Data-Based Service Networks - Technionie.technion.ac.il/serveng/References/0_SAMSI_Workshop.pdf ·...
Transcript of Data-Based Service Networks - Technionie.technion.ac.il/serveng/References/0_SAMSI_Workshop.pdf ·...
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
SAMSI Workshop, August 2012
I Overheads available at SAMSI and my Technion websites
1
Research PartnersI Students:
Aldor∗, Baron∗, Carmeli∗, Cohen∗, Feldman∗, Garnett∗, Gurvich∗,Khudiakov∗, Maman∗, Marmor∗, Reich∗, Rosenshmidt∗, Shaikhet∗,Senderovic, Tseytlin∗, Yom-Tov∗, Yuviler, Zaied∗, Zeltyn∗,Zychlinski∗, Zohar∗, Zviran∗, . . .
I Theory:Armony, Atar, Cohen, Gurvich, Huang, Jelenkovic, Kaspi, Massey,Momcilovic, Reiman, Shimkin, Stolyar, Trofimov, Wasserkrug,Whitt, Zeltyn, . . .
I Empirical/Statistical Analysis:Brown, Gans, Shen, Zhao; Zeltyn; Ritov, Goldberg; Gurvich, Huang,Liberman; Armony, Marmor, Tseytlin, Yom-Tov; Nardi, Plonsky;Gorfine, Ghebali; Pang, . . .
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, . . .
2
ContentsI Service Networks: Call Centers, Hospitals, Websites, · · ·I Redefine the paradigm of modeling/asymptotics via DataI ServNets: QNets, SimNets; FNets, DNets
I Ultimate Goal: Data-based creation and validationof ServNets, automatically in real-time
I Why be Optimistic? Pilot at the Technion SEELabI Lacking but Feasible: Dynamics, Durations, ProtocolsI Simple Models at the Service of Complex Realities
I State-Space Collapse (Queues, Waiting Times)I Congestion Laws (LN, Little, ImPatience, Staffing)I Universal Approximations: Simplifying the Asymptotic LandscapeI Stabilizing Time-Varying Performance (Offered-Load)
I Successes: Palm/Erlang-R (ED Feedback = FNet),Palm/Erlang-A (CC Abandonment = DNet)
I Elsewhere : Process Mining (Petri Nets, BPM), Networks (Social,Biological, Complex, . . .), Simulation-based, . . .
I Scenic Route : Open Problems, New Directions, Uncharted Territories
3
ContentsI Service Networks: Call Centers, Hospitals, Websites, · · ·I Redefine the paradigm of modeling/asymptotics via DataI ServNets: QNets, SimNets; FNets, DNets
I Ultimate Goal: Data-based creation and validationof ServNets, automatically in real-time
I Why be Optimistic? Pilot at the Technion SEELabI Lacking but Feasible: Dynamics, Durations, ProtocolsI Simple Models at the Service of Complex Realities
I State-Space Collapse (Queues, Waiting Times)I Congestion Laws (LN, Little, ImPatience, Staffing)I Universal Approximations: Simplifying the Asymptotic LandscapeI Stabilizing Time-Varying Performance (Offered-Load)
I Successes: Palm/Erlang-R (ED Feedback = FNet),Palm/Erlang-A (CC Abandonment = DNet)
I Elsewhere : Process Mining (Petri Nets, BPM), Networks (Social,Biological, Complex, . . .), Simulation-based, . . .
I Scenic Route : Open Problems, New Directions, Uncharted Territories
3
ContentsI Service Networks: Call Centers, Hospitals, Websites, · · ·I Redefine the paradigm of modeling/asymptotics via DataI ServNets: QNets, SimNets; FNets, DNets
I Ultimate Goal: Data-based creation and validationof ServNets, automatically in real-time
I Why be Optimistic? Pilot at the Technion SEELabI Lacking but Feasible: Dynamics, Durations, ProtocolsI Simple Models at the Service of Complex Realities
I State-Space Collapse (Queues, Waiting Times)I Congestion Laws (LN, Little, ImPatience, Staffing)I Universal Approximations: Simplifying the Asymptotic LandscapeI Stabilizing Time-Varying Performance (Offered-Load)
I Successes: Palm/Erlang-R (ED Feedback = FNet),Palm/Erlang-A (CC Abandonment = DNet)
I Elsewhere : Process Mining (Petri Nets, BPM), Networks (Social,Biological, Complex, . . .), Simulation-based, . . .
I Scenic Route : Open Problems, New Directions, Uncharted Territories3
On Asymptotic Research of Queueing Systems
Queueing asymptotics has grown to become a central researchtheme in Operations Research and Applied Probability, beyond justqueueing theory. Its claim to fame has been the deep insights that itprovides into the dynamics of Queueing Networks (QNets), andrightly so:
I Kingman’s invariance principle in conventional heavy-trafficI Whitt’s sample-path (functional) frameworkI Reiman’s network analysis via oblique reflectionI Bramson-Williams’ framework for state-space collapseI Laws’ resource poolingI Harrison’s paradigm for asymptotic control (Wein; van Mieghem’s Gcµ)I Dai’s fluid-based stabilityI Halfin-Whitt’s (QED regime) (√-staffing for many-server queues)I P = NP : Atar’s equivalence of Preemptive and Non-Preemptive SBR;
Stolyar, GurvichI Massey-Whitt’s research of time-varying queues
4
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 myapproach to make this good happen is by marrying theory with
data .
5
Models Approximations
QNets SimNets
Accuracy
Phenomenology
Prevalent Asymptotic Approximations
System (Data)
6
2
Models Approximations
QNets SimNets FNet DNet Models
Accuracy
Phenomenology
X
Data–Based Prevalent Asymptotic Approximations Models
System (Data)
X X
7
3
Models Approximations
QNets SimNets FNet DNet Models
Accuracy
Phenomenology Value
X X
Data–Based Prevalent Asymptotic Approximations Models
System (Data)
X X
8
4
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
Models Approximations
QNets SimNets FNet DNet Models
Accuracy
Phenomenology Value
X X
Data–Based Prevalent Asymptotic Approximations Models
System (Data)
X X
9
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
2
Models
Q F
Value
Data–Based Asymptotic Framework: Added Value
SimNets
D
Data
SubstitutionPrinciple
Congestion Laws
Inference: Patience, Predictors Performance: G/G/N, QED Control: Gc , SBR, FJ Design: - Staffing, Pooling Predictions
3
Models
Q F
Value
Ultimately: Automatic “Discovery, Conformance, Enhancement”
SimNets
D
Data
State-Space Collapse Multiple Scales (Regimes, UA) Snapshots Pathwise Little, ASTA, …
SubstitutionPrinciple
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
14
Call Centers, Then Hospitals, Now Internet
Call Centers - U.S. Stat.
I $200 – $300 billion annual expendituresI 100,000 – 200,000 call centersI “Window" into the company, for better or worseI Over 3 million agents = 2% – 4% workforce
Healthcare - similar, plus unique challenges:
I Cost-figures far more staggeringI Risks much higherI ED (initial focus) = hospital-windowI Over 3 million nurses
Internet - . . .
15
Call Centers, Then Hospitals, Now Internet
Call Centers - U.S. Stat.
I $200 – $300 billion annual expendituresI 100,000 – 200,000 call centersI “Window" into the company, for better or worseI Over 3 million agents = 2% – 4% workforce
Healthcare - similar, plus unique challenges:
I Cost-figures far more staggeringI Risks much higherI ED (initial focus) = hospital-windowI Over 3 million nurses
Internet - . . .
15
Call-Center Environment: Service Network
16
Operational Focus
Operational Measures:I Surrogates for overall performance: Financial, Psychological; Clinical
I Easiest to quantify, measure, track online, react upon / Research
17
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
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Call-Center Network: Gallery of ModelsCall-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
CustomersInterface Design
Computer-TelephonyIntegration - CTIMIS/CS
Marketing
Operations/BusinessProcessArchiveDatabaseDesignData Mining:MIS, Statistics, OperationsResearch,Marketing
InternetChatEmailFax
Lost Calls
Service Completion)75% in Banks (
( Waiting TimeReturn Time)
Logistics
CustomersSegmentation -CRM
Psychology, OperationsResearch,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
1619
Skills-Based Routing in Call CentersEDA and OR, with I. Gurvich and P. Liberman
Mktg. ⇒
OR ⇒
HRM ⇒
MIS ⇒
20
ER / ED Environment: Service Network
Acute (Internal, Trauma) Walking
Multi-Trauma
21
ED-Environment in Israel
22
Queueing in a “Good" HospitalTong-ren Hospital at 6am, Beijing
23
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 Control24
ED Patient Flow: The Physicians Viewwith J. Huang, B. Carmeli
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P 0(j,k)
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Triage-Patients
IP-Patients
ExitsS
Arrivals
C1(·) C2(·) C3(·) CK(·)
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· · ·
1
I Goal: Adhere to Triage-Constraints, then release In-Process PatientsI Model = Multi-class Q with Feedback: Min. convex congestion costs of
IP-Patients, s.t. deadline constraints on Triage-Patients.
I Solution: In conventional heavy-traffic, asymptotic least-cost s.t. asymptoticcompliance (as in Plambeck, Harrison, Kumar, who applied admission control):
I Triage or IP? former, if some deadline is “too" closeI Triage Priorities: Chose the closest deadlineI IP-Priorities: Gcµ, modified (simply) to account for feedback
25
ED Patient Flow: The Physicians Viewwith J. Huang, B. Carmeli
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· · ·
P 0(j,k)
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m01 m0
2 m0J
Triage-Patients
IP-Patients
ExitsS
Arrivals
C1(·) C2(·) C3(·) CK(·)
m1 m2 m3 mK
· · ·
1
I Goal: Adhere to Triage-Constraints, then release In-Process PatientsI Model = Multi-class Q with Feedback: Min. convex congestion costs of
IP-Patients, s.t. deadline constraints on Triage-Patients.I Solution: In conventional heavy-traffic, asymptotic least-cost s.t. asymptotic
compliance (as in Plambeck, Harrison, Kumar, who applied admission control):I Triage or IP? former, if some deadline is “too" closeI Triage Priorities: Chose the closest deadlineI IP-Priorities: Gcµ, modified (simply) to account for feedback
25
Emergency-Department Network: Flow Control
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 ∗Queueing-Science, w/ Armony, Marmor, Tseytlin, Yom-TovI ∗Fair ED-to-IW Routing (Patients vs. Staff), w/ Momcilovic, TseytlinI ∗Triage vs. InProcess/Release (Plambeck et al, van Mieghem) in EDs, w/
Carmeli, Huang; ShimkinI ∗Staffing Time-Varying Q’s with Re-Entrant Customers (de Vericourt &
Jennings), w/ Yom-TovI The Offered-Load in Fork-Join Nets (Adlakha & Kulkarni), w/ Kaspi, ZaeidI Synchronization Control of Fork-Join Nets, w/ Atar, Zviran
26
Prerequisite I: Data
Averages Prevalent (and could be useful / interesting).But I need data at the level of the Individual Transaction:For each service transaction (during a phone-service in a call center,or a patient’s visit in a hospital, or browsing in a website, or . . .), itsoperational history = time-stamps of events (events-log files).
27
Interesting Averages: The Human Factor, orEven “Doctors" Can Manage
Afternoon, by Case
Morning, by Hour
Operations Time In a Hospital
Operations Time Histogram: Operations Time - Morning (by Hour) vs. Afternoon (by Case): Ethical?
Even Doctors Can Manage!
