data-based science for service engineering and...

Data-Based Science forService Engineering and Management

or: Empirical Adventuresin Call-Centers and Hospitals

Avi Mandelbaum

Technion, Haifa, Israel

http://ie.technion.ac.il/serveng

IELM, Hong Kong UST, September 2011

1

Research PartnersI Students:

Aldor∗, Baron∗, Carmeli, Feldman∗, Garnett∗, Gurvich∗, Huang,Khudiakov∗, Maman∗, Marmor∗, Reich, Rosenshmidt∗, Shaikhet∗,Senderovic, Tseytlin∗, Yom-Tov∗, Zaied, Zeltyn∗, Zychlinski, Zohar∗,Zviran, . . .

I Theory:Armony, Atar, Gurvich, Jelenkovic, Kaspi, Massey, Momcilovic,Reiman, Shimkin, Stolyar, Wasserkrug, Whitt, Zeltyn, . . .

I Industry:IBM Research (OCR: Carmeli, Vortman, Wasserkrug, Zeltyn),Rambam Hospital, Hapoalim Bank, Mizrahi Bank, PelephoneCellular, . . .

I Technion SEE Center / Labaratory:Feigin; Trofimov, Nadjharov, Gavako, Kutsyy; Liberman, Koren,Rom, Plonsky; Research Assistants, . . .

I Empirical/Statistical Analysis:Brown, Gans, Zhao; Shen; Ritov, Goldberg; Allon, Ata, Bassamboo;Gurvich, Huang, Liberman; Armony, Marmor, Tseytlin, Yom-Tov;Zeltyn, Nardi, . . .

2

History, Resources (Downloadable)

I Math. + C.S. + Stat. + O.R. + Mgt. ⇒ IE (≥ 1990)

I Teaching: “Service-Engineering" Course (≥ 1995):http://ie.technion.ac.il/serveng - websitehttp://ie.technion.ac.il/serveng/References/teaching_paper.pdf

I Call-Centers Research (≥ 2000)e.g. <Call Centers> in Google-Scholar

I Healthcare Research (≥ 2005)e.g. OCR Project: IBM + Rambam Hospital + Technion

I The Technion SEE Center (≥ 2007)

3

The Case for Service Science / Engineering

I Service Science / Engineering (vs. Management) are emergingAcademic Disciplines. For example, universities (world-wide),IBM (SSME, a là Computer-Science), USA NSF (SEE), GermanyIAO (ServEng), ...

I Models that explain fundamental phenomena , which arecommon across applications:

- Call Centers- Hospitals- Transportation- Justice, Fast Food, Police, Internet, . . .

I Simple models at the Service of Complex Realities (Human)Note: Simple yet rooted in deep analysis.

I Mostly What Can Be Done vs. How To

4

Title: Expands the Scientific Paradigm

Physics, Biology, . . . : Measure, Model, Experiment, Validate, Refine.Human-complexity triggered above in Transportation, Economics.Starting with Data, expand to:Service Science/Engineering/Management

7. Feedback 1. Measurements / Data

6. Improvement 5. Implementation2. Modeling,

Analysis3. Validation

8. Novel needs, necessitating Science

4. Maturity enables Deployment

e.g. Validate, refute or discover congestion laws (Little, PASTA,SSC, ?, ?,...), in call centers and hospitals

5

Little’s Law: Call Center & Emergency Department

Time-Gap: # in System lags behind Piecewise-Little (L = λ × W )

Little’s Law and the Offered-Load: Empirical Adventures in Call Centers and Hospitals

The Black-Box View of a Call Center: Number in Q vs. Little shows a time-gap.

USBank Customers in queue(average), Telesales10.10.2001

0.00

20.00

40.00

60.00

80.00

100.00

120.00

140.00

160.00

180.00

200.00

07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00

Time (Resolution 30 min.)

Num

ber

of c

ases

Customers in queue(average) Little's law

HomeHospital Average patients in EDFebruary 2004, Wednesdays

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

00:00 02:00 04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00


Ave

rage

num

ber

of c

ases

Average patients in ED Lambda*E(S) smoothed

בהסתכלות ממוצעת על יום באמצע השבוע לאורך מספר חודשים

HomeHospital Average patients in EDTotal for February2004 March2004 April2004 May2004 June2004 July2004 August2004 September2004 October2004,Week days

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

00:00 02:00 04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00


Ave

rage

num

ber

of c

ases

Average patients in ED Lambda*E(S)

⇒ Time-Varying Little’s LawI Berstemas & Mourtzinou;I Fralix, Riano, Serfozo; . . .

6

QED Call Center: Staffing (N) vs. Offered-Load (R)IL Telecom; June-September, 2004; w/ Nardi, Plonski, Zeltyn

Empirical Analysis of a QED Call Center

• 2205 half-hour intervals of an Israeli call center • Almost all intervals are within R R− and 2R R+ (i.e. 1 2β− ≤ ≤ ),

implying: o Very decent forecasts made by the call center o A very reasonable level of service (or does it?)

• QED? o Recall: ( )GMR x is the asymptotic probability of P(Wait>0) as a

function of β , where x θµ=

o In this data-set, 0.35x θµ= =

o Theoretical analysis suggests that under this condition, for 1 2β− ≤ ≤ , we get 0 ( 0) 1P Wait≤ > ≤ …

2205 half-hour intervals in an Israeli Call Center7

QED Call Center: Performance

Large Israeli Bank

P{Wq > 0} vs. (R, N) R-Slice: P{Wq > 0} vs. N

• Intercepting plane of the previous plot for a "constant" offered load (actually

18 18.5R≤ ≤ ) • Smoother line was created using smoothing splines. • General shape of the smoothing line is as the Garnett delay functions! • In 2-d view, we get the following:

3 Operational Regimes:I QD: ≤ 25%

I QED: 25%− 75%

I ED: ≥ 75%

8

Empirical data: 2204 intervals of Technical category from Telecom call center

(13 summer weeks; week-days only; 30 min. resolution; excluding 6 outliers). Theoretical plot: based on approximations of Erlang-A for the proper rate.

Operational Regimes: Scaling, Performance,w/ I. Gurvich & J. HuangTable 1: Operational Regimes: Asymptotics, Scaling, Performance

Erlang-A Conventional scaling MS scaling NDS scaling

µ fixed Sub Critical Super QD QED ED ED+QED Sub Critical Super

Offered load per server 11+δ

< 1 1− β√n≈ 1 1

1−γ > 1 11+δ

1− β√n

11−γ

11−γ − β

√1

n(1−γ)31

1+δ1− β

n1

1−γ

Arrival rate λ µ1+δ

µ− β√nµ µ

1−γnµ1+δ

nµ− βµ√n nµ1−γ

nµ1−γ − βµ

√n

(1−γ)3nµ1+δ

nµ− βµ nµ1−γ

Number of servers 1 n n

Time-scale n 1 n

Abandonment rate θ/n θ θ/n

Staffing level λµ(1 + δ) λ

µ(1 + β√

n) λ

µ(1− γ) λ

µ(1 + δ) λ

µ+ β

√λµ

λµ(1− γ) λ

µ(1− γ) + β

√λµ

λµ(1 + δ) λ

µ+ β λ

µ(1− γ)

Utilization 11+δ

1−√

θµ

h(β)√n

1 11+δ

1−√

θµ

(1−α2)β+α2h(β)√n

1 1 11+δ

1−√

θµ

h(β)

n1

E(Q) α1

δ

√n

√µθ[h(β)− β] nµγ

θ(1−γ)1√2π

1+δδ2%n 1√

n

√n

√µθ[h(β)− β]α2

nµγθ(1−γ)

nµθ(1−γ)

(γ − β√n(1−γ)

) o(1) n√

µθ[h(β)− β] n2µγ

θ(1−γ)

P(Ab) 1n

1+δδ

θµα1

1√n

√θµ[h(β)− β] γ 1√

2π

θµ

(1+δ)2

δ2%n 1

n3/21√n

√θµ[h(β)− β]α2 γ γ − β

√1−γ√n

o( 1n2 ) 1

n

√θµ[h(β)− β] γ

P(Wq > 0) α1 ∈ (0,1) ≈ 1 1√2π

1+δδ%n 1√

n≈ 0 α2 ∈ (0,1) ≈ 1 ≈ 1 ≈ 0 ≈ 1

P(Wq > T ) α1e− δ

1+δµt 1 +O( 1√

n) 1 +O( 1

n) ≈ 0 G(T )1{G(T )<γ} α3, if G(T ) = γ ≈ 0 Φ(β+

√θµT )

Φ(β)1 +O( 1

n)

Congestion EWq

ES α11+δδ

√n

√µθ[h(β)− β] nµγ/θ 1√

2π

(1+δ)2

δ2%n 1

n3/21√n

√µθ[h(β)− β]α2 µ

∫∫∫ x∗

0G(s)ds µ

∫∫∫ x∗

0G(s)ds− µβ

√1−γ

hG(x∗)√n

o( 1n)

√µθ[h(β)− β] nµγ/θ

• δ > 0, γ ∈ (0, 1) and β ∈ (−∞,∞);

• QD:% = 11+δe

δ1+δ < 1;

• ED (ED+QED): G(x∗) = γ;

• QED:α2 = [1 +√

θµh(β)h(−β) ]−1, here β = β

√µθ and h(x) = φ(x)

Φ(x);

• ED+QED:α3 = G(T )Φ(β√

µg(T ));

• Conventional: critical: P(W > T ) = P( W√n> T√

n), super: P(W > T ) = P(Wn > T

n ); NDS: Super: P(W > T ) = P(Wn > Tn ).

