Data-Based Service Networks:A Research Framework for

Asymptotic Inference, Analysis & Controlof Service Systems

Avi Mandelbaum

Technion, Haifa, Israel

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

Gideon’s Party, June 2012

1

Research PartnersI Students:

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

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

I 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, . . .

I Technion SEE Center / Laboratory:Feigin; Trofimov, Nadjharov, Gavako, Kutsy; Liberman, Koren,Plonsky, Senderovic; Research Assistants, . . .

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

2

Contents

I Service Networks: Call Centers, Hospitals, Websites, · · ·

I Prevalent paradigm of modeling and approximations

I Redefining the role of Asymptotics, via Data

I ServNets: QNets, SimNets; FNets, DNets

I Goal: Data-based creation and validation of ServNets,automatically in real-time

I Why be Optimistic? Pilot at the Technion SEELab

I Successes: Erlang-A (CC Abandonment), Erlang-R (EDFeedback)

I Elsewhere: Process Mining (BPM), Networks (Social,Biological,· · · ), “Almost-Complex" Systems

3

Applying Asymptotic Research of Queueing Systems

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 myapproach to make this good happen is by marrying theory withdata.

4

Models Approximations

QNets SimNets

Accuracy

Phenomenology

Prevalent Asymptotic Approximations

System (Data)

5

3

Models Approximations

QNets SimNets FNet DNet Models

Accuracy

Phenomenology Value

X X

Data – Based Prevalent Asymptotic Approximations Models

System (Data)

X X

6

6

Models

QNets FNet DNet

Value

Data – Based Asymptotic Framework

System (Data)

SimNets

Redefine the Role of Asymptotics:I Data-based parsimonious FNets & DNets models

(vs. approximations)I Validation against data-based SimNets = Virtual Realities

(vs. originating stylized QNets)I Value = e.g. predictive power (robustness), without/with

interventions(vs. accuracy)

7

Call Centers, then Hospitals, now Internet

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

Internet - . . .

8

Call Centers, then Hospitals, now Internet

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

Internet - . . .

8

Call-Center Environment: Service Network

9

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 DesignFunctionScientific DisciplineMulti-Disciplinary

IndexCall Center Design

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

of the21th Century)

Tele-StressPsychology

10

ER / ED Environment: Service Network

Acute (Internal, Trauma) Walking

Multi-Trauma

12

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

Service Process Design

Quality

Efficiency

CustomersInterface Design

Medicine

(High turnoversMedical-Staff shortage)

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

StretcherWalking

Service Completion(sent to other department)

( Waiting TimeActive Dashboard )

PatientsSegmentation

Medicine,Psychology, Marketing

PsychologicalProcessArchive

Human FactorsEngineering(HFE)

InternalQueue

OrthopedicQueue

Arrivals

FunctionScientific DisciplineMulti-Disciplinary

Index

ED-StressPsychology

OperationsResearch, Medicine

Emergency-Department Network: Gallery of Models

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 DesignMIS, HFE,Operations Research

Psychology,Statistics

Home

I Forecasting, Abandonment = LWBS, SBR ≈ Flow Control13

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 (customers, servers) = time-stamps of events .

14

Pause for a Commercial:

The Technion SEE Center

15

Pause for a Commercial: The Technion SEE Center

15

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; access U.S. Bank, Israeli Bank, Rambam Hospital.

16

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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

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NonBusiness LineSubanco

NonBusiness LinePriority Service

NonBusiness Line NonCC ServiceCCO

NonCC ServiceQuick&Reilly

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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

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Logged Off

Available (7.36)

Consulting (562.54)

Consulting (208.00)

Consulting (175.42)

General Banking (218.31)

General Banking (136.89)

General Banking (97.22)

Idle (185.11)

Internet Support (313.66)

Internet Support (114.49)

Internet Support (166.50)

Investments (252.68)

Investments (211.41)

Logged In

Logged Off

Long Break (1,384.28)

Managers VIP (227.05)

Managers VIP (149.36)

Medium Break (507.26)

None (209.00)

Paperwork (140.57)

Paperwork (140.62)

Paperwork (140.97)

Paperwork (313.19)

Pension (758.00)

Pension (307.14)

Private Call (27.11)

Sales (177.14)

Sales (329.00)

Securities (278.20)

Securities (532.00)

Shifting (209.71)

Short Break (12.97)

Virtual Branches (195.46)

Virtual Branches (133.67)

Wrap Up (7.46)