0%2%4%6%8%
10%12%14%16%18%20%
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14
Hours
Freq
uenc
y
AVG: 2.08 HoursSTD: 4.12 HoursSample Size: 4347
CV >> 1
0
1
2
3
4
5
6
EEG Orthopedics Surgery Blood Surgery Plastic Surgery Heart/ChestSurgery
Neuro-Surgery Eyes E.I. Surgery
Department
Hou
rs
AM
PM
20
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: 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-durations (???, common, more later)
29
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-durations (???, common, more later)
29
Pause for a Commercial:
The Technion SEE Center
30
Pause for a Commercial: The Technion SEE Center
30
Technion SEE = Service Enterprise Engineering
SEELab: Data-repositories for research and teachingI For example:
I Bank Anonymous: 1 year, 350K calls by 15 agents - in 2000.Brown, Gans, Sakov, Shen, Zeltyn, Zhao (JASA),paved the way to:
I U.S. Bank: 2.5 years, 220M calls, 40M by 1000 agentsI Israeli Cellular: 2.5 years, 110M calls, 25M calls by 750 agentsI Israeli Bank: from January 2010, daily-deposit at a SEESafeI Home (Rambam) Hospital: 4 years, 1000 beds, ward-level flowI 5 EDs: gathered by the late David Sinreich, ED arrivals & LOS
SEEStat: Environment for graphical EDA in real-timeI Universal Design, Internet Access, Real-Time Response.
SEEServer: Free for academic useI RegisterI Access U.S. Bank, Bank Anonymous, Home Hospital
31
Technion SEE = Service Enterprise Engineering
SEELab: Data-repositories for research and teachingI For example:
I Bank Anonymous: 1 year, 350K calls by 15 agents - in 2000.Brown, Gans, Sakov, Shen, Zeltyn, Zhao (JASA),paved the way to:
I U.S. Bank: 2.5 years, 220M calls, 40M by 1000 agentsI Israeli Cellular: 2.5 years, 110M calls, 25M calls by 750 agentsI Israeli Bank: from January 2010, daily-deposit at a SEESafeI Home (Rambam) Hospital: 4 years, 1000 beds, ward-level flowI 5 EDs: gathered by the late David Sinreich, ED arrivals & LOS
SEEStat: Environment for graphical EDA in real-timeI Universal Design, Internet Access, Real-Time Response.
SEEServer: Free for academic useI RegisterI Access U.S. Bank, Bank Anonymous, Home Hospital
31
Technion SEE = Service Enterprise Engineering
SEELab: Data-repositories for research and teachingI For example:
I Bank Anonymous: 1 year, 350K calls by 15 agents - in 2000.Brown, Gans, Sakov, Shen, Zeltyn, Zhao (JASA),paved the way to:
I U.S. Bank: 2.5 years, 220M calls, 40M by 1000 agentsI Israeli Cellular: 2.5 years, 110M calls, 25M calls by 750 agentsI Israeli Bank: from January 2010, daily-deposit at a SEESafeI Home (Rambam) Hospital: 4 years, 1000 beds, ward-level flowI 5 EDs: gathered by the late David Sinreich, ED arrivals & LOS
SEEStat: Environment for graphical EDA in real-timeI Universal Design, Internet Access, Real-Time Response.
SEEServer: Free for academic useI RegisterI Access U.S. Bank, Bank Anonymous, Home Hospital
31
eg. RFID-Based Data: Mass Casualty Event (MCE)Drill: Chemical MCE, Rambam Hospital, May 2010
מאייר -קלים ודחק'מרתף פנימית ו-משפחותקרדיולוגיהעורנוירולוגיהבינוניים
נספח לנוהלקרדיולוגיה,עור,נוירולוגיה-בינונייםחדר אוכל-קשים
ד"מלר-משולבים
Focus on severely wounded casualties (≈ 40 in drill)Note: 20 observers support real-time control (helps validation)
32
Data Cleaning: MCE with RFID Support
Data-base Company report comment Asset id order Entry date Exit date Entry date Exit date
4 1 1:14:07 PM 1:14:00 PM 6 1 12:02:02 PM 12:33:10 PM 12:02:00 PM 12:33:00 PM 8 1 11:37:15 AM 12:40:17 PM 11:37:00 AM exit is missing
10 1 12:23:32 PM 12:38:23 PM 12:23:00 PM 12 1 12:12:47 PM 12:35:33 PM 12:35:00 PM entry is missing 15 1 1:07:15 PM 1:07:00 PM 16 1 11:18:19 AM 11:31:04 AM 11:18:00 AM 11:31:00 AM 17 1 1:03:31 PM 1:03:00 PM 18 1 1:07:54 PM 1:07:00 PM 19 1 12:01:58 PM 12:01:00 PM 20 1 11:37:21 AM 12:57:02 PM 11:37:00 AM 12:57:00 PM 21 1 12:01:16 PM 12:37:16 PM 12:01:00 PM
22 1 12:04:31 PM 12:20:40 PM first customer is missing
22 2 12:27:37 PM 12:27:00 PM 25 1 12:27:35 PM 1:07:28 PM 12:27:00 PM 1:07:00 PM 27 1 12:06:53 PM 12:06:00 PM
28 1 11:21:34 AM 11:41:06 AM 11:41:00 AM 11:53:00 AMexit time instead of entry time
29 1 12:21:06 PM 12:54:29 PM 12:21:00 PM 12:54:00 PM 31 1 11:40:54 AM 12:30:16 PM 11:40:00 AM 12:30:00 PM 31 2 12:37:57 PM 12:54:51 PM 12:37:00 PM 12:54:00 PM 32 1 11:27:11 AM 12:15:17 PM 11:27:00 AM 12:15:00 PM 33 1 12:05:50 PM 12:13:12 PM 12:05:00 PM 12:15:00 PM wrong exit time 35 1 11:31:48 AM 11:40:50 AM 11:31:00 AM 11:40:00 AM 36 1 12:06:23 PM 12:29:30 PM 12:06:00 PM 12:29:00 PM 37 1 11:31:50 AM 11:48:18 AM 11:31:00 AM 11:48:00 AM 37 2 12:59:21 PM 12:59:00 PM 40 1 12:09:33 PM 12:35:23 PM 12:09:00 PM 12:35:00 PM 43 1 12:58:21 PM 12:58:00 PM 44 1 11:21:25 AM 11:52:30 AM 11:52:00 AM entry is missing 46 1 12:03:56 PM 12:03:00 PM 48 1 11:19:47 AM 11:19:00 AM 49 1 12:20:36 PM 12:20:00 PM 52 1 11:21:29 AM 11:50:49 AM 11:21:00 AM 11:50:00 AM 52 2 12:10:07 PM 1:07:28 PM 12:10:00 PM 1:07:00 PM recorded as exit 53 1 12:24:26 PM 12:24:00 PM 57 1 11:32:02 AM 11:58:31 AM 11:58:00 AM entry is missing 57 2 12:59:41 PM 1:14:00 PM 12:59:00 PM 1:14:00 PM 60 1 12:27:12 PM 12:48:41 PM 12:27:00 PM 12:48:00 PM 63 1 12:10:04 PM 12:10:00 PM 64 1 11:30:29 AM 12:43:38 PM 11:30:00 AM 12:43:00 PM
- Imagine “Cleaning" 60,000+ customers per day (call centers) !
- “Psychology" of Data Trust and Transfer (e.g. 2 years till transfer)33
Event-Logs in a Call Center (Bank Anonymous)
5
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
AA0101 44777 37635950 2 PS 990901 18:15:47 18:15:52 5 18:15:52 18:16:57 65 AGENT 18:16:56 18:18:48 112 ANAT
AA0101 44778 0 0 NE 990901 18:30:43 18:30:52 9 00:00:00 00:00:00 0 AGENT 18:30:51 18:30:54 3 MORIAH
AA0101 44779 0 0 PS 990901 18:51:47 18:52:02 15 00:00:00 00:00:00 0 AGENT 18:52:02 18:55:30 208 TOVA
AA0101 44780 0 0 PS 990901 19:19:04 19:19:17 13 00:00:00 00:00:00 0 AGENT 19:19:15 19:20:20 65 MEIR
AA0101 44781 0 0 PS 990901 19:39:19 19:39:30 11 00:00:00 00:00:00 0 AGENT 19:39:29 19:41:42 133 BENSION
AA0101 44782 0 0 NW 990901 20:08:13 20:08:25 12 00:00:00 00:00:00 0 AGENT 20:08:28 20:08:41 13 NO_SERVER
AA0101 44783 0 0 PS 990901 20:23:51 20:24:05 14 00:00:00 00:00:00 0 AGENT 20:24:04 20:24:33 29 BENSION
AA0101 44784 0 0 NW 990901 20:36:54 20:37:14 20 00:00:00 00:00:00 0 AGENT 20:37:13 20:38:07 54 BENSION
AA0101 44785 0 0 PS 990901 20:50:07 20:50:16 9 00:00:00 00:00:00 0 AGENT 20:50:15 20:51:32 77 BENSION
AA0101 44786 0 0 PS 990901 21:04:41 21:04:51 10 00:00:00 00:00:00 0 AGENT 21:04:50 21:05:59 69 TOVA
AA0101 44787 0 0 PS 990901 21:25:00 21:25:13 13 00:00:00 00:00:00 0 AGENT 21:25:13 21:28:03 170 AVI
AA0101 44788 0 0 PS 990901 21:50:40 21:50:54 14 00:00:00 00:00:00 0 AGENT 21:50:54 21:51:55 61 AVI
AA0101 44789 9103060 2 NE 990901 22:05:40 22:05:46 6 22:05:46 22:09:52 246 AGENT 22:09:51 22:13:41 230 AVI
AA0101 44790 14558621 2 PS 990901 22:24:11 22:24:17 6 22:24:17 22:26:16 119 AGENT 22:26:15 22:27:28 73 VICKY
AA0101 44791 0 0 PS 990901 22:46:27 22:46:37 10 00:00:00 00:00:00 0 AGENT 22:46:36 22:47:03 27 AVI
AA0101 44792 67158097 2 PS 990901 23:05:07 23:05:13 6 23:05:13 23:05:30 17 AGENT 23:05:29 23:06:49 80 VICKY
AA0101 44793 15317126 2 PS 990901 23:28:52 23:28:58 6 23:28:58 23:30:08 70 AGENT 23:30:07 23:35:03 296 DARMON
AA0101 44794 0 0 PS 990902 00:10:47 00:12:05 78 00:00:00 00:00:00 0 HANG 00:00:00 00:00:00 0 NO_SERVER
AA0101 44795 0 0 PS 990902 07:16:52 07:17:01 9 00:00:00 00:00:00 0 AGENT 07:17:01 07:17:44 43 ANAT
AA0101 44796 0 0 PS 990902 07:50:05 07:50:16 11 00:00:00 00:00:00 0 AGENT 07:50:16 07:53:03 167 STEREN
- Unsynchronized transition times, consistently34
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
IdleBreak
ILDUBank January 24, 2010 from 06:00 to 23:59:59
Call-Backs Outgoing
Inbound
Available
IdleBreak
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
1
1 1 1 1
1
1
2
1
1
1
6
1
2
1
3
1 1
11
1
3
1
61
1
1 1
1
1
1
1
1
1
1 2
1
1
1
1
2 1
1 1
1
1
1
1
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1
11 11 1
1
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1
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1
1
1
2
23
1 1
1
1
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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.pdf seminar_presentation_boaz_estimatinger 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
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.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
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314 93
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64
3 3 4
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335 3 33 3
58 4 3
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44 35 7334 3
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14 93
3
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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.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
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3314
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547
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873
20863633 3536
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40
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198 35148
29
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3005
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3671
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19 356
126
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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)
OrderCheckbooks
Identification
ChangePassword
Agent
Fax
Query Transfer
Securities
StockMarket
PerformOperation
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-Rates2 Daily Peaks
CC: Dec. 1995, (USA, 700 Helpdesks) CC: May 1959 (England)
Q-Science
May 1959!
Dec 1995!
(Help Desk Institute)
Arrival Rate
Time 24 hrs
Time 24 hrs
% Arrivals
(Lee A.M., Applied Q-Th)
28
Q-Science
May 1959!
Dec 1995!
