boosting” hypothesis (EBH), according to which further infective exposures to VZV of varicella-immune individuals may boost their immune response, resulting in a protective effect against HZ. Inclusion of the EBH in VZV transmission dynamics models consequently predicts a large transient increase in HZ incidence following mass varicella immunization, whose fear is a main responsible of the current paralysis of varicella vaccination in Europe. In this talk, I will summarize the main results of an ECDC funded project on the subject I recently coordinated. First I show by the Poisson process that intensity and calendar of re-exposures to VZV heavily depend on the shape of age-specific contact patterns. Next I show the implications of a second fundamental hypothesis by Hope-Simpson, that we called the “progressive immunity” hypothesis, stating that that after each episode of re-exposure to VZV the level of CMI protection against HZ increases to levels higher than those conferred by previous ones. A model for VZV transmission and reactivation incorporating this hypothesis fits available European HZ data better than concurrent models, suggesting that progressive immunity may be critical in shaping HZ patterns. Counter-intuitive implications for immunization programs are also discussed. I conclude by a discussion on those I consider the main challenges in modeling VZV.

In this talk, a stochastic discrete‐time model is introduced to study the spread of an infectious disease in an n‐patch environment. The model includes an arbitrary distribution of the (random) infectious period �. Analytic results are used to investigate how the distribution of � may influence the model outcomes, particularly for the case �� patches. It is shown that the reproduction numbers corresponding to different distributions of � can be ordered based on their probability generating function. Specific distributions including Geometric, Negative Binomial, Poisson and Uniform are compared both numerically and analytically. The measures used in model comparisons are (i) the basic reproduction numbers �; (ii) probability of a minor epidemic (or disease extinction) ℙ�; (iii) final epidemic size; and (iv) duration of the epidemic.

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100


Stochastic Control - Alan Scheller-Wolff - Markov Decision Processes

TANDEM QUEUES WITH RENEGING - ANALYSIS AND INSIGHTS

Opher Baron

This paper considers tandem queueing systems with reneging. We develop a new technique to solve two dimensional Markov Chains with non-repeating structure. Our technique can be applied to additional settings and used to derive different service level measures. We demonstrate this technique on a two-station tandem queueing model with reneging, which has been considered analytically intractable. We provide exact numerical methods to derive various performance measures of this system; we demonstrate our method by calculating the customer loss rates from different stations. We use these numerical methods to consider a system design problem and investigate the effect of cross-trained servers in such a network. For example, we show that a small number of cross-trained servers can achieve most benefit of cross-training, and, somewhat surprisingly, cross-training too many servers can increase the percentage of lost customers.

SERVICE CENTER STAFFING WITH CROSS-TRAINED AGENTS, HETEROGENOUS CUSTOMERS, AND QUALITY GUARANTEES

Elvin Çoban, Aliza R. Heching, Alan Scheller-Wolf

We model a service center with cross-trained agents serving customer requests that are heterogeneous with respect to the skills they require and their priority: Higher priority requests preempt lower priority requests and less skilled agents can only serve less demanding requests, while highly skilled agents can serve all requests. We model this system as a Markov chain that is infinite in multiple dimensions and thus is not amenable to exact analysis. We therefore apply approximation and bounding techniques and develop a tractable algorithm that closely approximates the operations of our system under a simple but effective request-assignment policy. We numerically explore the performance of our policy comparing it to benchmark policies and investigate its sensitivity to various parameters such as the number of agents and agent utilization.

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101

CLEARING ANALYSIS ON PHASES: EXACT LIMITING PROBABILITIES FOR SKIP-FREE, UNIDIRECTIONAL, QUASI-BIRTH-DEATH PROCESSES

Sherwin Doroudi

A variety of problems in computing, service, and manufacturing systems can be modeled via infinite repeating Markov chains with an infinite number of levels and a finite number of phases. Many such chains are quasi-birth-death processes (QBDs) with level transitions that are “skip-free,” in that within a phase one can only transition to a state that is one level lower or one level higher than the current state (i.e., levels cannot be skipped), and “unidirectional” phase transitions, in that such transitions are only possible from lower-numbered phases to higher-numbered phases. We present a procedure, which we call Clearing Analysis on Phases (CAP), for determining the limiting probabilities of such Markov chains in closed form. The CAP method yields the limiting probability of each state in the repeating portion of the chain as a linear combination of scalar bases raised to a power corresponding to the level of the state. The weights in these linear combinations can be determined by solving a finite system of linear equations.

REPLENISHMENT AND FULFILLMENT BASED AGGREGATION FOR GENERAL ASSEMBLE-TO-ORDER SYSTEMS

Emre Nadar, Alp Akcay, Mustafa Akan, Alan Scheller-Wolf

We consider an assemble-to-order system with multiple products, multiple components which may be demanded in different quantities by different products, batch ordering of components, random lead times, and lost sales. We model the system as an infinite-horizon Markov decision process under the discounted cost criterion. A control policy specifies when a batch of components should be produced and whether an arriving demand for each product should be satisfied. As optimal solutions for such problems are computationally burdensome to calculate, we approximate the optimal cost function by reducing the state space of the original problem via a novel aggregation technique that uses knowledge of products’ component requirements and components’ replenishment batch sizes.

We establish that a lattice-dependent base-stock and lattice-dependent rationing policy is the optimal inventory replenishment and allocation policy for the aggregate problem under a disaggregation rule that disaggregates each aggregate state into its two extreme original states. This rule drastically reduces the per iteration computational complexity of the value iteration algorithm (without sacrificing much accuracy, according to our numerical experiments). We further alleviate the computational burden in the value iteration algorithm by eliminating suboptimal actions based on the optimal policy structure. For systems in which there is a product that has a fulfillment priority over all other products at optimality, we are also able to derive a finite error bound for the cost function of the aggregate problem.

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102


New Directions in Applied Probability - David Goldberg - New Directions in Applied Probability

UNBALANCED RANDOM MATCHING MARKETS: THE STARK EFFECT OF COMPETITION

Yashodhan Kanoria, Itai Ashlagi and Jacob Leshno

We study bilateral matching markets such as marriage markets, labor markets, and school-college admissions that allow participants to form partnerships with each other for mutual benefit. We study competition in matching markets with random heterogeneous preferences. First, we find that such markets are extremely competitive. In unbalanced markets, each agent on the short side of the market is matched to one of his top preferences and each agent on the long side does almost no better than being matched to a random partner. Second, we find that even the slightest imbalance leads to competition that yields an essentially unique stable matching. Our results suggest that any matching market is likely to have a small core, explaining why empirically small cores are ubiquitous.

NETWORK EPIDEMICS WITH EXTERNAL AGENTS

Siddhartha Banerjee, A. Chatterjee, A. Das, A. Gopalan, and S. Shakkottai

We study network epidemics (in particular, under the SI and SIS dynamics), whose spread is aided by external agents - sources unconstrained by the graph, but possessing a limited infection rate or virulence. Such a model captures many existing models of externally aided epidemics, and network use in many settings - epidemiology, marketing and advertising, network robustness, etc. We provide a detailed characterization of the impact of external agents on epidemic thresholds. In particular, we find conditions under which external agents can significantly affect the epidemic spreading time/lifetime; moreover, in many settings, we show that random external-spreading policies are near-optimal.

DECENTRALIZED SIGNAL CONTROL FOR URBAN ROAD NETWORKS

Neil Walton

We propose in this paper a decentralized traffic signal control policy for urban road networks. Our policy is an adaptation of a so-called BackPressure scheme which has been widely recognized in data network as an optimal throughput control policy. We have formally proved that our proposed BackPressure scheme, with fixed cycle time and cyclic phases, stabilizes the network for any feasible traffic demands. Simulation has been conducted to compare our BackPressure policy against other existing distributed control policies in various traffic and network scenarios. Numerical results suggest that the proposed policy can surpass other policies both in terms of network throughput and congestion.

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103

SPECTRAL METHODS FOR NONSTATIONARY QUEUES

Jamol Pender

We propose a new method for approximating the nonstationary moment dynamics of one dimensional Markovian birth-death processes. By expanding the transition probabilities of the Markov process in terms of Poisson-Charlier polynomials, we are able to estimate any moment of the Markov process even though the system of moment equations may not be closed. Using new weighted discrete Sobolev spaces,we derive explicit error bounds of the transition probabilities and new weak a priori estimates for approximating the moments of the Markov process using a truncated form of the expansion. Using our error bounds and estimates, we are able to show that our approximations converge to the true stochastic process as we add more terms to the expansion and give explicit bounds on the truncation error. As a result, we are the first paper in the queueing literature to provide error bounds and estimates on the performance of a moment closure approximation. Lastly, we perform several numerical experiments for some important models in the queueing theory literature and show that our expansion techniques are accurate at estimating the moment dynamics of these Markov process with only a few terms of the expansion.


Simulation - Ebru Angun - Stochastic Optimization

DECOMPOSITION ALGORITHMS FOR RISK-AVERSE MULTISTAGE STOCHASTIC PROGRAMS WITH APPLICATION TO WATER ALLOCATION UNDER UNCERTAINTY

Güzin Bayraksan

We study a risk-averse approach to multistage stochastic linear programming, where the conditional value-at-risk is incorporated into the objective function as the risk measure. We consider five decompositions of the resulting risk-averse model in order to solve it via the nested L-shaped method. We introduce separate approximations of the mean and the risk measure and also investigate the effectiveness of multiple cuts. As an application, we formulate a water allocation problem by risk-averse multistage programming, which has the advantage of controlling high-risk severe water shortage events. We apply the proposed formulation to the southeastern portion of Tucson, AZ to best use the limited water resources available to that region. In numerical experiments we (1) present a comparative computational study of the risk-averse nested L-shaped variants and (2) analyze the risk-averse approach to the water allocation problem.

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104

GLOBAL OPTIMIZATION FOR BLACK-BOX SIMULATION MODEL: INTRINSIC KRIGING METAMODELING

Ehsan Mehdad, Jack Kleijnen

We study the use of intrinsic Kriging as a new metamodel in global optimization of deterministic and random simulations. For deterministic simulation models, we combine the popular EGO method with our new intrinsic Kriging metamodel. For random simulation models, we study a state-of-the-art algorithm accounting for heteroscedastic variances of the simulation responses. We introduce a new variant of this algorithm, which has the following two distinguishing features: (1) The metamodel is a (stochastic) intrinsic Kriging model instead of a (stochastic) Kriging model. (2) To tackle the heteroscedastic variance of the simulation output, the algorithm uses a new method to allocate the available number of replications over observed input combinations. We use several numerical experiments with deterministic and random simulations for the following two comparisons: (1) Classic EGO versus EGO with intrinsic Kriging. (2) The original algorithm versus our modified version. We conclude that most experiments give the following empirical results: (1) EGO with intrinsic Kriging outperforms classic EGO. (2) The original algorithm and our modified version do not give significant differences.

ADAPTIVE SAMPLING TRUST-REGION OPTIMIZATION

Raghu Pasupathy

We develop derivative free algorithms for optimization contexts where the objective function is observable only through a stochastic simulation. The algorithms we develop follow the trust-region framework where a local model is constructed, used, and updated as the iterates evolve through the search space. The salient feature of our algorithms is the incorporation of adaptive sampling to keep the variance (statistical error) and the squared bias (model error) of the local model in lock step, in a bid to ensure optimal convergence rates. Such balancing is accomplished dynamically, through careful estimation of errors using function estimates at visited points. We will discuss convergence and efficiency.

STRATIFIED BAYESIAN OPTIMIZATION

Saul Toscano, Peter Frazier

We consider simulation optimization, and noisy derivative-free optimization of expensive functions, when most of the randomness in the objective is produced by a few influential scalar random inputs. We present a new Bayesian global optimization algorithm, called Stratified Bayesian Optimization (SBO), which uses this strong dependence to improve performance. Our algorithm is similar in spirit to stratification, a classical technique from simulation, which uses strong dependence on a categorical representation of the random input to reduce variance.

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105


Queues - Limit Theorems - Bruno Gaujal - When Limits Help in Large Scale Stochastic Systems

CONTROL OF PARALLEL NON-OBSERVABLE QUEUES: ASYMPTOTIC EQUIVALENCE AND OPTIMALITY OF PERIODIC POLICIES

Jonatha Anselmi

RANDOMIZED POLICIES FOR DEMAND DISPATCH

Ana Busic

Renewable energy sources such as wind and solar power have a high degree of unpredictability and time-variation, which makes balancing demand and supply challenging. One possible way to address this challenge is to harness the inherent flexibility in demand of many types of loads. We propose a technique for decentralized control for automated demand response that can be used by grid operators as ancillary service for maintaining demand-supply balance. It is assumed that there is one-way communication from the grid operator to each load. The loads are either on or off - their power consumption is not continuously variable. A randomized control architecture is proposed, motivated by the need for decentralized decision making, and the need to avoid synchronization that can lead to large and detrimental spikes in demand. A mean-field limit is used to obtain an input-output model, where the output is the aggregate power consumption. The main result is to show how the local transition laws can be designed so that the linearized mean field model is minimum phase.

We consider a queueing system composed of a dispatcher that routes deterministically jobs to a set of non‐observable queues, of different types, working in parallel. In this setting, the fundamental problem is which policy the dispatcher should implement to minimize the stationary mean waiting time of the incoming jobs. We present a structural property that holds in the classic scaling of the system where the network demand (arrival rate of jobs) grows proportionally with the number of queues. Assume that each queue of type is replicated times and consider the set of policies that are periodic with period and such that exactly jobs are sent in a period to each queue of type r. When k goes to infinity, our main result shows that all the policies in this set are equivalent, in the sense that they yield the same mean stationary waiting time, and optimal, in the sense that no other policy having the same aggregate arrival rate to all queues of a given type can do better in minimizing the stationary mean waiting time. This property holds in a strong probabilistic sense. Furthermore, the limiting mean waiting time achieved by our policies is a convex function of the arrival rate in each queue, which facilitates the development of a further optimization aimed at solving the fundamental problem above for large systems.

