Kellogg Operations Workshop , September 2012iew3.technion.ac.il/serveng/References/0_Kellogg_Workshop...Human Resources, Operations Research, MIS Customers Interface Design Computer-Telephony

Data-Based Service Networks:

A Research Framework for

Asymptotic Inference, Analysis & Control

of Service Systems

Avi Mandelbaum

Technion, Haifa, Israel

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

Kellogg Operations Workshop, September 2012

◮ Overheads available at my Technion websites

1

Research Partners

◮ Students:Aldor∗, Baron∗, Carmeli∗, Cohen∗, Feldman∗, Garnett∗, Gurvich∗,

Khudiakov∗, Maman∗, Marmor∗, Reich∗, Rosenshmidt∗, Shaikhet∗,

Senderovic, Tseytlin∗, Yom-Tov∗, Yuviler, Zaied∗, Zeltyn∗,

Zychlinski∗, Zohar∗, Zviran∗, . . .

◮ Theory:Armony, Atar, Cohen, Gurvich, Huang, Jelenkovic, Kaspi, Massey,

Momcilovic, Reiman, Shimkin, Stolyar, Trofimov, Wasserkrug,

Whitt, Zeltyn, . . .

◮ Empirical/Statistical Analysis:Brown, Gans, Shen, Zhao; Zeltyn; Ritov, Goldberg; Gurvich, Huang,

Liberman; Armony, Marmor, Tseytlin, Yom-Tov; Nardi, Plonsky;

Gorfine, Ghebali; Pang, . . .

◮ Industry:Mizrahi Bank (A. Cohen, U. Yonissi), Rambam Hospital (R. Beyar, S.

Israelit, S. Tzafrir), IBM Research (OCR Project), Hapoalim Bank (G.

Maklef, T. Shlasky), Pelephone Cellular, . . .

◮ Technion SEE Center / Laboratory:Feigin; Trofimov, Nadjharov, Gavako, Kutsy; Liberman, Koren,

Plonsky, Senderovic; Research Assistants, . . .2

Contents

◮ Service Networks: Call Centers, Hospitals, Websites, · · ·◮ Redefine the paradigm of modeling/asymptotics via Data

◮ ServNets: QNets, SimNets; FNets, DNets

◮ Ultimate Goal: Data-based creation and validation

of ServNets, automatically in real-time

3

Contents





◮ Why be Optimistic? Pilot at the Technion SEELab◮ Lacking but Feasible: Dynamics, Durations, Protocols◮ Simple Models at the Service of Complex Realities

◮ State-Space Collapse (Queues, Waiting Times)◮ Congestion Laws (LN, Little, ImPatience, Staffing)◮ Universal Approximations: Simplifying the Asymptotic Landscape◮ Stabilizing Time-Varying Performance (Offered-Load)

◮ Successes: Palm/Erlang-R (ED Feedback = FNet),Palm/Erlang-A (CC Abandonment = DNet)

3

Contents





◮ Why be Optimistic? Pilot at the Technion SEELab◮ Lacking but Feasible: Dynamics, Durations, Protocols◮ Simple Models at the Service of Complex Realities

◮ State-Space Collapse (Queues, Waiting Times)◮ Congestion Laws (LN, Little, ImPatience, Staffing)◮ Universal Approximations: Simplifying the Asymptotic Landscape◮ Stabilizing Time-Varying Performance (Offered-Load)

◮ Successes: Palm/Erlang-R (ED Feedback = FNet),Palm/Erlang-A (CC Abandonment = DNet)

◮ Elsewhere : Process Mining (Petri Nets, BPM), Networks (Social,Biological, Complex, . . .), Simulation-based, . . .

◮ Scenic Route : Open Problems, New Directions, Uncharted Territories

3

Applying Queueing Asymptotics

There are by now numerous insightful asymptotic queueingmodels at our disposal, and many arise from, and create, deepbeautiful theory:

Has it helped one approximate or simulate a service systemmore efficiently, estimate its parameter more accurately, teach itto our students more effectively, perhaps even manage thesystem better?

I am of the opinion that the answers to such questions havebeen too often negative, that positive answers must have theoryand applications nurture each other, which is good, and my

approach to make this good happen is by marrying theory with

data .

4

Models Approximations

QNets SimNets

Accuracy

Phenomenology

Prevalent Asymptotic Approximations

System (Data)

5


QNets SimNets FNet DNet Models

Accuracy

Phenomenology

X

Data–Based Prevalent Asymptotic Approximations Models

System (Data)

X X

6



Accuracy

Phenomenology Value

X

X


System (Data)

X X

7

System = Coin Tossing, Model = Binomial ; de Moivre 1738

Approx. / Models: SLLN (FNets), CLT (DNets) ; Laplace 1810

Value: Exceeds Value of originating stylized model

Normal, Brownian Motion ; Bachalier 1900

Poisson ; Poisson 1838



Accuracy

Phenomenology Value

X

X


System (Data)

X X

8

Models

Q F

Value

Data–Based Framework: (Almost) All Models Born Equal

SimNets

D

Data

2

Models

Q F

Value

Data–Based (Asymptotic) Framework: Simulation Mining

SimNets

D

Data

Data-Based

Real-Time

- Creation

- Application

- “Implies” other ServNets

- Virtual Realities

- Validation: Parsimony

- Theory: Reproducibility

- Application: Trust

- Data: Proprietary

3

Models

Q F

Value

Ultimately: Automatic “Discovery, Conformance, Enhancement”

SimNets

D

Data

State-Space Collapse

Multiple Scales (Regimes, UA)

Snapshots

Pathwise Little, ASTA, …

Substitution

Principle

Congestion

Laws

Service

Science

Inference: Patience, Predictors

Performance: G/G/N, QED

Control: Gc , SBR, FJ

Design: - Staffing, Pooling

Predictions

Scope of the Service Industry

Guangzhou Railway Station, Southern China

13

Call Centers, Then Hospitals, Now Internet

Call Centers - U.S. Stat.

◮ $200 – $300 billion annual expenditures

◮ 100,000 – 200,000 call centers

◮ “Window" into the company, for better or worse

◮ Over 3 million agents = 2% – 4% workforce

Healthcare - similar, plus unique challenges:

◮ Cost-figures far more staggering

◮ Risks much higher

◮ ED (initial focus) = hospital-window

◮ Over 3 million nurses

Internet - . . .

14

Call-Center Environment: Service Network

15

Operational Focus

Operational Measures:

◮ Surrogates for overall performance: Financial, Psychological; Clinical

◮ Easiest to quantify, measure, track online, react upon / Research

16

Call-Center Network: Gallery of Models

Agents

(CSRs)

Back-Office

Experts)(Consultants

VIP)Training (

Arrivals(Business Frontier

of the

21th Century)

Redial(Retrial)

Busy

)Rare(

Good

or

Bad

Positive: Repeat Business

Negative: New Complaint

Lost Calls

Abandonment

Agents

Service

Completion

Service Engineering: Multi-Disciplinary Process View

Forecasting

Statistics

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) Design

Marketing,

Human Resources,

Operations Research,

MIS

CustomersInterface Design

Computer-TelephonyIntegration - CTIMIS/CS

Marketing

Operations/

Business

Process

Archive

Database

Design

Data Mining:

MIS, Statistics,

Operations

Research,

Marketing

Internet

Chat

Email

Fax

Lost Calls

Service Completion)75% in Banks (

( Waiting Time

Return Time)

Logistics

CustomersSegmentation -CRM

Psychology,

Operations

Research,

Marketing

Expect 3 min

Willing 8 min

Perceive 15 min

Psychological

Process

Archive

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

Index

Call Center Design

(Turnover up to

200% per Year)

(Sweat Shops

of the

21th Century)

Tele-StressPsychology

17

Call-Center Network: Gallery of Models

Agents

(CSRs)

Back-Office

Experts)(Consultants

VIP)Training (

Arrivals(Business Frontier

of the

21th Century)

Redial(Retrial)

Busy

)Rare(

Good

or

Bad

Positive: Repeat Business

Negative: New Complaint

Lost Calls

Abandonment

Agents

Service

Completion

Service Engineering: Multi-Disciplinary Process View

Forecasting

Statistics


Operations,

Marketing


Parallel (Flat)

Sequential (Hierarchical)

Sociology/Psychology,

Operations Research

Human Resource Management

Service Process Design

To Avoid Delay

To Avoid Starvation Skill Based Routing

(SBR) Design

Marketing,

Human Resources,

Operations Research,

MIS

CustomersInterface Design

Computer-TelephonyIntegration - CTIMIS/CS

Marketing

Operations/

Business

Process

Archive

Database

Design

Data Mining:

MIS, Statistics,

Operations

Research,

Marketing

Internet

Chat

Email

Fax

Lost Calls

Service Completion)75% in Banks (

( Waiting Time

Return Time)

Logistics

CustomersSegmentation -CRM

Psychology,

Operations

Research,

Marketing

Expect 3 min

Willing 8 min

Perceive 15 min

Psychological

Process

Archive

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


Multi-Disciplinary

Index

Call Center Design

(Turnover up to

200% per Year)

(Sweat Shops

of the

21th Century)