(Help Desk Institute)
Arrival Rate
Time 24 hrs
Time 24 hrs
% Arrivals
(Lee A.M., Applied Q-Th)
28
CC: Nov. 1999 (Israel) ED: Jan.–July 2007 (Israel)
Arrival Process, in 1999
Yearly Monthly
Daily Hourly
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
Ave
rage
num
ber
of c
ases
Time (Resolution 60 min.)
HomeHospital Patients Arrivals to Emergency Department Total for January2007 February2007 March2007 April2007 May2007 June2007 July2007,Sundays
55
Dynamics: Parsimonious Models (Congestion Laws)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.)
Perc
enta
geMedium 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
I Area normalized to 100%I State-Space Collapse
56
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-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)
57
Durations: Answering MachineIsraeli 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
Tree-Network Tomography:Reconstruct topologyfrom root and leaves
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
58
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
Rel
ativ
e fre
quen
cies
%
Time(seconds)( resolution 30)
AnonymousBank Handled Wait Time, TOTAL Total 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
Rel
ativ
e fre
quen
cies
%
Time(seconds)( resolution 1)
USBank Wait time(waiting), Retail May 2002, Weekdays
Scheduling Priorities (sec.) [compare Hospital LOS (hours)]Medium Israeli Bank
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0.550
0.600
0.650
20.00 70.00 120.00 170.00 220.00 270.00 320.00 370.00
Rel
ativ
e fre
quen
cies
%
Time(seconds)( resolution 1)
ILBank Wait time (all) August 2007, Weekdays
59
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
60
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
60
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
60
Durations: (Im)Patience while Waiting (Psychology)Palm: (1943–53): Irritation ∝ Hazard Rate
Regular over VIP Customers – Israeli Bank
14
16
I Challenges: Un-Censoring, Dependence, Smoothing- requires Call-by-Call Data
I Here: VIP Customers are more Patient (Needy)I Peaks of abandonment at times of Announcements
61
Durations: (Im)Patience while Waiting (Psychology)Palm: (1943–53): Irritation ∝ Hazard Rate
Regular over VIP Customers – Israeli Bank
14
16I Challenges: Un-Censoring, Dependence, Smoothing- requires Call-by-Call Data
I Here: VIP Customers are more Patient (Needy)I Peaks of abandonment at times of Announcements
61
Durations: (Im)Patience while Waiting (Psychology)Palm: (1943–53): Irritation ∝ Hazard Rate
Regular over VIP Customers – Israeli Bank
14
16I Challenges: Un-Censoring, Dependence, Smoothing- requires Call-by-Call Data
I Here: VIP Customers are more Patient (Needy)I Peaks of abandonment at times of Announcements
61
Protocols + PsychologyPatient 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
Ha
zard
Fu
nct
ion
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)
62
Little’s Law: Call Center & Emergency Department
Time-Gap: # in System lags behind Piecewise-Little (L = λ×W )
Little’s Law and the Offered-Load: Empirical Adventures in Call Centers and Hospitals
The Black-Box View of a Call Center: Number in Q vs. Little shows a time-gap.
USBank Customers in queue(average), Telesales10.10.2001
0.0020.0040.0060.0080.00
100.00120.00140.00160.00180.00200.00
07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00
Time (Resolution 30 min.)
Num
ber o
f cas
es
Customers in queue(average) Little's law
HomeHospital Average patients in EDFebruary 2004, Wednesdays
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
00:00 02:00 04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00
Time (Resolution 30 min.)
Ave
rage
num
ber o
f cas
es
Average patients in ED Lambda*E(S) smoothed
בהסתכלות ממוצעת על יום באמצע השבוע לאורך מספר חודשים
HomeHospital Average patients in EDTotal for February2004 March2004 April2004 May2004 June2004 July2004 August2004 September2004 October2004,Week days
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
00:00 02:00 04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00
Time (Resolution 60 min.)
Ave
rage
num
ber o
f cas
es
Average patients in ED Lambda*E(S)
⇒ Time-Varying Little’s LawI Berstemas & Mourtzinou;I Fralix, Riano, Serfozo; . . .
63
Little’s Law: Call Center & Emergency Department
Time-Gap: # in System lags behind Piecewise-Little (L = λ×W )
Little’s Law and the Offered-Load: Empirical Adventures in Call Centers and Hospitals
The Black-Box View of a Call Center: Number in Q vs. Little shows a time-gap.
USBank Customers in queue(average), Telesales10.10.2001
0.0020.0040.0060.0080.00
100.00120.00140.00160.00180.00200.00
07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00
Time (Resolution 30 min.)
Num
ber o
f cas
es
Customers in queue(average) Little's law
HomeHospital Average patients in EDFebruary 2004, Wednesdays
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
00:00 02:00 04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00
Time (Resolution 30 min.)
Ave
rage
num
ber o
f cas
es
Average patients in ED Lambda*E(S) smoothed
בהסתכלות ממוצעת על יום באמצע השבוע לאורך מספר חודשים
HomeHospital Average patients in EDTotal for February2004 March2004 April2004 May2004 June2004 July2004 August2004 September2004 October2004,Week days
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
00:00 02:00 04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00
Time (Resolution 60 min.)
Ave
rage
num
ber o
f cas
es
Average patients in ED Lambda*E(S)
⇒ Time-Varying Little’s LawI Berstemas & Mourtzinou;I Fralix, Riano, Serfozo; . . .
63
Protocols: Staffing (N) vs. Offered-Load (R = λ× E(S))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 (13 summer weeks, week-days)64
Technion - Israel Institute of Technology
The William Davidson Faculty of Industrial Engineering and Management Center for Service Enterprise Engineering (SEE)
http://ie.technion.ac.il/Labs/Serveng/
SEEStat 3.0 Tutorial
August 23, 2012
41
HomeHospital Data Background: The data we rely on was collected at a large Israeli hospital. This hospital consists of about 1000 beds and 45 medical units. The data includes detailed information on patient flow throughout the hospital, over a period of several years (January 2004–October 2007). In particular, the data allows one to follow the paths of individual patients throughout their stay at the hospital, including admission, discharge, and transfers between hospital units. The data does not acknowledge resolutions within the ED or within wards.
Nd - average number of patients that arrived per weekday, for period January 1, 2004 - October, 31, 2007
Ny - average number of patients that arrived per year, for years 2004, 2005, 2006, all days (for year 2007 data not fully completed; missing two months - November and December).
N - average number of patients in ED/ED-to-Ward transfer/Wards, recorded at 12:00 per weekday, for period January 1, 2004 - October, 31, 2007
NX-Ray - average number of patients in X-Ray at 10:00 per weekday, for period January 1, 2004 - October, 31, 2007
LOS - length of stay in ED/ED-to-Ward transfer/Wards/X-Ray per weekday, for period January 1, 2004 - October, 31, 2007
Arrivals at the ED
Arrivals directly to Wards
Arrivals at the X-Ray
Ward
WardWard
Hospitalization
X-Ray Exits
Discharges
Discharges
Emergency Department
Hospital
X-Ray
ED-to-Ward transfer
Nd = 341 patients
Ny= 115997 patients
Nd = 228 patients
(67% from all arrivals at the ED)
N = 60 patientsLOS = 3 hours
N = 11 patientsLOS = 3 hours Nd = 113 patients
Ny = 37973 patients
(33% from all arrivals at the ED)
NX-Ray = 15 patients
LOS = 52 minutes
Nd = 225 patients
N = 905 patients
LOS = 3.7 days
Nd = 106 patients
Ny= 30416 patients (24% from all arrivals at hospital)
Nd = 127 patients
Ny= 32713 patients
10
3
6
9
12
15
18
21
24
27
30
33
36
39
42
45
48
51
00:00 02:00 04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00Time (Resolution 60 min.)
HomeHospital Number of Patients in Emergency Department (average), Emergency Internal Medicine Unit03.01.200510.01.200517.01.200524.01.200531.01.200507.02.200514.02.200521.02.200528.02.200507.03.200514.03.200521.03.200528.03.200504.04.200511.04.200518.04.200525.04.200502.05.200509.05.200516.05.200523.05.200530.05.200506.06.200520.06.200527.06.200504.07.200511.07.200518.07.200525.07.200501.08.200508.08.200515.08.200522.08.200529.08.200505.09.200512.09.200519.09.200526.09.200507.11.200514.11.200521.11.200528.11.200505.12.200512.12.200519.12.200526.12.2005Mondays
1
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
Freq
uenc
ies
Number of patients (resolution 1)
HomeHospital Time by ED Internal and Surgery state (sec.)January 2004-October 2007, all days
2
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
que
ncie
s
Number of patients (resolution 1)
HomeHospital Time by ED Internal and Surgery state (sec.)January 2004-October 2007, all days
[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
cases
Time (Resolution 60 min.)
HomeHospital Number of Patients in Emergency Department Internal and Surgery,January 2004-October 2007, all days
3
0
250
500
750
1000
1250
1500
1750
2000
2250
2500
0 10 20 30 40 50 60 70 80 90 100
Freq
uenc
ies
Number of patients (resolution 1)
HomeHospital Time by ED Internal and Surgery state (sec.)January 2004-October 2007, all days
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
4
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
5
0
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
3000
3250
3500
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Freq
uenc
ies
Number of patients (resolution 1)
HomeHospital Time by ED Internal state (sec.)January 2004-October 2007, all days
6
020406080
100120140160180200220240260280300320340
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Fre
que
ncie
s
Number of patients (resolution 1)
HomeHospital Time by ED Internal state (sec.)January 2004-October 2007, all days
[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)
10
15
20
25
30
35
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
cases
Time (Resolution 60 min.)
HomeHospital Number of Patients in Emergency Department (average), Emergency Internal Medicine Unit
January 2004-October 2007, all days
7
0
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
3000
3250
3500
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Freq
uenc
ies
Number of patients (resolution 1)
HomeHospital Time by ED Internal state (sec.)January 2004-October 2007, all days
Fitting Mixtures of Distributions
Normal (26.47%): location = 13 scale = 4.15Normal (24.17%): location = 20 scale = 6.09Normal (49.36%): location = 30 scale = 9.84Morning
Normal Intermediate hoursNormal Evening
Normal
8
Time by ED Internal state (sec.) Statistics N 120960000 N(average per day) 86400 Mean 23.63 Standard Deviation 10.7 Variance 114.4 Median 22 Minimum 0 Maximum 70 Skewness 0.557 Kurtosis -0.23178 Standard Error Mean 0.00097 Interquartile Range 16 Mean Absolute Deviation 8.808 Median Absolute Deviation(MAD) 8 Coefficient of Variation (CV) (%) 45.27 L-moment 2 (half of Gini's Mean Difference) 6.031 L-Skewness 0.121 L-Kurtosis 0.0813 Coefficient of L-variation (L-CV)(%) (Gini's Coefficient) 25.53
Parameter Estimates
Components Mixing Proportions (%) Location Scale Shape Mean Standard Deviation 1. Normal 26.47 13.01 4.15 13.01 4.15 2. Normal 24.17 20.00 6.09 20.00 6.09 3. Normal 49.36 30.04 9.84 30.04 9.84
Goodness-of-Fit Tests
Tests Statistic DF p Value Residuals Std 0.017 Kolmogorov-Smirnov 0.04 <.0001 Cramer-von Mises 34138.3 <.0001 Andersen-Darling 200657.9 <.0001 Chi-Square 584234.3 65 <.0001
51
Example 5.3: Number of patients in Internal ED Average per 10minute intervals, only on Mondays during 2005
Return to the "Statistical Models (Summaries)" window. Press the "New Model" button. Select "Time Series", then "Intraday". In the “Variable” tab, select "Number of Patients in Emergency Department (average)". In the "Select Categories" tab, select "Emergency Internal Medicine Unit". Click the "Dates->" button. Mark "Individual days for aggregated day. Select months from June 2005 to December 2005.
Open tab "Days" and select "Mondays". Click "OK".
03.10.2005
17.10.2005
0.005.00
10.0015.0020.0025.0030.0035.0040.0045.0050.0055.00
0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00
Num
ber o
f cas
es
Time (Resolution 10 min.)