1

9

Prerequisite I: Data

Averages Prevalent (and could be useful / interesting).But I need data at the level of the Individual Transaction:For each service transaction (during a phone-service in a call center,or a patient’s visit in a hospital, or browsing in a website, or . . .), itsoperational history = time-stamps of events .

Sources: “Service-floor" (vs. Industry-level, Surveys, . . .)

I Administrative (Court, via “paper analysis")I Face-to-Face (Bank, via bar-code readers)I Telephone (Call Centers, via ACD / CTI, IVR/VRU)I Hospitals (Emergency Departments, . . .)

I Expanding:I Hospitals, via RFIDI Operational + Financial + Contents (Marketing, Clinical)I Internet, Chat (multi-media)

10

Pause for a Commercial: The Technion SEE Center

11

Technion SEE = Service Enterprise Engineering

SEELab: Data-repositories for research and teaching

I For example:I Bank Anonymous: 1 years, 350K calls by 15 agents - in 2000.

Brown, Gans, Sakov, Shen, Zeltyn, Zhao (JASA), paved the wayfor:

I U.S. Bank: 2.5 years, 220M calls, 40M by 1000 agents.I Israeli Cellular: 2.5 years, 110M calls, 25M calls by 750 agents.I Israeli Bank: from January 2010, daily-deposit at a SEESafe.I Israeli Hospital: 4 years, 1000 beds; 8 ED’s- Sinreich’s data.

SEEStat: Environment for graphical EDA in real-time

I Universal Design, Internet Access, Real-Time Response.

SEEServer: Free for academic useRegister, then access (presently) U.S. Bank and Bank Anonymous.

Visitor: run mstsc, seeserver.iem.technion.ac.il ; Self-Tutorial

12

Technion - Israel Institute of Technology

The William Davidson Faculty of Industrial Engineering and Management

Center for Service Enterprise Engineering (SEE) http://ie.technion.ac.il/Labs/Serveng/

SEEStat 3.0 Tutorial

HKUST, Hong Kong, September 2011

Note: This tutorial is customized to HKUST. To become a regular user of SEEStat, please go to http://seeserver.iem.technion.ac.il/see-terminal/ click on “Register” (left menu), and follow the registration procedure.

2

, you are able to course-seminar/mini HKUSTAs a participant in the at the SEELab Serverlaptop) to the PC or (from your connect

Technion.

SEE taught -selfthe Once connected, you will be able to go through.that follows Tutorial

To start, you need a “User Name” and “Password”. Use the Number allocated to you in the seminar/mini-course (from 1-20):

Log In

On Page 3, you are asked to provide the following LogIn information:

Your User Name: visitor_1_20 Your Password: visitor_1_20

Log On to Windows

You will be needing the above also for the following step (Page 4):

Introduction: SEEStat is a environment for Exploratory Data Analysis (EDA) in real-time. It enables users to easily conduct statistical and performance analyses of massive datasets; in particular, datasets representing operational histories of large service operations (e.g. call centers, hospitals, internet sites), as available through the SEELab server. SEEStat can also automatically create sophisticated reports in Microsoft Excel, which support research and teaching. Both SEEStat and the SEELab Server were developed at the Faculty of Industrial Engineering and Management, Technion, Israel Institute of Technology. More information on the SEELab can be found at its homepage http://ie.technion.ac.il/Labs/Serveng/

1

SEEStat 3.0 Tutorial

Introduction ............................................................................................................................. 2

Connecting to SEEStat on the Technion SEELab Server ................................................... 3

SEEStat Tutorial ..................................................................................................................... 7

Part 1 ................................................................................................................................................. 7

Example 1.1: Distributions ............................................................................................................ 8

Example 1.2: Intraday time series ............................................................................................... 15

Example 1.3: Time series (Daily totals) ...................................................................................... 20

Part 2 ............................................................................................................................................... 24

Example 2.1: Distribution fitting ................................................................................................. 24

Example 2.2: Distribution mixture fitting .................................................................................... 26

Example 2.3: Survival analysis with smoothing of hazard rates ................................................. 28

Example 2.4: Smoothing of intraday time series ......................................................................... 30

Part 3 ............................................................................................................................................... 32

Example 3.1: Queue regulated by a protocol & annoucements .................................................. 32

Example 3.2: Queue length & state-space collapse .................................................................... 34

Example 3.3: Change-of_Shifts phenomena ................................................................................ 36

Example 3.4: Daily flow of calls .................................................................................................. 40

35

USBank Customers in queue(average)Total for May2002 June2002 July2002 August2002 September2002 October2002

November2002 December2002,Week days

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00


Ave

rage

num

ber

of c

ases

Business Platinum

Platinum is a small-scale service. You will now normalize the chart in order to identify patters. Click "Output" on the main menu and then "Modify Tables and Charts". Open the "Options" tab and select Percent to mean. Click "OK".

USBank Customers in queue(average)Total for May2002 June2002 July2002 August2002 September2002 October2002

November2002 December2002,Week days

0

100

200

300

400

500

600

700

800

900

0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00


Per

cent

to

mea

n

Business Platinum

Note the essentially overlapping patterns of the queue lengths of the two customer types. (This phenomenon is predicted by asymptotic analysis of queues in heavy traffic, where it is referred to as State-Space-Collapse.)

aviman

Highlight

aviman

Highlight

eg. RFID-Based Data: Mass Casualty Event (MCE)

Drill: Chemical MCE, Rambam Hospital, May 2010

מאייר -קלים ודחק'מרתף פנימית ו-משפחותקרדיולוגיהעורנוירולוגיהבינוניים

נספח לנוהלקרדיולוגיה,עור,נוירולוגיה-בינונייםחדר אוכל-קשים

ד"מלר-משולבים

Focus on severely wounded casualties (≈ 40 in drill)Note: 20 observers support real-time control (helps validation)

14

Data Cleaning: MCE with RFID Support

Data-base Company report comment

Asset id order Entry date Exit date Entry date Exit date 4 1 1:14:07 PM 1:14:00 PM 6 1 12:02:02 PM 12:33:10 PM 12:02:00 PM 12:33:00 PM 8 1 11:37:15 AM 12:40:17 PM 11:37:00 AM exit is missing

10 1 12:23:32 PM 12:38:23 PM 12:23:00 PM 12 1 12:12:47 PM 12:35:33 PM 12:35:00 PM entry is missing 15 1 1:07:15 PM 1:07:00 PM 16 1 11:18:19 AM 11:31:04 AM 11:18:00 AM 11:31:00 AM 17 1 1:03:31 PM 1:03:00 PM 18 1 1:07:54 PM 1:07:00 PM 19 1 12:01:58 PM 12:01:00 PM 20 1 11:37:21 AM 12:57:02 PM 11:37:00 AM 12:57:00 PM 21 1 12:01:16 PM 12:37:16 PM 12:01:00 PM

22 1 12:04:31 PM 12:20:40 PM first customer is missing

22 2 12:27:37 PM 12:27:00 PM 25 1 12:27:35 PM 1:07:28 PM 12:27:00 PM 1:07:00 PM 27 1 12:06:53 PM 12:06:00 PM

28 1 11:21:34 AM 11:41:06 AM 11:41:00 AM 11:53:00 AMexit time instead of entry time

29 1 12:21:06 PM 12:54:29 PM 12:21:00 PM 12:54:00 PM 31 1 11:40:54 AM 12:30:16 PM 11:40:00 AM 12:30:00 PM 31 2 12:37:57 PM 12:54:51 PM 12:37:00 PM 12:54:00 PM 32 1 11:27:11 AM 12:15:17 PM 11:27:00 AM 12:15:00 PM 33 1 12:05:50 PM 12:13:12 PM 12:05:00 PM 12:15:00 PM wrong exit time 35 1 11:31:48 AM 11:40:50 AM 11:31:00 AM 11:40:00 AM 36 1 12:06:23 PM 12:29:30 PM 12:06:00 PM 12:29:00 PM 37 1 11:31:50 AM 11:48:18 AM 11:31:00 AM 11:48:00 AM 37 2 12:59:21 PM 12:59:00 PM 40 1 12:09:33 PM 12:35:23 PM 12:09:00 PM 12:35:00 PM 43 1 12:58:21 PM 12:58:00 PM 44 1 11:21:25 AM 11:52:30 AM 11:52:00 AM entry is missing 46 1 12:03:56 PM 12:03:00 PM 48 1 11:19:47 AM 11:19:00 AM 49 1 12:20:36 PM 12:20:00 PM 52 1 11:21:29 AM 11:50:49 AM 11:21:00 AM 11:50:00 AM 52 2 12:10:07 PM 1:07:28 PM 12:10:00 PM 1:07:00 PM recorded as exit 53 1 12:24:26 PM 12:24:00 PM 57 1 11:32:02 AM 11:58:31 AM 11:58:00 AM entry is missing 57 2 12:59:41 PM 1:14:00 PM 12:59:00 PM 1:14:00 PM 60 1 12:27:12 PM 12:48:41 PM 12:27:00 PM 12:48:00 PM 63 1 12:10:04 PM 12:10:00 PM 64 1 11:30:29 AM 12:43:38 PM 11:30:00 AM 12:43:00 PM

Imagine “Cleaning" 60,000+ customers per day (call centers) !