Wrap Up (8.71)

Wrap Up (8.11)

Wrap Up (8.03)

ED

Pediatric

Neurology Nephrology Derrmatology

Internal

IntensiveCare

Cardiology

Rheumatology

UrologyOtorhinolaryngology Orthopedics

Surgery

Maternity

Gynecology

Psychiatry Ophthalmology

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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.pdf seminar_presentation_boaz_estimatinger load.ppt

project_ug_meirav.pdf see_report_output_june_2011.pdf

see_report_2010.pdf

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3 3 4

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5 4

3

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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

hazard_phase_type.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.pptintro_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

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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

hazard_phase_type.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.pptintro_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

8 AM - 9 AM

55888

1113 43320 160505194 7

36512 23633845 3792

845 103

120

314

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594 140

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Securities

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43123

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1

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5

1

Logged Off

Available (4.15)

General Banking (190.48)

General Banking (297.50)

General Banking (61.25)

Idle (69.87)

Investments (184.00) Investments (125.00)

Logged In

Logged Off Long Break (1,225.50)

Managers VIP (119.08)

Medium Break (273.50)

None (209.00)

Paperwork (140.57) Paperwork (77.48)

Paperwork (37.00)

Paperwork (97.20)

Private Call (31.90)

Sales (176.09)

Short Break (6.29)

Wrap Up (7.03) Wrap Up (5.53)

Wrap Up (2.50)

Wrap Up (14.53)

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

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

8 AM - 9 AM

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EntryEntry Entry Entry EntryEntry Entry

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8 AM - 9 AM

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10 AM - 11 AM

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Durations & “Protocols": 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-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)

20

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

21

Waiting Times in a Call Center⇒ Protocols

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) (compare 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, %

22

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

23

Psychology + Protocols

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

Haz

ard

Func

tion

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)

24

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: 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

25

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: 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

25

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

26

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

27

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

27

Congestion Laws: State-Space Collapse

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

ILBank Customers in queue (average), Private10.06.2007

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.)Pe

rcen

tage

Medium priority Low priority Unidentified

ILBank Customers in queue (average)10.06.2007

0.00

25.00

50.00

75.00

100.00

125.00

150.00

175.00

200.00

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 o

f cas

es

Medium priority Low priority Unidentified

Queue Shape

ILBank Customers in queue (average)10.06.2007

0.0050.00

100.00150.00200.00250.00300.00350.00400.00450.00500.00550.00600.00

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.)

Perc

ent t

o m

ean

Medium priority Low priority Unidentified

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%28

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).35

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 Center29

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 ↑ ∞:

• QED %{Cust Wait > 0} ≈ α, 0 < α < 1;

or %{Serv Idle > 0} ≈ 1− α

• Customers {Abandon} ≈ γ√N, 0 < γ;

• Agents OCC ≈ 1− β+γ√N

−∞ < β <∞ ;

• Managers N ≈ R + β√

R , R = λ× E(S) not small;

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.

38

QED Theory vs. Data: P(Wq > 0)

IL Telecom; June-September, 2004

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.

I 2205 30-min. intervals (13 summer weeks, week-days)I Erlang-A approximations for the appropriate θ/µ = 0.35

30

Asymptotic Landscape: 9 Operational Regimes, and then someErlang-A, w/ I. Gurvich & J. Huang

Mandelbaum Part B2 / Wednesday 15th February, 2012 / 17:38 ServiceNetworks

Table 1: (Part of the) Asymptotic Landscape of Erlang-A = 9 Operational Regimes

Erlang-A Conventional scaling Many-Server scaling NDS scalingµ & θ fixed Sub Critical Over QD QED ED Sub Critical OverOffered load 1

1+δ1− β√

n1

1−γ1

1+δ1− β√

n1

1−γ1

1+δ 1− βn

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) λµ

(1 + β√n

) λµ

(1− γ) λµ

(1 + δ) λµ

+ β√λµ

λµ

(1− γ) λµ

(1 + δ) λµ

+ β λµ

(1− γ)

Utilization 11+δ

1−√θµh(β)√n

1 11+δ

1−√θµh(β)√n

1 11+δ

1−√θµh(β)n

1

E(Q) 1δ(1+δ)

√ng(β) nµγ

θ(1−γ)1δ%n

√ng(β)α nµγ

θ(1−γ)o(1) ng(β) n2µγ

θ(1−γ)