Tele-StressPsychology

18

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

Mktg. ⇒

OR ⇒

HRM ⇒

MIS ⇒

19

ER / ED Environment: Service Network

Acute (Internal, Trauma) Walking

Multi-Trauma

20

ED-Environment in Israel

21

Queueing in a “Good" HospitalTong-ren Hospital at 6am, Beijing

22

Emergency-Department Network: Gallery of Models

Imaging

Laboratory

Experts

Interns

Returns (Old or New Problem)

“Lost” Patients

LWBS

Nurses

Statistics,

Human

Resource

Management

(HRM)


Operations,

Marketing,

MIS


Parallel (Flat) = ER

vs. a true ED

Sociology, Psychology,

Operations Research

Service Process

Design

Quality

Efficiency

Customers

Interface Design

Medicine

(High turnovers

Medical-Staff

shortage)

Operations/

Business

Process

Archive

Database

Design

Data Mining:

MIS, Statistics,

Operations

Research,

Marketing

Stretcher

Walking

Service Completion(sent to other department)

( Waiting Time

Active Dashboard )

Patients

Segmentation

Medicine,

Psychology,

Marketing

Psychological

Process

Archive

Human Factors

Engineering

(HFE)

Internal

Queue

Orthopedic

Queue

Arrivals

Function


Multi-Disciplinary

Index

ED-Stress

Psychology

Operations

Research, 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

Psychology,

Statistics

Home

◮ Forecasting, Abandonment = LWBS, SBR ≈ Flow Control

23

ED Patient Flow: The Physicians Viewwith J. Huang, B. Carmeli

✫✪✬✩

❄ ❄ ❄

❅❅❅❅❅❅❘

❈❈❈❈❈❈❲

✠

✒

✁✁✁✁✕

❆❆❆❆❑

❍❍❍❍

❍❍❍❍

❍❍❨✛

❄

✻✻ ✻✻

✲

λ0

1λ0

2λ0

J

· · ·

P 0(j,k)

P (k, l)

d1 d2 dJ

m0

1m0

2m0

J

Triage-Patients

IP-Patients

ExitsS

Arrivals

C1(·) C2(·) C3(·) CK(·)

m1 m2 m3 mK

· · ·

◮ Goal: Adhere to Triage-Constraints, then release In-Process Patients

◮ Model = Multi-class Q with Feedback: Min. convex congestion costs ofIP-Patients, s.t. deadline constraints on Triage-Patients.

24

ED Patient Flow: The Physicians Viewwith J. Huang, B. Carmeli

✫✪✬✩

❄ ❄ ❄

❅❅❅❅❅❅❘

❈❈❈❈❈❈❲

✠

✒

✁✁✁✁✕

❆❆❆❆❑

❍❍❍❍

❍❍❍❍

❍❍❨✛

❄

✻✻ ✻✻

✲

λ0

1λ0

2λ0

J

· · ·

P 0(j,k)

P (k, l)

d1 d2 dJ

m0

1m0

2m0

J

Triage-Patients

IP-Patients

ExitsS

Arrivals

C1(·) C2(·) C3(·) CK(·)

m1 m2 m3 mK

· · ·

◮ Goal: Adhere to Triage-Constraints, then release In-Process Patients

◮ Model = Multi-class Q with Feedback: Min. convex congestion costs ofIP-Patients, s.t. deadline constraints on Triage-Patients.

◮ Solution: In conventional heavy-traffic, asymptotic least-cost s.t. asymptoticcompliance (as in Plambeck, Harrison, Kumar, who applied admission control):

◮ Triage or IP? former, if some deadline is “too" close (least effort)◮ if Triage: Closest deadline (or Van Mieghem’s GLD)◮ if IP: Van Mieghem’s Gcµ, modified for feedback

24

Emergency-Department Network: Flow Control

Imaging

Laboratory

Experts

Interns

Returns (Old or New Problem)

“Lost” Patients

LWBS

Nurses

Statistics,

Human

Resource

Management

(HRM)


Operations,

Marketing,

MIS


Parallel (Flat) = ER

vs. a true ED

Sociology, Psychology,

Operations Research

Service Process

Design

Quality

Efficiency

Customers

Interface Design

Medicine

(High turnovers

Medical-Staff

shortage)

Operations/

Business

Process

Archive

Database

Design

Data Mining:

MIS, Statistics,

Operations

Research,

Marketing

Stretcher

Walking

Service Completion(sent to other department)

( Waiting Time

Active Dashboard )

Patients

Segmentation

Medicine,

Psychology,

Marketing

Psychological

Process

Archive

Human Factors

Engineering

(HFE)

Internal

Queue

Orthopedic

Queue

Arrivals

Function


Multi-Disciplinary

Index

ED-Stress

Psychology

Operations

Research, 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

Psychology,

Statistics

Home

◮ ∗Queueing-Science, w/ Armony, Marmor, Tseytlin, Yom-Tov

◮ ∗Fair ED-to-IW Routing (Patients vs. Staff), w/ Momcilovic, Tseytlin

◮ ∗Triage vs. InProcess/Release (Plambeck et al, van Mieghem) in EDs, w/Carmeli, Huang; Shimkin

◮ ∗Staffing Time-Varying Q’s with Re-Entrant Customers (de Vericourt &Jennings), w/ Yom-Tov

◮ The Offered-Load in Fork-Join Nets (Adlakha & Kulkarni), w/ Kaspi, Zaeid

◮ Synchronization Control of Fork-Join Nets, w/ Atar, Zviran

25

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 . . .), its

operational history = time-stamps of events (events-log files).

26

Interesting Averages: The Human Factor, or

Even “Doctors" Can Manage

Afternoon,

by Case

Morning,

by Hour

Operations Time - Morning (by Hour) vs. Afternoon (by Case):

Ethical?

Even Doctors Can Manage!

0

1

2

3

4

5

6

EEG Orthopedics Surgery Blood Surgery Plastic Surgery Heart/Chest

Surgery

Neuro-Surgery Eyes E.I. Surgery

Department

Ho

urs

AM

PM

27

Beyond Averages: The Human Factor

Histogram of Service-Time in an Israeli Call Center, 1999

January-OctoberJanuary-October

0

2

4

6

8

0 100 200 300 400 500 600 700 800 900

AVG: 185

STD: 238

0

2

4

6

8

?6.83%

November-December

0

2

4

6

8

0 100 200 300 400 500 600 700 800 900

November-December

0

2

4

6

8

0 100 200 300 400 500 600 700 800 900

AVG: 201

STD: 263

5.59%

%

Log-Normal

◮ 6.8% Short-Services:

28

Beyond Averages: The Human Factor

Histogram of Service-Time in an Israeli Call Center, 1999

January-OctoberJanuary-October

0

2

4

6

8

0 100 200 300 400 500 600 700 800 900

AVG: 185

STD: 238

0

2

4

6

8

?6.83%

November-December

0

2

4

6

8

0 100 200 300 400 500 600 700 800 900

November-December

0

2

4

6

8

0 100 200 300 400 500 600 700 800 900

AVG: 201

STD: 263

5.59%

%

Log-Normal

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

◮ Distributions must be measured (in seconds = natural scale)

◮ LogNormal service-durations (???, common, more later)

28

Pause for a Commercial:

Pause for a Commercial: The Technion SEE Center

29

Technion SEE = Service Enterprise Engineering

SEELab: Data-repositories for research and teaching

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

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

◮ U.S. Bank: 2.5 years, 220M calls, 40M by 1000 agents◮ Israeli Cellular: 2.5 years, 110M calls, 25M calls by 750 agents◮ Israeli Bank: from January 2010, daily-deposit at a SEESafe◮ Home (Rambam) Hospital: 4 years, 1000 beds, ward-level flow◮ 5 EDs: gathered by the late David Sinreich, ED arrivals & LOS

30






SEEStat: Environment for graphical EDA in real-time

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

30






SEEStat: Environment for graphical EDA in real-time

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

SEEServer: Free for academic use

◮ Register◮ Access U.S. Bank, Bank Anonymous, Home Hospital

30

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

Drill: Chemical MCE, Rambam Hospital, May 2010

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

31

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 PMfirst 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

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

- “Psychology" of Data Trust and Transfer (e.g. 2 years till transfer)32

Event-Logs in a Call Center (Bank Anonymous)

A Data Sample (Excel worksheet) vru+line call_id customer_id priority type date vru_entry vru_exit vru_time q_start q_exit q_time outcome ser_start ser_exit ser_time server