15

Beyond Averages: The Human Factor

Histogram of Service-Time in a (Small Israeli) Bank, 1999

January-OctoberJanuary-October

0

2

4

6

8

0 100 200 300 400 500 600 700 800 900

AVG: 185STD: 238

November-December

0

2

4

6

8

0 100 200 300 400 500 600 700 800 900

AVG: 201STD: 263

5.59%

?6.83%

Log-Normal

November-DecemberJanuary-October

0

2

4

6

8

0 100 200 300 400 500 600 700 800 900

AVG: 185STD: 238

November-December

0

2

4

6

8

0 100 200 300 400 500 600 700 800 900

AVG: 201STD: 263

5.59%

?6.83%

Log-Normal

I 6.8% Short-Services: Agents’ “Abandon" (improve bonus, rest),(mis)lead by incentives

I Distributions must be measured (in seconds = natural scale)I LogNormal service times common in call centers

16

Validating LogNormality of Service-Duration

Israeli Call Center, Nov-Dec, 1999

Log(Service Times)

0 2 4 6 8

0.0

0.1

0.2

0.3

0.4

Log(Service Time)

Pro

port

ion

LogNormal QQPlot

Log-normalS

ervi

ce ti

me

0 1000 2000 3000

010

0020

0030

00

I Practically Important: (mean, std)(log) characterizationI Theoretically Intriguing: Why LogNormal ? Naturally multiplicative

but, in fact, also Infinitely-Divisible (Generalized Gamma-Convolutions)

I Simple-model of a complex-reality? The Service Process:17

(Telephone) Service-Process = “Phase-Type" Model

RetailService(IsraeliBank)

StatisticsORIE

START

Hello14 / 20

I.D.24 / 23Request

15 / 8

Stock17 / 21

Question14 / 20

Password creation62 / 42

Processing49 / 24

Confirmation29 / 9

Answer32 / 19

END202/190

Dead time18 / 17

Paperwork22 / 12

End of call5 / 3

Credit34 / 32

Others21 / 21

Checking21 / 9

1

0.65

0.050.27

0.950.03

0.62

0.29

0.17

0.28

0.17

0.11

0.05

0.05 0.17

0.23 0.08

0.38

0.15

0.15

0.93

0.14

0.03

0.5

0.03 0.26

0. 4

0.23

0.590.09

0.05

0.67

0.11

0.17

0.67

0.330.7

0.25

0.66

0.43

0.57

0.130.2

0.93

0.04

0. 6

Password creation62 / 42

0.67

0.33

STDAVG

18

Individual Agents: Service-Duration, Variabilityw/ Gans, Liu, Shen & Ye

Agent 14115

Service-Time Evolution: 6 month Log(Service-Time)

I Learning: Noticeable decreasing-trend in service-durationI LogNormal Service-Duration, individually and collectively

19

Individual Agents: Learning, Forgetting, Switching

Daily-Average Log(Service-Time), over 6 monthsAgents 14115, 14128, 14136

Day Index

Mea

n Lo

g(S

ervi

ce T

ime)

0 20 40 60 80 100 120 1404.2

4.4

4.6

4.8

5.0

5.2

5.4

Day IndexM

ean

Log(

Ser

vice

Tim

e)

a 102−day break

0 20 40 60 80 100

4.4

4.6

4.8

5.0

5.2

5.4

5.6

Day Index

Mea

n Lo

g(S

ervi

ce T

ime)

switch to Online Banking after 18−day break

0 50 100 150

4.5

5.0

5.5

6.0

Weakly Learning-Curves for 12 Homogeneous(?) Agents

5 10 15 20

34

56

Tenure (in 5−day week)

Ser

vice

rat

e pe

r ho

ur

20

Why Bother?

In large call centers:+One Second to Service-Time implies +Millions in costs, annually

⇒ Time and "Motion" Studies (Classical IE with New-age IT)

I Service-Process Model: Customer-Agent InteractionI Work Design (w/ Khudiakov)

eg. Cross-Selling: higher profit vs. longer (costlier) services;Analysis yields (congestion-dependent) cross-selling protocols

I “Worker" Design (w/ Gans, Liu, Shen & Ye)eg. Learning, Forgetting, . . . : Staffing & individual-performanceprediction, in a heterogenous environment

I IVR-Process Model: Customer-Machine Interaction75% bank-services, poor design, yet scarce research;Same approach, automatic (easier) data (w/ Yuviler)

21

IVR-Time: HistogramsIsraeli Bank: IVR/VRU Only, May 2008

IVR_onlyMay 2008, Week days

0.000.100.200.300.400.500.600.700.800.901.001.101.201.301.40

00:00 00:30 01:00 01:30 02:00 02:30 03:00 03:30 04:00 04:30

Time(mm:ss) (Resolution 1 sec.)

Rel

ativ

e fr

eque

ncie

s %

mean=99st.dev.=101

Mixture: 7 LogNormals

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

00:00 00:30 01:00 01:30 02:00 02:30 03:00 03:30 04:00 04:30

Rel

ativ

e fr

eque

ncie

s %

Time(mm:ss) (Resolution 1 sec.)

Fitting Mixtures of Distributions for VRU only timeMay 2008, Week days

Empirical Total Lognormal Lognormal Lognormal

22

IVR-Process: “Phase-Type" Model

Start

3I.D.

7Query

5Agent

End

4828

49369

39634

43464

Change

127410

Bursa

11Poly

319

1110

30

15Error

15

157

4

41

2

2

2

68

16Kranot

4

1 2

1Pery

2Check

8Trans

9Nia

12Credit

6Fax

24

23

3

51

10

74

3

1

5

121

140

1

75

12

1157

308

140

2

99 549

1

5

10

219

14

2

1

10

37

8408

1233

655

29780

24

2

1

298

418

3,7,10,11P<<0.01

7P<<0.01

17380

23

Started with Call Centers, Expanded to Hospitals

Call Centers - U.S. (Netherlands) Stat.

I $200 – $300 billion annual expenditures (0.5)I 100,000 – 200,000 call centers (1500-2000)I “Window" into the company, for better or worseI Over 3 million agents = 2% – 4% workforce (100K)

Healthcare - similar and unique challenges:

I Cost-figures far more staggeringI Risks much higherI ED (initial focus) = hospital-windowI Over 3 million nurses

24

Call-Center Environment: Service Network

25

Call-Center Network: Gallery of Models

Agents(CSRs)

Back-Office

Experts)(Consultants

VIP)Training (

Arrivals(Business Frontier

of the21th Century)

Redial(Retrial)

Busy)Rare(

Goodor

Bad

Positive: Repeat BusinessNegative: New Complaint

Lost Calls

Abandonment

Agents

ServiceCompletion

Service Engineering: Multi-Disciplinary Process View

ForecastingStatistics

New Services Design (R&D)Operations,Marketing

Organization Design:Parallel (Flat)Sequential (Hierarchical)Sociology/Psychology,Operations Research

Human Resource Management

Service Process Design

To Avoid Delay

To Avoid Starvation Skill Based Routing

(SBR) DesignMarketing,Human Resources,Operations Research,MIS

Customers Interface Design

Computer-Telephony Integration - CTIMIS/CS

Marketing

Operations/BusinessProcessArchiveDatabaseDesignData Mining:MIS, Statistics, Operations Research, Marketing

InternetChatEmailFax

Lost Calls

Service Completion)75% in Banks (

( Waiting TimeReturn Time)

Logistics

Customers Segmentation -CRM

Psychology, Operations Research,Marketing

Expect 3 minWilling 8 minPerceive 15 min

PsychologicalProcessArchive

Psychology,Statistics

Training, IncentivesJob Enrichment

Marketing,Operations Research

Human Factors Engineering

VRU/IVR

Queue)Invisible (

VIP Queue

(If Required 15 min,then Waited 8 min)(If Required 6 min, then Waited 8 min)

Information Design

Function

Scientific Discipline

Multi-Disciplinary

IndexCall Center Design

(Turnover up to 200% per Year)(Sweat Shops

of the21th Century)

Tele-StressPsychology

27

Call-Center Network: Gallery of Models

Agents(CSRs)

Back-Office

Experts)(Consultants

VIP)Training (

Arrivals(Business Frontier

of the21th Century)

Redial(Retrial)

Busy)Rare(

Goodor

Bad

Positive: Repeat BusinessNegative: New Complaint

Lost Calls

Abandonment

Agents

ServiceCompletion

Service Engineering: Multi-Disciplinary Process View

ForecastingStatistics

New Services Design (R&D)Operations,Marketing

Organization Design:Parallel (Flat)Sequential (Hierarchical)Sociology/Psychology,Operations Research

Human Resource Management


To Avoid Delay

To Avoid Starvation Skill Based Routing

(SBR) DesignMarketing,Human Resources,Operations Research,MIS

Customers Interface Design

Computer-Telephony Integration - CTIMIS/CS

Marketing


InternetChatEmailFax

Lost Calls

Service Completion)75% in Banks (

( Waiting TimeReturn Time)

Logistics

Customers Segmentation -CRM

Psychology, Operations Research,Marketing

Expect 3 minWilling 8 minPerceive 15 min



Training, IncentivesJob Enrichment

Marketing,Operations Research

Human Factors Engineering

VRU/IVR

Queue)Invisible (

VIP Queue

(If Required 15 min,then Waited 8 min)(If Required 6 min, then Waited 8 min)

Information Design

Function


Multi-Disciplinary

IndexCall Center Design

(Turnover up to 200% per Year)(Sweat Shops

of the21th Century)