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106

RANDOM WALKS, RANDOM MEDIA AND POWER OPTIMIZATION IN RANDOM WIRELESS NETWORKS

Panayotis Mertikopoulos

Consider a large, random wireless network of N transmit-receive pairs where each transmitter adjusts his transmit power in order to maintain a target throughput value in the presence of interference from other users. As it turns out, the large N limit of this problem is equivalent to the so-called Anderson model of electron motion in dirty metals which has been used extensively in the analysis of diffusion processes in random, disordered media. A standard “mean-field” approximation to this model is the so-called coherent potential approximation (CPA) method which we apply to evaluate the first and second order intra-sample statistics of the optimal power vector in one- and two-dimensional systems.

This approach is equivalent to traditional techniques from random matrix theory and free probability but, while generally accurate, it does not fully describe the system: in particular, results obtained in this way fail to predict when power control becomes infeasible in the system. We find that the infinite system is always unstable beyond a certain value of the target throughput, but any finite system only has a small probability of becoming unstable. This instability probability is proportional to the tails of the eigenvalue distribution of the system, which we calculate to exponential accuracy using methodologies from the theory of random diffusions in random media. Finally, using the same techniques, we are also able to calculate the tails of the system’s power distribution.

A NON-UNIFORM MEAN FIELD MODEL FOR A CLASS OF LIST-BASED CACHE REPLACEMENT ALGORITHMS

Benny Van Houdt

In this talk we consider a family of cache replacement algorithms that decompose the cache into several lists. An item enters the cache via the first list and jumps to the next list whenever a hit on it occurs. We introduce a mean field model to approximate the transient behavior of the miss probability. This model can be proven to become exact as the cache size and number of items tends to infinity without the need to cluster the items in a finite number of classes. We discuss the uniqueness and existence of the fixed point and how it can be used to approximate the stationary miss probability. Numerical experiments that investigate the accuracy of the model for both the transient and stationary miss probability are presented.

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107


Stochastic Networks and Processes - Alessandro Zocca - Resource Sharing in Communication Networks

STORE-FORWARD AND ITS IMPLICATIONS FOR PROPORTIONAL SCHEDULING

Neil Walton

The Proportional Scheduler was recently proposed as a scheduling algorithm for multi-hop switch networks. For these networks, the BackPressure scheduler is the classical benchmark. For networks with fixed routing, the Proportional Scheduler is maximum stable, myopic and, furthermore, will alleviate certain scaling issued found in BackPressure for large networks. Nonetheless, the equilibrium and delay properties of the Proportional Scheduler have not been fully characterized. In this article, we postulate on the equilibrium behaviour of the Proportional Scheduler though the analysis of an analogous rule called the Store-Forward allocation. It has been shown that Store-Forward has asymptotically allocates according to the Proportional Scheduler. Further, for Store-Forward networks, numerous equilibrium quantities are explicitly calculable. For FIFO networks under Store-Forward, we calculate the policies stationary distribution and end-to-end route delay. We discussnetwork topologies when the stationary distribution is product-form, a phenomenon which we call product form resource pooling. We extend this product form notion to independent set scheduling on perfect graphs, where we show that non-neighbouring queues are statistically independent. Finally, we analyse the large deviations behaviour of the equilibrium distribution of Store-Forward networks in order to construct Lyapunov functions for FIFO switch networks.

MEAN-FIELD ANALYSIS OF RANDOM-ACCESS NETWORKS

Fabio Cecchi, Sem Borst, Johan van Leeuwarden

Distributed algorithms such as CSMA provide a popular mechanism for sharing the transmission medium among competing users in large-scale wireless networks. Conventional models for CSMA assume that users always have packets to transmit to allow tractable results. In contrast, when users do not compete for medium access when their buffers are empty, a complex interaction arises between the activity states and the buffer contents. We develop a mean-field analysis to investigate this dynamic interaction for networks with many users. We identify a time scale separation between the evolution of the activity states and the buffer contents, and obtain a deterministic dynamical system describing the network dynamics on a macroscopic scale. The fixed point of the dynamical system yields highly accurate approximations for the stationary distribution of the buffer content and packet delay, even when the number of users is relatively moderate.

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108

DYNAMIC RESOURCE ALLOCATION AND USER ASSOCIATION IN PICO-CELL NETWORKS

Bart Post, S. C. Borst, A. M. J. Koonen

Wireless networks have experienced immense growth in traffic loads over the last few years. A powerful approach to expand their capacity is to deploy pico-cells so as to cover areas with high traffic density. A crucial challenge in such ultra-dense wireless networks is to optimize the allocation of transmission resources and relevant performance metrics in the presence of highly dynamic user populations.

In this talk, we present a spatial stochastic model to examine the design of efficient resource allocation algorithms and establish fundamental performance limits. We consider a service area which is densely populated by antennas. Users arrive at random locations according to a spatial Poisson process, and have exponentially distributed service requirements. In order to be served, users need to be associated with an antenna within a certain communication range and be assigned a communication frequency. The total number of available frequencies is limited, and two antennas using the same frequency should be separated by a minimum distance. Due to these restrictions, users may be denied service (be blocked) because there are no reachable antennas or free frequencies available.

We develop novel loss network models with a continuum of routes to examine the performance in such networks, depending on the resource allocation algorithm, in terms of blocking probabilities and numbers of active users.

TEMPORAL STARVATION IN RANDOM-ACCESS NETWORKS

Alessandro Zocca, Sem C. Borst, Johan S.H. van Leeuwaarden, Francesca R. Nardi

We consider a stylized stochastic model for a wireless random-access network, which yields a product-form stationary distribution of the activity process for the various users and provides useful estimates for the user throughputs. Specifically, we model a CSMA network as an interacting particle system on a graph, called interference graph. This graph encodes the spatial structure of the network and the interference constraints: nodes that are neighbors in the interference graph are prevented from simultaneous activity, and thus the independent sets correspond to the feasible joint activity states. The resulting complex dynamic behavior is described by a continuous-time Markov process, which exhibits fascinating connections with the hard-core interaction between gas particles in chemistry and statistical mechanics.Even in scenarios where all users with an equal opportunity to be active in the long run or in symmetric scenarios where spatial fairness is automatically ensured, transient yet significant starvation effects can arise due to extremely slow transitions between high-likelihood configurations, (i.e. the maximum independent sets of the interference graph), yielding a form of meta-stability. Focusing on regular meshes, we use the above model to investigate the strong interplay in a high-load regime between these long transition times, poor throughput characteristics and starvation issues.

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109


Service Applications - Stella Kapodistria - State-of-the-art Applications of Service Systems

SOJOURN TIME IN A SINGLE SERVER WITH THRESHOLD SERVICE RATE CONTROL

Ivo Adan, Bernardo d’Auria

We study the sojourn time in a single-server exponential queueing system. Service is provided at low or high rate, which can be adapted at exponential inspection times depending on the number in the system. The state dependent changes in the service rate make the analysis of the sojourn time challenging, since the sojourn time now also depends on future arrivals. We determine the Laplace transform of the sojourn time by employing a new methodological tool, that is matrix generating functions. The power of this tool is that it can also be used to analyse generalisations to phase-type services and inspection times.

SHORTEST EXPECTED DELAY ROUTING

Jori Selen

We consider a system consisting of two non-identical exponential servers, each having its own waiting queue. Jobs arrive according to a Poisson process and join the queue promising the shortest expected delay. This system can be modelled as a three-dimensional Markov process. Using the compensation approach, we show that the equilibrium distribution can be expressed as an infinite sum of product forms.

NETWORKS OF FIXED-CYCLE TRAFFIC LIGHTS

Rick Boere

Networks of traffic intersections are typically too complex to analyze. We propose a discrete-time queueing model for traffic flows through a sequence of traffic lights. We discuss several technical issues, such as advanced root-finding techniques and numerical inversion of generating functions, which we have used to obtain the queue-length distributions at each of the intersections. Additionally, we presents novel insights on optimal settings for such traffic networks.

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110

QED SCALING FOR FIXED-CYCLE TRAFFIC SIGNALS

Marko Boon

The Quality-and-Efficiency Driven (QED) regime has proven its tremendous value in a wide range of application areas. We apply the QED principle of matching capacity with demand to urban road traffic. We develop a capacity sizing rule for determining fixed-cycle traffic signal settings, ensuring that the specified traffic flow will operate in the QED regime. As a consequence, we can let the system load tend to one, while retaining a strictly positive probability that arriving vehicles experience no delay.


Stochastic Applications - Philippe Chevalier -Stochastic Models for Operations Management

DYNAMIC ADMISSION CONTROL FOR TWO CUSTOMER CLASSES WITH STOCHASTIC DEMANDS AND STRICT DUE DATES

Tanja Mlinar, Philippe Chevalier

We study a dynamic capacity allocation problem with admission control decisions of a company that caters for two demand classes with random arrivals, capacity requirements, and strict due dates. We formulate the problem as a Markov decision process in order to find the optimal admission-control policy that maximizes the expected profit of the company. Such a formulation suffers a state-space explosion. Moreover, it involves an additional dimension arising from the multiple possible order sizes that customers can request which further increases the complexity of the problem. To reduce the cardinality of possible policies, and, thus, the computational requirements we propose a threshold-based policy. We formulate an MDP to generate such a policy. To deal with the curse of dimensionality we develop threshold-based approximate algorithms based on the state-reduction heuristics with aggregation proposed previously. Our results reveal that for the majority of instances considered the optimal policy has a threshold structure. We then demonstrate the superiority of the proposed threshold-based approximate algorithms over two benchmark policies in terms of the generated profits and the robustness of the solutions to changes in operational conditions. Finally, we show that our proposed policies are also robust to changes in actual demand from its estimation.

STOCHASTIC MODELLING FOR SALES AND OPERATIONS PLANNING

Nico Vandaele

Sales and Operations Planning is a key business process in many industrial and service industries. As it relates shorter term decisions as well as with strategic decisions, special care has to be taken to build relevant models for practice. We propose an approach where the core is a stochastic model for each planning period, embedded in a mixed integer planning model to link the periods for the planning horizon. Finally, a multi criteria decision model sorts out the relevant planning scenarios, stemming from risks related to externalities and discrete operations management decisions. We illustrate this approach with the outsourcing or contract manufacturing decision.

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111

ORDER ACCEPTANCE POLICIES FOR COMBINED MAKE-TO-ORDER AND MAKE-TO-STOCK ENVIRONMENTS

Alejandro Lamas

We study the case of a manufacturer that uses his scarce production capacity to satisfy orders from two different demand streams. Orders arrive stochastically and the demand streams differ in terms of unit profits, requested capacities, lead times and frequencies. The operation of the manufacturer combines make-to-order and make-to-stock environments. In a purely make-to-order environment, the manufacturer can gain from the absences of stocks, but the manufacturer must decide whether to accept or to reject less profitable orders with the aim of balancing the risk of rejecting more profitable orders and the risk of idle capacity. In order to reduce such risks, the manufacturer can also produce in advance (make-to-stock), but extra holding costs have to be paid. Thus, in addition to the acceptance/rejection decision, the manufacturer should decide when to produce stocks of the different demand streams in order to maximize his profit. We propose a Markov Decision Process for obtaining the optimal acceptance/rejection and stock policy. Constructing such policy, however, requires the full track of the already accepted orders and the current stock level, so a high computational effort may be needed. With the aim of reducing the computational time and keeping a high efficiency, we develop heuristic approaches based on the partial aggregation of the information related to the already accepted orders.

TAKING FORECAST RELIABILITY INTO ACCOUNT FOR SCHEDULING

Philippe Chevalier

Forecast sharing is one of the major activities in supply chain collaboration. Ideally, it would lead to a win-win situation: with the forecast information, the supplier builds production capacity to cope with potential demand, and in turn, the buyer develops a dependable supply source. However, in practice this merit is often compromised due to the forecast imprecision. Our objective is to study the impact of the forecast (in)accuracy for the supplier. If a supplier fully relies on the forecasts, he has great chance to experience poor capacity utilisation and overwork. As a consequence, a supplier would like to serve buyers that provide credible forecast earlier than those communicating “cheap talk”. Nevertheless, the link between forecast error and the induced delay for the cash inflow is not evident. In this presentation, we explore the interface between supply chain information flow and cash flow. We consider a global supply chain with a supplier serving multiple buyers sequentially. The supplier decides the sequence of order fulfilment and quotes due dates in response to the buyers’ forecasts. To ensure a high service level, the supplier will have to include more safety time in the quoted lead-time for a less reliable order. We want to address the following questions: what is the effect of forecast variability on the supply chain cash inflow and optimal scheduling? How can a supplier improve the cash inflows by scheduling the due dates and orders? Should a reliable buyer be served with priority?

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112


Stochastic Applications - Jing Dong - Asymptotics in Queues and Matching Graphs

SCALING AND STAFFING FOR CALL CENTER MANAGEMENT

Xiaowei Zhang, L. Jeff Hong, Jiheng Zhang

The Poisson process has been an integral part of many models for the arrival process to a telephone call centers. However, various publications in recent years suggest the presence of a significant “overdispersion” relative to the Poisson process in real-life call center arrival data. In this paper, we study the overdispersion in the context of heavy traffic and identify a critical factor that characterizes the stochastic variability of the arrivals relative to their averages. We refer to such a factor as the scaling parameter as it determines the appropriate way to scale the arrival process in heavy traffic. Indeed, data exhibits that the square-root scaling as in the conventional central limit theorem may not truly reflect the reality. We propose a new stochastic model to capture the scaling parameter and develop the associated staffing rule in the QED (quality-and-efficiency driven) regime. The new staffing rule stipulates that in order to achieve a balance between quality-of-service and agent efficiency, the safety margin of the staffing level ought to have an order of magnitude that depends on the scaling parameter. In particular, in the presence of overdispersion the safety margin has a higher order of magnitude than what the square-root staffing rule suggests. We apply the new staffing rule to real data and find out it significantly outperforms the square-root staffing rule.