AA0101 44749 27644400 2 PS 990901 11:45:33 11:45:39 6 11:45:39 11:46:58 79 AGENT 11:46:57 11:51:00 243 DORIT

AA0101 44750 12887816 1 PS 990905 14:49:00 14:49:06 6 14:49:06 14:53:00 234 AGENT 14:52:59 14:54:29 90 ROTH

AA0101 44967 58660291 2 PS 990905 14:58:42 14:58:48 6 14:58:48 15:02:31 223 AGENT 15:02:31 15:04:10 99 ROTH

AA0101 44968 0 0 NW 990905 15:10:17 15:10:26 9 15:10:26 15:13:19 173 HANG 00:00:00 00:00:00 0 NO_SERVER

AA0101 44969 63193346 2 PS 990905 15:22:07 15:22:13 6 15:22:13 15:23:21 68 AGENT 15:23:20 15:25:25 125 STEREN

AA0101 44970 0 0 NW 990905 15:31:33 15:31:47 14 00:00:00 00:00:00 0 AGENT 15:31:45 15:34:16 151 STEREN

AA0101 44971 41630443 2 PS 990905 15:37:29 15:37:34 5 15:37:34 15:38:20 46 AGENT 15:38:18 15:40:56 158 TOVA

AA0101 44972 64185333 2 PS 990905 15:44:32 15:44:37 5 15:44:37 15:47:57 200 AGENT 15:47:56 15:49:02 66 TOVA

AA0101 44973 3.06E+08 1 PS 990905 15:53:05 15:53:11 6 15:53:11 15:56:39 208 AGENT 15:56:38 15:56:47 9 MORIAH

AA0101 44974 74780917 2 NE 990905 15:59:34 15:59:40 6 15:59:40 16:02:33 173 AGENT 16:02:33 16:26:04 1411 ELI

AA0101 44975 55920755 2 PS 990905 16:07:46 16:07:51 5 16:07:51 16:08:01 10 HANG 00:00:00 00:00:00 0 NO_SERVER

AA0101 44976 0 0 NW 990905 16:11:38 16:11:48 10 16:11:48 16:11:50 2 HANG 00:00:00 00:00:00 0 NO_SERVER

AA0101 44977 33689787 2 PS 990905 16:14:27 16:14:33 6 16:14:33 16:14:54 21 HANG 00:00:00 00:00:00 0 NO_SERVER

AA0101 44978 23817067 2 PS 990905 16:19:11 16:19:17 6 16:19:17 16:19:39 22 AGENT 16:19:38 16:21:57 139 TOVA

AA0101 44764 0 0 PS 990901 15:03:26 15:03:36 10 00:00:00 00:00:00 0 AGENT 15:03:35 15:06:36 181 ZOHARI

AA0101 44765 25219700 2 PS 990901 15:14:46 15:14:51 5 15:14:51 15:15:10 19 AGENT 15:15:09 15:17:00 111 SHARON

AA0101 44766 0 0 PS 990901 15:25:48 15:26:00 12 00:00:00 00:00:00 0 AGENT 15:25:59 15:28:15 136 ANAT

AA0101 44767 58859752 2 PS 990901 15:34:57 15:35:03 6 15:35:03 15:35:14 11 AGENT 15:35:13 15:35:15 2 MORIAH

AA0101 44768 0 0 PS 990901 15:46:30 15:46:39 9 00:00:00 00:00:00 0 AGENT 15:46:38 15:51:51 313 ANAT

AA0101 44769 78191137 2 PS 990901 15:56:03 15:56:09 6 15:56:09 15:56:28 19 AGENT 15:56:28 15:59:02 154 MORIAH

AA0101 44770 0 0 PS 990901 16:14:31 16:14:46 15 00:00:00 00:00:00 0 AGENT 16:14:44 16:16:02 78 BENSION

AA0101 44771 0 0 PS 990901 16:38:59 16:39:12 13 00:00:00 00:00:00 0 AGENT 16:39:11 16:43:35 264 VICKY

AA0101 44772 0 0 PS 990901 16:51:40 16:51:50 10 00:00:00 00:00:00 0 AGENT 16:51:49 16:53:52 123 ANAT

AA0101 44773 0 0 PS 990901 17:02:19 17:02:28 9 00:00:00 00:00:00 0 AGENT 17:02:28 17:07:42 314 VICKY

AA0101 44774 32387482 1 PS 990901 17:18:18 17:18:24 6 17:18:24 17:19:01 37 AGENT 17:19:00 17:19:35 35 VICKY

AA0101 44775 0 0 PS 990901 17:38:53 17:39:05 12 00:00:00 00:00:00 0 AGENT 17:39:04 17:40:43 99 TOVA

AA0101 44776 0 0 PS 990901 17:52:59 17:53:09 10 00:00:00 00:00:00 0 AGENT 17:53:08 17:53:09 1 NO_SERVER

- Unsynchronized transition times, consistently

33

33975

4221

2006

2886

2741

361

354697

1111

43

1049

994

319

162

2281

3908

160

504

266

193

122

158

118

7

1

32246

35916

1259

2006

2823

3798

876 352

370

107 491

545

566

127

48

48

2052

47

59

33

307

19

17

390

547

14

3

192

221

20

12

60

282

174

355

1405

169 91

23 1

1022

73

35

112

27

3

119

4

34

243

2935

3908

7

23

17

1

44

71

2

516

267229

266

11

78

8 132

132

172

5

11

41

67

7

17

8

8

118

45

27

13

22

8

135

150

24

1

8

3

3

37

24

16

19

26

67

32

10

3

4

2516

1 11

11

11

8

35

6

7

11

6

4

13

6

3

1

6

13

1

11

2

18

1

7

1

9

33

2 7

2

4

2

6

6

4

3

4

1

1

11

6

1

3

8

1

2

1

4

2

2

1

1

1

1

1

1

1

1

1

2

2

3

1

1

1

1

1

1

1

34

3

1

1

26102

735

4489

1704

2232

2240

53

782 95

10

604

393

24134

105

32

2194

82

1369

1362

33

71

501

22

57

205

757

545

3

208

4

1497

2561

34

254

176

25

82

115

153

2843

3071 560

327

14

1

17

1058

26

18932

15

28

1215

34

46

31 94

2

58

15

49

39

9

94 36

52

5916

70

5

1

8

27

8

4

4

2

10

9

6

1

71

3

5

86

7

5

4

1

1

1

4

21

1

1

1

1

3

1

2

3

1

1

Entry

VRURetail

VRUPremier

VRUBusiness

VRUConsumer Loans

VRUSubanco

VRUSummit

Business LineRetail

Business LinePremier

Business LineBusiness

Business LinePlatinum

Business LineConsumer Loans

Business LineOnline Banking

Business LineEBO

Business LineTelesales

Business LineSubanco

Business LineSummit

AnnouncementRetail

AnnouncementPremier

AnnouncementPlatinum

AnnouncementConsumer Loans

AnnouncementOnline Banking

AnnouncementTelesales

AnnouncementSubanco

MessageRetail

MessagePremier

MessageBusiness

MessagePlatinum

MessageConsumer Loans

MessageTelesales

NonBusiness LineBusiness

NonBusiness LineSubanco

NonBusiness LinePriority Service

NonBusiness Line NonCC ServiceCCO

NonCC ServiceQuick&Reilly

Overnight ClosedRetail

Overnight ClosedConsumer Loans

Overnight ClosedEBO

TrunkRetail

TrunkPremier

TrunkBusiness

TrunkConsumer Loans

TrunkOnline Banking

TrunkEBO

TrunkTelesales

TrunkPriority Service

Trunk

Trunk

Trunk

Trunk

Trunk

InternalTotal

OutgoingTotal

NormalTermination

NormalTermination

UndeterminedTermination

AbandonedShort

Abandoned

OtherUnhandled

Transfer

Transfer

1 Day in a Call Center

- Customers

USBank

April 2, 2001

33975

4221

2006

2886

2741

361

354697

1111

43

1049

994

319

162

2281

3908

160

504

266

193

122

158

118

7

1

32246

35916

1259

2006

2823

3798

876 352

370

107 491

545

566

127

48

48

2052

47

59

33

307

19

17

390

547

14

3

192

221

20

12

60

282

174

355

1405

169 91

23 1

1022

73

35

112

27

3

119

4

34

243

2935

3908

7

23

17

1

44

71

2

516

267229

266

11

78

8 132

132

172

5

11

41

67

7

17

8

8

118

45

27

13

22

8

135

150

24

1

8

3

3

37

24

16

19

26

67

32

10

3

4

2516

1 11

11

11

8

35

6

7

11

6

4

13

6

3

1

6

13

1

11

2

18

1

7

1

9

33

2 7

2

4

2

6

6

4

3

4

1

1

11

6

1

3

8

1

2

1

4

2

2

1

1

1

1

1

1

1

1

1

2

2

3

1

1

1

1

1

1

1

34

3

1

1

26102

735

4489

1704

2232

2240

53

782 95

10

604

393

24134

105

32

2194

82

1369

1362

33

71

501

22

57

205

757

545

3

208

4

1497

2561

34

254

176

25

82

115

153

2843

3071 560

327

14

1

17

1058

26

18932

15

28

1215

34

46

31 94

2

58

15

49

39

9

94 36

52

5916

70

5

1

8

27

8

4

4

2

10

9

6

1

71

3

5

86

7

5

4

1

1

1

4

21

1

1

1

1

3

1

2

3

1

1

Entry

VRURetail

VRUPremier

VRUBusiness

VRUConsumer Loans

VRUSubanco

VRUSummit

Business LineRetail

Business LinePremier

Business LineBusiness

Business LinePlatinum

Business LineConsumer Loans

Business LineOnline Banking

Business LineEBO

Business LineTelesales

Business LineSubanco

Business LineSummit

AnnouncementRetail

AnnouncementPremier

AnnouncementPlatinum

AnnouncementConsumer Loans

AnnouncementOnline Banking

AnnouncementTelesales

AnnouncementSubanco

MessageRetail

MessagePremier

MessageBusiness

MessagePlatinum

MessageConsumer Loans

MessageTelesales

NonBusiness LineBusiness

NonBusiness LineSubanco

NonBusiness LinePriority Service

NonBusiness Line NonCC ServiceCCO

NonCC ServiceQuick&Reilly

Overnight ClosedRetail

Overnight ClosedConsumer Loans

Overnight ClosedEBO

TrunkRetail

TrunkPremier

TrunkBusiness

TrunkConsumer Loans

TrunkOnline Banking

TrunkEBO

TrunkTelesales

TrunkPriority Service

Trunk

Trunk

Trunk

Trunk

Trunk

InternalTotal

OutgoingTotal

NormalTermination

NormalTermination

UndeterminedTermination

AbandonedShort

Abandoned

OtherUnhandled

Transfer

Transfer

1 Day in a Call Center

- Customers

USBank

April 2, 2001

ILDUBank

January 24, 2010

from 06:00 to 23:59:59

Call-Backs Outgoing

Inbound

Available

Idle

Break

ILDUBank

January 24, 2010

from 06:00 to 23:59:59

Call-Backs Outgoing

Inbound

Available

Idle

Break

ED

Pediatric

Neurology Nephrology Derrmatology

Internal

IntensiveCare

Cardiology

Rheumatology

UrologyOtorhinolaryngology Orthopedics

Surgery

Maternity

Gynecology

Psychiatry Ophthalmology

R

L

D

Out

R

R

R

R

R R

R

RR RR

R

R

R

R

R

L

LL

L

L

L

L

L

L

D

D

D

D

D DD

D

DD

D

D

Out

Out

Out

Out Out

Out

OutOut Out OutOut

Out

Out

1 Day in a Hospital

- Patients

Rambam

August, 2004

Released (Home)