Tele-StressPsychology

16

aviman

Oval

aviman

Oval

aviman

Oval

aviman

Oval

aviman

Oval

Skills-Based Routing in Call CentersEDA and OR, with I. Gurvich and P. Liberman

Mktg. ⇒

OR ⇒

HRM ⇒

MIS ⇒

29

SBR Topologies: I; V, Reversed-V; N, X; W, M

Israeli Cellular, March 2008Implication

Itai Gurvich (Kellogg/MEDS) Call Centers as Queueing Systems May, 2010 41 / 52

30

SBR: Class-Dependent Services

“Reduction" to V-Topology (Equivalent Brownian Control)“Many-server approximations” simplification

The class-dependent case

In what sense does the reduction work? Same Brownian controlproblem.Itai Gurvich (Kellogg/MEDS) Call Centers as Queueing Systems May, 2010 43 / 52

PhD’s: Tezcan, Dai; Shaikhet, w/ Atar; Gurvich, Whitt31

SBR: Pool-Dependent Services

“Reduction" to Reversed-V and I (Equivalent Brownian Control)Many-server approximations simplification

The pool-dependent case

e.g. Tezcan and Dai(09’), G. & Whitt (08’), Atar et. al. (10’)Itai Gurvich (Kellogg/MEDS) Call Centers as Queueing Systems May, 2010 42 / 52

PhD’s: Tezcan, Dai; Shaikhet, w/ Atar; Gurvich, Whitt32

Waiting Times in a Call Center (Theory?)

Exponential in Heavy-Traffic (min.)Small Israeli Bank

quantiles of waiting times to those of the exponential (the straight line at the right plot). The �t is reasonableup to about 700 seconds. (The p-value for the Kolmogorov-Smirnov test for Exponentiality is however 0 {not that surprising in view of the sample size of 263,007).

Figure 9: Distribution of waiting time (1999)

Time

0 30 60 90 120 150 180 210 240 270 300

29.1 %

20 %

13.4 %

8.8 %

6.9 %5.4 %

3.9 %3.1 %

2.3 % 1.7 %

Mean = 98SD = 105

Waiting time given agent

Exp

qua

ntile

s

0 200 400 600

020

040

060

0

Remark on mixtures of independent exponentials: Interestingly, the means and standard deviations in Table19 are rather close, both annually and across all months. This suggests also an exponential distributionfor each month separately, as was indeed veri�ed, and which is apparently inconsistent with the observerdannual exponentiality. The phenomenon recurs later as well, hence an explanation is in order. We shall besatis�ed with demonstrating that a true mixture W of independent random varibles Wi, all of which havecoeÆcients of variation C(Wi) = 1, can also have C(W ) � 1. To this end, let Wi denote the waiting time inmonth i, and suppose it is exponentially distributed with meanmi. Assume that the months are independentand let pi be the fraction of calls performed in month i (out of the yearly total). If W denotes the mixtureof these exponentials (W =Wi with probability pi, that is W has a hyper-exponential distribution), then

C2(W ) = 1 + 2C2(M);

where M stands for a �ctitious random variable, de�ned to be equal mi with probability pi. One concludesthat if themi's do not vary much relative to their mean (C(M) << 1), which is the case here, then C(W ) � 1,allowing for approximate exponentiality of both the mixture and its constituents.

6.2.1 The various waiting times, and their rami�cations

We �rst distinguished between queueing time and waiting time. The latter does not account for zero-waits,and it is more relevant for managers, especially when considered jointly with the fraction of customers thatdid wait. A more fundamental distinction is between the waiting times of customer that got served and thosethat abandoned. Here is it important to recognize that the latter does not describe customers' patience,which we now explain.

A third distinction is between the time that a customer needs to wait before reaching an agent vs. the timethat a customer is willing to wait before abandoning the system. The former is referred to as virtual waitingtime, since it amounts to the time that a (virtual) customer, equipped with an in�nite patience, would havewaited till being served; the latter will serve as our operational measure of customers' patience. While bothmeasures are obviously of great importance, note however that neither is directly observable, and hence mustbe estimated.

25

Routing via Thresholds (sec.)Large U.S. Bank

Chart1

Page 1

0

2

4

6

8

10

12

14

16

18

20

2 5 8 11 14 17 20 23 26 29 32 35

Time

Relative frequencies, %

Scheduling Priorities (sec) (later: Hospital LOS, hr.)Medium Israeli Bankwaitwait

Page 1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380

Time (Resolution 1 sec.)

Relative frequencies, %

33

ER / ED Environment: Service Network

Acute (Internal, Trauma) Walking

Multi-Trauma

34

Emergency-Department Network: Gallery of Models

ImagingLaboratory

Experts

Interns

Returns (Old or New Problem)

“Lost” Patients

LWBS

Nurses

Statistics,HumanResourceManagement(HRM)

New Services Design (R&D)

Operations,Marketing,MIS

Organization Design:Parallel (Flat) = ERvs. a true ED Sociology, Psychology,Operations Research


Quality

Efficiency

CustomersInterface Design

Medicine

(High turnoversMedical-Staff shortage)


StretcherWalking

Service Completion(sent to other department)

( Waiting TimeActive Dashboard )

PatientsSegmentation

Medicine,Psychology, Marketing


Human FactorsEngineering(HFE)

InternalQueue

OrthopedicQueue

Arrivals

Function


Multi-Disciplinary

Index

ED-StressPsychology

OperationsResearch, Medicine


Returns

TriageReception

Skill Based Routing (SBR) DesignOperations Research,HRM, MIS, Medicine

IncentivesGame Theory,Economics

Job EnrichmentTrainingHRM

HospitalPhysiciansSurgical

Queue

Acute,Walking

Blocked(Ambulance Diversion)

Forecasting

Information Design

MIS, HFE,Operations Research


Home

I Forecasting, Abandonment = LWBS, SBR ≈ Flow Control36


ImagingLaboratory

Experts

Interns


“Lost” Patients

LWBS

Nurses






Quality

Efficiency


Medicine



StretcherWalking







InternalQueue

OrthopedicQueue

Arrivals

Function


Multi-Disciplinary

Index

ED-StressPsychology



Returns

TriageReception





Queue

Acute,Walking


Forecasting

Information Design



Home

I Fork-Join Q’s, eg. After Physician: Nurse, Lab-Tests, X-RayI Synchronization Control, with R. Atar and A. ZviranI ED-to-IW Routing

25

aviman

Oval

aviman

Line

aviman

Line

aviman

Line

aviman

Line

aviman

Oval

aviman

Oval

aviman

Text Box

ED Design, with B. Golany, Y. Marmor, S. Israelit

Routing: Triage (Clinical), Fast-Track (Operational), . . . (via DEA)eg. Fast Track most suitable when elderly dominate

Triage

ED Area 1

“Hospital”

ED Area 2 ED Area 3

Patient Arrival

Patient Departure

(a) Triage Model

Triage

Fast TackLane*

“Hospital”

ED Area 1 ED Area 2

Patient Arrival

Patient Departure* operational criteria (short treatments time) –acute or walking patient

(b) Fast-Track Model

Admission

ED Area 1

“Hospital”

ED Area 2 ED Area 3

Wrong ED placement

Wrong ward placementPatient Arrival

Patient Departure

(c) Illness-based Model (d) Walking-Acute Model

ED Area 1 ED Area 2Room1 Room2

Walking Area Acute Area

“Hospital”Wrong ED placement

Wrong ward placement

Patient Arrival

Patient Departure

Admission

38

Emergency-Department Network: Flow Control

ImagingLaboratory

Experts

Interns


“Lost” Patients

LWBS

Nurses






Quality

Efficiency


Medicine



StretcherWalking







InternalQueue

OrthopedicQueue

Arrivals

Function


Multi-Disciplinary

Index

ED-StressPsychology



Returns

TriageReception





Queue

Acute,Walking


Forecasting

Information Design



Home

I Queueing-Science, w/ Armony, Marmor, Tseytlin, Yom-TovI Fair ED-to-IW Routing (Patients vs. Staff), w/ Momcilovic, TseytlinI Triage vs. In-Process / Release in EDs, w/ Carmeli, Huang, ShimkinI Workload and Offered-Load in Fork-Join Networks, w/ Kaspi, ZaeidI Synchronization Control of Fork-Join Networks, w/ Atar, ZviranI Staffing Time-Varying Q’s with Re-Entrant Customers, w/ Yom-Tov

39

ED Patient Flow: The Physicians View

&%'$

? ? ?

@@@@@@R

CCCCCCW

��

��

��

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AAAAK

HHHH

HHHH

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66 66

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λ01 λ0

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· · ·

P 0(j,k)

P (k, l)

d1 d2 dJ

m01 m0

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Triage-Patients

IP-Patients

ExitsS

Arrivals

C1(·) C2(·) C3(·) CK(·)

m1 m2 m3 mK

· · ·

1

I Goal: Adhere to Triage-Constraints, then process/release In-Process PatientsI Model = Multi-class Q with Feedback: Min. convex congestion costs of

IP-Patients, s.t. deadline constraints on Triage-Patients.I Solution: In conventional heavy-traffic, asymptotic least-cost s.t. asymptotic

compliance, via threshold (w/ B. Carmeli, J. Huang, S. Israelit, N. Shimkin; asin Plambeck, Harrison, Kumar, who applied admission control).