STABILITY OF THE STOCHASTIC MATCHING PROBLEM

Pascal Moyal, J. Mairesse

Consider a model in which items arrive one by one, and depart from it as soon as possible, but by pairs. The items of a departing pair are said to be matched. There is a finite set of classes V for the items, and the allowed matchings depend on the classes, according to a matching graph on V. The matching policy determines how to match an arriving item when there is more than one possibility. We investigate the stability of this discrete-time model. We first investigate the structural properties of the matching graphs leading to simple succinct stability conditions, and then derive a general stability result in the case of iid sequences of classes, in function of the matching policy. We also address the generalization of our results to the general stationary ergodic framework, and the connections with the corresponding matching model for bipartite graphs Customers/Servers.

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113

ROUTING POLICIES FOR A PARTIALLY OBSERVABLE TWO-SERVER QUEUEING SYSTEM

Peter Kovacs, Wendy Ellens, Rudesindo Nunez Queija, Hans van den Berg

APPROXIMATIONS FOR QUEUEING DYNAMICS IN HOSPITAL INPATIENT WARDS OPERATIONS

Jing Dong

When viewed as a queueing systems in which beds are servers handling patients in impatient wards, a hospital constitutes a complex time-inhomogeneous stochastic network with several unique features. In this work, we propose new types of deterministic (fluid) approximations, aiming to capture the most significant aspects of queueing dynamics, including periodicity, server blocking and the bulk departure. In particular, the fluid approximation is a time-inhomogeneous dynamical system with jumps. We propose notion of stability associated with the asymptotic periodicity. Our results facilitate long-run performance analysis, and provide insights into the effects of admission delay, discharge polices and discharge delay that are prevalent in practice.

We consider a queueing system controlled by decisions based on partial information. The motivation for this work stems from road traffic, in which drivers may, or may not, be subscribed to a smartphone application for dynamic route planning.

Our model consists of two queues, both operating in a FIFO manner with independent exponential service times, serving two types of jobs. Arrivals occur according to a Poisson process of rate a fraction of jobs (type X) are observable and controllable. At all times the number of X jobs in each queue and their individual positions are known. Upon its arrival a router decides which queue the next X job should join.

Y jobs (fraction ) are non‐observable, not even the number of Y jobs in the queues, and non‐controllable. They join queue with static routing probabilities .

We address the following research questions: 1) what penetration level (percentage of X jobs) is needed for effective control, 2) which policy should be implemented at the router, and 3) what is the added value of having more system information (e.g., average service times and the values of )? We investigate these questions through extensive simulations. This simulation study reveals that for heavily loaded systems a low penetration level suffices and that the performance of a simple policy that relies on little system information is close to the optimal (weighted join‐the‐shortest‐queue policy). The latter result is confirmed by analysis of a dynamical system that approximates the stochastic evolution in heavy traffic.

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Stochastic Control - Sheng Qiang - Price and Portfolio Optimization

INSIDE INFORMATION AND ROBUST PORTFOLIO OPTIMIZATION

Ioannis Baltas, Athanasios Yannacopoulos

We study a robust-entropic optimal control problem in the presence of inside information. To be more precise, we consider an economic agent who is allowed to invest her wealth in a classical Black-Scholes type financial market. From the beginning of the trading interval, the agent exclusively possesses some inside information concerning the future realization of the stock price process. However, we assume that she is uncertain as to the validity of this information, thus introducing in this way robust aspects to our model. The aim of the economic agent is to solve an expected utility maximization problem under the worst-case scenario, taking into account her enlarged information set. By formulating this problem as a two-player, zero sum stochastic differential game, we are able to provide closed form solutions for the optimal robust strategies and the robust value function, which is also the solution of the associated Bellman-Isaacs partial differential equation, in the case of the exponential and power utility functions.

INVESTIGATION OF INVESTOR BEHAVIORS IN FINANCIAL MARKETS AND THE SIMPLIFICATION IN PORTFOLIO SELECTION

Efe Çötelioğlu, Süleyman Özekici

Our aim is to analyze and explain the investor behaviour in different financial markets. We consider the multi-period optimal portfolio selection problem where the investor maximizes his expected utility of the terminal wealth. The utility function belongs to the HARA (hyperbolic absolute risk aversion) family which contains exponential, logarithmic and power utility functions. Returns of the risky assets and the utility function all depend on an external process that represents a regime-switching market. We assume that the random changes in the market states are depicted by a Markov chain. Dynamic programming is used to obtain an explicit characterization of the optimal policy.In the numerical illustration part, we compare the out-of-sample performance of the models with different utility function types across several empirical datasets of monthly returns. We consider HFRI Hedge Fund Indices, Credit Suisse Hedge Fund Indices and ten industry portfolios formed by Kenneth French. The states of the market are classified by using the Chicago Fed National Activity Index (CFNAI) and NBER’s US Business Cycle Expansions and Contractions data and the transition probability matrix of the Markov chain is estimated. For each dataset, we find the model that has the closest out-of-sample Sharpe ratio to the value-weighted market portfolio. By this, we aim to determine the utility function that represents the investor behaviour in each financial market.

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115

DYNAMIC PRICING WITH DEMAND COVARIATES

Sheng Qiang, Mohsen Bayati, Michael Harrison

We consider a generic problem in which a firm sells products over periods without knowing the demand function. The firm sequentially sets prices to earn revenue and to learn the underlying demand function simultaneously. In addition, in each period before setting the prices, the firm has access to some demand covariates, which may be correlated with the demand. Demand covariates can include marketing expenditures, geographical information, consumers socio-economic attributes, macroeconomic indices, etc. The performance is measured by the regret, which is the expected revenue deviation from the optimal (oracle) pricing policy when demand function is known. We study the role of demand covariates in deriving asymptotically optimal algorithms for optimizing the regret.

INTEGRATED REQUIREMENT PLANNING AND SHIFT SCHEDULING FORCALL CENTERS WITH UNCERTAIN ARRIVAL RATES

Andrej Saweljew, Raik Stolletz, Ger Koole

We consider a call center that can be described as a Gt / G / ct / K+G system with redials. Arrivals follow a doubly stochastic process. We determine cost-minimal shift schedules while ensuring that aggregated performance goals are met.We make a two-step decision: with knowledge of the arrival rate distribution for each period t, we decide on shift plans for regular and outsourced agents. As we receive more information about the arrival rates, we decide on how many planned outsourced agents to staff.To integrate the staffing, scheduling, and recourse decisions, we make use of the analytic center cutting-plane method which combines simulation and optimization.The numerical study suggests that taking into account the double stochastic aspects of the arrival process is a necessity for achieving sufficient performance levels. The increased staffing costs that result from acknowledging a higher variability in the arrival process can be reduced by recourse. We can further show that integrating the recourse decision further decreases the staffing costs while maintaining a sufficient performance.

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116


Stochastic Networks and Processes - Andreas Lopker -Stochastic Processes

FIRST PASSAGE TIME OF SKIP-FREE MARKOV CHAINS WITH APPLICATION TO RUIN THEORY

Michael Chek Hin Choi, Pierre Patie

In this talk, we develop the potential theory of skip-free Markov chains and characterize the distribution of their first passage times, with potential overshoot, to a fixed level. We recall that a Markov chain on a denumerable state space is said to be, for instance, upward skip-free if it has upward jump of unit size, yet it can have downward transition of arbitrary magnitude. Karlin and McGregor have shown that the first passage time upward from state x to state y, where x < y, is a convolution of geometric random variables for birth-and-death processes, that is for Markov chains which are both upward and downward skip-free. This fascinating result has been extended to upward skip-free Markov chains by two different interesting methods, namely a spectral approach by Abate and Whitt and an intertwining approach by Fill and Diaconis and Fill. Unfortunately, little is known about the more delicate case when x > y, that is in the presence of overshoot as the skip-free property no longer apply. To overcome this difficulty, we suggest an original and comprehensive approach based on a combination of potential theory and the theory of Martin boundary. The motivation underlying our approach is to take advantage of the upward skip-free property to characterize explicitly the upper Martin boundary, and, by means of tools from potential theory, we develop methodologies to study the first passage time downward. Finally, we shall explain how our approach allows one to get detailed information, regarding the (finite-time) ruin probability as well as the so-called Gerber-Shiu function in the framework of general discrete-time risk models.

STOCHASTIC MODELS FOR NONLINEAR TIME SCALES

Peter Straka, Sergei Fedotov, Hans-Peter Scheffler, Boris Baeumer, Mihaly Kovacs

In many complex systems, inter-event times are drawn from a heavy-tailed distribution. Scaling limits of such dynamics can be compactly modelled by a time change with the inverse of a Levy process (stochastic subordination). This talk presents various applications of this result: In disordered media, trapping times are heavy-tailed, leading to subordinated diffusion processes [1]; Sound wave propagation in human tissue is nicely modelled by a subordinated wave equation [2]; And extremes of events occurring in bursts are modelled by subordinated extremal processes.

[1] Straka, P. & Fedotov, S. (2015). Transport equations for subdiffusion with nonlinear particle interaction. Journal of Theoretical Biology, 366, 71-83. doi:10.1016/j.jtbi.2014.11.012[2] Baeumer, B., Kovacs, M. & Straka, P. (in preparation)

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117

MULTIFRACTAL ANALYSIS USING THE CROSSING TREE

Geoffrey Decrouez, Pierre-Olivier Amblard

The crossing-tree is a very general concept. It provides a representation of a one-dimensional continuous stochastic process in terms of a random tree. Its construction relies on an ad-hoc decomposition of a signal adapted to its dynamics, and thus represents a natural tool for the analysis of its local fluctuations. The crossing-tree is used here in the context of the multifractal analysis of H-sssi processes. We present a new multifractal formalism and a novel approach for estimating the spectrum of singularities of H-sssi processes using the crossing-tree. The performance of the crossing-tree based method is demonstrated in a numerical study, and compared with state of the art wavelet-based techniques, including wavelet leaders.

ON A GENERALIZATION OF THE STATIONARY EXCESS OPERATOR

Andreas Löpker, Yoav Kerner

We show that overshoots over Erlang random variables give rise to a natural generalization of the stationary excess operator and its iterates. The new operators can be used to derive expansions for expectations of the form E(g(X)) for a non-negative random variable X, similar to Taylor-like expansions encountered when studying stationary excess operators.


Service Applications - Philipp Afèche - Strategic Behavior in Queues

AN EQUILIBRIUM ANALYSIS OF A MULTICLASS QUEUE WITH ENDOGENOUS ABANDONMENTS

Xiaoshan Peng

This paper studies a multiclass queueing system with endogenous abandonments where the congestion affects customers’ abandonment behavior and vice versa. Our model captures this interaction by developing two closely related models: an abandonment model and a queueing model. In the abandonment model, customers take the virtual waiting time distribution as given. Customers receive a reward from service and incur a cost per period of waiting. Customers are forward looking and make wait or abandon decisions dynamically to maximize their expected discounted utilities. The queueing model takes the customers’ abandonment time distribution as an input and studies the resulting virtual waiting time distribution. Because the multiclass queueing system is not amenable to exact analysis, we resort to an approximate analysis in the conventional heavy traffic limit (under the hazard rate scaling). Leveraging the so-called state-space collapse property, we provide a characterization of the system performance. Combining the results for the two models, we show that there exists a unique equilibrium in which the customers’ abandonment time and the virtual waiting time for the various classes are consistent in the two models. Lastly, we provide a computational scheme that calculates the equilibrium numerically.

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118

OBSERVATIONAL LEARNING AND ABANDONMENT IN CONGESTED SYSTEMS

John Yao

Demand systems used in operations management and service operations settings often assume that system parameters that may affect user decisions, e.g., to join a system or purchase a service, are known or accurately communicated to the market. In several practical settings, this need not be the case, but users may still form estimates of these system parameters through their own observations or experiences in the system. In this talk, we study the effect of observational learning on user behavior and equilibrium system performance in the context of a queueing model. Specifically, we analyze a congested service system in which delay-sensitive customers have no a priori knowledge of the service rate, but instead join the system and observe their progress through the queue in order to learn the system’s service rate, estimate remaining waiting times, and make abandonment decisions.

WHAT TO INFORM CUSTOMERS TO MAKE THEM JOIN A SERVICE SYSTEM

Nahum Shimkin

We consider a service system, such as a queueing system, to which customers arrive sequentially. Upon arrival, each customer receives from the system manager some information about his or her expected quality of service (for example, the expected waiting time in the queue, based on the current queue size which is unobservable by the arriving customer), and may then decide whether to balk or join the system. The manager is committed to truth telling, but can provide partial information (e.g., a range of possible waiting times, which must include the true one). We ask what information should be provided to arriving customers to maximize the throughput, namely the fraction of customers that choose to join. This question is formulated as an optimization problem, in terms of the service demand curve and the probability distribution of the service quality. Concrete solutions are derived, whose form depends on the convexity or concavity properties of the demand curve.

RATIONAL ABANDONMENT FROM PRIORITY QUEUES: EQUILIBRIUM STRATEGY AND PRICING IMPLICATIONS

Philipp Afèche

The literature on the economics of queues predominantly focuses on the queue-joining decisions of customers and ignores subsequent abandonment decisions. Such abandonment behavior is particularly important in priority queues, which are quite prevalent in practice.We characterize the equilibrium joining and abandonment behavior of utility-maximizing customers for an observable two-class priority queue and identify novel pricing implications.

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119


Stochastic Applications - Tolga Tezcan - Application of Stochastic Processes

PROBABILISTIC MATCHING SYSTEMS

Burak Büke, Hanyi Chen

In this work we introduce a novel queueing model with two classes of users where users wait in the system to match with a candidate from the other class, instead of accessing a resource. The users are selective and the matchings occur probabilistically. This new model is useful for analyzing the traffic in web portals that match suppliers of a service/product with customers. Examples of such systems are employment portals, matrimonial and dating sites and rental portals. We provide a Markov chain model for analyzing the stability of these systems and derive the probability distribution of the number of matches up to some finite time given the number of arrivals. Further, to gain more insight into the behavior of probabilistic matching systems, we propose approximation methods based on fluid and diffusion limits using different scalings. We also discuss the optimal control mechanisms to maximize the profit.