Left with Medical File

Deceased

Transfer

115

17

5

43

2

1

1

1

8

6

2

1

1

2

23

3

15

1

7

1

11

1

2

1

6

1

1

1 13

1

1

1

1

1

1

1 3

1

1

1

1

1

1 1

1 1

1

1

11

1 1

1 11

1

11 11 1

1

3

1

1

1

1

1

2

1

3

1 1

2

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2

1

1

1

1

1

1

1

1

1

1

1

11

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

11

1 1

1

1

1

1

1

1

1

1

1

1

2

11

1

1

1

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2

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6

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3

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61

1

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1

1

1

1

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1

1

1

1

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1 1

1

1

1

1

1

1 11

1

11 11 1

1

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1

1

1

1

1

2

23

1 1

1

1

1

1

1

1

11

1

1

1

1

1

1 11

1 1

2 2

1

2

1

1 1 1 11

2

1

1

serveng

lectures

references

predictionmay1809.pdf

homeworks

hw1_amusement_park_mgt.pdf

recitations

hazard_phase_type.pdf

course2011winter

icpr_service_engineering.pdfon_cc_data_frommsom.pdf pivot_table.pdf

thesis_polyna.pdf

proposal_polina.pdf

moss.pdf

files

nurse_proj_psu_tech.pdf

national_cranberry_1.pdf

hw9par_sol_2012w.pdf

syllabus8.pdfexams

statanalysis.pdf

agent-heterogeneity-brown-book-final.pdf

1_1_qed_lecture_introduction_2011w.pdf

course2004

retail.pdf hall_6.pdf

jasa_callcenter.pdf dantzig.pdf

bacceli.pdf

avishai-class-april-2003.ppt

hw7_2011w_par_sol_part2.pdf

ibmrambam technion srii presentation final.ppt

dimensioning.pdf

palmannoyance.pdf

yair_sergurywaitingtimeanalysis.pdf

nejmsb1002320.pdf

qed_qs_scientific_generic_caschina.pdf

web_summary13_2011s.pdf

4callcenters_coverpage.pdf

documentation.pdf rec9part2.pdf

web_summary9_2010w.pdf

qed_qs_scientific_wharton_stat_seminar.pdf

l2_measurements_class_2009s.pdf

ccreview.pdfvdesign_ecompanion.pdf

wharton_lecture_avi_printout.pdflarson_atoa.pdf

proposal_michael.pdf

stanford_seminar_avi_printout.pdf

solver_guide.pdf

fitz.pdf

rec10_part2.pdf rec5.pdf

paxson.pdf

connect_to_see_terminal.pdf

project_ug_simulationinterface.pdf revenue_manage.pdf

hw7_2009s_forecast.xls

syllabus2.pdf

infocom04.pdf

hall_transportation.pdf

whitt_handout_chapter1.pdf

fromprojecttoprocess.pdf

nytimes.2007-06-23.pdf ds_pert_lec1.pdf syllabus6.pdf hw9_2011w.pdf hw7_2011w.pdf

mmng_constraint_supp_or.pdf

awam-april_2011.pdfwhitt_scaling_slides.pdfmmng_seminar.pdf

syllabus4.pdf

syllabus12_2012w.pdf

syllabus5.pdf

mm1_strong_apprximations.pdf seestat_tutorial.ppt

kaplan_porter_2011-9_how-to-solve-the-cost-crisis-in-health-care_hbr.pdf

rec9_part2.pdf

callcenterdata

bocaraton.ps

gccreport 20-04-07 uk version.pdf

hall_manpower_planning.pdf

bi18.pdf

syllabus12.pdf fluidview_2010w_short_version.pdf hw7_2009s_par_sol_part2.pdf

moed_b_2007w_solution.pdf

newell.pdf

rec11.pdf

moedb_2009w.pdf

rec3.pdf

hw10_par_sol_2011s.pdf

thirdlevel support ijtm1_cs.pdf

hw4_full.pdf

moedb_2010s_sol.pdf

chandra.pdf

gurvich_seminar.pdf

ed_sinreich_marmor.ppt regimes_supplement.pdf

web_summary10_2011s.pdf hw6par_sol_2009s.xls

exam_2005s_sol.pdf

servicefull.pdf

witor09_empirical_adventures_sep2009.pdf

web_summary12_2011s.pdf

heb_summary_yulia.pdf

hw9par_sol_2011s.pdf

seminar_yulia_9_2.pdf

bpr_2003.pdf staffing_italian_us_2011w.xlsdesignofqueues.pdf thepsychologyofwaitinglines.pdf

erlangabc.pdf

studentsevaluations.pdf hw8_2011s.pdf

hw9_2012w.pdf

edie.pdf

syllabus9.pdf

ds_pert_lec2.pdf

zohar_seminar.ppt

hist-normal.pdf

vandergraft.pdf

rec13.pdf

desighn_ivr.pdf

cohen_aircraft_failures.pdf

sbr.pdf

nano.pdf

hw2_2011w.pdfseminar_presentation_boaz_estimating

er load.ppt

project_ug_meirav.pdf see_report_output_june_2011.pdf

see_report_2010.pdf

1 Day in a Website

- Spiders (e.g. Googlebot)

ServEng

May 19, 2012

470

228

5

101

8 4 3

30

53

44 35

6 33

7334 3

4

4

321

3

314 93

3

4 43

19

4

3 3

5

48 34

64

3 3 4

14

7 3

5 4

3

6

7

335 3 33 3

58 4 3

53

44 35 7334 3

4

3

3

5

14 93

3

4 43

4

3

5

48 34

64

3 3 45 4

3

6 335 3 33 3

serveng

lectures

hall_flows_hospitals_chapter1text.pdf

references

4callcenterscourse2004 homeworks

icpr_service_engineering.pdf keycorp.pdf

files

revenue_manage.pdfhw1_amusement_park_mgt.pdf

recitations

setup.exe

garnett.pdf abandon02.pdf erlang_a.pdf ccbib.pdf

us7_cc_avi.pdf

jasa_callcenter.pdfsbr.pdf


mazeget_noa_yul.pptexams technion-call-center-report-sept-2004.pdfbrandt.pdf statanalysis.pdfmjp_into_erlang_a.pdf loch.pdf sbr_stolyar.pdf

callcenterdata

larson_atoa.pdf sivanthesisfinal.pdf columbia05.pdf crmis.pptfluid_systems.rar

januarytxt.zip

sbr_wharton_ccforum_short_may03.pdf predictingwaitingtime.pdf avishaicourse_munichor.ppt vdesign_pub.pdf hierarchical_modelling_chen_mandelbaum_part1.pdf

solver_guide.pdf

datamocca_february_2008_cc_forum_lecture.ppt seminar_presentation_galit.pptntro_to_serveng_teaching_note.pdf

pivot_table.pdf

project_ug_simulationinterface.pdf

course2010winter

ed_simulation_modeling_supp.pdf avim_cv.pdfdantzig.pdf yariv_phd.pdf

rec10_part2.pdf

fr6.pdfnejmsb1002320.pdf

hebrew_syllabus_2011s.pdf

patientflow main.pdf

hw9_2012w.pdf

avim_cv_18_12_2011.pdf

1 Day in a Website

- Clickstream Structure

- Menus, Files

ServEng

January 5, 2012

470

228

5

101

8 4 3

30

53

44 35

6 33

7334 3

4

4

321

3

314 93

3

4 43

19

4

3 3

5

48 34

64

3 3 4

14

7 3

5 4

3

6

7

335 3 33 3

58 4 3

53

44 35 7334 3

4

3

3

5

14 93

3

4 43

4

3

5

48 34

64

3 3 45 4

3

6 335 3 33 3

serveng

lectures

hall_flows_hospitals_chapter1text.pdf

references

4callcenterscourse2004 homeworks

icpr_service_engineering.pdf keycorp.pdf

files

revenue_manage.pdfhw1_amusement_park_mgt.pdf

recitations

setup.exe

garnett.pdf abandon02.pdf erlang_a.pdf ccbib.pdf

us7_cc_avi.pdf

jasa_callcenter.pdfsbr.pdf


mazeget_noa_yul.pptexams technion-call-center-report-sept-2004.pdfbrandt.pdf statanalysis.pdfmjp_into_erlang_a.pdf loch.pdf sbr_stolyar.pdf

callcenterdata

larson_atoa.pdf sivanthesisfinal.pdf columbia05.pdf crmis.pptfluid_systems.rar

januarytxt.zip

sbr_wharton_ccforum_short_may03.pdf predictingwaitingtime.pdf avishaicourse_munichor.ppt vdesign_pub.pdf hierarchical_modelling_chen_mandelbaum_part1.pdf

solver_guide.pdf

datamocca_february_2008_cc_forum_lecture.ppt seminar_presentation_galit.pptntro_to_serveng_teaching_note.pdf

pivot_table.pdf

project_ug_simulationinterface.pdf

course2010winter

ed_simulation_modeling_supp.pdf avim_cv.pdfdantzig.pdf yariv_phd.pdf

rec10_part2.pdf

fr6.pdfnejmsb1002320.pdf

hebrew_syllabus_2011s.pdf

patientflow main.pdf

hw9_2012w.pdf

avim_cv_18_12_2011.pdf

1 Day in a Website

- Clickstream Structure

- Menus, Files

ServEng

January 5, 2012

Menus:

Files:

8 AM - 9 AM

55888

1113 43320 160505194 7

36512 23633845 3792

845 103

120

314

3314

155

43

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4

34

594 140

547

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3

Entry

Entry Entry Entry Entry EntryEntry Entry

VRU

Retail Premier Business PlatinumConsumer Loans EBOTelesales Subanco Summit

VRU

Retail PremierBusiness PlatinumConsumer LoansEBO Telesales SummitOut

VRU

Retail

C

A

C

CC C

C

C

C

C

C

C

C

C C

C

C

C

C

C

C

AA

A

A AA A A

A

A

OutOut Out Out OutOut OutOut Out

Out OutOut

1 Hour in a Call Center

- Customers

- Hierarchical

USBank

April 2, 2001

Abandoned Continued

(Branch,

Another ID)Completed

Zoom Out: Call Center Network Inter-Queues

USBank April 2, 2001

Philadelphia NYC

Boston

C C

AA

C

A

USBank April 2, 2001

Zoom Out: Call Center Network Inter-Queues

Daily

(Deposit)

Order

Checkbooks

Identification

Change

Password

Agent

Fax

Query Transfer

Securities

Stock

Market

Perform

Operation

Credit

Error

43123

234

270

1352

35315

3900

7895

830

68

65

273

1063

231

768

133

1032

897

40

49

54

21

71

30

165

53

114

29

16

35

10

9

12

107

6

5

21

8

26091

4115

12317

962

1039

147

42

18

28

114

14

72

20

Zoom In: Interactive Voice Response (IVR)

ILBank

May 2, 2008

ILDUBank

January 24, 2010

Agent #043

Shift: 16:00 - 23:45

Inbound Call-Backs

Outgoing

Available Idle

Private Call

Break

Private Prepaid

PrivatePrivate VIP Private Customer RetentionBusiness Business VIP Business Customer Retention

Language 2 Prepaid Language 2 Prepaid Low PriorityLanguage 2 PrivateLanguage 3 Internet Surfing Internet

Overseas

1 24 5 810 11 12 1315161718 20 21 24 25

2104

2208 26264592 445 3411432049 1484 8734785 4658 411 444480 43 847 181 974

1253823 843343 1390109630 4454 1240792 3793 49 735

1433 193

Zoom In: Skills - Based Routing (SBR)

Network Structure (Protocol)

Israeli Telecom

February 10, 2008

N -Design

L -Design

Agent Pool

Customer Class

Connected Component

1 24 5 810 11 12 1315161718 20 21 24 25

2104

2208 26264592 445 3411432049 1484 8734785 4658 411 444480 43 847 181 974

1253823 843343 1390109630 4454 1240792 3793 49 735

1433 193

Goal: Data-Based Real-Time Simulation (SimNet)

Private Prepaid Language 3 Language 2 Private Language 2 Prepaid Language 2 Prepaid Low Priority Internet Surfing Internet

Private VIP Private Business Business VIP Business Customer Retention Private Customer Retention Overseas

What is lacking?

1. Dynamics

2. Durations

3. Protocols

Israeli Telecom

February 10, 2008

8 AM - 9 AM

USBank

April 2, 2001

8 AM - 9 AM

9 AM - 10 AM

10 AM - 11 AM

11 AM - 12 PM

Dynamics: Time-Varying Arrival-Rates

2 Daily Peaks

CC: Dec. 1995, (USA, 700 Helpdesks) CC: May 1959 (England)

Dec 1995!

(Help Desk Institute)

Arrival

Time

Time

24 hrs

% Arrivals

Q-Science

May 1959!

Arrival

Rate

Time

24 hrs

CC: Nov. 1999 (Israel) ED: Jan.–July 2007 (Israel)

Daily

0.00

2.50

5.00

7.50

10.00

12.50

15.00

17.50

20.00

22.50

25.00

27.50

30.00

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

Avera

ge n

um

be

r of

cases

Time (Resolution 60 min.)

HomeHospital Patients Arrivals to Emergency DepartmentTotal for January2007 February2007 March2007 April2007 May2007 June2007 July2007,Sundays

54

Durations: Phone Calls (2 Surprises)

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-normal

Serv

ice t

ime

0 1000 2000 3000

01000

2000

3000

◮ Practically Important: (mean, std)(log) characterization

◮ Theoretically Intriguing: Why LogNormal ? Naturally multiplicative

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

56

Durations: Answering Machine

Israeli Bank: IVR/VRU Only, May 2008

IVR_only

May 2008, Week days

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

1.40

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.)

Rela

tive f

req

uen

cie

s %

mean=99st.dev.=101

Tree-Network Tomography:Reconstruct topologyfrom root and leaves

Mixture: 7 LogNormals

0.00

0.10

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0.40

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

Rela

tiv

e f

req

uen

cie

s %

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

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

Time(mm:ss) (Resolution 1 sec.)Empirical Total Lognormal Lognormal Lognormal

57

Durations: Waiting Times in a Call Center

⇒ Protocols

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

0.00

2.50

5.00

7.50

10.00

12.50

15.00

17.50

20.00

22.50

25.00

27.50

30.00

0.00 100.00 200.00 300.00 400.00 500.00 600.00

Rela

tive f

requencie

s %

Time(seconds)( resolution 30)

AnonymousBank Handled Wait Time, TOTALTotal for January1999 February1999 March1999 April1999 May1999 June1999 July1999 August1999 September1999 October1999

November1999 December1999,Weekdays

Empirical Exponential (scale=106.67)

Mean = 106 SD = 109

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

0.00

2.50

5.00

7.50

10.00

12.50

15.00

17.50

20.00

2.00 7.00 12.00 17.00 22.00 27.00 32.00 37.00

Rela

tive f

requencie

s %


USBank Wait time(waiting), RetailMay 2002, Weekdays

Scheduling Priorities (sec.) [compare Hospital LOS (hours)]Medium Israeli Bank

0.000

0.050

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0.150

0.200

0.250

0.300

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0.400

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0.550

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20.00 70.00 120.00 170.00 220.00 270.00 320.00 370.00

Rela

tive f

requencie

s %


ILBank Wait time (all)August 2007, Weekdays

58

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

59



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

59



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 ?◮ Hourly Scale: Staffing,. . .

◮ Daily: Flow / Bed Control,. . .

Arrivals, Departures, # Patients in Ward A, by Hour

59

Protocols + Psychology

Patient Customers, Announcements, Priority Upgrades

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0 60 120 180 240 300 360 420 480 540 600 660 720 780 840 900

Hazard

Function

Time (seconds)

USBank December 2002, Week days, Quick&Reilly

time willing to wait (hazard rate)

time willing to wait (Pspline)

virtual wait (hazard rate)

virtual wait (Pspline)

61

Dynamics: Parsimonious Models (Congestion Laws)

3 Queue-Lengths at 30 sec. resolution (ILBank, 10/6/2007)

ILBank Customers in queue (average)

10.06.2007

0.00

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225.00

06:00 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

Time (Resolution 30 sec.)

Num

ber

of

cases

Medium priority Low priority Unidentified

Queue “Shape"

ILBank Customers in queue (average)

10.06.2007

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

06:00 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

Time (Resolution 30 sec.)

Perc

enta

ge

Medium priority Low priority Unidentified

◮ Area normalized to 100%

◮ State-Space Collapse

55

Little’s Law: Call Center & Emergency Department

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

USBank Customers in queue(average), Telesales

10.10.2001

0.00

20.00

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120.00

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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


Nu

mb

er

of ca

se

s

Customers in queue(average) Little's law

HomeHospital Average patients in ED

February 2004, Wednesdays

0.00

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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

rag

e n

um

be

r o

f ca

se

s

Average patients in ED Lambda*E(S) smoothed

62

Little’s Law: Call Center & Emergency Department

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

USBank Customers in queue(average), Telesales

10.10.2001

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


Nu

mb

er

of ca

se

s

Customers in queue(average) Little's law

HomeHospital Average patients in ED

February 2004, Wednesdays

0.00

10.00

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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

rag

e n

um

be

r o

f ca

se

s

Average patients in ED Lambda*E(S) smoothed

⇒ Time-Varying Little’s Law

◮ Berstemas & Mourtzinou;

◮ Fralix, Riano, Serfozo; . . .