HomeHospital Number of Patients in Emergency Department (average), Emergency Internal Medicine Unit
June2005 July2005 August2005 September2005 October2005 November2005 December2005,Mondays
06.06.2005 20.06.2005 27.06.2005 04.07.2005 11.07.2005 18.07.2005 25.07.2005 01.08.2005 08.08.2005 15.08.2005 22.08.2005 29.08.2005 05.09.2005 12.09.2005 19.09.2005 26.09.2005 03.10.2005 10.10.2005 17.10.2005 24.10.2005 31.10.2005 07.11.2005 14.11.2005 21.11.2005 28.11.2005 05.12.2005 12.12.2005 19.12.2005 26.12.2005 Mondays
12
0
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
15 20 25 30 35 40 45 50
Freq
uenc
ies
Number of patients (resolution 1)
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)
13
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
“Recall":I Arrivals processes to EDs “are" inhomogeneous Poisson
(≈ piecewise Poisson)I M/M/∞ (ample-server queue) has Poisson steady-stateI Birth-Death processes are time-reversible hence steady-state
invariant under state-conditioning
Our EDA “implies" (w/ Armony, Marmor, Tseytlin, Yom-Tov):I Israeli ED census d
= M/M/∞I ED d
= Reversible Birth-Death process, hence conditioning onfinite capacity, say B, gives rise to M/M/B/B (Erlang-B)
I U.S. ED census d= M/M/B/B, which can be (has been) used to
analyze Ambulance Diversion (ED Blocking)I 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
79
Internal ED, non-holiday Mondays, 13:00-23:00
“Recall":I Arrivals processes to EDs “are" inhomogeneous Poisson
(≈ piecewise Poisson)I M/M/∞ (ample-server queue) has Poisson steady-stateI Birth-Death processes are time-reversible hence steady-state
invariant under state-conditioning
Our EDA “implies" (w/ Armony, Marmor, Tseytlin, Yom-Tov):I Israeli ED census d
= M/M/∞
I ED d= Reversible Birth-Death process, hence conditioning on
finite capacity, say B, gives rise to M/M/B/B (Erlang-B)I U.S. ED census d
= M/M/B/B, which can be (has been) used toanalyze Ambulance Diversion (ED Blocking)
I 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
79
Internal ED, non-holiday Mondays, 13:00-23:00
“Recall":I Arrivals processes to EDs “are" inhomogeneous Poisson
(≈ piecewise Poisson)I M/M/∞ (ample-server queue) has Poisson steady-stateI Birth-Death processes are time-reversible hence steady-state
invariant under state-conditioning
Our EDA “implies" (w/ Armony, Marmor, Tseytlin, Yom-Tov):I Israeli ED census d
= M/M/∞I ED d
= Reversible Birth-Death process, hence conditioning onfinite capacity, say B, gives rise to M/M/B/B (Erlang-B)
I U.S. ED census d= M/M/B/B, which can be (has been) used to
analyze Ambulance Diversion (ED Blocking)
I 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
79
Internal ED, non-holiday Mondays, 13:00-23:00
“Recall":I Arrivals processes to EDs “are" inhomogeneous Poisson
(≈ piecewise Poisson)I M/M/∞ (ample-server queue) has Poisson steady-stateI Birth-Death processes are time-reversible hence steady-state
invariant under state-conditioning
Our EDA “implies" (w/ Armony, Marmor, Tseytlin, Yom-Tov):I Israeli ED census d
= M/M/∞I ED d
= Reversible Birth-Death process, hence conditioning onfinite capacity, say B, gives rise to M/M/B/B (Erlang-B)
I U.S. ED census d= M/M/B/B, which can be (has been) used to
analyze Ambulance Diversion (ED Blocking)I 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 realities79
Emergency-Department Network
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
80
Prerequisite II: Models (FNets)“Laws of Large Numbers" capture Predictable Variability
Deterministic Models: Scale Averages-out Stochastic Individualism
# Severely-Wounded Patients, 11:00-13:00 (Censored LOS)
Cleaning Data – An Example: RFID data in an MCE Drill
0
10
20
30
40
50
60
11:09 11:16 11:24 11:31 11:38 11:45 11:52 12:00 12:07 12:14 12:21 12:28 12:36 12:43 12:50 12:57 13:04 13:12 13:19 13:26
entriesexitsentries (original)exits (original)
0
5
10
15
20
25
30
11:09 11:16 11:24 11:31 11:38 11:45 11:52 12:00 12:07 12:14 12:21 12:28 12:36 12:43 12:50 12:57 13:04 13:12 13:19 13:26
number of patientsnumber of patients (original)
I Transient Q’s:I Control of Mass Casualty Events (w/ I. Cohen, N. Zychlinski)I Staffing Chemical MCE (w/ G. Yom-Tov)
81
Prerequisite II: Models (FNets)“Laws of Large Numbers" capture Predictable Variability
Deterministic Models: Scale Averages-out Stochastic Individualism
# Severely-Wounded Patients, 11:00-13:00 (Censored LOS)
Cleaning Data – An Example: RFID data in an MCE Drill
0
10
20
30
40
50
60
11:09 11:16 11:24 11:31 11:38 11:45 11:52 12:00 12:07 12:14 12:21 12:28 12:36 12:43 12:50 12:57 13:04 13:12 13:19 13:26
entriesexitsentries (original)exits (original)
0
5
10
15
20
25
30
11:09 11:16 11:24 11:31 11:38 11:45 11:52 12:00 12:07 12:14 12:21 12:28 12:36 12:43 12:50 12:57 13:04 13:12 13:19 13:26
number of patientsnumber of patients (original)
I Transient Q’s:I Control of Mass Casualty Events (w/ I. Cohen, N. Zychlinski)I Staffing Chemical MCE (w/ G. Yom-Tov)
81
The Basic Service-Network Model: Erlang-R
w/ G. Yom-Tov
Erlang-R:
Needy (s-servers)
Content (Delay)
1-p
p
1
2
Arrivals Patient discharge
Needy (st-servers)
rate- μ
Content (Delay) rate - δ
1-p
p
1
2
Arrivals Poiss(λt) Patient discharge
Erlang-R (IE: Repairman Problem 50’s; CS: Central-Server 60’s) =2-station “Jackson" Network = (M/M/S, M/M/∞) :
I λt – Time-Varying Arrival rateI St – Number of Servers (Nurses / Physicians)I µ – Service rate (E [Service] = 1
µ)
I p – ReEntrant (Feedback) fractionI δ – Content-to-Needy rate (E [Content] = 1
δ)
82
Fluid Model↔ (Time-Varying) Erlang-R System
Erlang-R:
Needy (s-servers)
Content (Delay)
1-p
p
1
2
Arrivals Patient discharge
Needy (st-servers)
rate- μ
Content (Delay) rate - δ
1-p
p
1
2
Arrivals Poiss(λt) Patient discharge
FNet of a 2-station “Jackson" Network:
ddt
q1t = λt − µ ·
(q1
t ∧ st)
+ δ · q2t ,
ddt
q2t = p · µ ·
(q1
t ∧ st)− δ · q2
t .
(1)
83
Fluid Model↔ (Time-Varying) Erlang-R System
Erlang-R:
Needy (s-servers)
Content (Delay)
1-p
p
1
2
Arrivals Patient discharge
Needy (st-servers)
rate- μ
Content (Delay) rate - δ
1-p
p
1
2
Arrivals Poiss(λt) Patient discharge
FNet of a 2-station “Jackson" Network:
ddt
q1t = λt − µ ·
(q1
t ∧ st)
+ δ · q2t ,
ddt
q2t = p · µ ·
(q1
t ∧ st)− δ · q2
t .
(1)
83
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
Tota
l N
um
be
r o
f P
ati
en
ts
Time
Cumulative Arrivals
Cumulative Departures
15
20
25
30
er of M
CE Patients in ED
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
Num
be
Time
I Recurrent/Repeated services in MCE Events: eg. Injection every 15 minutes
I Fluid (Sample-path) Modeling, via Functional Strong Laws of Large NumbersI Stochastic Modeling, via Functional Central Limit Theorems
I ED in MCE: Confidence-interval, usefully narrow for ControlI ED in normal (time-varying) conditions: Personnel Staffing
84
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
Tota
l N
um
be
r o
f P
ati
en
ts
Time
Cumulative Arrivals
Cumulative Departures
15
20
25
30
er of M
CE Patients in ED
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
Num
be
Time
I Recurrent/Repeated services in MCE Events: eg. Injection every 15 minutesI Fluid (Sample-path) Modeling, via Functional Strong Laws of Large NumbersI Stochastic Modeling, via Functional Central Limit Theorems
I ED in MCE: Confidence-interval, usefully narrow for ControlI ED in normal (time-varying) conditions: Personnel Staffing84
An Asymptotic Framework: Erlang-R in the ED
System = Emergency Department (eg. Rambam Hospital)I SimNet = Customized ED-Simulator (Marmor & Sinreich)I QNet = Erlang-R (time-varying 2-station Jackson; w/ Yom-Tov)I FNets = 2-dim dynamical system (Massey & Whitt)I DNets = 2-dim Markovian Service Net (w/ Massey and Reiman)
Asymptotic FrameworkI Data and MeasurementsI Fit a simple model (time-varying Erlang-R) to a complex reality
(ED Physicians)I Develop FNets (Offered-Load of Physicians) and (relevant) DNet
(if needed)I Use FNet / DNet for Design (√-Staffing), Analysis, . . .I Simulate reality (ED with √-staffing of Physicians)I Validation: stable performance, confidence intervals, . . .
85
An Asymptotic Framework: Erlang-R in the ED
System = Emergency Department (eg. Rambam Hospital)I SimNet = Customized ED-Simulator (Marmor & Sinreich)I QNet = Erlang-R (time-varying 2-station Jackson; w/ Yom-Tov)I FNets = 2-dim dynamical system (Massey & Whitt)I DNets = 2-dim Markovian Service Net (w/ Massey and Reiman)
Asymptotic FrameworkI Data and MeasurementsI Fit a simple model (time-varying Erlang-R) to a complex reality
(ED Physicians)I Develop FNets (Offered-Load of Physicians) and (relevant) DNet
(if needed)I Use FNet / DNet for Design (√-Staffing), Analysis, . . .I Simulate reality (ED with √-staffing of Physicians)I Validation: stable performance, confidence intervals, . . .
85
Case Study: Emergency Ward Staffing
Many-Server (↑ ∞) Approximations for Small Systems (1-7)
I Staffing resolution: 1 hourI Lower bound: 1 doctor per typeI Flexible (time-varying square-root) staffing: Yunan’s LectureI Rounding effects⇒ Not all performance levels achievable
90 Servers 1-7 Doctors
Challenges in Applications
Many server approximations; Are they good for smallsystems?
P(wait) in ED using time-varying staffing and MOLapproximation
Large systems (90 servers) Small Systems (1-7 servers)
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
TimeBeta=0.1 Beta=0.5 Beta=1 Beta=1.5
Average of 100 replications.
Galit Yom-Tov INFORMS 2010
Challenges in Applications
Many server approximations; Are they good for smallsystems?
P(wait) in ED using time-varying staffing and MOLapproximation
Large systems (90 servers) Small Systems (1-7 servers)
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
TimeBeta=0.1 Beta=0.5 Beta=1 Beta=1.5
Average of 100 replications.
Galit Yom-Tov INFORMS 2010
86
ED Patient Flow: The Physicians Viewwith J. Huang, B. Carmeli; N. Shimkin
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66 66
-
λ01 λ0
2 λ0J
· · ·
P 0(j,k)
P (k, l)
d1 d2 dJ
m01 m0
2 m0J
Triage-Patients
IP-Patients
ExitsS
Arrivals
C1(·) C2(·) C3(·) CK(·)
m1 m2 m3 mK
· · ·
1
Goal: Adhere to Triage-Constraints, then release In-Process PatientsModels: Time-Varying (FNet) - in process; Stationary (DNet) - v. soon
87
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.