SERVICE SYSTEMS WITH UNCERTAIN QUALITY AND ANECDOTAL REASONING CUSTOMERS

Kenan Arifoğlu

We study service systems with uncertain service quality and boundedly-rational customers who form their belief about the service quality based on word of mouth (anectodes). We characterize customers’ equilibrium joining behaviour and study the impact of rationality level on the monopolistic firm’s profit, consumer surplus and social welfare. We show that firm’s optimal pricing policy with boundedly-rational customers is different from the fully rational benchmark: High price does not always signal high quality and the ‘’law of demand” in queuing theory fails. The firm may price lower when higher rationality level induces higher demand. We find that the firm should either disclose or suppress all information about the service quality. The firm with a low service quality should promote service quality uncertainty by customizing service while the firm with high service quality should adopt service standardization and reduce the quality uncertainty. Interestingly, we find that bounded rationality can benefit customers and improve consumer surplus when the service quality is high: Higher rationality level leads customers to make better decision and cause congestion, which in turn reduces consumer surplus. Finally, we show that even though high level of rationality and service uncertainty benefit a low-quality server, both are harmful for the social welfare.

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120

ANALYSIS OF TRIAGE SYSTEMS IN EMERGENCY ROOMS

Tolga Tezcan

In this paper we study the triage decisions in emergency rooms. Particularly we provide a procedure to determine when to apply provider triage (PT) based on the operational and financial considerations using a queueing framework. We obtain closed-form expressions for the range of arrival rates in which PT economically outperforms the traditional nurse triage using a steady-state many-server fluid approximation. We show that the proposed solution methodology based on this approximation procedure is asymptotically optimal under a many-server asymptotic regime. Using patient data from a large teaching hospital, we show via simulation experiments that the performance of the proposed policy is within 0.82% of the best solution that was found by total enumeration, and it is computationally much more efficient than total enumeration.

PULL-BASED LOAD DISTRIBUTION IN LARGE-SCALE HETEROGENEOUS SERVICE SYSTEMS

Sasha Stolyar

We consider a heterogeneous service system, consisting of several (different) large server pools, and study an asymptotic regime in which the customer arrival rate and pool sizes scale to infinity simultaneously. We introduce a ‘pull-based’ scheme (called PULL), for routing arriving customers to servers and prove that, under subcritical load, both waiting times and blocking probabilities asymptotically vanish. In particular, the performance of PULL is vastly superior to that of the celebrated ‘power-of-d-choices’ (JSQ(d)) routing algorithm.


Stochastic Applications - Mustafa Hayri Tongarlak - Sustainable Operations Management

SUSTAINABILITY, NO MORE A NIGHTMARE BUT A DREAMCOME TRUE: EMPIRICAL EVIDENCE OF CAUSALITY

Damla Usar, Mehmet Ali Soytaş, Meltem Denizel, Nil Deniz

There are different opinions about the link between corporate sustainability performance and corporate financial performance. The direction of the relationship between and corporate sustainability and corporate financial performance, and even whether there is such a relationship is still an unsolved problem. There are few academic papers, studying this question and the results are mixed due to the endogeneity problem. We address the endogeneity problem with IV methodology and present empirical evidence that corporate sustainability is positively associated with corporate financial performance using a sample of 8,523 firms and 34,092 firm-year observations. Furthermore we find that sustainability investments are more costly for productive companies.

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121

HOW TO USE DATA ANALYTICS FOR SMARTER ENERGY MANAGEMENT

Özge İşleğen

The electricity industry has recently enjoyed the influx of “big data.” Through smart grid technologies, many firms now have access to the consumption behavior of their customers in unprecedented detail. This talk demonstrates how firms use this data to design effective demand side management programs to change the consumption behavior of their customers and, in turn, support the peak load reduction and energy conservation efforts in electricity supply chains.

CONVERTING RETAIL WASTE INTO BY-PRODUCT

Mustafa Hayri Tongarlak, Deishin Lee

By-product synergy (BPS) is a form of joint production that uses the waste stream from one (primary) process as useful input into another (secondary) process. The synergy is derived from avoiding waste disposal cost in the primary process and virgin raw material cost in the secondary process. BPS increases profit and can have a positive environmental impact by reducing waste. We investigate how BPS can mitigate food waste in a retail grocery setting, and how it interacts with other mechanisms for reducing waste (i.e., waste disposal fee and tax credit for food donation). We derive the retailer’s optimal order policy under BPS and compare it to the policy when the two processes operate independently, showing how it affects the amount of waste. We find that BPS can reduce waste when waste disposal cost and tax benefit for donation are low, but increases waste when they are high. Ironically, by using waste productively, BPS can actually increase total waste. We show that BPS can increase the threshold tax benefit required to induce donation because BPS competes with donation for excess primary units. We find that tax credit and disposal fee are substitute mechanisms for inducing food donation. We propose a hybrid approach to implementing BPS that preserves managerial autonomy, increases profit compared to independent ordering, and reduces food waste.

TRADEOFF BETWEEN STORAGE AND TRANSPORT IN MERCHANT ENERGY TRADING ON A NETWORK

Nicola Secomandi, Selvaprabu Nadarajah

The operations of merchant energy trading in wholesale markets across different geographical locations and current and future dates can be represented as a network where storage and transport trades compete for the capacity of storage and transport assets. We study the tradeoff between storage and transport trading for a network with a single storage asset and multiple transport assets, a realistic situation that we model as a Markov decision problem (MDP). Due to the intractability of computing an optimal policy of this MDP, we leverage our structural analysis of this model to modify a least squares Monte Carlo method to obtain a heuristic policy, also computing both lower and upper bounds on the market value of an optimal policy. On a realistic natural gas application, we document a substantial tradeoff between storage and transport trading. This tradeoff is difficult to manage, as sequential storage and transport trading is considerably suboptimal, especially when prioritizing transport over storage. In contrast, our joint policy is near optimal. A practice-based method based on sequentially reoptimizing a deterministic model is also near optimal, but, even after simplification, is computationally more intensive than our approach. Beyond natural gas, our research has relevance for managing the merchant trading operations of other energy sources, natural resources, and other storable commodities.

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122


Stochastic Control - Bora Çekyay -Stochastic Control and Network Problems

A PARTITIONING ALGORITHM FOR MARKOV DECISION PROCESSES AND ITS APPLICATION TO LIMIT ORDER BOOKS

Ningyuan Chen, Steven Kou, Chun Wang

We propose a partitioning algorithm for a class of linear-quadratic Markov decision processes (MDPs) with inequality constraints and non-convex stagewise cost. Within each region of the partitioned state space, the value function and the optimal solution can be computed analytically. As an application, we present a model for limit order books with stochastic market depth, consistent with empirical studies. Stochastic market depth is necessary to accommodate various order activities, such as limit order submission at and outside the best quotes and order cancellation, which may account for a large proportion of limit order activities. The optimal order execution policy is solved by the algorithm and significantly outperforms the policy of a deterministic market depth model in numerical examples.

CUSTOMIZED MARKOV DECISION PROCESS ANALYSIS UNDER DISCOUNTED COST CRITERION

Bora Çekyay

The uniformization technique is a widely used method in establishing the existence of optimal policies with certain monotonicity properties. This technique converts a semi-Markov decision process with exponential sojourn times (ESMDP) into an equivalent discrete-time Markov decision process by defining some bogus jumps. In this study, a new device, called customization, is proposed, which can be used to convert a given ESMDP into an equivalent one which is more suitable to show the existence of certain monotonicity properties of the optimal policy. The customization technique uses the bogus jump idea to establish the equivalence under stationary deterministic policies just like the uniformization technique. However, it can be applied even when the transition rates are unbounded, where the uniformization is not applicable. We analyze a simple infinite server queue problem with unbounded transition rates to demonstrate the superiority of the customization technique.

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123

OPTIMAL ADMISSION CONTROL FOR MANY-SERVER SYSTEMS WITH QED-DRIVEN REVENUES

Jaron Sanders, S.C. Borst, A.J.E.M. Janssen, J.S.H. van Leeuwaarden

We consider Markovian many-server systems with admission control operating in a Quality-and-Efficiency-Driven (QED) regime, where the relative utilization approaches unity while the number of servers grows large, providing natural Economies-of-Scale. In order to determine the optimal admission control policy, we adopt a revenue maximization framework, and suppose that the revenue rate attains a maximum when no customers are waiting and no servers are idling. When the revenue function scales properly with the system size, we show that a nondegenerate optimization problem arises in the limit. Detailed analysis demonstrates that the revenue is maximized by nontrivial policies that bar customers from entering when the queue length exceeds a certain threshold of the order of the typical square-root level variation in the system occupancy. We identify a fundamental equation characterizing the optimal threshold, which we extensively leverage to provide broadly applicable upper/lower bounds for the optimal threshold, establish its monotonicity, and examine its asymptotic behavior, all for general revenue structures. For linear and exponential revenue structures, we present explicit expressions for the optimal threshold.

PATHWISE DIFFERENTIABILITY OF SEMIMARTINGALE REFLECTED BROWNIAN MOTIONS IN CONVEX POLYHEDRONS

David Lipshutz, Kavita Ramanan

We consider the pathwise differentiability of semimartingale reflected Brownian motions (SRBMs) in convex polyhedrons. In particular, we consider derivatives with respect to input parameters that determine the initial condition and the drift of the SRBMs. We characterize these derivatives as solutions to time-dependent Skorokhod type problems associated with the SRBMs. We use these results to characterize derivatives of stochastic flows for SRBMs, extending previous results which considered stochastic flows of SRBMs in convex polyhedrons stopped at the first hitting time of the non-smooth part of the boundary. As a second application, we obtain results on the sensitivity of SRBMs to small perturbations of the drift. Since SRBMs arise as diffusion approximations to queueing networks in heavy traffic, functionals of SRBMs are used to approximate performance measures of queueing networks. We demonstrate how our sensitivity results are relevant for assessing the sensitivity of performance measures to system parameters and have the potential to simplify computations used to estimate these sensitivities.

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124


New Directions in Applied Probability - Alessandro Arlotto - New Directions in Applied Probability

SEQUENTIAL KNAPSACKS AND ONLINE MONOTONE SEQUENCES: CONNECTIONS, DIFFERENCES AND CENTRAL LIMIT THEOREMS

Alessandro Arlotto

In this talk we prove a central limit theorem for the sequential selection of a monotone increasing subsequence of maximal expected length and discuss how this result relates to limit theorems for sequential knapsack problems.

MANAGING CONGESTION IN MATCHING MARKETS

Yash Kanoria, Nick Arnosti, Ramesh Johari

We consider a decentralized two-sided matching market in which agents arrive and depart asynchronously. As a result, it is possible that an agent on one side of the market (an “employer”) identifies an agent on the other side of the market (an “applicant”) who is a suitable match, only to find that the applicant is already matched. We find using a mean field approach that lack of knowledge about availability can create large welfare losses to both employers and applicants. We consider a simple intervention available to the platform: limiting visibility of applicants. We find that this intervention can significantly improve the welfare of agents on both sides of the market; applicants pay lower application costs, while employers are less likely to find that the applicants they screen have already matched. Somewhat counterintuitively, the benefits of showing fewer applicants to each employer are greatest in markets in which there is a shortage of applicants.

STRATEGIC TIMING OF CONTENT IN ONLINE SOCIAL NETWORKS

Tauhid Zaman

In online social networks users generate content for which they wish to maximize engagement by other users. Engagement consists of the users interacting with the content through various mechanisms. For instance, in Twitter where the content is a 140 character message known as a tweet, engagement can be a user replying to the tweet. Engagement can only occur if a user actually sees the content, which is referred to as an impression. Therefore, if one can increase the number of impressions their content receives, this should also increase engagement. While many studies have looked at how to engineer the content to maximize engagement, to the best of our knowledge there has been no work done on how to maximize the number of impressions. We show that through strategic timing of content in social networks such as Twitter or Instagram, one can increase the number of impressions it receives by as much as 80%. Our work transforms the timing of content into an operational lever that can be used to optimize social media campaigns for essentially no cost.

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125

NECESSITY OF FUTURE INFORMATION FOR EFFECTIVE ADMISSION CONTROL

Kuang Xu

Can predictions and knowledge about the future significantly improve queueing performance? If so, how much future information do we need? We investigate these questions in the context of queueing admission control, where a system manager is allowed to divert incoming jobs up to a fixed rate, in order to minimize the queueing delay experienced by the admitted jobs. The manager also has access to a look-ahead window containing (noisy) predictions of future arrival and service times.

In the heavy-traffic regime, we establish a lower bound on the length of the look-ahead window, below which any prediction-aided policy cannot improve delay by more than a constant multiplicative factor over an optimal online policy. Our lower bound matches an upper bound of [Spencer et al. 2014] up to a constant multiplicative factor. Our result hence demonstrates that the system’s delay performance depends sharply on the amount of future information available.

The proof is based on analyzing certain excursion probabilities of the input sample paths, and exploiting a connection between a policy’s diversion decisions and subsequent server idling.


Simulation - Shane Henderson -Simulation Optimization

QUANTIFY UNCERTAINTY IN STOCHASTIC OPTIMIZATION

Enlu Zhou, Henry Lam

We consider stochastic optimization problems in which the input probability distribution is not fully known, and can only be observed through data. Common procedures handle such problems by optimizing an empirical counterpart, namely via using an empirical distribution of the input. The optimal solutions obtained through such procedures are hence subject to uncertainty of the data. We explore techniques to quantify this uncertainty that have potentially good finite-sample performance. We consider three approaches: the empirical likelihood method, nonparametric Bayesian approach and the bootstrap approach. They are designed to approximate the confidence intervals or posterior distributions of the optimal values or the optimality gaps. We present computational procedures for each of the approaches and discuss their relative benefits. A numerical example on conditional value-at-risk is used to demonstrate these methods.