62

Prerequisite II: Models (FNets)

“Laws of Large Numbers" capture Predictable Variability

Deterministic Models: Scale Averages-out Stochastic Individualism

70

The Basic Service-Network Model: Erlang-R

w/ G. Yom-Tov Needy

(st-servers)

rate-

Content

(Delay)

rate -

1-p

p

1

2

Arrivals

Poiss( t)Patient discharge

Erlang-R (IE: Repairman Problem 50’s; CS: Central-Server 60’s) =

2-station “Jackson" Network = (M/M/S, M/M/∞) :

◮ λt – Time-Varying Arrival rate

◮ St – Number of Servers (Nurses / Physicians)

◮ µ – Service rate (E [Service] = 1µ

)

◮ p – ReEntrant (Feedback) fraction

◮ δ – Content-to-Needy rate (E [Content] = 1δ)

71

Fluid Model ↔ (Time-Varying) Erlang-R System

Needy

(st-servers)

rate-

Content

(Delay)

rate -

1-p

p

1

2

Arrivals

Poiss( t)Patient discharge

FNet of a 2-station “Jackson" Network:

d

dtq1

t = λt − µ ·(

q1t ∧ st

)

+ δ · q2t ,

d

dtq2

t = p · µ ·(

q1t ∧ st

)

− δ · q2t .

(1)

72

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

Chemical MCE Drill (Israel, May 2010)

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

0

10

20

30

40

50

60

11:02 11:16 11:31 11:45 12:00 12:14 12:28 12:43 12:57 13:12 13:26

To

ta

l N

um

be

r o

f P

atie

nts

Time

Cumulative Arrivals

Cumulative Departures

15

20

25

30

erofMCEPatientsinED

Actual Q(t)

Fluid Q(t)

Lower Envelope Q(t) (Theoretical)

Upper Envelope Q(t) (Theoretical)

Fluid Q1

0

5

10

11:02 11:16 11:31 11:45 12:00 12:14 12:28 12:43 12:57 13:12 13:26

Numbe

Time

◮ Recurrent/Repeated services in MCE Events: eg. Injection every 15 minutes

73

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

Chemical MCE Drill (Israel, May 2010)

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

0

10

20

30

40

50

60

11:02 11:16 11:31 11:45 12:00 12:14 12:28 12:43 12:57 13:12 13:26

To

ta

l N

um

be

r o

f P

atie

nts

Time

Cumulative Arrivals

Cumulative Departures

15

20

25

30

erofMCEPatientsinED

Actual Q(t)

Fluid Q(t)

Lower Envelope Q(t) (Theoretical)

Upper Envelope Q(t) (Theoretical)

Fluid Q1

0

5

10

11:02 11:16 11:31 11:45 12:00 12:14 12:28 12:43 12:57 13:12 13:26

Numbe

Time

◮ Recurrent/Repeated services in MCE Events: eg. Injection every 15 minutes

◮ Fluid (Sample-path) Modeling, via Functional Strong Laws of Large Numbers

◮ Stochastic Modeling, via Functional Central Limit Theorems

◮ ED in MCE: Confidence-interval, usefully narrow for Control◮ ED in normal (time-varying) conditions: Personnel Staffing73

An Asymptotic Framework: Erlang-R in the ED

System = Emergency Department (eg. Rambam Hospital)

◮ SimNet = Customized ED-Simulator (Marmor & Sinreich)

◮ QNet = Erlang-R (time-varying 2-station Jackson; w/ Yom-Tov)

◮ FNets = 2-dim dynamical system (Massey & Whitt)

◮ DNets = 2-dim Markovian Service Net (w/ Massey and Reiman)

74

An Asymptotic Framework: Erlang-R in the ED

System = Emergency Department (eg. Rambam Hospital)

◮ SimNet = Customized ED-Simulator (Marmor & Sinreich)

◮ QNet = Erlang-R (time-varying 2-station Jackson; w/ Yom-Tov)

◮ FNets = 2-dim dynamical system (Massey & Whitt)

◮ DNets = 2-dim Markovian Service Net (w/ Massey and Reiman)

Asymptotic Framework

◮ Data and Measurements

◮ Fit a simple model (time-varying Erlang-R) to a complex reality(ED Physicians)

◮ Develop FNets (Offered-Load of Physicians) and (relevant) DNet(if needed)

◮ Use FNet / DNet for Design (√

-Staffing), Analysis, . . .

◮ Simulate reality (ED with√

-staffing of Physicians)

◮ Validation: stable performance, confidence intervals, . . .

74

Case Study: Emergency Ward Staffing

Many-Server (↑ ∞) Approximations for Small Systems (1-7)

◮ Staffing resolution: 1 hour◮ Lower bound: 1 doctor per type◮ Flexible (time-varying square-root) staffing: Yunan’s Lecture◮ Rounding effects ⇒ Not all performance levels achievable

90 Servers 1-7 Doctors

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P(W

>0)

0

0.1

0.2

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100105110115

Time [Hour]

beta=0.1 beta=0.3 beta=0.5 beta=0.7 beta=1 beta=1.5

0.3

0.4

0.5

0.6

0.7

0.8

0.9

P(W

>0)

0

0.1

0.2

0 1000 2000 3000 4000 5000 6000 7000

Time

Beta=0.1 Beta=0.5 Beta=1 Beta=1.5

75

Protocols: Staffing (N) vs. Offered-Load (R = λ× E(S))

IL Telecom; June-September, 2004; w/ Nardi, Plonski, Zeltyn

2205 half-hour intervals (13 summer weeks, week-days)63

Number of patients in ED

0

250

500

750

1000

1250

1500

1750

2000

2250

2500

0 10 20 30 40 50 60 70 80 90 100

Fre

que

ncie

s

Number of patients (resolution 1)

HomeHospital Time by ED Internal and Surgery state (sec.)January 2004-October 2007, all days

0

20

40

60

80

100

120

140

160

180

200

220

240

260

0 10 20 30 40 50 60 70 80 90 100

Fre

quencie

s



[00:00:00 - 01:00:00)

[01:00:00 - 02:00:00)

[02:00:00 - 03:00:00)

[03:00:00 - 04:00:00)

[04:00:00 - 05:00:00)

[05:00:00 - 06:00:00)

[06:00:00 - 07:00:00)

[07:00:00 - 08:00:00)

[08:00:00 - 09:00:00)

[09:00:00 - 10:00:00)

[10:00:00 - 11:00:00)

[11:00:00 - 12:00:00)

[12:00:00 - 13:00:00)

[13:00:00 - 14:00:00)

[14:00:00 - 15:00:00)

[15:00:00 - 16:00:00)

[16:00:00 - 17:00:00)

[17:00:00 - 18:00:00)

[18:00:00 - 19:00:00)

[19:00:00 - 20:00:00)

[20:00:00 - 21:00:00)

[21:00:00 - 22:00:00)

[22:00:00 - 23:00:00)

[23:00:00 - 24:00:00)

15

20

25

30

35

40

45

50

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

Avera

ge n

um

ber

of c

ases


HomeHospital Number of Patients in Emergency Department Internal and Surgery,January 2004-October 2007, all days

0

250

500

750

1000

1250

1500

1750

2000

2250

2500

0 10 20 30 40 50 60 70 80 90 100

Fre

que

ncie

s



Fitting Mixtures of Distributions

Normal (21.45%): location = 17.59 scale = 5.39Gamma (40.19%): scale = 7 shape = 4.55Weibull (38.36%): scale = 49.28 shape = 3.61

MorningNormal

EveningWeibull

Intermediate hoursGamma

Time by ED Internal and Surgery state (sec.) Statistics

N 120960000

N(average per day) 86400

Mean 34.1

Standard Deviation 16.34

Variance 266.87

Median 32

Minimum 0

Maximum 99

Skewness 0.524

Kurtosis -0.44452

Standard Error Mean 0.00149

Interquartile Range 25

Mean Absolute Deviation 13.7

Median Absolute Deviation(MAD) 12

Coefficient of Variation (CV) (%) 47.9

L-moment 2 (half of Gini's Mean Difference) 9.25

L-Skewness 0.121

L-Kurtosis 0.0561

Coefficient of L-variation (L-CV)(%) (Gini's Coefficient) 27.12

Parameter Estimates

Components Mixing Proportions (%) Location Scale Shape Mean Standard Deviation

1. Normal 21.45 17.59 5.39 17.59 5.39

2. Gamma 40.19 7.00 4.55 31.83 14.99

3. Weibull 38.36 49.28 3.61 44.42 13.679

Goodness-of-Fit Tests

Tests Statistic DF p Value

Residuals Std 0.011

Kolmogorov-Smirnov 0.028 <.0001

Cramer-von Mises 14953.04 <.0001

Andersen-Darling 96156.09 <.0001

Chi-Square 460741.3 94 <.0001

0

250

500

750

1000

1250

1500

1750

2000

2250

2500

2750

15 20 25 30 35 40 45 50

Fre

quen

cie

s


HomeHospital Time by ED Internal state (sec.), [13:00-23:00)Total for January2005 February2005 March2005 April2005 May2005 June2005 July2005 August2005 September2005

November2005 December2005,Mondays

Empirical Normal (mu=33.22 sigma=5.76)