Yet, heavily-loaded queueing systems with Congestion Index = 0.1(Waiting one order of magnitude less than Service) are prevalent:
I Call Centers: Wait “seconds" for minutes service;I Transportation: Search “minutes" for hours parking;I Hospitals: Wait “hours" in ED for days hospitalization in IW’s.
Moreover, a significant fraction not delayed in queue: e.g. in well-runI CCs: 50% served “immediately" & 90% utilization⇒ QEDI EDs + IWs: ? Multiple scales! IW-“Beds" (10’s) are QED while
IW-Doctors (1‘s) are in conventional heavy-traffic (hours wait forminutes service), hence the bottlenecks
88
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.
Yet, heavily-loaded queueing systems with Congestion Index = 0.1(Waiting one order of magnitude less than Service) are prevalent:
I Call Centers: Wait “seconds" for minutes service;I Transportation: Search “minutes" for hours parking;I Hospitals: Wait “hours" in ED for days hospitalization in IW’s.
Moreover, a significant fraction not delayed in queue: e.g. in well-runI CCs: 50% served “immediately" & 90% utilization⇒ QEDI EDs + IWs: ? Multiple scales! IW-“Beds" (10’s) are QED while
IW-Doctors (1‘s) are in conventional heavy-traffic (hours wait forminutes service), hence the bottlenecks
88
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.
Yet, heavily-loaded queueing systems with Congestion Index = 0.1(Waiting one order of magnitude less than Service) are prevalent:
I Call Centers: Wait “seconds" for minutes service;I Transportation: Search “minutes" for hours parking;I Hospitals: Wait “hours" in ED for days hospitalization in IW’s.
Moreover, a significant fraction not delayed in queue: e.g. in well-runI CCs: 50% served “immediately" & 90% utilization⇒ QEDI 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 bottlenecks
88
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.
Yet, heavily-loaded queueing systems with Congestion Index = 0.1(Waiting one order of magnitude less than Service) are prevalent:
I Call Centers: Wait “seconds" for minutes service;I Transportation: Search “minutes" for hours parking;I Hospitals: Wait “hours" in ED for days hospitalization in IW’s.
Moreover, a significant fraction not delayed in queue: e.g. in well-runI CCs: 50% served “immediately" & 90% utilization⇒ QEDI EDs + IWs: ? Multiple scales! IW-“Beds" (10’s) are QED while
IW-Doctors (1‘s) are in conventional heavy-traffic (hours wait forminutes service), hence the bottlenecks88
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).89
Erlang-A: Practical Relevance?
Experience:I Arrival process not pure Poisson (time-varying, σ2 too large)I Service times not Exponential (typically close to LogNormal)I Patience times not Exponential (various patterns observed).
I Building Blocks need not be independent (eg. long waitassociated with long service; w/ M. Reich and Y. Ritov)
I Customers and Servers not homogeneous (classes, skills)I Customers return for service (after busy, abandonment;
dependently; P. Khudiakov, M. Gorfine, P. Feigin)I · · · , and more.
Question: Is Erlang-A Relevant?
YES ! Fitting a Simple Model to a Complex Reality, bothTheoretically and Practically
90
Erlang-A: Practical Relevance?
Experience:I Arrival process not pure Poisson (time-varying, σ2 too large)I Service times not Exponential (typically close to LogNormal)I Patience times not Exponential (various patterns observed).
I Building Blocks need not be independent (eg. long waitassociated with long service; w/ M. Reich and Y. Ritov)
I Customers and Servers not homogeneous (classes, skills)I Customers return for service (after busy, abandonment;
dependently; P. Khudiakov, M. Gorfine, P. Feigin)I · · · , and more.
Question: Is Erlang-A Relevant?
YES ! Fitting a Simple Model to a Complex Reality, bothTheoretically and Practically
90
Erlang-A: Practical Relevance?
Experience:I Arrival process not pure Poisson (time-varying, σ2 too large)I Service times not Exponential (typically close to LogNormal)I Patience times not Exponential (various patterns observed).
I Building Blocks need not be independent (eg. long waitassociated with long service; w/ M. Reich and Y. Ritov)
I Customers and Servers not homogeneous (classes, skills)I Customers return for service (after busy, abandonment;
dependently; P. Khudiakov, M. Gorfine, P. Feigin)I · · · , and more.
Question: Is Erlang-A Relevant?
YES ! Fitting a Simple Model to a Complex Reality, bothTheoretically and Practically
90
Erlang-A: Fitting a Simple Model to a Complex Reality
Hourly Performance vs. Erlang-A Predictions (1 year)
% Abandon E[Wait] %{Wait > 0}
0 0.1 0.2 0.3 0.4 0.5 0.60
0.1
0.2
0.3
0.4
0.5
Probability to abandon (Erlang−A)
Pro
babi
lity
to a
band
on (
data
)
0 50 100 150 200 2500
50
100
150
200
250
Waiting time (Erlang−A), sec
Wai
ting
time
(dat
a), s
ec
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Probability of wait (Erlang−A)
Pro
babi
lity
of w
ait (
data
)
I Empirically-Based & Theoretically-Supported Estimation of(Im)Patience: θ = P{Ab}/E[Wq])
I Small Israeli Bank (more examples in progress)I Hourly performance vs. Erlang-A predictions, 1 year: aggregated
groups of 40 similar hours91
QED Theory (Erlang ’13; Halfin & Whitt ’81; w/Garnett & Reiman ’02)
Consider a sequence of steady-state M/M/N + M 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;
Here R = Offered Loadeg. R = 25 call/min. × 4 min./call = 100
92
Erlang-A: QED Approximations (Examples)
Assume Offered Load R not small (λ→∞).
Let β = β
õ
θ, h(·) =
φ(·)1− Φ(·) = hazard rate of N (0,1).
I Delay Probability:
P{Wq > 0} ≈[
1 +
√θ
µ· h(β)
h(−β)
]−1
.
I Probability to Abandon:
P{Ab|Wq > 0} ≈ 1√N·√θ
µ·[h(β)− β
].
I P{Ab} ∝ E[Wq] , both order 1√N
:
P{Ab}E[Wq]
= θ (≈ g(0) > 0).
93
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 half-hour intervals (13 summer weeks, week-days)I Erlang-A approximations for the appropriate µ/θ ≈ 3
94
Process Limits (Queueing, Waiting)• QN = {QN(t), t ≥ 0} : stochastic process obtained by
centering and rescaling:
QN =QN −N√
N
• QN(∞) : stationary distribution of QN
• Q = {Q(t), t ≥ 0} : process defined by: QN(t)d→ Q(t).
��
�
�
� �
QN(t) QN(∞)
Q(t) Q(∞)
t → ∞
t → ∞
N → ∞ N → ∞
Approximating (Virtual) Waiting Time
VN =√N VN ⇒ V =
[1
μQ
]+(Puhalskii, 1994)
stochastic process
Waiting Time
95
QED Intuition: Why P{Wq > 0} ∈ (0, 1) ?
1. Why subtle: Consider a large service system (e.g. call center).I Fix λ and let N ↑ ∞: P{Wq > 0} ↓ 0, P(I > 0) ↑ 1.
I Fix N and let λ ↑ ∞: P{Wq > 0} ↑ 1, P(I > 0) ↓ 0.I ⇒ Must have both λ and N increase simultaneously:I ⇒ (CLT) Square-root staffing: N ≈ R + β
√R(
λ ≈ Nµ− β√
Nµ)
2. Erlang-A (M/M/N+M), with parameters λ, µ, θ; N, in which µ = θ:(Im)Patience and Service-times are equally distributed.
I Steady-state: L(M/M/N + M)d= L(M/M/∞)
d= Poisson(R), with
R = λ/µ (Offered-Load)I Poisson(R)
d≈ R + Z
√R, with Z d
= N(0, 1).
I P{Wq(M/M/N + M) > 0} PASTA= P{L(M/M/N + M) ≥ N} µ=θ
=
P{L(M/M/∞) ≥ N} ≈ P{R + Z√
R ≥ N} =
P{Z ≥ (N − R)/√
R}√· staffing≈ P{Z ≥ β} = 1− Φ(β).
3. QED Excursions
96
QED Intuition: Why P{Wq > 0} ∈ (0, 1) ?
1. Why subtle: Consider a large service system (e.g. call center).I Fix λ and let N ↑ ∞: P{Wq > 0} ↓ 0, P(I > 0) ↑ 1.I Fix N and let λ ↑ ∞: P{Wq > 0} ↑ 1, P(I > 0) ↓ 0.
I ⇒ Must have both λ and N increase simultaneously:I ⇒ (CLT) Square-root staffing: N ≈ R + β
√R(
λ ≈ Nµ− β√
Nµ)
2. Erlang-A (M/M/N+M), with parameters λ, µ, θ; N, in which µ = θ:(Im)Patience and Service-times are equally distributed.
I Steady-state: L(M/M/N + M)d= L(M/M/∞)
d= Poisson(R), with
R = λ/µ (Offered-Load)I Poisson(R)
d≈ R + Z
√R, with Z d
= N(0, 1).
I P{Wq(M/M/N + M) > 0} PASTA= P{L(M/M/N + M) ≥ N} µ=θ
=
P{L(M/M/∞) ≥ N} ≈ P{R + Z√
R ≥ N} =
P{Z ≥ (N − R)/√
R}√· staffing≈ P{Z ≥ β} = 1− Φ(β).
3. QED Excursions
96
QED Intuition: Why P{Wq > 0} ∈ (0, 1) ?
1. Why subtle: Consider a large service system (e.g. call center).I Fix λ and let N ↑ ∞: P{Wq > 0} ↓ 0, P(I > 0) ↑ 1.I Fix N and let λ ↑ ∞: P{Wq > 0} ↑ 1, P(I > 0) ↓ 0.I ⇒ Must have both λ and N increase simultaneously:I ⇒ (CLT) Square-root staffing: N ≈ R + β
√R(
λ ≈ Nµ− β√
Nµ)
2. Erlang-A (M/M/N+M), with parameters λ, µ, θ; N, in which µ = θ:(Im)Patience and Service-times are equally distributed.
I Steady-state: L(M/M/N + M)d= L(M/M/∞)
d= Poisson(R), with
R = λ/µ (Offered-Load)I Poisson(R)
d≈ R + Z
√R, with Z d
= N(0, 1).
I P{Wq(M/M/N + M) > 0} PASTA= P{L(M/M/N + M) ≥ N} µ=θ
=
P{L(M/M/∞) ≥ N} ≈ P{R + Z√
R ≥ N} =
P{Z ≥ (N − R)/√
R}√· staffing≈ P{Z ≥ β} = 1− Φ(β).
3. QED Excursions
96
QED Intuition: Why P{Wq > 0} ∈ (0, 1) ?
1. Why subtle: Consider a large service system (e.g. call center).I Fix λ and let N ↑ ∞: P{Wq > 0} ↓ 0, P(I > 0) ↑ 1.I Fix N and let λ ↑ ∞: P{Wq > 0} ↑ 1, P(I > 0) ↓ 0.I ⇒ Must have both λ and N increase simultaneously:I ⇒ (CLT) Square-root staffing: N ≈ R + β
√R(
λ ≈ Nµ− β√
Nµ)
2. Erlang-A (M/M/N+M), with parameters λ, µ, θ; N, in which µ = θ:(Im)Patience and Service-times are equally distributed.
I Steady-state: L(M/M/N + M)d= L(M/M/∞)
d= Poisson(R), with
R = λ/µ (Offered-Load)I Poisson(R)
d≈ R + Z
√R, with Z d
= N(0, 1).
I P{Wq(M/M/N + M) > 0} PASTA= P{L(M/M/N + M) ≥ N} µ=θ
=
P{L(M/M/∞) ≥ N} ≈ P{R + Z√
R ≥ N} =
P{Z ≥ (N − R)/√
R}√· staffing≈ P{Z ≥ β} = 1− Φ(β).
3. QED Excursions
96
QED Intuition: Why P{Wq > 0} ∈ (0, 1) ?