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126

DISCRETE OPTIMIZATION VIA SIMULATION USING COORDINATE SEARCH

L. Jeff Hong

We propose a coordinate search algorithm to quickly find a local optimal solution for optimization-via-simulation problems with integer-ordered decision variables. Furthermore, to apply the coordinate search algorithm iteratively to find global optimal solutions, we design a test of convexity to identify starting points that may lead to different local optimal solutions. We test our algorithm extensively through numerical studies and find it quite competitive when compared to existing algorithms.

CNORTA FOR CONSTRAINED ESTIMATION USING GAUSSIAN COPULAS

Raghu Pasupathy

We consider the question of efficiently estimating a performance measure that is expressed as the expectation of a function of a constrained NORTA random vector. Motivating contexts seem numerous and include the estimation of portfolio performance in finance and loss modeling in epidemics, in addition to more standard contexts arising in manufacturing and production systems. The main idea we propose is an adaptive tilting scheme that progressively refines the (Gaussian) measure that drives simulation sampling. We show that the proposed method is particularly effective in contexts where naive simulation is rendered inefficient due to the existence of the constraint set. We will discuss the asymptotic optimality and finite-time efficiency of the proposed estimator.

EFFICIENT RANKING AND SELECTION IN PARALLEL COMPUTING ENVIRONMENTS

Susan R. Hunter

We develop an algorithm for ranking and selection (i.e., simulation optimization on finite sets) in a parallel computing environment that provides a probability of good selection guarantee on the returned solution. We explain why care is required in implementing screening in parallel settings, and why a probability of good selection guarantee is preferable to a probability of correct selection guarantee when the number of systems is large. Finally, we demonstrate our algorithm on a problem with over one million systems.

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127


Queues - Limit Theorems - Itai Gurvich - Limit Theorems and Approximations

HEAVY TAILS IN THE HALFIN-WHITT REGIME

David Goldberg, Yuan Li

In this talk, we analyze multi-server queues in the Halfin-Whitt regime, when the processing time distribution is heavy-tailed. We prove tightness of the corresponding sequence of (normalized) steady-state queue-length variables, and analyze the associated large-deviations behavior. Interestingly, we find that the large-deviations behavior is qualitatively different from that in the light-tailed case. Our proofs combine a stochastic-comparison approach with results for heavy-tailed renewal processes, and bounds for the suprema of random walks. Time permitting, we will also discuss the setting in which the inter-arrival times are heavy-tailed, as well as a novel scaling regime in which both the inter-arrival and processing times are heavy-tailed.

A UNIFIED APPROACH TO DIFFUSION ANALYSIS OF QUEUES WITH GENERAL PATIENCE-TIME DISTRIBUTIONS

Junfei Huang, Jiheng Zhang and Hanqin Zhang

We propose a unified approach to establishing diffusion approximations for queues with impatient customers within a general framework of scaling customer patience time. The approach consists of two steps. The first step is to show that the diffusion-scaled abandonment process is asymptotically close to a function of the diffusion-scaled queue length process under appropriate conditions. The second step is to construct a continuous mapping not only to characterize the system dynamics using the system primitives, but also to help verify the conditions needed in the first step. The diffusion approximations can then be obtained by applying the continuous mapping theorem. The approach has two advantages: (i) it provides a unified procedure to establish the diffusion approximations regardless of the structure of the queueing model or the type of patience time scaling; (ii) and it makes the diffusion analysis of queues with customer abandonment essentially the same as the diffusion analysis of queues without customer abandonment. We demonstrate the application of the approach via the single server system with Markov-modulated service speeds in the traditional heavy-traffic regime and the many-server system in the Halfin-Whitt regime and NDS regimes.

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128

NON-MARKOVIAN STATE-DEPENDENT NETWORKS IN CRITICAL LOADING

Yunan Liu

We establish a heavy traffic limit theorem for the queue-length process in a critically loaded single class queueing network with state-dependent arrival and service rates. A distinguishing feature of our model is non-Markovian state dependence. The limit stochastic process is a continuous-path reflected process on the nonnegative orthant. We give an application to a generalised Jackson network with state-dependent rates.

THE JAMMING CONSTANT OF UNIFORM RANDOM GRAPHS

Pascal Moyal, P. Bermolen and M.Jonckheere

We propose a differential equation method in infinite dimension, allowing to assess the jamming constant, i.e. the size of the maximal independent set of random graphs having a given degree distribution, under the large graph asymptotics. This is done by constructing the independent set (i.e. a set of nodes including no couple of neighbors) together with the random (multi-) graph, by matching the half-edges of the nodes uniformly at random, as in the Configuration Model. This algorithm induces a Markov representation in the space of point measures, and the fluid limit of the latter process leads to an asymptotic characterization of the jamming constant. We thereby retrieve and generalize existing results for regular graphs, and to asymptotic Poisson degree distributions, as is the case for the Erdos-Renyi graph. An application to the CSMA protocol is also proposed.

July 8th - Wednesday 09:00-10:40Room No: ENG B05

Service Applications - Douglas Down - Queues with Computer Applications

DISPATCHING JOBS TO PARALLEL SERVERS WITH NON-LINEAR COST STRUCTURES

Esa Hyytia

In traditional job dispatching problems the arriving jobs are assigned to servers with the aim of minimizing the mean delay, i.e., the mean number of jobs in the system. However, this objective may be ill-suited if, e.g., fairness needs to be taken into account. Therefore, we consider the job dispatching problem with two different cost structures. First, we assume that users become gradually more impatient and their holding cost rate increases linearly in time, yielding a total cost that is the square of the sojourn time. Second, we consider a cost structure that is based on deadlines. The latter is a relevant model, e.g., in the context of quality of experience (QoE), where customers perceive a service as good as long as it is completed within a reasonable time, and otherwise the service is classified as unsatisfactory. Also service level agreements (SLA) can be defined in terms of deadlines. We develop efficient dispatching policies for both scenarios in the framework of Markov decision processes. First, we analyze single M/G/1 -queues and derive value functions with respect to different cost structures, and then obtain efficient cost-aware dispatching policies by means of policy iteration and lookahead. The performance of the resulting policies is evaluated by numerical simulations. Furthermore, we observe a peculiar behavior regarding stability when the cost structure is based on deadlines.

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STOCHASTIC AND FLUID INDEX POLICIES FOR RESOURCE ALLOCATION PROBLEMS

Maialen Larranaga

We develop a unifying framework to obtain efficient index policies for restless multi-armed bandit problems with birth-and-death state evolution. This is a broad class of stochastic resource allocation problems whose objective is to determine efficient policies to share resources among competing projects.In a seminal work, Whittle developed a methodology to derive well-performing (Whittle’s) index policies that are obtained by solving a relaxed version of the original problem. Our first main contribution is the derivation of a closed-form expression for Whittle’s index as a function of the steady-state probabilities. It can be efficiently calculated, however, it requires several technical conditions to be verified, and in addition, it does not provide qualitative insights into Whittle’s index. We therefore formulate a fluid version of the relaxed optimization problem and in our second main contribution we develop a fluid index policy. The latter does provide qualitative insights and is close to Whittle’s index. The applicability of our approach is illustrated by two important problems: optimal class selection and optimal load balancing. Allowing state-dependent capacities we can model important phenomena: e.g. power-aware server-farms and opportunistic scheduling in wireless systems. Numerical simulations show that Whittle’s index and our fluid index policy are both nearly optimal.

DYNAMIC MATCHING IN OVERLOADED WAITING LISTS

Jacob Leshno

We consider waiting lists as dynamic allocation mechanisms and study their efficiency in matching agents to the right items. The waiting list dynamically assigns items that arrive over time, and the randomness of arrivals causes waiting times to fluctuate. Welfare is maximized when agents decline mismatched items, but impatient agents will take a mismatched item if they face too long a wait for their preferred item. This is socially inefficient, as waiting time is not eliminated, but transferred to other agents. We set up a model that allows us to analyze welfare and derive policies that increase welfare by reducing misallocation.

PARAMETER TUNING FOR SIZE-BASED ROUTING POLICIES

Douglas Down

Size-based routing policies are one means to mitigate the effects of high service time variance in systems of parallel queues. Most of the literature assumes that the service time distribution is known, allowing one to make exact calculations of ranges of service times which are to be assigned to each queue. In the face of errors in estimating the service time distribution, performance of such schemes could dramatically degrade. In this talk, we propose a simple scheme to dynamically adjust these ranges, using the notion of a proportional controller with dead zones.

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130


Queues - Barış Balcıoğlu - New Perspectives in Queueing Models

CUSTOMER EQUILIBRIUM POLICIES IN FEED-FORWARD OPEN JACKSON NETWORKS

Apostolos Burnetas

Consider a feed-forward open Jackson network with given external arrival rates in each queue. We assume that customers make join/balk as well as routing decisions based on a service reward/delay cost trade-off. External customers arriving to a queue decide whether to join or balk. Customers who finish service in a queue decide to join one of the immediately following queues or balk. The number of servers, the exponential service rate, the service reward and the delay cost rate are generally queue-dependent. We consider the unobservable case, where customers do not have any information on the system state, but know the network structure as well as all system parameters. Each customer’s objective is to maximize the expected total net benefit from the entire route they follow in the system.

We consider customer equilibrium strategies for the unobservable case. We develop recursive equations for the customer’s benefit function and show that under fairly general conditions there exist unique, generally mixed, equilibrium strategies.

Finally, we discuss the problem of optimally admitting/routing customers within the network so as to maximize the expected total benefit of all customers. In order to compare equilibrium and optimal strategies, we also consider the unobservable, i.e., state-independent case of the social optimization problem. We present analogous recursive expressions that lead to a dynamic programming formulation and discuss the relationship between equilibrium and optimal strategies.

EXPLORING AND EXPLOITING MDP FORMULATIONS OF OPTIMAL SCHEDULING OF CUSTOMERS WITH ABANDONMENT

Peter Jacko

Many real-world situations involve queueing systems in which customers may abandon while waiting, which causes a damage to such customers and/or the operator. Two notorious examples are scheduling of customers in contact centres and scheduling of patients for a treatment. We use the Markov decision processes (MDP) framework to formulate and study a comprehensive model for multi-class queue scheduling accounting for customer abandonment, with the objective of minimizing the sum of waiting costs and abandonment penalties. We are interested in the optimal policy, but due to its complexity and intractability we also develop simple “index rule” heuristics by a number of approaches, which turn out to have elegant and novel closed-form characterizations.

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131

OPTIMIZING ARRIVAL PATTERNS IN TIME-DEPENDENT QUEUES

Raik Stolletz

Demand management mechanisms for distribution centers aim at smoothing demand by shifting truck arrivals from peak to off-peak periods in order to improve the system’s operational performance. We provide a general, reliable, and fast methodology to evaluate and optimize the arrival pattern for the time-dependent G(t)/G/c queueing system of truck handling operations.

Our optimization approach is based on the stationary backlog-carryover approach to analyze the system’s performance. The time-dependent arrival rates serve as decision variables, i.e., the decision model’s outcome are changes to an originally preferred or forecasted demand pattern. Two objectives are considered in this non-linear optimization model: Minimizing total waiting times and minimizing the related and penalized shift in the arrival pattern. A numerical study compares the performance measures of original and optimized arrival patterns for truck handling operations of a distribution center and at an air cargo terminal. It shows that a significant reduction in waiting times can be reached even with minor shifts in time-dependent arrival rates.

MULTICLASS M/GI/1 QUEUE WITH STATE-DEPENDENT ARRIVAL RATES

Barış Balcıoğlu

In this study, we consider a single server queue attending to multiple classes of customers arriving one at a time according to independent Poisson processes. The arrival rate of each class depends on how many customers of its type are in the system. Customer classes are prioritized and served according to the preemptive-resume priority policy. We assume that if a specific class were the only class served, its service time would follow a general distribution specific to that class. However, if the presence or new arrivals of higher priority customers make the server unavailable/interrupted for a customer type, the total time it spends on the server may get longer. The total amount of time spent on the server, which includes periods of server interruption, is referred to as the process completion time in the literature. Using a busy period analysis, we characterize the process completion time random variable and eventually employ the single class M/GI/1 queue results to obtain the steady-state system size distribution for each class. We then use the model in a make-to-stock queue where dynamic policies govern the customer arrival rates.

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132


Stochastic Applications - Özge Büyükdağlı - Inventory and Production

A HEURISTIC METHOD FOR STORAGE LOCATION ASSIGNMENT PROBLEM FOR A DISTRIBUTION CENTER

Zeynep Turgay, Necati Aras

Storage location assignment (addressing) of products is an important research topic in warehouse design. In this study, we solve storage location assignment problem for a distribution center providing day-to-day service for a retail chain. Orders are received from the stores and each order has a random number of products with random quantities. Order pickers collect the products ordered by the stores from their addresses and deliver to the order shipment area. The objective is to optimize the total effort spent by the order pickers in terms of the total traveling distance (or time). The most widely used method in the literature is the cube-per-order index policy (COI). Even though it is simple to implement, the COI policy relies on strict assumptions. Since, a distribution center of a retail chain may have significantly different characteristics from these assumptions, the use of COI policy may lead to suboptimal results. In this study we propose a different indexing policy based on relative ordering frequency and joint relative ordering frequency of items, and also develop improvement heuristics for the same problem. We demonstrate the effectiveness of our proposed method on real data and also show that it improves the results obtained by classical methods such as COI and ABC analysis by up to 25%.

A QUEUEING-INVENTORY MODEL

Rim Essifi

We study an ‐type queueing model with the following additional feature.The server works continuously, at fixed speed, even if there are no service requirements. In the latter case, it is building up inventory, which can be interpreted as negative workload. At random times, with an intensity when the inventory is at level , the present inventory is removed instantaneously reducing the inventory to zero. We study the steady‐state distribution of the (positive and negative) workload levels for the cases is constant and . The key tool is the Wiener‐Hopf factorisation technique. When is constant, no specific assumptions will be made on the service requirement distribution. However, in the linear case, we need some algebraic hypotheses concerning the Laplace‐Stieltjes transform of the service requirement distribution.

Finally, we shall study a similar model coming from insurance risk theory.