Time by ED Internal state (sec.), [13:00-23:00) Statistics

N 1656000

N(average per day) 36000

Mean 33.22

Standard Deviation 5.756

Variance 33.13

Median 33

Minimum 16

Maximum 54

Skewness 0.282

Kurtosis -0.31791

Standard Error Mean 0.00447

Interquartile Range 8

Mean Absolute Deviation 4.712

Median Absolute Deviation(MAD) 4

Coefficient of Variation(CV) (%) 17.33

L-moment 2 (half of Gini's Mean Difference) 3.263

L-Skewness 0.0601

L-Kurtosis 0.0924

Coefficient of L-variation(L-CV)(%) (Gini's Coefficient) 9.82

Parameters for Normal Distribution

Parameter Estimate

mu 33.22

sigma 5.76

mean 33.22

std 5.756

Goodness-of-Fit Tests for Normal Distribution

Test Statistic DF p Value

Residuals Std 0.025

Kolmogorov-Smirnov 0.069 <.0001

Cramer-von Mises 1068.72 <.0001

Anderson-Darling 6193.27 <.0001

Chi-Square >1000 34 <.0001

Internal ED, non-holiday Mondays, 13:00-23:00

Our EDA “implies" (w/ Armony, Marmor, Tseytlin, Yom-Tov):

◮ Israeli ED censusd= M/M/∞: “Secret" behind

√-Staffing

68




√-Staffing

◮ EDd= Reversible Birth-Death process, hence conditioning on

finite capacity, say B, gives rise to M/M/B/B (Erlang-B)

◮ U.S. ED censusd= M/M/B/B, which can be (has been) used to

analyze Ambulance Diversion (ED Blocking)

68




√-Staffing

◮ EDd= Reversible Birth-Death process, hence conditioning on

finite capacity, say B, gives rise to M/M/B/B (Erlang-B)

◮ U.S. ED censusd= M/M/B/B, which can be (has been) used to

analyze Ambulance Diversion (ED Blocking)

◮ Puzzle (getting ahead): Is the ED an Erlang-A system withµ = θ? then the “effective number of servers" can be, perhaps,deduced via those LWBS = Left Without Being Seen (or LAMA)

Simple models at the service of complex realities

68

Prerequisite II: Models (DNets, QED 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.

77





E [Service]= 10,


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

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

77





E [Service]= 10,




Moreover, a significant fraction not delayed in queue: e.g. in well-run

◮ CCs: 50% served “immediately" & 90% utilization ⇒ QED

◮ EDs + IWs: ?

77





E [Service]= 10,




Moreover, a significant fraction not delayed in queue: e.g. in well-run

◮ CCs: 50% served “immediately" & 90% utilization ⇒ QED

◮ EDs + IWs: ? Multiple scales! IW-“Beds" (10’s) are QED whileIW-Doctors (1‘s) are in conventional heavy-traffic (hours wait forminutes service), hence the bottlenecks77

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

agents

arrivals

abandonment

1

2

n

…

queue

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

◮ λ – Arrival rate (Poisson)

◮ µ – Service rate (Exponential; E [S] = 1µ )

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

◮ N – Number of Servers (Agents).

78

Erlang-A: Practical Relevance?

Experience:

◮ Arrival process not pure Poisson (time-varying, σ2 too large)

◮ Service times not Exponential (typically close to LogNormal)

◮ Patience times not Exponential (various patterns observed).

79


Experience:




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

◮ Customers and Servers not homogeneous (classes, skills)

◮ Customers return for service (after busy, abandonment;dependently; P. Khudiakov, M. Gorfine, P. Feigin)

◮ · · · , and more.

79


Experience:




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

◮ Customers and Servers not homogeneous (classes, skills)

◮ Customers return for service (after busy, abandonment;dependently; P. Khudiakov, M. Gorfine, P. Feigin)

◮ · · · , and more.

Question: Is Erlang-A Relevant?

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

79

Asymptotic Landscape: 9 Operational Regimes, and then some

Erlang-A, w/ I. Gurvich & J. HuangErlang-A Conventional scaling Many-Server scaling NDS scaling

µ & θ fixed Sub Critical Over QD QED ED Sub Critical OverOffered load 1

1+δ1− β

√n

11−γ

11+δ

1− β√

n

11−γ

11+δ 1− β

n

11−γper server

Arrival rate λ µ1+δ

µ− β√

nµ µ

1−γnµ1+δ

nµ− βµ√

n nµ1−γ

nµ1+δ

nµ− βµ nµ1−γ

# servers 1 n nTime-scale n 1 n

Impatience 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 + δ) λ

µ+ β λ

µ(1− γ)

Utilization 11+δ

1−√

θµ

h(β)√

n1 1

1+δ1−

√

θµ

h(β)√

n1 1

1+δ1−

√

θµ

h(β)n

1

E(Q) 1δ(1+δ)

√ng(β) nµγ

θ(1−γ)1δn

√ng(β)α nµγ

θ(1−γ)o(1) ng(β) n2µγ

θ(1−γ)

P(Ab) 1n

1δ

θµ

θ√

nµg(β) γ 1

n

(1+δ)δ

θµn

θ√

nµg(β)α γ o( 1

n2 )θ

nµg(β) γ

P(Wq > 0) 11+δ

≈ 1 n α ∈ (0,1) ≈ 1 ≈ 0 ≈ 1

P(Wq > T ) 11+δ

e−

δ1+δ

µT 1 +O( 1√

n) 1 +O( 1

n) ≈ 0 f(T ) ≈ 0 Φ(β+

√

θµT )

Φ(β)1 +O( 1

n)

CongestionEWq

ES1δ

√ng(β) nµγ/θ 1

n

(1+δ)δ

nα√

ng(β) µγ

θo( 1

n) g(β) nµγ/θ

◮ Conventional: Ward & Glynn (03, G/G/1 + G)

◮ Many-Server:◮ QED: Halfin-Whitt (81), Garnett-M-Reiman (02)◮ ED: Whitt (04)◮ NDS: Atar (12)

91

Asymptotic Landscape: 9 Operational Regimes, and then some

Erlang-A, w/ I. Gurvich & J. HuangErlang-A Conventional scaling Many-Server scaling NDS scaling

µ & θ fixed Sub Critical Over QD QED ED Sub Critical OverOffered load 1

1+δ1− β

√n

11−γ

11+δ

1− β√

n

11−γ

11+δ 1− β

n

11−γper server

Arrival rate λ µ1+δ

µ− β√

nµ µ

1−γnµ1+δ

nµ− βµ√

n nµ1−γ

nµ1+δ

nµ− βµ nµ1−γ

# servers 1 n nTime-scale n 1 n

Impatience 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 + δ) λ

µ+ β λ

µ(1− γ)

Utilization 11+δ

1−√

θµ

h(β)√

n1 1

1+δ1−

√

θµ

h(β)√

n1 1

1+δ1−

√

θµ

h(β)n

1

E(Q) 1δ(1+δ)

√ng(β) nµγ

θ(1−γ)1δn

√ng(β)α nµγ

θ(1−γ)o(1) ng(β) n2µγ

θ(1−γ)

P(Ab) 1n

1δ

θµ

θ√

nµg(β) γ 1

n

(1+δ)δ

θµn

θ√

nµg(β)α γ o( 1

n2 )θ

nµg(β) γ

P(Wq > 0) 11+δ

≈ 1 n α ∈ (0,1) ≈ 1 ≈ 0 ≈ 1

P(Wq > T ) 11+δ

e−

δ1+δ

µT 1 +O( 1√

n) 1 +O( 1

n) ≈ 0 f(T ) ≈ 0 Φ(β+

√

θµT )

Φ(β)1 +O( 1

n)

CongestionEWq

ES1δ

√ng(β) nµγ/θ 1

n

(1+δ)δ

nα√

ng(β) µγ

θo( 1

n) g(β) nµγ/θ

◮ Conventional: Ward & Glynn (03, G/G/1 + G)

◮ Many-Server:◮ QED: Halfin-Whitt (81), Garnett-M-Reiman (02)◮ ED: Whitt (04)◮ NDS: Atar (12)

◮ “Missing": ED+QED; Hazard-rate scaling (M/M/N+G); Time-Varying,Non-Parametric; Moderate- and Large-Deviation; Networks; Control

91

Universal Approximations: Erlang-A (M/M/N + M)

agents

arrivals

abandonment

1

2

n

…

queuew/ I. Gurvich & J. Huang

92


agents

arrivals

abandonment

1

2

n

…


◮ QNet: Birth & Death Queue, with B - D rates

F (q) = λ− µ · (q ∧ n)− θ · (q − n)+, q = 0, 1, . . .

92


agents

arrivals

abandonment

1

2

n

…



F (q) = λ− µ · (q ∧ n)− θ · (q − n)+, q = 0, 1, . . .

◮ FNet: Dynamical (Deterministic) System – ODE

dxt = F (xt)dt , t ≥ 0

92


agents

arrivals

abandonment

1

2

n

…



F (q) = λ− µ · (q ∧ n)− θ · (q − n)+, q = 0, 1, . . .



◮ DNet: Universal (Stochastic) Approximation – SDE

dYt = F (Yt)dt +√

2λ dBt , t ≥ 0

92


agents

arrivals

abandonment

1

2

n

…



F (q) = λ− µ · (q ∧ n)− θ · (q − n)+, q = 0, 1, . . .