1. Why subtle: Consider a large service system (e.g. call center).I Fix λ and let N ↑ ∞: P{Wq > 0} ↓ 0, P(I > 0) ↑ 1.I Fix N and let λ ↑ ∞: P{Wq > 0} ↑ 1, P(I > 0) ↓ 0.I ⇒ Must have both λ and N increase simultaneously:I ⇒ (CLT) Square-root staffing: N ≈ R + β
√R(
λ ≈ Nµ− β√
Nµ)
2. Erlang-A (M/M/N+M), with parameters λ, µ, θ; N, in which µ = θ:(Im)Patience and Service-times are equally distributed.
I Steady-state: L(M/M/N + M)d= L(M/M/∞)
d= Poisson(R), with
R = λ/µ (Offered-Load)
I Poisson(R)d≈ R + Z
√R, with Z d
= N(0, 1).
I P{Wq(M/M/N + M) > 0} PASTA= P{L(M/M/N + M) ≥ N} µ=θ
=
P{L(M/M/∞) ≥ N} ≈ P{R + Z√
R ≥ N} =
P{Z ≥ (N − R)/√
R}√· staffing≈ P{Z ≥ β} = 1− Φ(β).
3. QED Excursions
96
QED Intuition: Why P{Wq > 0} ∈ (0, 1) ?
1. Why subtle: Consider a large service system (e.g. call center).I Fix λ and let N ↑ ∞: P{Wq > 0} ↓ 0, P(I > 0) ↑ 1.I Fix N and let λ ↑ ∞: P{Wq > 0} ↑ 1, P(I > 0) ↓ 0.I ⇒ Must have both λ and N increase simultaneously:I ⇒ (CLT) Square-root staffing: N ≈ R + β
√R(
λ ≈ Nµ− β√
Nµ)
2. Erlang-A (M/M/N+M), with parameters λ, µ, θ; N, in which µ = θ:(Im)Patience and Service-times are equally distributed.
I Steady-state: L(M/M/N + M)d= L(M/M/∞)
d= Poisson(R), with
R = λ/µ (Offered-Load)I Poisson(R)
d≈ R + Z
√R, with Z d
= N(0, 1).
I P{Wq(M/M/N + M) > 0} PASTA= P{L(M/M/N + M) ≥ N} µ=θ
=
P{L(M/M/∞) ≥ N} ≈ P{R + Z√
R ≥ N} =
P{Z ≥ (N − R)/√
R}√· staffing≈ P{Z ≥ β} = 1− Φ(β).
3. QED Excursions
96
QED Intuition: Why P{Wq > 0} ∈ (0, 1) ?
1. Why subtle: Consider a large service system (e.g. call center).I Fix λ and let N ↑ ∞: P{Wq > 0} ↓ 0, P(I > 0) ↑ 1.I Fix N and let λ ↑ ∞: P{Wq > 0} ↑ 1, P(I > 0) ↓ 0.I ⇒ Must have both λ and N increase simultaneously:I ⇒ (CLT) Square-root staffing: N ≈ R + β
√R(
λ ≈ Nµ− β√
Nµ)
2. Erlang-A (M/M/N+M), with parameters λ, µ, θ; N, in which µ = θ:(Im)Patience and Service-times are equally distributed.
I Steady-state: L(M/M/N + M)d= L(M/M/∞)
d= Poisson(R), with
R = λ/µ (Offered-Load)I Poisson(R)
d≈ R + Z
√R, with Z d
= N(0, 1).
I P{Wq(M/M/N + M) > 0} PASTA= P{L(M/M/N + M) ≥ N} µ=θ
=
P{L(M/M/∞) ≥ N} ≈ P{R + Z√
R ≥ N} =
P{Z ≥ (N − R)/√
R}√· staffing≈ P{Z ≥ β} = 1− Φ(β).
3. QED Excursions
96
QED Intuition: Why P{Wq > 0} ∈ (0, 1) ?
1. Why subtle: Consider a large service system (e.g. call center).I Fix λ and let N ↑ ∞: P{Wq > 0} ↓ 0, P(I > 0) ↑ 1.I Fix N and let λ ↑ ∞: P{Wq > 0} ↑ 1, P(I > 0) ↓ 0.I ⇒ Must have both λ and N increase simultaneously:I ⇒ (CLT) Square-root staffing: N ≈ R + β
√R(
λ ≈ Nµ− β√
Nµ)
2. Erlang-A (M/M/N+M), with parameters λ, µ, θ; N, in which µ = θ:(Im)Patience and Service-times are equally distributed.
I Steady-state: L(M/M/N + M)d= L(M/M/∞)
d= Poisson(R), with
R = λ/µ (Offered-Load)I Poisson(R)
d≈ R + Z
√R, with Z d
= N(0, 1).
I P{Wq(M/M/N + M) > 0} PASTA= P{L(M/M/N + M) ≥ N} µ=θ
=
P{L(M/M/∞) ≥ N} ≈ P{R + Z√
R ≥ N} =
P{Z ≥ (N − R)/√
R}√· staffing≈ P{Z ≥ β} = 1− Φ(β).
3. QED Excursions96
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.
97
QED Intuition via Excursions: Asymptotics
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
õ/
Both apply as√
N(1− ρN)→ β,−∞ < β <∞.
Hence, P(Customer Wait > 0) ∼[1 +
h(δ)/δ
h(−β)/β)
]−1
, and
P(Server Wait > 0) = P(Customer Wait = 0)
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.
98
QED Intuition via Excursions: Asymptotics
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
õ/
Both apply as√
N(1− ρN)→ β,−∞ < β <∞.
Hence, P(Customer Wait > 0) ∼[1 +
h(δ)/δ
h(−β)/β)
]−1
, and
P(Server Wait > 0) = P(Customer Wait = 0)
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} ≈ 1/2.98
QED Erlang-X (Markovian Q’s: Performance Analysis)I Pre-History, 1914: Erlang (Erlang-B = M/M/n/n, Erlang-C = M/M/n)I Pre-History, 1974: Jagerman (Erlang-B)I History Milestone, 1981: Halfin-Whitt (Erlang-C, GI/M/n)I Erlang-A (M/M/N+M), 2002: w/ Garnett & ReimanI Erlang-A with General (Im)Patience (M/M/N+G), 2005: w/ ZeltynI Erlang-C (ED+QED), 2009: w/ ZeltynI Erlang-B with Retrial, 2010: Avram, Janssen, van LeeuwaardenI Refined Asymptotics (Erlang A/B/C), 2008-2011: Janssen, van Leeuwaarden,
Zhang, ZwartI Production Q’s, 2011: Reed & ZhangI Universal Erlang-A, ongoing: w/ Gurvich & HuangI Queueing Networks:
I (Semi-)Closed: Nurse Staffing (Jennings & de Vericourt), CCs with IVR (w/Khudiakov), Erlang-R (w/ Yom-Tov)
I CCs with Abandonment and Retrials: w. Massey, Reiman, Rider, StolyarI Markovian Service Networks: w/ Massey & Reiman
I Leaving out:I Non-Exponential Service Times: M/D/n (Erlang-D), G/Ph/n, · · · , G/GI/n+GI,
Measure-Valued DiffusionsI Dimensioning (Staffing): M/M/n, · · · , time-varying Q’s, V- and Reversed-V, · · ·I Control: V-network, Reversed-V, · · · , SBRNets
99
Operational Regimes: Q(uality) vs. E(fficiency)
CC in a Large Israeli Bank
P{Wq > 0} vs. (R, N) R-Slice: P{Wq > 0} vs. N
• Intercepting plane of the previous plot for a "constant" offered load (actually
18 18.5R≤ ≤ ) • Smoother line was created using smoothing splines. • General shape of the smoothing line is as the Garnett delay functions! • In 2-d view, we get the following:
3 Operational Regimes:I QD: ≤ 25%
I QED: 25%− 75%
I ED: ≥ 75%
100
Operational Regimes: Conceptual Framework
R: Offered LoadDef. R = Arrival-rate × Average-Service-Time = λ
µ
eg. R = 25 calls/min. × 4 min./call = 100
N = #Agents ? Intuition, as R or N increase unilaterally.
QD Regime: N ≈ R + δR , 0.1 < δ < 0.25 (eg. N = 115)I Framework developed in O. Garnett’s MSc thesisI Rigorously: (N − R)/R → δ, as N, λ ↑ ∞, with µ fixed.I Performance: Delays are rare events
ED Regime: N ≈ R − γR , 0.1 < γ < 0.25 (eg. N = 90)I Essentially all customers are delayedI Wait same order as service-time; γ% Abandon (10-25%).
QED Regime: N ≈ R + β√
R , −1 < β < +1 (eg. N = 100)I Erlang 1913-24, Halfin & Whitt 1981 (for Erlang-C)I %Delayed between 25% and 75%I E[Wait] ∝ 1√
N× E[Service] (sec vs. min); 1-5% Abandon.
101
Operational Regimes: Conceptual Framework
R: Offered LoadDef. R = Arrival-rate × Average-Service-Time = λ
µ
eg. R = 25 calls/min. × 4 min./call = 100
N = #Agents ? Intuition, as R or N increase unilaterally.
QD Regime: N ≈ R + δR , 0.1 < δ < 0.25 (eg. N = 115)I Framework developed in O. Garnett’s MSc thesisI Rigorously: (N − R)/R → δ, as N, λ ↑ ∞, with µ fixed.I Performance: Delays are rare events
ED Regime: N ≈ R − γR , 0.1 < γ < 0.25 (eg. N = 90)I Essentially all customers are delayedI Wait same order as service-time; γ% Abandon (10-25%).
QED Regime: N ≈ R + β√
R , −1 < β < +1 (eg. N = 100)I Erlang 1913-24, Halfin & Whitt 1981 (for Erlang-C)I %Delayed between 25% and 75%I E[Wait] ∝ 1√
N× E[Service] (sec vs. min); 1-5% Abandon.
101
Operational Regimes: Conceptual Framework
R: Offered LoadDef. R = Arrival-rate × Average-Service-Time = λ
µ
eg. R = 25 calls/min. × 4 min./call = 100
N = #Agents ? Intuition, as R or N increase unilaterally.
QD Regime: N ≈ R + δR , 0.1 < δ < 0.25 (eg. N = 115)I Framework developed in O. Garnett’s MSc thesisI Rigorously: (N − R)/R → δ, as N, λ ↑ ∞, with µ fixed.I Performance: Delays are rare events
ED Regime: N ≈ R − γR , 0.1 < γ < 0.25 (eg. N = 90)I Essentially all customers are delayedI Wait same order as service-time; γ% Abandon (10-25%).
QED Regime: N ≈ R + β√
R , −1 < β < +1 (eg. N = 100)I Erlang 1913-24, Halfin & Whitt 1981 (for Erlang-C)I %Delayed between 25% and 75%I E[Wait] ∝ 1√
N× E[Service] (sec vs. min); 1-5% Abandon.
101
Operational Regimes: Conceptual Framework
R: Offered LoadDef. R = Arrival-rate × Average-Service-Time = λ
µ
eg. R = 25 calls/min. × 4 min./call = 100
N = #Agents ? Intuition, as R or N increase unilaterally.
QD Regime: N ≈ R + δR , 0.1 < δ < 0.25 (eg. N = 115)I Framework developed in O. Garnett’s MSc thesisI Rigorously: (N − R)/R → δ, as N, λ ↑ ∞, with µ fixed.I Performance: Delays are rare events
ED Regime: N ≈ R − γR , 0.1 < γ < 0.25 (eg. N = 90)I Essentially all customers are delayedI Wait same order as service-time; γ% Abandon (10-25%).