We study an ‐type queueing model with the following additional feature.The server works continuously, at fixed speed, even if there are no service requirements. In the latter case, it is building up inventory, which can be interpreted as negative workload. At random times, with an intensity when the inventory is at level , the present inventory is removed instantaneously reducing the inventory to zero. We study the steady‐state distribution of the (positive and negative) workload levels for the cases is constant and . The key tool is the Wiener‐Hopf factorisation technique. When is constant, no specific assumptions will be made on the service requirement distribution. However, in the linear case, we need some algebraic hypotheses concerning the Laplace‐Stieltjes transform of the service requirement distribution.

Finally, we shall study a similar model coming from insurance risk theory.

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133

CONTROL OF A MULTI-SERVER MAKE-TO-STOCK PRODUCTION SYSTEM WITH SETUP COSTS

Sinem Özkan, Önder Bulut

We study production and inventory control problems for an M/M/s make-to-stock queue with production setup costs, several customer classes and lost sales. We model and analyze the system using optimal control and dynamic programming techniques. At any system state, production decision is to specify whether to activate new production channels or to continue with the currently active ones. If the decision is to activate new channels, a fixed/setup cost is incurred per channel. At the decision epochs where the system experiences demand from any customer class, the controller should also decide whether to satisfy the arriving demand or to reject it. We extend the literature of the control of make-to-stock queues by considering fixed system costs and multiple servers at the same time. With numerical studies we first characterize the structure of the optimal production and rationing policies and then propose new/alternative policies that have well-defined structures and are easier to apply compared to the optimal ones. Numerical and theoretical studies are carried out to assess the performances of the proposed policies.

CONTROL OF M/Ek/s MAKE-TO-STOCK PRODUCTION SYSTEM WITH SEVERAL DEMAND CLASSES

Özge Büyükdağlı, Önder Bulut, Murat Fadıloğlu

This study considers production control and stock rationing problem of a make‐to‐stock system with parallel production channels producing a single product demanded by different customer classes. If an arriving demand cannot be met, it is assumed to be lost. In this setting, at any system state that provides the inventory level and production status information, the controller ought to consider the following: how many more channels should be activated and how the current inventory should be allocated among different customer classes. Today’s information technology enables us to monitor such real life systems so as to observe the production status and inventory level. We aim to define and model a general setting to make use of the available system information. For this reason, we model the system as an make‐to‐stock queue. Erlangian processing times allow us not only to monitor system as frequent as needed by changing the value of k‐parameter (at the two extremes our model can be converted to and queues) but also benefit from Markovian structure and corresponding MDP analysis techniques. These together with the flexibility of changing the value of the parameter s, the number of channels/servers, would allow us to depict the structural characteristics of optimal policies and to propose easy‐to‐apply and good‐performing production and rationing policies compared to the ones widely studied in literature and used in practice for different settings ‐including typical lost sales inventory systems‐.

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134


Stochastic Models - Peter Taylor -Markov Modulated Stochastic Models

STICKY MARKOV-MODULATED BROWNIAN MOTIONS

Guy Latouche, Giang Nguyen

We consider Markov-modulated Brownian motions with sticky boundaries and analyse the effects of these boundaries in the limiting behaviour of the processes.

FLUID SYSTEMS WITH REACTIVE BARRIERS

Eleonora Deiana, Marie-Ange Remiche, Guy Latouche

We consider a Markov modulated fluid queue with two boundaries, at level 0 and level b>0. Each time the fluid hits one of the two boundaries, the process immediately enters a new phase so as to move away from the boundaries. This set of new phases is partitioned into two parts: a set of negative phases where the system enters only upon hitting the maximum level, a set of positive phases where the system enters only upon hitting the level 0.The reason to introduce this extra set of phases is to allow the system to react in case it is idle or it reaches the maximum capacity. When the maximum capacity is reached, the negative phases forces the level to go down for a certain amount of time, before coming back to the regular behaviour. In the same way, when the system is idle, instead of staying at level 0 waiting for some new jobs, it can for example do some extra work during a certain interval of time and then continue with regular work.As the only way to enter into this set of phases is to hit a boundary, the generator of the phase process is not irreducible. We show in which measure the introduction of this extra set of phases influences the structure of the stationary distribution of the system.

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135

MARKOV-MODULATED BROWNIAN MOTIONS WITH PHASE-TYPE JUMPS

Matthieu Simon, Guy Latouche

Markov-Modulated Brownian Motions (MMBMs) have received considerable attention in the literature over the last decades. Different ways have been used for their analysis, including spectral methods and time-reversion.Recently a new method was proposed, which consists in approximating a MMBM by a family of Markov-Modulated Fluid Flows (MMFFs).In this talk we pursue the analysis of MMBMs through the last method. We consider a MMBM such that the variance in each state of the Markov-modulating process can be either positive or equal to zero, and such that two-sided jumps with phase-type distributions can occur. We define a MMFF-approximation to derive the stationary distribution and some first passage times of our MMBM. Then we use the powerful tools developed within the framework of MMFF to highlight the probabilistic meaning of our results. The approach we follow has the advantage that it gives a large importance to the physical and probabilistic behaviour of the MMBM. As a consequence, further constraints on the process can be added without needing to begin the analysis all over again.Moreover, the addition of phase-type jumps neither changes the structure of the results nor their interpretation, and does not complicate at all the analysis.

PERTURBATION ANALYSIS OF MARKOV MODULATED FLUID MODELS

Sarah Dendievel, Guy Latouche

We consider �� a Markov modulated fluid model. Our purpose is to describe the effect on Ψ, the matrix of first return probabilities to the initial level from above, of a perturbation on the parameters of the fluid model, as � is one of the key matrices for those Markov models. The model is described as follows: �� is a Markov chain, with finite state space � and infinitesimal generator A, called the phase process; ��, called the level, may be expressed as

�� . We partition � into �� with �� ,

�� and �� . Similarly, we define the rate matrix � � �� , where �∗ � �� ∗�. Formally, we define the matrix of first return probabilities from above as the matrix �, with dimension |��| � |��|, with components �� |�� , where �� and � � �� . When � is perturbed into �� , the analysis follows the usual path: �� the perturbed first return probability matrix is easily shown to be differentiable and computable equations are smoothly derived for the derivatives of ��. When � is perturbed into �� , the analysis may become involved if phases of �� in �� are transformed into phases of �� or �� in the perturbed model. Indeed, the dimensions of �� the perturbed first return probability matrix are not the same anymore as those of � the first return probability matrix of the initial model. The comparison between these matrices requires more care. This leads us to the construction of a substitute for the matrix of first return probabilities which enables the analysis of the effect of the perturbation under consideration.

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136


Stochastic Networks - Guodong Pang - Asymptotics in Stochastic Networks

ON THE TAIL ASYMPTOTICS IN GENERALISED JACKSON NETWORKS WITH HEAVY-TAILED DISTRIBUTIONS OF SERVICE TIMES

Sergey Foss

We consider a generalised Jackson network with a non-acyclic routing and heavy-tailed distributions of service times. We will present new results on the exact tail asymptotics for the distribution of a stationary sojourn time in a number of cases. The talk is partially based on a joint paper with Masakiyo Miyazawa (JAP, 2014) and on ongoing work with Masakiyo Miyazawa and Dima Korshunov.

DYNAMIC SCHEDULING FOR MARKOV MODULATED SINGLE-SERVER MULTICLASS QUEUEING SYSTEMS IN HEAVY TRAFFIC.

Arka Ghosh

Queueing networks arise as models in various areas including computer systems, telecommunications, manufacturing, and service industry. One of the key objectives in the queueing network settings is to obtain the “good” (or nearly optimal) control policies for scheduling, sequencing, and routing of jobs in the system. In this talk, I’ll present a recent study on scheduling control problem for a single-server multiclass queueing network in heavy traffic, operating in a changing environment. The changing environment is modeled as a finite state Markov process that modulates the arrival and service rates in the system. Various cases are considered: fast changing environment, fixed environment and slow changing environment. In each of the cases, using weak convergence analysis, in particular functional limit theorems for renewal processes and ergodic Markov processes, it is shown that an appropriate “averaged” version of the classical cμ-policy (the priority policy that favors classes with higher values of the product of holding cost c and service rate μ) is asymptotically optimal for an infinite horizon discounted cost criterion.

DIFFUSION PARAMETERS OF FLOWS IN STABLE QUEUEING NETWORKS AND THEIR ROLE IN DECOMPOSITION HEURISTICS

Yoni Nazarathy, Werner Scheinhardt

We consider open multi-class queueing networks with general arrival processes, general processing time sequences and Bernoulli routing. The network is assumed to be operating under an arbitrary work-conserving scheduling policy that makes the system stable. An example is a generalized Jackson network with load less than unity and any work conserving policy. We find a simple diffusion limit for the inter-queue flows with an explicit computable expression for the covariance matrix. Specifically, we present a simple computable expression for the asymptotic variance of arrivals (or departures) of each of the individual queues and each of the flows in the network. Our formulas may be put to use in improving approximate queueing network decomposition heuristics such as the well-known Queueing Network Analyzer (QNA) framework and related methods.

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137

QUEUES IN CELLS

Justin Dean

The central dogma of molecular biology states that DNA makes RNA which makes protein. Stochastic models of this process describe the transcription of DNA into RNA, and the translation of RNA into protein using Markov models. Questions of biological interest pertain to fluctuations in protein molecule numbers within cells, and their temporal dynamics. In this talk, we present models of transcription and translation as infinite server queues. We use these models to derive autocovariance functions for the numbers of molecules in terms of parameters of the input process and the service time distribution. Finally, we present some asymptotic results in the limiting regime corresponding to high transcription rates.


Healthcare Applications - Burhaneddin Sandıkçı - Stochastic Models in Healthcare

MAXIMIZING FILL RATE FOR ONLINE APPOINTMENT SCHEDULING UNDER PATIENT CHOICES

Peter van de Ven, Nan Liu, Bo Zhang

With population aging, the imbalance between health service capacity and patient demand is becoming a global problem. However, a significant amount of provider time is wasted due to the inefficiency of appointment scheduling systems currently in use; this is particularly reflected in the fluctuating, and relatively low, fill rate (the ratio between number of patients scheduled and number of available slots). A recent development in patient scheduling is the emergence of online appointment scheduling, which offer more flexibility in patient choices and give patients an Amazon-like “shopping” experience of accessing care. Although online scheduling makes it possible to better accommodate patient preferences, it is not likely to eliminate the problem of low fill rates if not managed well. In this work we investigate how to design online scheduling to achieve high fill rates.We develop models that account for heterogeneous patient preferences in appointment choice offering: there are different types of patients who accept different (sets of) appointment slot types. We observe that by only offering a strict subset of all available slots to patients, we can direct patients into appointment slots that are beneficial for achieving a high fill rate. We first obtain the optimal “blocking” policies of such for certain instances using MDPs. We then derive heuristics for the general cases, based on fluid models, and show that these significantly increase the fill rate.

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138

IMPACT OF BREAST DENSITY ON DESIGNING MAMMOGRAPHY SCREENING POLICIES

Mücahit Çevik

Mammography screening is the golden standard in the practice of breast cancer screening, but it is also known to be less accurate in women with dense breasts. Because of its imperfect nature, supplemental screening methods such as Magnetic Resonance Imaging and ultrasound have been considered to achieve better detection accuracy. There are ongoing debates about the cost-effectiveness of the use of supplemental tests, but no consensus of opinion exists in their use.In addition to influencing the accuracy of screening methods, breast density is also identified in the medical literature as a strong risk factor for breast cancer. Therefore, we incorporate the breast density information while evaluating supplemental screening tests to maximize a patient’s total quality-adjusted life years. In particular, we allow the accuracy of each test to be a function of the breast density of the patient, which itself may change with the age of the patient, and therefore, further complicates the decision process. We formalize the optimal breast cancer screening problem using a discrete-time partially observable Markov decision process model. The state space of our model is composed of the patient’s health states and her breast density states. At each decision epoch, the physician first decides whether or not the patient should undergo mammography screening, and then uses the results of this screening to follow up with supplemental screening. Our numerical study demonstrates that incorporating breast density to the design of breast cancer screening policies can significantly improve the effectiveness of the screening recommendations.

SERVICE SYSTEMS WITH HETEROGENEOUS CUSTOMERS: INVESTIGATING THE EFFECT OF TELEMEDICINE ON PATIENT CARE

Tolga Tezcan

Medical specialists treating chronic conditions typically face a very heterogeneous set of patients. Such heterogeneity arises because of difference in patients’ medical conditions as well as the travel burden each of them faces when trying to reach the clinic. In this paper, we investigate the impact of patient heterogeneity on the strategic behavior of medical specialists in terms of their operating decisions. In addressing this problem, we expand current results in queuing theory related to the service speed-quality trade-off for both revenue-maximizing and welfare-maximizing servers in the context of customers with a general and heterogeneous utility function. We find that as the relative travel burden increases for the patients, it also has a negative effect on the productivity of the specialist and his expected income. Comparing the optimal service characteristics of revenue-maximizing and welfare-maximizing specialists we see that the former will overall serve a smaller patient population, will have shorter waiting times, and will operate at a lower utilization. We also analyze the impact of the newly introduced telemedicine technology on patient utility and the specialists’ operating decisions. We prove that with the introduction of telemedicine the revenue- maximizing service rate moves closer to the socially optimal one. While the enhanced access to specialist care increases the overall social welfare, we explain why some patients, unexpectedly, will be even worse-off with the introduction of this technology. Our analytical results lead to some important policy implications for facilitating the further deployment of telemedicine in the care of chronically ill patients.