◮ DNet: Universal (Stochastic) Approximation – SDE

dYt = F (Yt)dt +√

2λ dBt , t ≥ 0

eg. µ = θ : x = λ− µ · x , Y = OU process

Accuracy increases as λ ↑ ∞ (no additional assumptions)92

Value of Universal Approximation

◮ Tractable - closed-form stable expressions

◮ Accurate - more than heavy traffic limits

◮ Robust - all many-server regimes, and beyond, with hardly anyassumptions

◮ Value

◮ Performance Analysis◮ Optimization (Staffing)◮ Inference (w/ G. Pang)◮ Simulation (w/ J. Blanchet)

◮ Limitation: Steady-State (but working on it)

Why does it work so well?

Coupling “Busy" + “Idle" Excursions of B&D and the correspondingDiffusion (durations order 1√

λ)

93

Universal Diffusion: Tractability

◮ Density function of Y (∞)− n:

π(x) =

√µ

√λ

φ(√µ(x/

√λ+β/µ))

Φ(β/√µ) p(β, µ, θ), if x ≤ 0,

√θ√λ

φ(√θ(x/

√λ+β/θ))

1−Φ(β/√θ)

(1− p(β, µ, θ)), if x > 0,

Here β := (nµ− λ)/√λ

and

p(β, µ, θ) =

[

1 +

√

µ

θ

φ(β/√µ)

Φ(β/√µ)

1− Φ(β/√θ)

φ(β/√θ)

]−1

.

94

Universal Approximation: Accuracy

◮ ∆λ is the “balancing" state, obtained by solving

λ = µ(n ∧∆λ) + θ(∆λ − n)+.

Solution: ∆λ = λµ −

(

λµ − n

)+(

1− µθ

)

.

Specifically: QD = λµ ; ED = n + 1

θ (λ− nµ); QED = n +O(√λ))

◮ Centered processes (excursions):

Qλ(·) = Q(·)−∆λ, Yλ(·) = Y (·)−∆λ.

TheoremFor f bounded by an m-degree polynomial (m ≥ 0):

Ef (Qλ(∞))− Ef (Yλ(∞)) = O(√λ

m−1).

◮ Accuracy: higher than heavy-traffic limits

95

Universal Approximation: Why 2λ?

◮ Semi-martingale representation of the B&D process:Fluid + Martingale

◮ Predictable quadratic variation:

∫ t

0

[λ+ µ(Qs ∧ n) + θ(Qs − n)+]ds

◮ In steady-state, arrival rate ≡ departure rate:

λ = E[µ(Qs ∧ n) + θ(Qs − n)+]

◮ Expectation of the predictable quadratic variation:

E

∫ t

0

[λ+ µ(Qs ∧ n) + θ(Qs − n)+]ds = 2λt

◮ dMartingalet ≈√

2λ · dBrowniant

96

Reconciling Steady-State and Time-Varying Models

◮ Challenge: Accommodate time-varying demand (routine)◮ Prerequisite: Flexible Capacity

◮ As in Call Centers and to a degree in Healthcare,◮ In contrast to rigid (fixed) staffing level during a shift: doomed to

alternate between overloading and underloading

97





◮ Idea/Goal: In the face of time-varying demand, designtime-varying staffing which accommodates demand such thatperformance is stable over time

◮ Solution: In fact, a time-varying system with Steady-Stateperformance, at all times, via (Modified) Offered-Load(Square-Root) Staffing.

97





◮ Idea/Goal: In the face of time-varying demand, designtime-varying staffing which accommodates demand such thatperformance is stable over time

◮ Solution: In fact, a time-varying system with Steady-Stateperformance, at all times, via (Modified) Offered-Load(Square-Root) Staffing.

◮ History:◮ Jennings, M., Reiman, Whitt (1996): Emergence of the

phenomenon, via infinite-server heuristics◮ Feldman, M., Massey, Whitt (2008): Stabilize delay probability via

QED staffing (justified theoretically only for Erlang-A with µ = θ)◮ Liu and Whitt (ongoing): Stabilize abandonment probability by ED

staffing, via a corresponding network, theoretically and empirically◮ Huang, Gurvich, M. (ongoing): QED theory

97

The Offered-Load R(t), t ≥ 0 (R(t)↔ R)

Empirically (in SEEStat):◮ Process: L(·) = Least number of servers that guarantees no delay.◮ Offered-Load Function R(·) = E [L(·)]

98

The Offered-Load R(t), t ≥ 0 (R(t)↔ R)

Empirically (in SEEStat):◮ Process: L(·) = Least number of servers that guarantees no delay.◮ Offered-Load Function R(·) = E [L(·)]

ILTelecom , Private

12.05.2004

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

50.00

55.00

60.00

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


Nu

mb

er

of ca

se

s

Offered load Abandons Av_agents_in_system

98

Time-Varying Arrival Rates

Square-Root Staffing:N(t) = R(t) + β

√

R(t) , −∞ < β <∞.

R(t) is the Offered-Load at time t ( R(t) 6= λ(t)× E[S] )

Arrivals, Offered-Load and Staffing

0

50

100

150

200

2500 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

99

Time-Stable Performance of Time-Varying Systems

Delay Probability = as in the Stationary Erlang-A / R

Delay Probability

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 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

100

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

(λ(t),N(t) = R(t) + β√

R(t))

- 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, modeled as Erlang-R).

101

Calculating the Offered-Load R(t), Theoretically

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

◮ Offered-Load Function R(t) = E [L(t)], t ≥ 0.

Think Mt/G/N?t + G vs. Mt/G/∞: Ample-Servers.

102





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−S

λ(u)du

]

= E [λ(t − Se)] · E [S] ≈ ... .

◮ {A(t), t ≥ 0} Arrival-Process, rate λ(·);◮ S (Se) generic Service-Time (Residual Service-Time).

102






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

∫ t

−∞

λ(u) · P(S > t − u)du = E

[

A(t)− A(t − S)

]

=

= E

[∫ t

t−S

λ(u)du

]

= E [λ(t − Se)] · E [S] ≈ ... .


◮ Relating L, λ,S (“W”): Time-Varying Little’s Formula.Stationary models: λ(t) ≡ λ then R(t) ≡ λ× E[S].

102






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

∫ t

−∞

λ(u) · P(S > t − u)du = E

[

A(t)− A(t − S)

]

=

= E

[∫ t

t−S

λ(u)du

]

= E [λ(t − Se)] · E [S] ≈ ... .


◮ 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).

102

Extending the Notion of the “Offered-Load"

◮ Business (Banking Call-Center): Offered Revenues

◮ Healthcare (Maternity Wards): Fetus in stress

◮ 2 patients (Mother + Child) = high operational and cognitive load◮ Fetus dies ⇒ emotional load dominates

◮ ⇒◮ Offered Operational Load

◮ Offered Cognitive Load

◮ Offered Emotional Load

◮ ⇒ 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

106

ServNets: Data-Based Online Automatic Creation

w/ V. Trofimov, E. Nadjharov, I. Gavako = Technion SEELab

◮ ServNets = QNets, SimNets, FNets, DNets

◮ SimNets of Service Systems = Virtual Realities

◮ SimNets also of QNEts, FNets, DNets

eg. ED MD (Physics): Where are the Differential Equations?

112







◮ Ultimately, Research Labs will become necessary (hence mustbe funded!): offering universal access to data and ServNets, andtheir analysis

112








◮ Data-based Research: Tradition in Physics, Chemistry, Biology;Psychology (now also in Transportation (Science) and(Behavioral) Economics)

◮ Why not in Service Science / Engineering / Management ?

112








◮ Data-based Research: Tradition in Physics, Chemistry, Biology;Psychology (now also in Transportation (Science) and(Behavioral) Economics)

◮ Why not in Service Science / Engineering / Management ?

◮ Moreover, address the Reproducibility and ProprietaryCrisis in Scientific Research

112

Data-Based Creation ServNets: some Technicalities


◮ Graph Layout: Adapted from but significantly extends Graphviz(AT&T, 90’s); eg. edge-width, which must be restricted topoly-lines, since there are “no parallel Bezier (Cubic) curves(Bn(p) = EpF [B(n, p)], 0 ≤ p ≤ 1)

◮ Algorithm: Dot Layout (but with cycles), based on Sugiyama,Tagawa, Toda (’81): “Visual Understanding of HierarchicalSystem Structures"

113

Data-Based Creation ServNets: some Technicalities


◮ Graph Layout: Adapted from but significantly extends Graphviz(AT&T, 90’s); eg. edge-width, which must be restricted topoly-lines, since there are “no parallel Bezier (Cubic) curves(Bn(p) = EpF [B(n, p)], 0 ≤ p ≤ 1)

◮ Algorithm: Dot Layout (but with cycles), based on Sugiyama,Tagawa, Toda (’81): “Visual Understanding of HierarchicalSystem Structures"

◮ Draws data directly from SEELab data-bases:◮ Relational DBs (Large! eg. USBank Full Binary = 37GB, Summary

Tables = 7GB)◮ Structure: Sequence of events/states, which (due to size)

partitioned (yet integrated) into days (eg. call centers) or months(eg. hospitals)

◮ Differs from industry DBs (in call centers, hospitals, websites)

113

Documents

Kellogg Operations Workshop , September 2012iew3.technion.ac.il/serveng/References/0_Kellogg_Workshop...Human Resources, Operations Research, MIS Customers Interface Design Computer-Telephony