QED Regime: N ≈ R + β√
R , −1 < β < +1 (eg. N = 100)I Erlang 1913-24, Halfin & Whitt 1981 (for Erlang-C)I %Delayed between 25% and 75%I E[Wait] ∝ 1√
N× E[Service] (sec vs. min); 1-5% Abandon.101
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
102
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
102
Universal Approximations: Erlang-A (M/M/N + M)agents
arrivals
abandonment
λ
µ
1
2
n
…
queue
θ
w/ I. Gurvich & J. Huang
I QNet: Birth & Death Queue, with B - D rates
F (q) = λ− µ · (q ∧ n)− θ · (q − n)+, q = 0,1, . . .
I FNet: Dynamical (Deterministic) System – ODEdxt = F (xt )dt , t ≥ 0
I 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)
103
Universal Approximations: Erlang-A (M/M/N + M)agents
arrivals
abandonment
λ
µ
1
2
n
…
queue
θ
w/ I. Gurvich & J. Huang
I QNet: Birth & Death Queue, with B - D rates
F (q) = λ− µ · (q ∧ n)− θ · (q − n)+, q = 0,1, . . .
I FNet: Dynamical (Deterministic) System – ODEdxt = F (xt )dt , t ≥ 0
I 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)
103
Universal Approximations: Erlang-A (M/M/N + M)agents
arrivals
abandonment
λ
µ
1
2
n
…
queue
θ
w/ I. Gurvich & J. Huang
I QNet: Birth & Death Queue, with B - D rates
F (q) = λ− µ · (q ∧ n)− θ · (q − n)+, q = 0,1, . . .
I FNet: Dynamical (Deterministic) System – ODEdxt = F (xt )dt , t ≥ 0
I 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)
103
Universal Approximations: Erlang-A (M/M/N + M)agents
arrivals
abandonment
λ
µ
1
2
n
…
queue
θ
w/ I. Gurvich & J. Huang
I QNet: Birth & Death Queue, with B - D rates
F (q) = λ− µ · (q ∧ n)− θ · (q − n)+, q = 0,1, . . .
I FNet: Dynamical (Deterministic) System – ODEdxt = F (xt )dt , t ≥ 0
I 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)
103
Universal Approximations: Erlang-A (M/M/N + M)agents
arrivals
abandonment
λ
µ
1
2
n
…
queue
θ
w/ I. Gurvich & J. Huang
I QNet: Birth & Death Queue, with B - D rates
F (q) = λ− µ · (q ∧ n)− θ · (q − n)+, q = 0,1, . . .
I FNet: Dynamical (Deterministic) System – ODEdxt = F (xt )dt , t ≥ 0
I 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)103
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 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?
Coupling “Busy" + “Idle" Excursions of B&D and the correspondingDiffusion (durations order 1√
λ)
104
Universal Diffusion: Tractability
I 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 +
õ
θ
φ(β/√µ)
Φ(β/√µ)
1− Φ(β/√θ)
φ(β/√θ)
]−1
.
105
Universal Approximation: Accuracy
I ∆λ is the “balancing" state, obtained by solving
λ = µ(n ∧∆λ) + θ(∆λ − n)+.
Solution: ∆λ = λµ −
(λµ − n
)+ (1− µ
θ
).
Specifically: QD = λµ ; ED = n + 1
θ (λ− nµ); QED = n +O(√λ))
I 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).
I Accuracy: higher than heavy-traffic limits
106
Universal Approximation: Accuracy
I ∆λ is the “balancing" state, obtained by solving
λ = µ(n ∧∆λ) + θ(∆λ − n)+.
Solution: ∆λ = λµ −
(λµ − n
)+ (1− µ
θ
).
Specifically: QD = λµ ; ED = n + 1
θ (λ− nµ); QED = n +O(√λ))
I 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).
I Accuracy: higher than heavy-traffic limits106
Universal Approximation: Why 2λ?
I Semi-martingale representation of the B&D process:Fluid + Martingale
I Predictable quadratic variation:∫ t
0[λ+ µ(Qs ∧ n) + θ(Qs − n)+]ds
I In steady-state, arrival rate ≡ departure rate:
λ = E[µ(Qs ∧ n) + θ(Qs − n)+]
I Expectation of the predictable quadratic variation:
E∫ t
0[λ+ µ(Qs ∧ n) + θ(Qs − n)+]ds = 2λt
I dMartingalet ≈√
2λ · dBrowniant
107
Reconciling Steady-State and Time-Varying ModelsI Challenge: Accommodate time-varying demand (routine)I Prerequisite: Flexible Capacity
I As in Call Centers and to a degree in Healthcare,I In contrast to rigid (fixed) staffing level during a shift: doomed to
alternate between overloading and underloading
I Idea/Goal: In the face of time-varying demand, designtime-varying staffing which accommodates demand such thatperformance is stable over time
I Solution: In fact, a time-varying system with Steady-Stateperformance, at all times, via (Modified) Offered-Load(Square-Root) Staffing.
I History:I Jennings, M., Reiman, Whitt (1996): Emergence of the
phenomenon, via infinite-server heuristicsI Feldman, M., Massey, Whitt (2008): Stabilize delay probability via
QED staffing (justified theoretically only for Erlang-A with µ = θ)I Liu and Whitt (ongoing): Stabilize abandonment probability by ED
staffing, via a corresponding network, theoretically and empiricallyI Huang, Gurvich, M. (ongoing): QED theory
108
Reconciling Steady-State and Time-Varying ModelsI Challenge: Accommodate time-varying demand (routine)I Prerequisite: Flexible Capacity
I As in Call Centers and to a degree in Healthcare,I In contrast to rigid (fixed) staffing level during a shift: doomed to
alternate between overloading and underloadingI Idea/Goal: In the face of time-varying demand, design
time-varying staffing which accommodates demand such thatperformance is stable over time
I Solution: In fact, a time-varying system with Steady-Stateperformance, at all times, via (Modified) Offered-Load(Square-Root) Staffing.
I History:I Jennings, M., Reiman, Whitt (1996): Emergence of the
phenomenon, via infinite-server heuristicsI Feldman, M., Massey, Whitt (2008): Stabilize delay probability via
QED staffing (justified theoretically only for Erlang-A with µ = θ)I Liu and Whitt (ongoing): Stabilize abandonment probability by ED
staffing, via a corresponding network, theoretically and empiricallyI Huang, Gurvich, M. (ongoing): QED theory
108
Reconciling Steady-State and Time-Varying ModelsI Challenge: Accommodate time-varying demand (routine)I Prerequisite: Flexible Capacity
I As in Call Centers and to a degree in Healthcare,I In contrast to rigid (fixed) staffing level during a shift: doomed to
alternate between overloading and underloadingI Idea/Goal: In the face of time-varying demand, design
time-varying staffing which accommodates demand such thatperformance is stable over time
I Solution: In fact, a time-varying system with Steady-Stateperformance, at all times, via (Modified) Offered-Load(Square-Root) Staffing.
I History:I Jennings, M., Reiman, Whitt (1996): Emergence of the
phenomenon, via infinite-server heuristicsI Feldman, M., Massey, Whitt (2008): Stabilize delay probability via
QED staffing (justified theoretically only for Erlang-A with µ = θ)I Liu and Whitt (ongoing): Stabilize abandonment probability by ED
staffing, via a corresponding network, theoretically and empiricallyI Huang, Gurvich, M. (ongoing): QED theory
108
The Offered-Load R(t), t ≥ 0 (R(t)↔ R)
Empirically (in SEEStat):I Process: L(·) = Least number of servers that guarantees no delay.I Offered-Load Function R(·) = E [L(·)]
ILTelecom , Private12.05.2004
0.005.00
10.0015.0020.0025.0030.0035.0040.0045.0050.0055.0060.00
07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00
Time (Resolution 30 min.)
Num
ber o
f cas
es
Offered load Abandons Av_agents_in_system
109
The Offered-Load R(t), t ≥ 0 (R(t)↔ R)
Empirically (in SEEStat):I Process: L(·) = Least number of servers that guarantees no delay.I Offered-Load Function R(·) = E [L(·)]
ILTelecom , Private12.05.2004
0.005.00
10.0015.0020.0025.0030.0035.0040.0045.0050.0055.0060.00
07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00
Time (Resolution 30 min.)
Num
ber o
f cas
es
Offered load Abandons Av_agents_in_system
109
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
250
0 1 2 4 5 6 7 8
10
11
12
13
14
16
17
18
19
20
22
23
0
500
1000
1500
2000
Arr
iva
ls p
er
ho
ur
beta 1.2 beta 0 beta -1.2 Offered Load Arrivals
QDQED
ED
110
Time-Stable Performance of Time-Varying Systems
Delay Probability = as in the Stationary Erlang-A / RDelay Probability
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
10 1 2 4 5 6 7 8
10
11
12
13
14
16
17
18
19
20
22
23
beta 2 beta 1.6 beta 1.2 beta 0.8 beta 0.4 beta 0
beta -0.4 beta -0.8 beta -1.2 beta -1.6 beta -2
111
Time-Stable Performance of Time-Varying SystemsWaiting 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)
Waiting Time given Wait > 0:
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
Simulated Theoretical (N=175)
Waiting Time given Wait > 0:
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
Simulated Theoretical (N=160)
- 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).
112
Calculating the Offered-Load R(t), Theoretically
I Offered-Load Process: L(·) = Least number of servers thatguarantees no delay.
I Offered-Load Function R(t) = E [L(t)], t ≥ 0.Think Mt/G/N?
t + G vs. Mt/G/∞: Ample-Servers.
Four (all useful) representations, capturing “workload before t":
R(t) = E [L(t)] =
∫ t
−∞λ(u) · P(S > t − u)du = E
[A(t)− A(t − S)
]=
= E[∫ t
t−Sλ(u)du
]= E [λ(t − Se)] · E [S] ≈ ... .
I {A(t), t ≥ 0} Arrival-Process, rate λ(·);I S (Se) generic Service-Time (Residual Service-Time).I Relating L, λ,S (“W ”): Time-Varying Little’s Formula.
Stationary models: λ(t) ≡ λ then R(t) ≡ λ× E[S].
QED-c: Nt = Rt + βRct , 1/2 ≤ c < 1; (c = 1 separate analysis).
113
Calculating the Offered-Load R(t), Theoretically
I Offered-Load Process: L(·) = Least number of servers thatguarantees no delay.
I Offered-Load Function R(t) = E [L(t)], t ≥ 0.Think Mt/G/N?
t + G vs. Mt/G/∞: Ample-Servers.
Four (all useful) representations, capturing “workload before t":
R(t) = E [L(t)] =
∫ t
−∞λ(u) · P(S > t − u)du = E
[A(t)− A(t − S)
]=
= E[∫ t
t−Sλ(u)du
]= E [λ(t − Se)] · E [S] ≈ ... .
I {A(t), t ≥ 0} Arrival-Process, rate λ(·);I S (Se) generic Service-Time (Residual Service-Time).
I Relating L, λ,S (“W ”): Time-Varying Little’s Formula.Stationary models: λ(t) ≡ λ then R(t) ≡ λ× E[S].
QED-c: Nt = Rt + βRct , 1/2 ≤ c < 1; (c = 1 separate analysis).
113
Calculating the Offered-Load R(t), Theoretically
I Offered-Load Process: L(·) = Least number of servers thatguarantees no delay.
I Offered-Load Function R(t) = E [L(t)], t ≥ 0.Think Mt/G/N?
t + G vs. Mt/G/∞: Ample-Servers.
Four (all useful) representations, capturing “workload before t":
R(t) = E [L(t)] =
∫ t
−∞λ(u) · P(S > t − u)du = E
[A(t)− A(t − S)
]=
= E[∫ t
t−Sλ(u)du
]= E [λ(t − Se)] · E [S] ≈ ... .
I {A(t), t ≥ 0} Arrival-Process, rate λ(·);I S (Se) generic Service-Time (Residual Service-Time).I Relating L, λ,S (“W ”): Time-Varying Little’s Formula.
Stationary models: λ(t) ≡ λ then R(t) ≡ λ× E[S].
QED-c: Nt = Rt + βRct , 1/2 ≤ c < 1; (c = 1 separate analysis).
113
Calculating the Offered-Load R(t), Theoretically
I Offered-Load Process: L(·) = Least number of servers thatguarantees no delay.