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139

COMPETITION VERSUS COOPERATION OF PUBLIC AND PRIVATE HEALTHCARE CENTERS

Saloumeh Sadeghzadeh, Pelin G. Canbolat, Lerzan Örmeci

We consider a health care system where a public center and a private center coexist. The public center is assumed to offer free service whereas the private center charges the patients a positive fee and offers a faster service. Exponentiality assumption on service and interarrival times enables a tractable analysis of the system. We analyze this system under varying assumptions on observability and decision makers. In the so-called decentralized setting, patients choose which center to attend by comparing their own expected costs, whereas in the centralized setting, a gate keeper assigns patients to centers with the objective of minimizing the total expected costs. In the observable setting, patients choose the center they will receive service after observing the numbers of patients in each center, whereas in the unobservable setting, they decide without knowing how many patients are at each center.In the observable setting, we prove that there exists an optimal policy of threshold type in both decentralized and centralized systems. More explicitly, for a given number of patients in the private (public) center, patients go to the private (public) center if the number in the public (private) center is higher than a threshold. Interestingly, the thresholds for the centralized system turn out to be smaller than the thresholds for the decentralized system, suggesting a more balanced allocation through centralization.In the unobservable setting, we characterize the (symmetric) Nash equilibrium for the decentralized model and the optimal solution for the centralized one. Each problem reduces to solving a polynomial equation.


Queues Performance - Harsha Honappa - Queues Performance

RELATING BUSY PERIOD DURATION AND SINGLE BIG JUMP PRINCIPLE IN HEAVY TRAFFIC

Bart Kamphorst, Bert Zwart, Nikhil Bansal

The Principle of a Single Big Jump (PSBJ) for M/G/1 queues with heavy-tailed job size distributions and fixed load is known for several years. However, in practice the load of servers is often increasing due to growing demands in both the number and the size of jobs. As the system approaches heavy traffic, the notion of a single big jump is poorly defined since both the mean workload and the number of jobs in a busy period tend to infinity. In this presentation I will introduce a notion for a big job for which the PSBJ holds uniformly as the load increases. Similarly, queueing literature shows only a few results on the duration of the busy period in heavy traffic. For a fixed load it has been shown that there is an intimate relation between the duration of the busy period and the PSBJ. I will discuss this relation and consequently provide intuition for the proof, which is based on a sample path analysis.

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140

A TWO-CLASS RETRIAL SYSTEM WITH COUPLED ORBIT QUEUES

Ioannis Dimitriou

PERFORMANCE ANALYSIS OF TIME-DEPENDENT QUEUEING SYSTEMS: SURVEY AND CLASSIFICATION

Justus Arne Schwarz, Raik Stolletz

Many queueing systems are subject to time-dependent changes of system parameters such as the arrival rate or the number of servers. Examples include time-dependent call volumes and agents at inbound call centers, time-varying air traffic at airports, time-dependent truck arrival rates at seaports, and cyclic message volumes in computer systems.Several approaches for the performance analysis of time-dependent queueing systems exist in the literature. We establish a classification scheme which groups the approaches according to the underlying key ideas into (i) numerical and analytical solutions, (ii) approaches based on models with piecewise constant parameters, and (iii) approaches based on modified system characteristics. Links between different approaches are identified. Furthermore, we provide a survey of applications that are categorized into service systems, road and air traffic, and IT systems.

We consider a single server system which accepts two types of retrial customers. Type � customer, � � �� arrive according to a Poisson process with rate �� and if it finds the server busy, it is blocked and routed to a separate type� orbit queue of infinite capacity. Customers from the orbits try to access the server according to the constant retrial policy. We consider coupled orbit queues, and when both orbit queues are non‐empty, the orbit queue � tries to re‐dispatch a blocked customer of type � to the main service station after an exponentially distributed time with rate ��. If an orbit queue empties, the other orbit queue changes its re‐dispatch rate from �� to ��∗. Customer's service time in the main service station is exponentially distributed with rate �. Such a system serves as a model for competing job streams in interference‐limited wireless carrier sense multiple access systems with two coupled relay nodes. An arriving job that finds the main transmitter idle, occupies it and the transmission begins. Otherwise, it is rerouted to a relay node according to its type. The relay node is responsible for transmitting the blocked jobs to the main transmitter and its rate depends on the number of jobs in the other relay node. Using generating function analysis, we show that the probability generating function of the joint stationary orbit queue length distribution can be determined using the theory of Riemann‐Hilbert boundary value problems. Performance metrics are obtained and computational issues are discussed.

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141

RARE EVENTS IN A SINGLE SERVER QUEUE WITH LONG RANGE DEPENDENT AND NON-STATIONARY TRAFFIC

Harsha Honnappa, Peter Glynn


Stochastic Processes - Masayiko Miyazawa - Queues and Stochastic Processes

COUPLING METHOD FOR MONOTONICITY PROPERTIES OF THE GI/G/1-QUEUE WITH ABANDONMENTS AND RETRIALS

Dwi Ertiningsih

A UNIFIED APPROACH FOR A LARGE QUEUE IN A HETEROGENEOUS MULTI-SERVER SYSTEM

Masakiyo Miyazawa

We are interested in a large queue in a GI/G/k queue with heterogeneous servers. For this, we consider tail asymptotics and a heavy traffic approximation for the stationary distribution of its queue length process under the stability condition, where service time distributions are assumed to have light tails. They are basically known. For example, the tail asymptotics are obtained in Neuts and Takahashi (1981) and Sadowsky and Szpankowski (1995) under some extra conditions, and a diffusion approximation is obtained for the queue length process in Chen and Ye (2011). We aim to give a unified approach for them using a piecewise deterministic Markov process and its extended generator. A key step of this approach is to drop jump components to get the extended generator in terms of the logarithmic moment generating functions, so called rate functions, of the point processes for arrivals and services. This idea is an extension of a similar one for the GI/G/1 queue in the author’s recent work.

We present large deviations principles (LDP) for the workload and waiting time processes of a queue, where traffic is assumed to follow a 'random scattering' model. A finite population

of users sample arrival times from a common distribution function and enter the queue in order of the sampled times. The LDP is unlike those derived for standard models, since the traffic model exhibits long range dependence and possible non‐stationarity which complicates the analysis. We characterize the most typical paths to overflow and most likely time of overflow, by developing the LDP in a large population acceleration regime.

We consider a ‐queueing system with generally distributed retrials. This is a FIFO ‐queue where customers may renege after an exponentially distributed time. A

reneging customer will retry with a given probability after a generally distributed time. The retrial process behaves as an ‐queue. Each customer has a static joining rule that determines if the customer enters the system upon arrival. We are interested in the structural properties of the value function of the system as a function of the joining rule. The derivation of structural properties cannot be done using standard mathematical tools, since the analysis is hindered due to the fact that the system is not uniformisable. We present a coupling method that overcomes the limitations of standard techniques and allows for the derivation of structural properties.

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142

SOME IMPORTANT RESULTS FOR LOG-PREINVEX STOCHASTIC PROCESSES

Nurgül Okur Bekar, İmdat İşcan, Hande Günay Akdemir

The well‐known Hermite‐Hadamard integral inequality:

� �� 2 � � 1

� � �� 2

�

�

is used to provide estimations of the mean value of a continuous convex function �� , �� .

In recent years, there has been an extensive interest in providing inequalities involving variety of convexity generalizations. Two of the significant extensions are invex and preinvex functions introduced by Ben‐Israel and Mond. We established these extensions for stochastic processes and provided a Hermite‐Hadamard inequality for preinvex stochastic processes in the study as follows:

Let �� be a stochastic process on ‐invex index set �. ��, � is called preinvex with respect to if

�� , ��, � � �1 � ��, � � ��, ��. �� for all �, �� and �0,1�. Throughout this paper, we assume that ��, � � ��, �� ⊆ �0,∞�.

Let �� be a preinvex stochastic process with respect to , under some assumptions, then Hermite‐Hadamard inequality for preinvex stochastic processes for any �, �� is obtained as follows:

� ��,�� , � � ��,�� , �� ,��,��

��,�� .

In this paper, we introduce another extension of preinvexity for stochastic processes, which is called log‐preinvex. Besides, we obtain a Kuhn‐type result and a Hermite‐Hadamard integral type inequality for these processes.

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143

LIMITING DISTRIBUTION OF RANDOM FIELD IN ISING MODEL UNDER RENORM-GROUP TRANSFORMATION

Farida Kachapova, Ilias Kachapov


Service Applications - Jiheng Zhang - Applications of Approximate Queueing Analysis

PATIENT FLOW MANAGEMENT IN EMERGENCY DEPARTMENTS

Junfei Huang, Boaz Carmeli and Avishai Mandelbaum

Motivated by its significant impact on quality of care and patient satisfaction, we consider the control of patient flow through physicians in emergency departments (EDs). The physicians must choose between catering to patients right after triage, who are yet to be checked, and those that are in-process (IP), who are occasionally returning to be checked. Physician capacity is thus modeled as a queueing system with multi-class customers, where some of the classes face deadline constraints on their time-till-first-service, while the other classes feedback through service while incurring congestion costs. We consider two types of such costs: first, costs that are incurred at queue-dependent rates, and second, costs that are functions of IP sojourn time. The former is our base-model, which paves the way for the latter (perhaps more ED-realistic). In both cases, we propose and analyze scheduling policies that, asymptotically in conventional heavy-traffic, minimize congestion costs while adhering to all deadline constraints. For our proposed policies, we develop some congestion laws (snapshot principles) that support forecasting of waiting and sojourn times. Simulation shows that these policies outperform some commonly-used ones. It also validates our laws and demonstrates that some ED features, the complexity of which reaches beyond our model (e.g., time-varying arrival rates) do not lead to significant performance degradation.

We consider an Ising model on n‐dimensional integer lattice with a constant strength of interaction and no external field. It is well known that there exists the thermodynamic limit of the Ising model where the limiting Gibbs measure is defined. A random variable with Bernoulli distribution is assigned to every point in the lattice, and these variables are weakly dependent. This models a physical system with many particles that have weak interaction characterized by parameter . The renormalization group with parameters and transforms the random field into a random field .

We prove a theorem that for small absolute values of the parameter the random field converges in distribution to an independent normal field as tends to infinity. The proof is based on estimations of semi‐invariants of with respect to the limiting Gibbs measure.

Ising model is used in statistical mechanics to describe a physical system with many particles, such as ferromagnetic system, the random field describes some property of the particles and renormalization group models scale transformations of the system. Thus, our theorem states that in systems with weak interaction the normalized sums of over large regions have approximately normal independent distribution.

We consider an Ising model on n‐dimensional integer lattice with a constant strength of interaction and no external field. It is well known that there exists the thermodynamic limit of the Ising model where the limiting Gibbs measure is defined. A random variable with Bernoulli distribution is assigned to every point in the lattice, and these variables are weakly dependent. This models a physical system with many particles that have weak interaction characterized by parameter . The renormalization group with parameters and transforms the random field into a random field .

We prove a theorem that for small absolute values of the parameter the random field converges in distribution to an independent normal field as tends to infinity. The proof is based on estimations of semi‐invariants of with respect to the limiting Gibbs measure.

Ising model is used in statistical mechanics to describe a physical system with many particles, such as ferromagnetic system, the random field describes some property of the particles and renormalization group models scale transformations of the system. Thus, our theorem states that in systems with weak interaction the normalized sums of over large regions have approximately normal independent distribution.

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144

CAPACITY SIZING IN QUEUES WITH AN UNCERTAIN NUMBER OF SERVERS

Rouba Ibrahim

We study the problem of staffing many-server queues with general abandonment and a random number of servers. For example, uncertainty in the number of servers may arise in virtual call centers where agents are free to set their own schedules. We rely on a fluid model to determine optimal staffing levels, and demonstrate the asymptotic accuracy of the fluid prescription. We also characterize the optimal staffing policy with self-scheduling servers.

OPTIMAL PRIORITY CONTROL FOR MULTICLASS MANY-SERVER QUEUES WITH GENERAL PATIENCE DISTRIBUTIONS

Zhenghua Long

We consider the problem of server scheduling in an overloaded multiclass queueing system with multiple homogeneous servers and customer abandonment. In the case of exponential reneging, an indexed priority policy, called the cμ/θ rule, is known to be asymptotically optimal in the many-server heavy traffic regime, where the arrival rates and number of servers increase proportionally. For general patience distributions, we aim to find an asymptotically optimal control policy among the class of fixed priority rules. We first describe a fluid model, which is known to be the limit of the stochastic system under priority rules, and show its convergence to an equilibrium state in the case of exponential service. We then formulate an optimization problem in terms of these equilibrium states, which leads to a nonlinear program of a certain type which we term as the Fractional 0-1 Knapsack Problem. A dynamic programming algorithm is developed to efficiently solve this new type of knapsack-like problem.

PROVISIONING LARGE SCALE SYSTEMS: INTERPLAY BETWEEN NETWORK EFFECTS AND STRATEGIC BEHAVIOR IN THE USER BASE

Bert Zwart

We consider the problem of capacity provisioning for an online service supported by advertising.We analyse the strategic interaction between the service provider and the user base in this setting, modeling positive network effects, as well as congestion sensitivity in the user base. We focus specifically on the influence of positive network effects, as well as the impact of non-cooperative behavior in the user base on the firm’s capacity provisioning decision and its profit. Our analysis reveals that stronger positive network effects, as well as non-cooperation in the user base, drive the service into a more congested state and lead to increased profit for the service provider. The impact of non-cooperation in the user base strongly dominates the impact of network effects.

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145


Queues - Antonis Economou - Optimization and Strategic Behavior in Queueing

THE EFFECTS OF INFORMATION IN TRANSPORTATION SYSTEMS WITH STRATEGIC CUSTOMERS

Athanasia Manou, Pelin G. Canbolat, Fikri Karaesmen

In many transportation systems the service provider is able to obtain information about the expected delays due to congestion and transmit it to customers. Such information affects the behavior of customers and consequently the behavior of the service provider. So, different levels of delay information have different effects on the overall system. We explore these effects considering a transportation system under three levels of delay information: unobservable, partially observable (the queue length is observed) and observable (the exact waiting time is observed).We consider a transportation station, where customers arrive according to a Poisson process. A transportation facility with unlimited capacity visits the station according to a renewal process and at each visit it serves all present customers. We assume that the arriving customers are free decide whether to join or balk based on a natural cost-reward structure, which is imposed on the system. The service provider also makes a decision about the price of the fee he imposes on customers that join. So, this situation can be considered as a game among the customers and the service provider.For each level of information, we obtain the equilibrium behavior of the customers and the service provider, we measure their satisfaction computing appropriate performance measures and comparing the three levels of information we conclude which level is preferable for the customers, for the service provider and for the society.