40

Operational Fairness

1. “Punishing" fast wards in ED-to-IW Routing:

I Parallel IWs: similar clinically , differ operationallyI Problem: Short Length-of-Stay goes hand in hand with high

bed-occupancy, bed-turnover, yet clinically apt: unfair!I Solution: Both nurses and managers content, w/ P. Momcilovic

and Y. Tseytlin (3 time-scales: hour, day, week; “compare" withcall-centers SBR)

2. Balancing Load across Maternity Wards:

I 2 Maternity Wards: 1 = pre-birth, 2 = post-birth complications

I Problem: Nurses think the “others-work-less": unfair!I Goal: Balance workload, mostly via normal birthsI Challenge: Workload is Operational, Cognitive, Emotional

I Operational: Work content of a task, in time-unitsI Emotional: e.g. Mother and fetus-in-stress, suddenly fetus dies

⇒ Need help: A. Rafaeli & students (Psychology) - Ongoing41

LogNormal & Beyond: Length-of-Stay in a Hospital

Israeli Hospital, in Days: LN

0 2 4 6 8 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Israeli Hospital, in Hours: Mixture

0 .2.4 .6 .8 11.21.51.82.12.42.7 33.23.53.84.14.44.7 55.25.55.86.16.46.7 7 7.37.67.98.28.58.89.19.49.7 10

Explanation: Patients releasedaround 3pm (1pm in Singapore)

Why Bother ?I Hourly Scale: Staffing,. . .I Daily: Flow / Bed Control,. . .

Workload at the Internal Ward (In Progress): Arrivals, Departures, # Patients in Ward A, by Hour

42

Prerequisite II: Models (Fluid Q’s)“Laws of Large Numbers" capture Predictable Variability

Deterministic Models: Scale Averages-out Stochastic Individualism

# Severely-Wounded Patients, 11:00-13:00 (Censored LOS)

Cleaning Data – An Example: RFID data in an MCE Drill

0

10

20

30

40

50

60

11:09 11:16 11:24 11:31 11:38 11:45 11:52 12:00 12:07 12:14 12:21 12:28 12:36 12:43 12:50 12:57 13:04 13:12 13:19 13:26

entries

exits

entries (original)

exits (original)

0

5

10

15

20

25

30

11:09 11:16 11:24 11:31 11:38 11:45 11:52 12:00 12:07 12:14 12:21 12:28 12:36 12:43 12:50 12:57 13:04 13:12 13:19 13:26

number of patients

number of patients (original)

I Paths of doctors, nurses, patients (100+, 1 sec. resolution)eg. (could) Help predict “What if 150+ casualties severely wounded ?"

I Transient Q’s:I Control of Mass Casualty Events (w/ I. Cohen, N. Zychlinski)I Chemical MCE = Needy-Content Cycles (w/ G. Yom-Tov)

43

The Basic Service-Network Model: Erlang-R

Erlang-R:

Needy (s-servers)

Content (Delay)

1-p

p

1

2

Arrivals Patient discharge

Needy (st-servers)

rate- μ

Content (Delay) rate - δ

1-p

p

1

2

Arrivals Poiss(λt) Patient discharge

Erlang-R (IE: Repairman Problem 50’s; CS: Central-Server 60’s) =2-station “Jackson" Network = (M/M/S, M/M/∞) :

I λ(t) – Time-Varying Arrival rateI S(·) – Number of Servers (Nurses / Physicians).I µ – Service rate (E [Service] = 1

µ)

I p – ReEntrant (Feedback) fractionI δ – Content-to-Needy rate (E [Content] = 1

δ)

44

Erlang-R: Fitting a Simple Model to a Complex Reality

Chemical MCE Drill (Israel, May 2010)

Arrivals & Departures (RFID) Erlang-R (Fluid, Diffusion)

!

"!

#!

$!

%!

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""(!# ""("' ""($" ""(%& "#(!! "#("% "#(#) "#(%$ "#(&* "$("# "$(#'

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+,-,./01234839/60,637

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50.67123-4

89:;<1=>?;09@;123-413AB;9<;-6,/04

C@@;<1=>?;09@;123-413AB;9<;-6,/04

50.6712#

I Recurrent/Repeated services in MCE Events: eg. Injection every 15 minutesI Fluid (Sample-path) Modeling, via Functional Strong Laws of Large Numbers

I Stochastic Modeling, via Functional Central Limit Theorems

I ED in MCE: Confidence-interval, usefully narrow for ControlI ED in normal (time-varying) conditions: Personnel Staffing

45

Prerequisite II: Models (Diffusion/QED’s Q’s)

Traditional Queueing Theory predicts that Service-Quality andServers’ Efficiency must be traded off against each other.

For example, M/M/1 (single-server queue): 91% server’s utilizationgoes with

Congestion Index =E [Wait ]

E [Service]= 10,

and only 9% of the customers are served immediately upon arrival.

Yet, heavily-loaded queueing systems with Congestion Index = 0.1(Waiting one order of magnitude less than Service) are prevalent:

I Call Centers: Wait “seconds" for minutes service;I Transportation: Search “minutes" for hours parking;I Hospitals: Wait “hours" in ED for days hospitalization in IW’s;

and, moreover, a significant fraction are not delayed in queue. (Forexample, in well-run call-centers, 50% served “immediately", alongwith over 90% agents’ utilization, is not uncommon ) ? QED

44

The Basic Staffing Model: Erlang-A (M/M/N + M)

agents

arrivals

abandonment

λ

µ

1

2

n

…

queue

θ

Erlang-A (Palm 1940’s) = Birth & Death Q, with parameters:

I λ – Arrival rate (Poisson)I µ – Service rate (Exponential; E [S] = 1

µ )

I θ – Patience rate (Exponential, E [Patience] = 1θ )

I n – Number of Servers (Agents).45

Testing the Erlang-A Primitives

I Arrivals: Poisson?I Service-durations: Exponential?I (Im)Patience: Exponential?

I Primitives independent (eg. Impatience and Service-Durations)?I Customers / Servers Homogeneous?I Service discipline FCFS?I . . . ?

Validation: Support? Refute?

46

Arrivals to ServiceArrival-Rates to Three Call Centers

Dec. 1995 (U.S. 700 Helpdesks) May 1959 (England)

Q-Science

May 1959!

Dec 1995!

(Help Desk Institute)

Arrival Rate

Time 24 hrs

Time 24 hrs

% Arrivals

(Lee A.M., Applied Q-Th)

28

Q-Science

May 1959!

Dec 1995!

(Help Desk Institute)

Arrival Rate

Time 24 hrs

Time 24 hrs

% Arrivals

(Lee A.M., Applied Q-Th)

28

November 1999 (Israel)

Arrival Process, in 1999

Yearly Monthly

Daily Hourly

Random Arrivals “must be"(Axiomatically)Time-Inhomogeneous Poisson

47

Arrivals to Service: only Poisson-Relatives

Arrival-Counts: Coefficient-of-Variation (CV), per 30 min.Israeli-Bank Call-Center, 263 regular days (4/2007 - 3/2008)Coefficient of Variation Per 30 Minutes, seperated weekdays

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Time

Coe

ffici

ent o

f Var

iatio

n

Sundays Mondays Tuesdays Wednesdays Thursdays

I Poisson CV (Dashed Line) = 1/√

mean arrival-rateI Poisson CV’s� Sampled CV’s (Solid) ⇒ Over-Dispersion

⇒ Modeling (Poisson-Mixture) of and Staffing ( >√· ) against

Time-Varying Over-Dispersed Arrivals (w/ S. Maman & S. Zeltyn)48

Service Durations: LogNormal Prevalent

Israeli Bank Service-ClassesLog-Histogram Survival-Functions

0

100

200

300

400

500

600

700

800

900

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

Log(service time)

Freq

uenc

y

frequency normal curve

Average = 2.24St.dev. = 0.42

31

Service TimeSurvival curve, by Types

Time

Surv

ival

Means (In Seconds)

NW (New) = 111

PS (Regular) = 181

NE (Stocks) = 269

IN (Internet) = 381

34- New Customers: 2 min (NW);

- Regulars: 3 min (PS);

- Stock: 4.5 min (NE);

- Tech-Support: 6.5 min (IN).

I Service Durations are LogNormal (LN) and Heterogeneous

49

(Im)Patience while Waiting (Palm 1943-53)Hazard Rate of (Im)Patience Distribution ∝ Irritation

Regular over VIP Customers – Israeli Bank 14

16

I VIP Customers are more Patient (Needy)I Peaks of abandonment at times of AnnouncementsI Challenges: Un-Censoring, Dependence (vs. KM), Smoothing

- requires Call-by-Call Data50

Dependent Primitives: Service- vs. Waiting-Time

Average Service-Time as a function of Waiting-TimeU.S. Bank, Retail, Weedays, January-June, 2006

Introduction Relationship Between Service Time and Patience Workload and Offered-Load Empirical Results Future Research

Motivation

Mean Service-Time as a Function of Waiting-TimeU.S. Bank - Retail Banking Service - Weekdays - January-June, 2006

190

210

230

250

270

290

150

170

190

210

230

250

270

290

0 50 100 150 200 250 300

Waiting Time

Fitted Spline Curve E(S|τ>W=w)

⇒ Focus on ( Patience, Service-Time ) jointly , w/ Reich and Ritov.E [S |Patience = w ], w ≥ 0: Service-Time of the Unserved.

51

Erlang-A: Practical Relevance?