I Offered-Load Function R(t) = E [L(t)], t ≥ 0.Think Mt/G/N?
t + G vs. Mt/G/∞: Ample-Servers.
Four (all useful) representations, capturing “workload before t":
R(t) = E [L(t)] =
∫ t
−∞λ(u) · P(S > t − u)du = E
[A(t)− A(t − S)
]=
= E[∫ t
t−Sλ(u)du
]= E [λ(t − Se)] · E [S] ≈ ... .
I {A(t), t ≥ 0} Arrival-Process, rate λ(·);I S (Se) generic Service-Time (Residual Service-Time).I Relating L, λ,S (“W ”): Time-Varying Little’s Formula.
Stationary models: λ(t) ≡ λ then R(t) ≡ λ× E[S].
QED-c: Nt = Rt + βRct , 1/2 ≤ c < 1; (c = 1 separate analysis).
113
Estimating / Predicting the Offered-Load
Must account for “service times of abandoning customers".
I Prevalent Assumption: Services and (Im)Patience independent.I But recall Patient VIPs: Willing to wait longer for more services.
Survival Functions by Type (Small Israeli Bank)
31
Service Time (cont’)Survival curve, by types
Time
Surv
ival
Service times stochastic order: SNew
st< SReg
st< SVIP
Patience times stochastic order: τNew
st< τReg
st< τVIP
114
Dependent Primitives: Service- vs. Waiting-Time
Average Service-Time as a function of Waiting-TimeU.S. Bank, Retail, Weedays, January-June, 2006
Introduction Relationship Between Service Time and Patience Workload and Offered-Load Empirical Results Future Research
Motivation
Mean Service-Time as a Function of Waiting-TimeU.S. Bank - Retail Banking Service - Weekdays - January-June, 2006
190
210
230
250
270
290
150
170
190
210
230
250
270
290
0 50 100 150 200 250 300
Waiting Time
Fitted Spline Curve E(S|τ>W=w)
⇒ Focus on ( Patience, Service-Time ) jointly , w/ Reich and Ritov.E [S |Patience = w ], w ≥ 0: Service-Time of the Unserved.
115
Dependent Primitives: Service- vs. Waiting-Time
Average Service-Time as a function of Waiting-TimeU.S. Bank, Retail, Weedays, January-June, 2006
Introduction Relationship Between Service Time and Patience Workload and Offered-Load Empirical Results Future Research
Motivation
Mean Service-Time as a Function of Waiting-TimeU.S. Bank - Retail Banking Service - Weekdays - January-June, 2006
190
210
230
250
270
290
150
170
190
210
230
250
270
290
0 50 100 150 200 250 300
Waiting Time
Fitted Spline Curve E(S|τ>W=w)
⇒ Focus on ( Patience, Service-Time ) jointly , w/ Reich and Ritov.E [S |Patience = w ], w ≥ 0: Service-Time of the Unserved.
115
(Imputing) Service-Times of Abandoning Customers
w/ M. Reich, Y. Ritov:
1. Estimate g(w) = E [S |Patience > Wait = w ], w ≥ 0:Mean service time of those served after waiting exactly w unitsof time (via non-linear regression: Si = g(Wi ) + εi );
2. Calculate
E [S |Patience = w ] = g(w)− g′(w)
hτ (w);
hτ (w) = hazard-rate of (im)patience (via un-censoring);
3. Offered-load calculations: Impute E [S |Patience = w ](or the conditional distribution).
Challenges: Stable and accurate inference of g,g′,hτ .
116
Extending the Notion of the “Offered-Load"
I Business (Banking Call-Center): Offered Revenues
I Healthcare (Maternity Wards): Fetus in stressI 2 patients (Mother + Child) = high operational and cognitive loadI Fetus dies⇒ emotional load dominates
I ⇒I Offered Operational Load
I Offered Cognitive Load
I Offered Emotional Load
I ⇒ Fair Division of Load (Routing) between 2 Maternity Wards:One attending to complications before birth, the other tocomplications after birth, and both share normal birth
117
Server Networksw/ A. Senderovic: Planning
then joined A. Gal, M. Weid, Y. Goldberg: Process Mining
ILDUBank January 24, 2010 Agent #043 Shift: 16:00 - 23:45
Inbound Call-Backs
Outgoing
Available Idle
Private Call
Break
I As Challenging - in theory and practice - as Customer NetworksI Uncharted territory: e.g. Gnedenko/Palm’s Machine Repair model,
Armony and Ward (Asymptotic Little & ASTA)118
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
119
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
120
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
120
Tele-Service Process (Beyond present SEE Data)
RetailService(IsraeliBank)
StatisticsORIEPsychol.MIS
START
Hello14 / 20
I.D.24 / 23Request
15 / 8
Stock17 / 21
Question14 / 20
Password creation62 / 42
Processing49 / 24
Confirmation29 / 9
Answer32 / 19
END202/190
Dead time18 / 17
Paperwork22 / 12
End of call5 / 3
Credit34 / 32
Others21 / 21
Checking21 / 9
1
0.65
0.050.27
0.950.03
0.62
0.29
0.17
0.28
0.17
0.11
0.05
0.05 0.17
0.23 0.08
0.38
0.15
0.15
0.93
0.14
0.03
0.5
0.03 0.26
0. 4
0.23
0.590.09
0.05
0.67
0.11
0.17
0.67
0.330.7
0.25
0.66
0.43
0.57
0.130.2
0.93
0.04
0. 6
Password creation62 / 42
0.67
0.33
STDAVG
121
Why Bother?I Server Networks: Unchartered Research TerritoryI In large call centers:
+One Second to Service-Time implies +Millions in costs,annually
⇒ Time and "Motion" Studies (Classical IE with New-age IT)
I Service-Process Model: Customer-Agent InteractionI Work Design (w/ Khudiakov)
eg. Cross-Selling: higher profit vs. longer (costlier) services;Analysis yields (congestion-dependent) cross-selling protocols
I “Worker" Design (w/ Gans, Liu, Shen & Ye)eg. Learning, Forgetting, . . . : Staffing & individual-performanceprediction, in a heterogenous environment
I IVR-Process Model: Customer-Machine Interaction75% bank-services, poor design, yet scarce research;Same approach, automatic (easier) data (w/ N. Yuviler)
122
Why Bother?I Server Networks: Unchartered Research TerritoryI In large call centers:
+One Second to Service-Time implies +Millions in costs,annually
⇒ Time and "Motion" Studies (Classical IE with New-age IT)
I Service-Process Model: Customer-Agent InteractionI Work Design (w/ Khudiakov)
eg. Cross-Selling: higher profit vs. longer (costlier) services;Analysis yields (congestion-dependent) cross-selling protocols
I “Worker" Design (w/ Gans, Liu, Shen & Ye)eg. Learning, Forgetting, . . . : Staffing & individual-performanceprediction, in a heterogenous environment
I IVR-Process Model: Customer-Machine Interaction75% bank-services, poor design, yet scarce research;Same approach, automatic (easier) data (w/ N. Yuviler)
122
Why Bother?I Server Networks: Unchartered Research TerritoryI In large call centers:
+One Second to Service-Time implies +Millions in costs,annually
⇒ Time and "Motion" Studies (Classical IE with New-age IT)
I Service-Process Model: Customer-Agent InteractionI Work Design (w/ Khudiakov)
eg. Cross-Selling: higher profit vs. longer (costlier) services;Analysis yields (congestion-dependent) cross-selling protocols
I “Worker" Design (w/ Gans, Liu, Shen & Ye)eg. Learning, Forgetting, . . . : Staffing & individual-performanceprediction, in a heterogenous environment
I IVR-Process Model: Customer-Machine Interaction75% bank-services, poor design, yet scarce research;Same approach, automatic (easier) data (w/ N. Yuviler)
122
ServNets: Data-Based Online Automatic Creationw/ V. Trofimov, E. Nadjharov, I. Gavako = Technion SEELab
I ServNets = QNets, SimNets, FNets, DNets
I SimNets of Service Systems = Virtual Realities
I SimNets also of QNEts, FNets, DNetseg. ED MD (Physics): Where are the Differential Equations?
I Ultimately: Research Labs, offering universal access to dataand ServNets, will become necessary (hence must be funded!)
I Data-based Research: Tradition in Physics, Chemistry, Biology;Psychology (now also in Transportation (Science) and(Behavioral) Economics)
I Why not in Service Science / Engineering / Management ?I Moreover, address the Reproducibility Crisis in Scientific
Research (Computation, Massive-data, . . .)
123
ServNets: Data-Based Online Automatic Creationw/ V. Trofimov, E. Nadjharov, I. Gavako = Technion SEELab
I ServNets = QNets, SimNets, FNets, DNets
I SimNets of Service Systems = Virtual Realities
I SimNets also of QNEts, FNets, DNetseg. ED MD (Physics): Where are the Differential Equations?
I Ultimately: Research Labs, offering universal access to dataand ServNets, will become necessary (hence must be funded!)
I Data-based Research: Tradition in Physics, Chemistry, Biology;Psychology (now also in Transportation (Science) and(Behavioral) Economics)
I Why not in Service Science / Engineering / Management ?I Moreover, address the Reproducibility Crisis in Scientific
Research (Computation, Massive-data, . . .)
123
ServNets: Data-Based Online Automatic Creationw/ V. Trofimov, E. Nadjharov, I. Gavako = Technion SEELab
I ServNets = QNets, SimNets, FNets, DNets
I SimNets of Service Systems = Virtual Realities
I SimNets also of QNEts, FNets, DNetseg. ED MD (Physics): Where are the Differential Equations?
I Ultimately: Research Labs, offering universal access to dataand ServNets, will become necessary (hence must be funded!)
I Data-based Research: Tradition in Physics, Chemistry, Biology;Psychology (now also in Transportation (Science) and(Behavioral) Economics)
I Why not in Service Science / Engineering / Management ?
I Moreover, address the Reproducibility Crisis in ScientificResearch (Computation, Massive-data, . . .)
123
ServNets: Data-Based Online Automatic Creationw/ V. Trofimov, E. Nadjharov, I. Gavako = Technion SEELab
I ServNets = QNets, SimNets, FNets, DNets
I SimNets of Service Systems = Virtual Realities
I SimNets also of QNEts, FNets, DNetseg. ED MD (Physics): Where are the Differential Equations?
I Ultimately: Research Labs, offering universal access to dataand ServNets, will become necessary (hence must be funded!)
I Data-based Research: Tradition in Physics, Chemistry, Biology;Psychology (now also in Transportation (Science) and(Behavioral) Economics)
I Why not in Service Science / Engineering / Management ?I Moreover, address the Reproducibility Crisis in Scientific
Research (Computation, Massive-data, . . .)123
Data-Based Creation ServNets: some Technicalities
I ServNets = QNets, SimNets, FNets, DNetsI 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)
I Algorithm: Dot Layout (but with cycles), based on Sugiyama,Tagawa, Toda (’81): “Visual Understanding of HierarchicalSystem Structures"
I Draws data directly from SEELab data-bases:I Relational DBs (Large! eg. USBank Full Binary = 37GB, Summary
Tables = 7GB)I Structure: Sequence of events/states, which (due to size)
partitioned (yet integrated) into days (eg. call centers) or months(eg. hospitals)
I Differs from industry DBs (in call centers, hospitals, websites)
124
Data-Based Creation ServNets: some Technicalities
I ServNets = QNets, SimNets, FNets, DNetsI 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)
I Algorithm: Dot Layout (but with cycles), based on Sugiyama,Tagawa, Toda (’81): “Visual Understanding of HierarchicalSystem Structures"
I Draws data directly from SEELab data-bases:I Relational DBs (Large! eg. USBank Full Binary = 37GB, Summary
Tables = 7GB)I Structure: Sequence of events/states, which (due to size)
partitioned (yet integrated) into days (eg. call centers) or months(eg. hospitals)
I Differs from industry DBs (in call centers, hospitals, websites)
124