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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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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
33975
4221
2006
2886
2741
361
354697
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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
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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
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
R
L
D
Out
R
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R R
R
RR RR
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R
R
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L
LL
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Out Out
Out
OutOut Out OutOut
Out
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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
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
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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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
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
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1113 43320 160505194 7
36512 23633845 3792
845 103
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3314
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594 140
547
191
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3052 3908
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20863633 3536
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260
294
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666
198 35148
29
213
3005
6
3671
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19 356
126
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254
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8742
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1618 171426
10126
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14688
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193
Daily(Deposit)
OrderCheckbooks
Identification
ChangePassword
Agent
Fax
Query Transfer
Securities
StockMarket
PerformOperation
Credit
Error
43123
234
270
1352
35315
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273
1063
231
768
133
1032
897
40
49
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26091
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44 15
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1 11233
1
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1
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1 1 21 3
12
1
3
2
2
2
1
33
7
18 32
1
10
14 23
3
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
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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