Experience:I Arrival process not pure Poisson (time-varying, σ2 too large)I Service times not Exponential (typically close to LogNormal)I Patience times not Exponential (various patterns observed).

I Building Blocks need not be independent (eg. long waitassociated with long service; with w/ M. Reich and Y. Ritov)

I Customers and Servers not homogeneous (classes, skills)I Customers return for service (after busy, abandonment;

dependently; P. Khudiakov, M. Gorfine, P. Feigin)I · · · , and more.

Question: Is Erlang-A Relevant?

YES ! Fitting a Simple Model to a Complex Reality, bothTheoretically and Practically

52

Estimating (Im)Patience: via P{Ab} ∝ E[Wq]

“Assume" Exp(θ) (im)patience. Then, P{Ab} = θ · E[Wq] .

% Abandonment vs. Average Waiting-TimeBank Anonymous (JASA): Yearly Data

Hourly Data Aggregated

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Average waiting time, sec

Pro

bab

ility

to

ab

and

on

0 50 100 150 200 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55


Pro

bab

ility

to

ab

and

on

Graphs based on 4158 hour intervals.

Estimate of mean (im)patience: 250/0.55 sec. ≈ 7.5 minutes.

53

Erlang-A: Fitting a Simple Model to a Complex Reality

I Bank Anonymous Small Israeli Call-Center

I (Im)Patience (θ) estimated via P{Ab} / E[Wq]

I Graphs: Hourly Performance vs. Erlang-A Predictions,during 1 year (aggregating groups with 40 similar hours).

P{Ab} E[Wq] P{Wq > 0}

0 0.1 0.2 0.3 0.4 0.5 0.60

0.1

0.2

0.3

0.4

0.5

Probability to abandon (Erlang−A)

Pro

babi

lity

to a

band

on (

data

)

0 50 100 150 200 2500

50

100

150

200

250

Waiting time (Erlang−A), sec

Wai

ting

time

(dat

a), s

ec

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of wait (Erlang−A)

Pro

babi

lity

of w

ait (

data

)

54

Erlang-A: Fitting a Simple Model to a Complex Reality

Large U.S. Bank

Retail. P{Wq > 0} Telesales. E[Wq]

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of wait (QED: aggregated)

Pro

babi

lity

of w

ait (

data

: agg

rega

ted)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Average wait (QED: aggregated), sec

Ave

rage

wai

t (da

ta: a

ggre

gate

d), s

ec

Partial success – in some cases Erlang-A does not work well(Networking, SBR).

Ongoing Validation Project, w/ Y. Nardi, O. Plonsky, S. Zeltyn

55

Erlang-A: Simple, but Not Too Simple

Practical (Data-Based) questions, started in Brown et al. (JASA):1. Fitting Erlang-A (Validation, w/ Nardi, Plonsky, Zeltyn).2. Why does it practically work? justify robustness.3. When does it fail? chart boundaries.4. Generate needs for new theory.

Theoretical Framework: Asymptotic Analysis, as load- andstaffing-levels increase, which reveals model-essentials:

I Efficiency-Driven (ED) regime: Fluid models (deterministic)I Quality- and Efficiency-Driven (QED): Diffusion refinements.

Motivation: Moderate-to-large service systems (100’s - 1000’sservers), notably Call-Centers.

Results turn out accurate enough to also cover <10 servers:I Practically Important: Relevant to Healthcare

(First: F. de Véricourt and O. Jennings; w/ G. Yom-Tov; Y. Marmor, S.Zeltyn; H. Kaspi, I. Zaeid)

I Theoretically Justifiable: Gap-Analysis by A. Janssen, J. vanLeeuwaarden, B. Zhang, B. Zwart.

56

Operational Regimes: Conceptual Framework

R: Offered LoadDef. R = Arrival-rate × Average-Service-Time = λ

µ

eg. R = 25 calls/min. × 4 min./call = 100

N = #Agents ? Intuition, as R or N increase unilaterally.

QD Regime: N ≈ R + δR , 0.1 < δ < 0.25 (eg. N = 115)I Framework developed in O. Garnett’s MSc thesisI Rigorously: (N − R)/R → δ, as N, λ ↑ ∞, with µ fixed.I Performance: Delays are rare events

ED Regime: N ≈ R − γR , 0.1 < γ < 0.25 (eg. N = 90)I Essentially all customers are delayedI Wait same order as service-time; γ% Abandon (10-25%).

QED Regime: N ≈ R + β√

R , −1 < β < +1 (eg. N = 100)I Erlang 1913-24, Halfin & Whitt 1981 (for Erlang-C)I %Delayed between 25% and 75%I E[Wait] ∝ 1√

N× E[Service] (sec vs. min); 1-5% Abandon.

57

Operational Regimes: Rules-of-Thumb, w/ S. ZeltynOperational Regimes in Practice

Constraint P{Ab} E[W ] P{W > T}Tight Loose Tight Loose Tight Loose

1-10% ≥ 10% ≤ 10%E[τ ] ≥ 10%E[τ ] 0 ≤ T ≤ 10%E[τ ] T ≥ 10%E[τ ]

Offered Load 5% ≤ α ≤ 50% 5% ≤ α ≤ 50%

Small (10’s) QED QED QED QED QED QED

Moderate-to-Large QED ED, QED ED, QED ED+QED

(100’s-1000’s) QED QED if τ d= exp

ED: n ≈ R − γR.

QD: n ≈ R + δR.

QED: n ≈ R + β√

R.

ED+QED: n ≈ (1 − γ)R + β√

R.

1

ED: N ≈ R − γR (0.1 ≤ γ ≤ 0.25 ).

QD: N ≈ R + δR (0.1 ≤ δ ≤ 0.25 ).

QED: N ≈ R + β√

R (−1 ≤ β ≤ 1 ).

ED+QED: N ≈ (1− γ)R + β√

R (γ, β as above).

WFM: How to determine specific staffing level N ? e.g. β.

58

Operational Regimes: Scaling, Performance,w/ I. Gurvich & J. HuangTable 1: Operational Regimes: Asymptotics, Scaling, Performance

Erlang-A Conventional scaling MS scaling NDS scaling

µ fixed Sub Critical Super QD QED ED ED+QED Sub Critical Super

Offered load per server 11+δ

< 1 1− β√n≈ 1 1

1−γ > 1 11+δ

1− β√n

11−γ

11−γ − β

√1

n(1−γ)31

1+δ1− β

n1

1−γ

Arrival rate λ µ1+δ

µ− β√nµ µ

1−γnµ1+δ

nµ− βµ√n nµ1−γ

nµ1−γ − βµ

√n

(1−γ)3nµ1+δ

nµ− βµ nµ1−γ

Number of servers 1 n n

Time-scale n 1 n

Abandonment rate θ/n θ θ/n

Staffing level λµ(1 + δ) λ

µ(1 + β√

n) λ

µ(1− γ) λ

µ(1 + δ) λ

µ+ β

√λµ

λµ(1− γ) λ

µ(1− γ) + β

√λµ

λµ(1 + δ) λ

µ+ β λ

µ(1− γ)

Utilization 11+δ

1−√

θµ

h(β)√n

1 11+δ

1−√

θµ

(1−α2)β+α2h(β)√n

1 1 11+δ

1−√

θµ

h(β)

n1

E(Q) α1

δ

√n

√µθ[h(β)− β] nµγ

θ(1−γ)1√2π

1+δδ2%n 1√

n

√n

√µθ[h(β)− β]α2

nµγθ(1−γ)

nµθ(1−γ)

(γ − β√n(1−γ)

) o(1) n√

µθ[h(β)− β] n2µγ

θ(1−γ)

P(Ab) 1n

1+δδ

θµα1

1√n

√θµ[h(β)− β] γ 1√

2π

θµ

(1+δ)2

δ2%n 1

n3/21√n

√θµ[h(β)− β]α2 γ γ − β

√1−γ√n

o( 1n2 ) 1

n

√θµ[h(β)− β] γ

P(Wq > 0) α1 ∈ (0,1) ≈ 1 1√2π

1+δδ%n 1√

n≈ 0 α2 ∈ (0,1) ≈ 1 ≈ 1 ≈ 0 ≈ 1

P(Wq > T ) α1e− δ

1+δµt 1 +O( 1√

n) 1 +O( 1

n) ≈ 0 G(T )1{G(T )<γ} α3, if G(T ) = γ ≈ 0 Φ(β+

√θµT )

Φ(β)1 +O( 1

n)

Congestion EWq

ES α11+δδ

√n

√µθ[h(β)− β] nµγ/θ 1√

2π

(1+δ)2

δ2%n 1

n3/21√n

√µθ[h(β)− β]α2 µ

∫∫∫ x∗

0G(s)ds µ

∫∫∫ x∗

0G(s)ds− µβ

√1−γ

hG(x∗)√n

o( 1n)

√µθ[h(β)− β] nµγ/θ

1 59

QED Call Center: Staffing (N) vs. Offered-Load (R)IL Telecom; June-September, 2004; w/ Nardi, Plonski, Zeltyn

Empirical Analysis of a QED Call Center

• 2205 half-hour intervals of an Israeli call center • Almost all intervals are within R R− and 2R R+ (i.e. 1 2β− ≤ ≤ ),

implying: o Very decent forecasts made by the call center o A very reasonable level of service (or does it?)