PRICING AND PRIORITIZING TIME-SENSITIVE CUSTOMERS WITH HETEROGENEOUS DEMAND RATES

Opher Baron

We consider the pricing/lead-time menu design problem for a monopoly service where time-sensitive customers have demand on multiple occasions. Customers differ in their demand rates and valuations per use. We compare a model where the demand rate is the private information of the buyers to a model where the firm has full information. The model assumes that customers queue for a finite-capacity service under a general pricing structure. Customers choose a plan from the menu to maximize their expected utility. In contrast to previous work, we assume customers do not differ in their waiting cost. Yet we show that in the private information case prioritizing customers may be optimal as a result of demand rate heterogeneity. We provide necessary and sufficient conditions for this result. In particular, we show that for intermediate capacity, more frequent-use customers that hold a lower marginal value per use should be prioritized. Further, less frequent-use customers may receive a consumer surplus. We demonstrate the applicability of these results to relevant examples. The structure of the result implies that in some cases it may be beneficial for the firm to prioritize a customer class with a lower marginal cost of waiting.

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146

VALUE DRIVEN LOAD BALANCING

Sherwin Doroudi

To date, the study of dispatching or load balancing in server farms has primarily focused on the minimization of response time. Server farms are typically modeled by a front-end router that employs a dispatching policy to route jobs to one of several servers, with each server scheduling all the jobs in its queue via Processor-Sharing. However, the common assumption has been that all jobs are equally important or valuable, in that they are equally sensitive to delay. Our work departs from this assumption: we model each arrival as having a randomly distributed value parameter, independent of the arrival’s service requirement (job size). Given such value heterogeneity, the correct metric is no longer the minimization or response time, but rather, the minimization of value-weighted response time. In this context, we ask “what is a good dispatching policy to minimize the value-weighted response time metric?” We propose a number of new dispatching policies that are motivated by the goal of minimizing the value-weighted response time. Via a combination of exact analysis, asymptotic analysis, and simulation, we are able to deduce many unexpected results regarding dispatching.

A BAYESIAN MODEL FOR CUSTOMER IMPATIENCE IN UNOBSERVED QUEUES

Yoav Kerner

We consider a queueing system in which the service rate is unknown to the customers, but they have a common prior on it. While waiting each customer updates his prior. Furthermore, this update is toward slower service rates. Customers may leave the queue due to too high expected waiting time. However, the individual, while knowing that those in front of him might abandon, may consider resuming he waiting. We seek for an impatience distribution, such that if all follow it, then the individual has no better response than following it himself. That is, a mixed Nash equilibrium. In the talk I will survey the literature about strategic abandonment from queues and also will present extensions of our model that are ongoing works.


Stochastic Applications - Balaji Prabhakar - Algorithms and Big Data Methods in some Real World Systems

HEAVY-TRAFFIC BEHAVIOR OF THE MAXWEIGHT ALGORITHM IN A SWITCH WITH UNIFORM TRAFFIC

Shiva Theja

We consider a switch with uniform traffic operating under the MaxWeight scheduling algorithm. This traffic pattern is interesting to study in the heavy-traffic regime since the queue lengths exhibit a multi-dimensional state-space collapse. We use a Lyapunov-type drift technique to characterize the heavy-traffic behavior of the expectation of the sum queue lengths in steady-state. Specifically, in the case of Bernoulli arrivals, we show that the heavy-traffic scaled queue length is n−32+12n. Our result implies that the MaxWeight algorithm has optimal queue-length scaling behavior in the heavy-traffic regime with respect to the size of a switch with a uniform traffic pattern. This settles the heavy-traffic version of an open conjecture.

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DESIGNING LARGE-SCALE NUDGE ENGINES: THEORY AND PRACTICE OF ROAD TRAFFIC CONGESTION REDUCTION USING INCENTIVES

Chinmoy Mandayam

Congestion is a widespread problem in modern urban transportation networks; hundreds of billions of dollars are lost each year due to wasted time, extra fuel consumption, traffic accidents, etc. We explore the feasibility of reducing peak-hour road traffic congestion using incentives to shift drivers’ commute times. We first discuss a practical implementation of such an incentive program — CAPRI (Congestion And Parking Relief Incentives). This program aimed to reduce peak-hour vehicular traffic into and out of Stanford University. Commuters who sign up for CAPRI earn points for the “good trips” they make, and these points can be redeemed for rewards (both monetary and in-kind). CAPRI also includes the capability to personalize incentives based on users’ historical behavior.To complement the implementation, we develop a theoretical model for optimally reducing the cost of peak-hour congestion with targeted incentives. We study the evolution of congestion on a highway under time-varying demands using fluid models of traffic flow. We then examine the effect of shifting user’ commute times on congestion. We show that ideas from the theory of optimal transport of measures can be used to develop cost-effective incentive schemes to reduce congestion. Specifically, we show that the “cost of congestion” and the “cost of nudging” are closely related to the Wasserstein distance between measures. We use this relationship to formulate linear programming problems to compute personalized recommendations and incentives to nudge drivers to the off-peak hour. We find that the resultant reduction in the cost of congestion is quite significant.

A BIG DATA SYSTEM FOR THE INTERNET OF MOVING THINGS

Balaji Prabhakar

The world consists of many interesting things that move: people go to work, home, school, and shop in public transit buses and trains or in cars and taxis; goods move on these networks and by trucks or by air each day; and food items travel a very large distance to meet their eater. Thus, massive movement processes are underway in the world every day and it is critical to ensure their safe, timely and efficient operation. Towards this end, low-cost sensing and acquisition of the movement data is being achieved: from GPS devices, RFID and barcode scanners, to smart commuter cards and smartphones, snapshots of the movement process are becoming available. In this talk, I will present a system for stitching together these snapshots and reconstructing urban mobility at a very fine-grained level. The system, which we call the Space-Time Engine, provides an interactive dashboard and a querying engine for answering questions such as: what is the crowding at a train station? Where’re packages held up and how can their delivery be sped up? How can the available supply of transport capacity be better used to address daily demand as well as the demand on exceptional days (such as rallies and severe weather events)? I will describe the STE’s capabilities for operational and planning purposes, and as a learning system.

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VISUALIZING AND INTERACTING WITH BIG DATA FROM THE EVERYDAY WORLD

Damon Wischik

With small data, it takes a skilled statistician to understand which questions can give meaningful answers; and it is the job of the statistician to build models and perform tests, and give conclusions (plus the odd box-plot) to the non-expert end user. With big data, there are so many more questions that can give meaningful answers, and the answers can often be read directly off well-constructed visualizations. Here, it is the job of the data scientist to build systems which help the end user to formulate and answer his or her own questions. These systems must be built on good aesthetics, good interactivity, and good guidance.

I will describe work by myself and colleagues at Urban Engines, on visualizing and interacting with big data about things that move: cars, buses, and people. I will use visualizations to answer questions about the underlying networks, and about the people who use them.


Stochastic Models - Michel Mandjes - Mixed Poisson Models and Related Queueing Systems

ROBUST HEAVY-TRAFFIC APPROXIMATIONS FOR SERVICE SYSTEMS FACING OVERDISPERSED DEMAND

Britt Mathijsen

An increasing number of empirical studies suggest that service systems face arrival processes that are profoundly more volatile than expected under the classical Poisson assumption, a phenomenon called overdispersion. We therefore analyze a model with mixed Poisson arrivals, which can capture overdispersion, and develop an extension of the QED heavy-traffic regime to balance quality-of-service and efficiency. This leads to an adaptation of the famous square-root staffing rule that can deal with overdispersion. We illustrate the implications of this new staffing rule using real-life data.

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SCALING LIMITS FOR AN OVERDISPERSED SERVICE SYSTEM

Mariska Heemskerk, M.R.H. Mandjes, J.S.H. van Leeuwaarden

DEVIATIONS OF INFINITE-SERVER QUEUES AND ORNSTEIN-UHLENBECK PROCESSES IN A RANDOM ENVIRONMENT

Marijn Jansen

We will consider an infinite-server queue in a random environment and investigate deviations from its mean behavior. A random environment is represented by a stochastic background process, which modulates both the arrival process and the service process of the queue. This means that changes in the environment influence the behavior of the queue. This leads to overdispersion, in the sense that a modulated infinite-server queue exhibits more variability than its non-modulated counterpart. We are interested to see how the random environment influences deviations of the infinite-server queue from its typical (mean) behavior on different scales. It turns out that, under a linear scaling of the arrival rates and an appropriate scaling of the background process, we may derive several limit theorems. We will present a central limit theorem, a moderate deviations principle and a large deviations principle for the number of jobs in the system. In addition, we will compare estimates for overflow probabilities using these limit theorems. Finally, we will argue that the results obtained for the modulated infinite-server queue carry over to the modulated Ornstein-Uhlenbeck process.

In this talk I consider an �� model in a random environment, where the arrival rate is sampled

in an i.i.d. fashion at sampling frequency��.

We prove that the (transient as well as stationary) number of clients is of mixed Poisson type (i.e., Poisson with random parameter).

The main results cover the situation in which both � and �� are scaled; we establish scaling limits such

as a (functional) central limit theorem.

The nature of the limiting Gaussian process depends on the specific scaling imposed.

In particular, if �'s rate of increase is higher than ��'s rate of increase, then the system is

overdispersed.

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MARKOV-MODULATED ORNSTEIN-UHLENBECK PROCESSES

Michel Mandjes, Gang Huang, Marijn Jansen, Peter Spreij and Koen de Turck

This paper deals with the problem of inferring short time-scale fluctuations of a system’s behavior from periodic state measurements. In particular, we devise a novel, efficient procedure to compute four interesting performance metrics for a transient birth-death process on an interval of fixed length with given begin and end states: the probability to exceed a predefined (critical) level m, and the expectation of the time, area, and number of arrivals above level m. Moreover, our procedure allows to compute the variances and cross-correlations of the latter three metrics. The asymptotic behavior of the metrics for small and large measurement intervals is also derived.

An extensive numerical study in the context of communication networks reveals the impact of important system parameters on the considered performance metrics, and shows that the three latter metrics are very highly correlated. We also illustrate through this numerical study how our analysis can be used in practical situations to support e.g., capacity management and sla verification.


Stochastic Networks - Marko Boon - Single-Server Networks: Applications, Analysis and Asymptotics

STOCHASTIC ANALYSIS AND OPTIMISATION OF OPTICAL ACCESS NETWORKS

Stella Kapodistria

In this talk we illustrate how to model Ethernet Passive Optical Networks (EPON) uploading direction using single server systems. Our objective is two-fold: on the one hand we are interested in the performance analysis of the stochastic model under investigation, and on the other hand we are interested in developing an algorithmic approach for the maximization of throughput, by optimizing the time-division multiplexing (the length of the time slot each optical network unit can transmit packets).

More concretely, we initially consider both the marginal queue length distributions and the marginal workload distributions of the service system at hand and derive close form expressions, which we subsequently use for the derivation of the optimal time-division multiplex. In addition, we look at the joint queue length distribution using a generating function approach. The approach amounts to translating the determination of the generating function to the determination of the solution of a Dirichlet boundary value problem of mathematical physics. Our approach seems to work equivalently well for the Laplace transform of the joint workload distribution, even in the case that the service times have a general distribution.

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MARKOVIAN POLLING SYSTEMS AND THEIR APPLICATION TO WIRELESS RANDOM-ACCESS NETWORKS

Jan-Pieter Dorsman, Sem Borst, Onno Boxma, Rob van der Mei and Maria Vlasiou

Polling systems are queueing systems that consist of a number of queues, attended by a single server. These systems find their origin in a wealth of real-life applications, such as manufacturing environments, computer-communication systems and traffic systems. Motivated by an application in wireless random-access networks, we study polling systems with so-called Markovian routing, where the order in which the server visits the queues is governed by a discrete-time Markov chain. This class of polling systems is in general much harder to analyse than the well-studied class of systems where the server visits the queues in a cyclic order.We discuss several new results for the analysis of Markovian polling systems. Using these results, we can derive expressions for certain system parameters which minimize the total expected amount of work or nearly minimize the weighted sum of mean waiting times in systems that arise from the wireless random-access network setting. Based on these expressions, we present an adaptive control algorithm for finding the optimal parameter values in a distributed fashion, which is particularly relevant in the context of wireless random-access networks.

HEAVY TRAFFIC ANALYSIS OF ROVING SERVER NETWORKS

Erik Winands

We study the heavy-traffic (HT) behaviour of queueing networks with a single roving server. External customers arrive at the queues according to independent renewal processes and after completing service, a customer either leaves the system or is routed to another queue. This type of customer routing in queueing networks arises very naturally in many application areas (in production systems, computer- and communication networks, maintenance, etc.). In these networks, the single most important characteristic of the system performance is oftentimes the path time, i.e. the total time spent in the system by an arbitrary customer traversing a specific path. We present the first HT asymptotic for the path-time distribution in queueing networks with a roving server under general renewal arrivals. By combining this result with novel light-traffic asymptotics we derive an approximation of the mean path-time for arbitrary values of the load and renewal arrivals.

RANDOM FLUID LIMIT OF AN OVERLOADED POLLING MODEL

Maria Remerova, Sergey Foss and Bert Zwart

There are only a few examples of queueing models in the literature that have random fluid (or law-of-large-numbers) limits. In this talk, we discuss such an unusual example. We consider the classical cyclic polling model. The assumptions that lead to the random fluid limit are overload and empty initial state, and the cornerstone of our analysis is a connection with branching processes. The trajectories of the fluid limit have an interesting structure. They oscillate infinitely often in the neighborhood of zero, and all of them can be mapped into the same deterministic function by a linear time-space scaling. Additionally, we prove that f-moments of the busy period in an M/G/1 queue are finite for a wide class of functions f including power and logarithmic functions.

ORGANIZING SECRETARIAT

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E-mail: [email protected]

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Fax: + 90 212 338 1205

Email: [email protected]

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