P(Ab) 1n

1δθµ

θ√nµg(β) γ 1

n(1+δ)δ

θµ%n

θ√nµg(β)α γ o( 1

n2 ) θnµg(β) γ

P(Wq > 0) 11+δ

≈ 1 %n α ∈ (0,1) ≈ 1 ≈ 0 ≈ 1

P(Wq > T ) 11+δ

e− δ

1+δµT 1 +O( 1√

n) 1 +O( 1

n) ≈ 0 f(T ) ≈ 0 Φ(β+

√θµT )

Φ(β)1 +O( 1

n)

CongestionEWqES

1δ

√ng(β) nµγ/θ 1

n(1+δ)δ

%nα√ng(β) µγ

θo( 1n

) g(β) nµγ/θ

1

I Conventional: Ward & Glynn (03, G/G/1 + G)I Many-Server:

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

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

37

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

arrivals

abandonment

λ

µ

1

2

n

…

queue

θ

w/ J. Huang & I. Gurvich

I QNets: Birth & Death Queue, with B - D rates

Fλ(q) = λ− µ · (q ∧ nλ)− θ · (q − nλ)+, q = 0,1, . . .

I FNets: Dynamical (Deterministic) System – ODE

dxλt = Fλ(xλ

t )dt , t ≥ 0

I DNets: Universal (Stochastic) Approximation – SDE

dYλt = Fλ(Yλ

t )dt +√

2λ dBt , t ≥ 0

Accuracy increases as λ ↑ ∞

41

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

arrivals

abandonment

λ

µ

1

2

n

…

queue

θ

w/ J. Huang & I. Gurvich

I QNets: Birth & Death Queue, with B - D rates

Fλ(q) = λ− µ · (q ∧ nλ)− θ · (q − nλ)+, q = 0,1, . . .

I FNets: Dynamical (Deterministic) System – ODE

dxλt = Fλ(xλ

t )dt , t ≥ 0

I DNets: Universal (Stochastic) Approximation – SDE

dYλt = Fλ(Yλ

t )dt +√

2λ dBt , t ≥ 0

Accuracy increases as λ ↑ ∞

41

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

arrivals

abandonment

λ

µ

1

2

n

…

queue

θ

w/ J. Huang & I. Gurvich

I QNets: Birth & Death Queue, with B - D rates

Fλ(q) = λ− µ · (q ∧ nλ)− θ · (q − nλ)+, q = 0,1, . . .

I FNets: Dynamical (Deterministic) System – ODE

dxλt = Fλ(xλ

t )dt , t ≥ 0

I DNets: Universal (Stochastic) Approximation – SDE

dYλt = Fλ(Yλ

t )dt +√

2λ dBt , t ≥ 0

Accuracy increases as λ ↑ ∞

41

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

arrivals

abandonment

λ

µ

1

2

n

…

queue

θ

w/ J. Huang & I. Gurvich

I QNets: Birth & Death Queue, with B - D rates

Fλ(q) = λ− µ · (q ∧ nλ)− θ · (q − nλ)+, q = 0,1, . . .

I FNets: Dynamical (Deterministic) System – ODE

dxλt = Fλ(xλ

t )dt , t ≥ 0

I DNets: Universal (Stochastic) Approximation – SDE

dYλt = Fλ(Yλ

t )dt +√

2λ dBt , t ≥ 0

Accuracy increases as λ ↑ ∞41

Value of Universal Approximation

I Tractable - closed-form stable expressionsI Accurate - more than heavy traffic limitsI Robust - all many-server regimes, and beyond, with hardly any

assumptionsI Value

I Performance AnalysisI Universal Optimization (Staffing)I Inference (w/ G. Pang)I Simulation (w. J. Blanchet)

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

Why does it work so well?I Steady-state embodied in a single excursion: ‘Busy"→ “Idle"→I Successful coupling of an Excursion, which is order 1√

λ

42

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

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

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

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

Then P (Wait > 0) =TN,N−1

TN,N−1+TN−1,N=[1 +

TN−1,NTN,N−1

]−1.

39

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 +

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

Special case: µ = θ (Impatient):

Then Q d= M/M/∞, since sojourn-time is exp(µ = θ).

If also β = 0 (Prevalent): P{Wait > 0} ≈ 1/2.

40

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 +

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

Special case: µ = θ (Impatient):

Then Q d= M/M/∞, since sojourn-time is exp(µ = θ).

If also β = 0 (Prevalent): P{Wait > 0} ≈ 1/2.

40