Stability using fluid limits: Illustration through an example "Push-Pull" queuing network

Finite Buffer Fluid Networks with OverflowsYoni Nazarathy,Swinburne University of Technology, Melbourne.Stijn Fleuren and Erjen Lefeber,Eindhoven University of Technology, the Netherlands.The University of Sydney,Operations Management and Econometrics Seminar,July 29, 2011.Almost Discrete Sojourn Time Phenomena

Taken from seminar of Avi Mandelbaum, MSOM 2010 (slide 82).OutlineBackground: Open Jackson networksIntroducing overflowsFluid networks as limiting approximationsTraffic equations and their solutionDiscrete sojourn times Open Jackson NetworksJackson 1957, Goodman & Massey 1984, Chen & Mandelbaum 1991

Traffic Equations (Stable Case):Traffic Equations (General Case):

Problem Data:Assume: open, no dead nodesOpen Jackson NetworksJackson 1957, Goodman & Massey 1984, Chen & Mandelbaum 1991

Traffic Equations (Stable Case):

Problem Data:Assume: open, no dead nodes

Product Form Miracle:Modification: Finite Buffers and Overflows

Exact Traffic Equations:

Problem Data:

Explicit Solutions:Generally No

Generally NoAssume: open, no dead nodes, no jam (open overflows)A practical (important) model:We say YesWhen K is Big, Things are Simpler

Scaling Yields a Fluid System

A sequence of systems:Make the jobs fast and the buffers big by taking

The proposed limiting model is a deterministic fluid system:

Fluid Trajectories as an Approximation

Not proved in this current work, yet similar statement appears in a different model (and rigorously proved). Come to 14:00 Stats Seminar, Carslaw 173.Traffic Equations

or

orLCP

(Linear Complementarity Problem)Min-Linear Equations as LCP

Existence, Uniqueness and Solution

Immediate naive algorithm with 2M stepsWe essentially assume that our matrix ( ) is a P-MatrixWe have an algorithm (for our G) taking M2 steps

Back To Sojourn Times.

Taken from seminar of Avi Mandelbaum, MSOM 2010 (slide 82).

Sojourn Times Scale to a Discrete Distribution!!!

The Fast Chain and Slow Chain1234120

Fast chain on {0, 1, 2, 1, 2, 3, 4}:Slow chain on {0, 1, 2}start

DPH distribution (hitting time of 0)transitions based on Fast chainE.g: Moshe Haviv (soon) book: Queues, Section on Shortcutting statesThe DPH Parameters (Details)

Fast chain Slow chain Sojourn Times Scale to a Discrete Distribution!!!

SummaryTrend in queueing networks in past 20 years: When dont have product-form. dont give up: try asymptoticsLimiting traffic equations and trajectoriesMolecule sojourn times (asymptotic) Discrete!!!Future work on the limits:More standard: E.g. convergence of trajectories (2:00 talk)Hi-tech (I dont know how to approach): Weak convergence of sojourn times (we will leave it as a conjecture for now)Molecule Sojourn Times

Observe,

For job at entrance of buffer :

A fast chain and slow chainA job at entrance of buffer : routed almost immediately according to