• QED? o Recall: ( )GMR x is the asymptotic probability of P(Wait>0) as a

function of β , where x θµ=

o In this data-set, 0.35x θµ= =

o Theoretical analysis suggests that under this condition, for 1 2β− ≤ ≤ , we get 0 ( 0) 1P Wait≤ > ≤ …

2205 half-hour intervals in an Israeli Call Center60

QED Call Center: Performance

Large Israeli Bank

P{Wq > 0} vs. (R, N) R-Slice: P{Wq > 0} vs. N

• Intercepting plane of the previous plot for a "constant" offered load (actually

18 18.5R≤ ≤ ) • Smoother line was created using smoothing splines. • General shape of the smoothing line is as the Garnett delay functions! • In 2-d view, we get the following:

3 Operational Regimes:I QD: ≤ 25%

I QED: 25%− 75%

I ED: ≥ 75%

61

QED Theory (Erlang ’13; Halfin-Whitt ’81; Garnett MSc; Zeltyn PhD)

Consider a sequence of steady-state M/M/N + G queues, N = 1, 2, 3, . . .Then the following points of view are equivalent, as N ↑ ∞:

Consider a sequence of M/M/N+G models, N=1,2,3,…

Then the following points of view are equivalent:

� QED %{Wait > 0} � � , 0 < � < 1 ;

� Customers %{Abandon} �N� , 0 < � ;

� Agents OCCN��

�� 1 � < � < ;

� Managers RRN �� , � �R E(S) not small;

QED performance (ASA, ...) is easily computable, all in terms

of � (the square-root safety staffing level) – see later.

I QED performance: Laplace Method (asymptotics of integrals).I Parameters: Arrivals and Staffing - β, Services - µ,

(Im)Patience - g(0) = patience density at the origin.

62

Erlang-A: QED Approximations (Examples)

Assume Offered Load R not small (λ→∞).

Let β = β

√µ

θ, h(·) =

φ(·)1− Φ(·) = hazard rate of N (0,1).

I Delay Probability:

P{Wq > 0} ≈[

1 +

√θ

µ· h(β)

h(−β)

]−1

.

I Probability to Abandon:

P{Ab|Wq > 0} ≈ 1√n·√θ

µ·[h(β)− β

].

I P{Ab} ∝ E[Wq] , both order 1√n :

P{Ab}E[Wq]

= θ.

63

Garnett / Halfin-Whitt Functions: P{Wq > 0}HW/GMR Delay Functions

α vs. β

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

Beta

Del

ay P

roba

bilit

y

Halfin-Whitt Garnett(0.1) Garnett(0.5) Garnett(1)

Garnett(2) Garnett(5) Garnett(10) Garnett(20)

Garnett(50) Garnett(100)

θ/µ

Halfin-Whitt

QED Erlang-A

64

Empirical data: 2204 intervals of Technical category from Telecom call center

(13 summer weeks; week-days only; 30 min. resolution; excluding 6 outliers). Theoretical plot: based on approximations of Erlang-A for the proper rate.

QED Intuition: Why P{Wq > 0} ∈ (0, 1) ?

1. Why subtle: Consider a large service system (e.g. call center).I Fix λ and let n ↑ ∞: P{Wq > 0} ↓ 0.I Fix n and let λ ↑ ∞: P{Wq > 0} ↑ 1.I ⇒ Must have both λ and n increase simultaneously:I ⇒ (CLT) Square-root staffing: n ≈ R + β

√R.

2. Erlang-A (M/M/n+M), with parameters λ, µ, θ; n, in which µ = θ:(Im)Patience and Service-times are equally distributed.

I Steady-state: L(M/M/n + M)d= L(M/M/∞)

d= Poisson(R), with

R = λ/µ (Offered-Load)I Poisson(R)

d≈ R + Z

√R, with Z d

= N(0, 1).

I P{Wq(M/M/n + M) > 0} PASTA= P{L(M/M/n + M) ≥ n} µ=θ

=

P{L(M/M/∞) ≥ n} ≈ P{R + Z√

R ≥ n} =

P{Z ≥ (n − R)/√

R}√· staffing≈ P{Z ≥ β} = 1− Φ(β).

3. QED Excursions65

QED Intuition via Excursions: Busy-Idle CyclesM/M/N+M (Erlang-A) with Many Servers: N ↑ ∞M/M/N+M (Erlang-A) with Many Servers: N ↑ ∞

0 1 N-1 N N+1

Busy Period

µ 2µNµ(N-1)µ Nµ +

Q(0) = N : all servers busy, no queue.

Let TN,N−1 = Busy Period (down-crossing N ↓ N − 1 )

TN−1,N = Idle Period (up-crossing N − 1 ↑ N )

Then P (Wait > 0) =TN,N−1

TN,N−1 + TN−1,N=

[1 +

TN−1,N

TN,N−1

]−1

.

Calculate TN−1,N =1

λNE1,N−1∼ 1

Nµ× h(−β)/√

N∼ 1√

N· 1/µ

h(−β)

TN,N−1 =1

Nµπ+(0)∼ 1√

N· β/µ

h(δ) /δ, δ = β

√µ/θ

Both apply as√

N (1− ρN) → β, −∞ < β < ∞.

Hence, P (Wait > 0) ∼[1 +

h(δ)/δ

h(−β)/β

]−1

.

1


Let TN,N−1 = E[Busy Period] down-crossing N ↓ N − 1

TN−1,N = E[Idle Period] up-crossing N − 1 ↑ N )


TN,N−1+TN−1,N=[1 +

TN−1,NTN,N−1

]−1.

66

QED Intuition via Excursions: Asymptotics

M/M/N+M (Erlang-A) with Many Servers: N ↑ ∞

0 1 N-1 N N+1

Busy Period

µ 2µNµ(N-1)µ Nµ +


Let TN,N−1 = Busy Period (down-crossing N ↓ N − 1 )

TN−1,N = Idle Period (up-crossing N − 1 ↑ N )


TN,N−1 + TN−1,N=

[1 +

TN−1,N

TN,N−1

]−1

.

Calculate TN−1,N =1

λNE1,N−1∼ 1

Nµ× h(−β)/√

N∼ 1√

N· 1/µ

h(−β)

TN,N−1 =1

Nµπ+(0)∼ 1√

N· β/µ

h(δ) /δ, δ = β

√µ/θ

Both apply as√

N (1− ρN) → β, −∞ < β < ∞.

Hence, P (Wait > 0) ∼[1 +

h(δ)/δ

h(−β)/β

]−1

.

1

67

Process Limits (Queueing, Waiting)• QN = {QN(t), t ≥ 0} : stochastic process obtained by

centering and rescaling:

QN =QN −N√

N

• QN(∞) : stationary distribution of QN

• Q = {Q(t), t ≥ 0} : process defined by: QN(t)d→ Q(t).

��

�

�

� �

QN(t) QN(∞)

Q(t) Q(∞)

t →∞

t →∞

N →∞ N →∞

Approximating (Virtual) Waiting Time

VN =√

N VN ⇒ V =

[1

μQ

]+

(Puhalskii, 1994)

stochastic process

Waiting Time

68

QED Erlang-X (Markovian Q’s: Performance Analysis)I Pre-History, 1914: Erlang (Erlang-B = M/M/n/n, Erlang-C = M/M/n)I Pre-History, 1974: Jagerman (Erlang-B)I History Milestone, 1981: Halfin-Whitt (Erlang-C, GI/M/n)I Erlang-A (M/M/N+M), 2002: w/ Garnett & ReimanI Erlang-A with General (Im)Patience (M/M/N+G), 2005: w/ ZeltynI Erlang-C (ED+QED), 2009: w/ ZeltynI Erlang-B with Retrial, 2010: Avram, Janssen, van LeeuwaardenI Refined Asymptotics (Erlang A/B/C), 2008-2011: Janssen, van Leeuwaarden,

Zhang, ZwartI NDS Erlang-C/A, 2009: AtarI Production Q’s, 2011: Reed & ZhangI Universal Erlang-R, ongoing: w/ Gurvich & HuangI Queueing Networks:

I (Semi-)Closed: Nurse Staffing (Jennings & de Vericourt), CCs with IVR (w/Khudiakov), Erlang-R (w/ Yom-Tov)

I CCs with Abandonment and Retrials: w. Massey, Reiman, Rider, StolyarI Markovian Service Networks: w/ Massey & Reiman

I Leaving out:I Non-Exponential Service Times: M/D/n (Erlang-D), G/Ph/n, · · · , G/GI/n+GI,

Measure-Valued DiffusionsI Dimensioning (Staffing): M/M/n, · · · , time-varying Q’s, V- and Reversed-V, · · ·I Control: V-network, Reversed-V, · · · , SBRNets

69

Back to “Why does Erlang-A Work?"

Theoretical (Partial) Answer:

M?,Jt /G∗/Nt + G

d≈ (M/M/N + M)t , t ≥ 0.

I Over-Dispersed Arrivals: R + βRc , c-Staffing (c ≥ 1/2).

I General Patience: Behavior at the origin matters most (only).

I General Services: Empirical insensitivity beyond the mean.

I Heterogeneous Customers / Servers: State-Collapse.

I Time-Varying Arrivals: Modified Offered-Load approximations.

I Dependent Building-Blocks: eg. When (Im)Patience andService-Times correlated (positively):

I Predict performance with E [S |Served].I Calculate offered-load with E [S] = E [S |Wait = 0].I Note: staffing← service-times← waiting / abandonment← staffing

70

“Why does Erlang-A Work?" General PatienceIsraeli Bank: Yearly Data

Hourly Data Aggregated

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


Pro

bab

ility

to

ab

and

on

0 50 100 150 200 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55


Pro

bab

ility

to

ab

and

on

Theory:Erlang-A: P{Ab} = θ · E[Wq]; M/M/N+G: P{Ab} ≈ g(0) · E[Wq].

g(0) = Patience-density at originRecipe:In both cases, use Erlang-A, with θ = P{Ab}/E[Wq] (slope above).References on g(0):

- Stationary M/M/N+GI, w/ Zeltyn- Process G/GI/N+GI: w/ Momcilovic; Dai & He;

71

“Why does Erlang-A Work?" Over-Dispersion

ln(STD) vs. ln(AVG) (Israeli Bank, 4/2007-3/2008)

Tue-Wed, 30 min resolutionln(sd) vs ln(average) per 30 minutes. Sundays

y = 0.8027x - 0.1235R2 = 0.9899

y = 0.8752x - 0.8589R2 = 0.9882

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8

ln(Average Arrival)

ln(S

tand

ard

Dev

iatio

n)

00:00-10:30 10:30-00:00

Tue-Wed, 5 min resolutionln(Standard Deviation) vs ln(Average) per 5 minutes, thu-wed

y = 0.7228x - 0.0025R2 = 0.9937

y = 0.7933x - 0.5727R2 = 0.9783

0

1

2

3

4

5

0 1 2 3 4 5 6

ln(Average)

ln(S

tand

ard

Dev

iatio

n)

00:00-10:30 10:30-00:00

Significant linear relations (w/ Aldor & Feigin; then w/Maman & Zeltyn ):

ln(STD) = c · ln(AVG) + a

(Poisson: STD = AVG1/2, hence c = 1/2, a = 0.)

72

Over-Dispersion: Random Arrival-Rates

Linear relation between ln(STD) and ln(AVG) gives rise to:

Poisson-Mixture (Doubly-Poisson, Cox) model for Arrivals:Poisson(Λ) with Random-Rate of the form

Λ = λ + λc · X , c ≤ 1 ;

I c determines magnitude of over-dispersion (λc)c = 1, proportional to λ; c ≤ 1/2, Poisson-level;

- In Call Centers: c ≈ 0.75− 0.85 (significant over-dispersion).- In Emergency Departments, c ≈ 0.5 (Poisson).

I X random-variable with E [X ] = 0 (E [Λ] = λ), capturing themagnitude of stochastic deviation from mean arrival-rate:under conventional Gamma prior (λ large), X can be takenNormal with std. derived from the intercept.

QED-c Regime: Erlang-A, with Poisson(Λ) arrivals, amenable toasymptotic analysis (with S. Maman & S. Zeltyn)

73

Over-Dispersion: The QED-c Regime

QED-c Staffing: Under offered-load R = λ · E[S],

N = R + β · Rc , 0.5 < c < 1

Performance measures (M/M/N + G):

- Delay probability: P{Wq > 0} ∼ 1−G(β)

- Abandonment probability: P{Ab} ∼ E [X − β]+

n1−c

- Average offered wait: E [V ] ∼ E [X − β]+

n1−c · g0

- Average actual wait: EΛ,N [W ] ∼ EΛ,N [V ]

74

Why Does Erlang-A Work? Time-Varying Arrival Rates

Square-Root Staffing: Nt = Rt + β√

Rt , −∞ < β <∞What is Rt , the Offered-Load at time t ? ( Rt 6= λt × E[S] )

Arrivals, Offered-Load and Staffing

0

50

100

150

200

250

0 1 2 4 5 6 7 8

10

11

12

13

14

16

17

18

19

20

22

23

0

500

1000

1500

2000

Arr

iva

ls p

er

ho

ur

beta 1.2 beta 0 beta -1.2 Offered Load Arrivals

QDQED

ED

75

Time-Stable Performance of Time-Varying Systems

Delay Probability = As in the Stationary Erlang-A (Garnett)Delay Probability

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10 1 2 4 5 6 7 8

10

11

12

13

14

16

17

18

19

20

22

23

beta 2 beta 1.6 beta 1.2 beta 0.8 beta 0.4 beta 0

beta -0.4 beta -0.8 beta -1.2 beta -1.6 beta -2

76

Time-Stable Performance of Time-Varying Systems

Waiting Time, Given Waiting:Empirical vs. Theoretical Distribution

Waiting Time given Wait > 0:

beta = 1.2 QD ( 0.1)

0

0.05

0.1

0.15

0.2

0.25

0.0

00

0.0

02

0.0

04

0.0

06

0.0

08

0.0

10

0.0

12

0.0

14

0.0

16

0.0

18

0.0

20

ho

urs

Simulated Theoretical (N=191)


beta = 0 QED ( 0.5)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.0

00

0.0

02

0.0

04

0.0

06

0.0

08

0.0

10

0.0

12

0.0

14

0.0

16

0.0

18

0.0

20

0.0

22

0.0

24

0.0

26

0.0

28

ho

urs



beta = -1.2 ED ( 0.9)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.0

00

0.0

04

0.0

08

0.0

12

0.0

16

0.0

20

0.0

24

0.0

28

0.0

32

0.0

36

0.0

40

0.0

44

ho

urs


- Empirical: Simulate time-varying Mt/M/Nt + M (λt ,Nt = Rt + β√

Rt )

- Theoretical: Naturally-corresponding stationary Erlang-A, with QEDβ-staffing (some Averaging Principle?)

- Generalizes up to a single-station within a complex network (eg.Doctors in an Emergency Department).

77

What is the Offered-Load R(t)?

I Offered-Load Process: L(·) = Least number of servers thatguarantees no delay.

I Offered-Load Function R(t) = E [L(t)], t ≥ 0.Think Mt/G/N?

t + G vs. Mt/G/∞: Ample-Servers.

Four (all useful) representations, capturing “workload before t":

R(t) = E [L(t)] =

∫ t

−∞λ(u) · P(S > t − u)du = E

[A(t)− A(t − S)

]=

= E[∫ t

t−Sλ(u)du

]= E [λ(t − Se)] · E [S] ≈ ... .

I {A(t), t ≥ 0} Arrival-Process, rate λ(·);I S (Se) generic Service-Time (Residual Service-Time).I Relating L, λ,S (“W ”): Time-Varying Little’s Formula.

Stationary models: λ(t) ≡ λ then R(t) ≡ λ× E[S].

QED-c: Nt = Rt + βRct , 1/2 ≤ c < 1; (c = 1 separate analysis).

78

The Offered-Load R(t), t ≥ 0I Backbone of time-varying staffing:

I Practically robust: up to a station within a complex network (ED).I Theoretically challenging: only Erlang-A with µ = θ tractable.

I Process: L(·) = Least number of servers that guarantees no delay.I Offered-Load Function R(·) = E [L(·)] (Mt/G/N?

t + G ↔ Mt/G/∞).

ILTelecom , Private12.05.2004

0.005.00

10.0015.0020.0025.0030.0035.0040.0045.0050.0055.0060.00

07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00


Num

ber

of c

ases

Offered load Abandons Av_agents_in_system

79

Estimating / Predicting the Offered-Load

Must account for “service times of abandoning customers".

I Prevalent Assumption: Services and (Im)Patience independent.I But recall Patient VIPs: Willing to wait more for longer services.

Survival Functions by Type (Small Israeli Bank)

31

Service Time (cont’)Survival curve, by types

Time

Surv

ival

Service times stochastic order: SNew

st< SReg

st< SVIP

Patience times stochastic order: τNew

st< τReg

st< τVIP

80

Dependent Primitives: Service- vs. Waiting-Time

Average Service-Time as a function of Waiting-TimeU.S. Bank, Retail, Weedays, January-June, 2006

Introduction Relationship Between Service Time and Patience Workload and Offered-Load Empirical Results Future Research

Motivation

Mean Service-Time as a Function of Waiting-TimeU.S. Bank - Retail Banking Service - Weekdays - January-June, 2006

190

210

230

250

270

290

150

170

190

210

230

250

270

290

0 50 100 150 200 250 300

Waiting Time

Fitted Spline Curve E(S|τ>W=w)

⇒ Focus on ( Patience, Service-Time ) jointly , w/ Reich and Ritov.E [S |Patience = w ], w ≥ 0: Service-Time of the Unserved.

81

(Imputing) Service-Times of Abandoning Customers

w/ M. Reich, Y. Ritov:

1. Estimate g(w) = E [S |Patience > Wait = w ], w ≥ 0:Mean service time of those served after waiting exactly w unitsof time (via non-linear regression: Si = g(Wi ) + εi );

2. Calculate

E [S |Patience = w ] = g(w)− g′(w)

hτ (w);

hτ (w) = hazard-rate of (im)patience (via un-censoring);

3. Offered-load calculations: Impute E [S |Patience = w ](or the conditional distribution).

Challenges: Stable and accurate inference of g,g′,hτ .

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Extending the Notion of the “Offered-Load"

I Business (Banking Call-Center): Offered Revenues

I Healthcare (Maternity Wards): Fetus in stressI 2 patients (Mother + Child) = high operational and cognitive loadI Fetus dies⇒ emotional load dominates

I ⇒I Offered Operational Load

I Offered Cognitive Load

I Offered Emotional Load

I ⇒ Fair Division of Load (Routing) between 2 Maternity Wards:One attending to complications before birth, the other tocomplications after birth, and both share normal birth

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The Technion SEE Center / LaboratoryData-Based Service Science / Engineering

2

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data-based science for service engineering and...

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