Data-Based Service Networks:
A Research Framework for
Asymptotic Inference, Analysis & Control
of Service Systems
Avi Mandelbaum
Technion, Haifa, Israel
http://ie.technion.ac.il/serveng
Kellogg Operations Workshop, September 2012
◮ Overheads available at my Technion websites
1
Research Partners
◮ Students:Aldor∗, Baron∗, Carmeli∗, Cohen∗, Feldman∗, Garnett∗, Gurvich∗,
Khudiakov∗, Maman∗, Marmor∗, Reich∗, Rosenshmidt∗, Shaikhet∗,
Senderovic, Tseytlin∗, Yom-Tov∗, Yuviler, Zaied∗, Zeltyn∗,
Zychlinski∗, Zohar∗, Zviran∗, . . .
◮ Theory:Armony, Atar, Cohen, Gurvich, Huang, Jelenkovic, Kaspi, Massey,
Momcilovic, Reiman, Shimkin, Stolyar, Trofimov, Wasserkrug,
Whitt, Zeltyn, . . .
◮ Empirical/Statistical Analysis:Brown, Gans, Shen, Zhao; Zeltyn; Ritov, Goldberg; Gurvich, Huang,
Liberman; Armony, Marmor, Tseytlin, Yom-Tov; Nardi, Plonsky;
Gorfine, Ghebali; Pang, . . .
◮ Industry:Mizrahi Bank (A. Cohen, U. Yonissi), Rambam Hospital (R. Beyar, S.
Israelit, S. Tzafrir), IBM Research (OCR Project), Hapoalim Bank (G.
Maklef, T. Shlasky), Pelephone Cellular, . . .
◮ Technion SEE Center / Laboratory:Feigin; Trofimov, Nadjharov, Gavako, Kutsy; Liberman, Koren,
Plonsky, Senderovic; Research Assistants, . . .2
Contents
◮ Service Networks: Call Centers, Hospitals, Websites, · · ·◮ Redefine the paradigm of modeling/asymptotics via Data
◮ ServNets: QNets, SimNets; FNets, DNets
◮ Ultimate Goal: Data-based creation and validation
of ServNets, automatically in real-time
3
Contents
◮ Service Networks: Call Centers, Hospitals, Websites, · · ·◮ Redefine the paradigm of modeling/asymptotics via Data
◮ ServNets: QNets, SimNets; FNets, DNets
◮ Ultimate Goal: Data-based creation and validation
of ServNets, automatically in real-time
◮ Why be Optimistic? Pilot at the Technion SEELab◮ Lacking but Feasible: Dynamics, Durations, Protocols◮ Simple Models at the Service of Complex Realities
◮ State-Space Collapse (Queues, Waiting Times)◮ Congestion Laws (LN, Little, ImPatience, Staffing)◮ Universal Approximations: Simplifying the Asymptotic Landscape◮ Stabilizing Time-Varying Performance (Offered-Load)
◮ Successes: Palm/Erlang-R (ED Feedback = FNet),Palm/Erlang-A (CC Abandonment = DNet)
3
Contents
◮ Service Networks: Call Centers, Hospitals, Websites, · · ·◮ Redefine the paradigm of modeling/asymptotics via Data
◮ ServNets: QNets, SimNets; FNets, DNets
◮ Ultimate Goal: Data-based creation and validation
of ServNets, automatically in real-time
◮ Why be Optimistic? Pilot at the Technion SEELab◮ Lacking but Feasible: Dynamics, Durations, Protocols◮ Simple Models at the Service of Complex Realities
◮ State-Space Collapse (Queues, Waiting Times)◮ Congestion Laws (LN, Little, ImPatience, Staffing)◮ Universal Approximations: Simplifying the Asymptotic Landscape◮ Stabilizing Time-Varying Performance (Offered-Load)
◮ Successes: Palm/Erlang-R (ED Feedback = FNet),Palm/Erlang-A (CC Abandonment = DNet)
◮ Elsewhere : Process Mining (Petri Nets, BPM), Networks (Social,Biological, Complex, . . .), Simulation-based, . . .
◮ Scenic Route : Open Problems, New Directions, Uncharted Territories
3
Applying Queueing Asymptotics
There are by now numerous insightful asymptotic queueingmodels at our disposal, and many arise from, and create, deepbeautiful theory:
Has it helped one approximate or simulate a service systemmore efficiently, estimate its parameter more accurately, teach itto our students more effectively, perhaps even manage thesystem better?
I am of the opinion that the answers to such questions havebeen too often negative, that positive answers must have theoryand applications nurture each other, which is good, and my
approach to make this good happen is by marrying theory with
data .
4
Models Approximations
QNets SimNets
Accuracy
Phenomenology
Prevalent Asymptotic Approximations
System (Data)
5
Models Approximations
QNets SimNets FNet DNet Models
Accuracy
Phenomenology
X
Data–Based Prevalent Asymptotic Approximations Models
System (Data)
X X
6
Models Approximations
QNets SimNets FNet DNet Models
Accuracy
Phenomenology Value
X
X
Data–Based Prevalent Asymptotic Approximations Models
System (Data)
X X
7
System = Coin Tossing, Model = Binomial ; de Moivre 1738
Approx. / Models: SLLN (FNets), CLT (DNets) ; Laplace 1810
Value: Exceeds Value of originating stylized model
Normal, Brownian Motion ; Bachalier 1900
Poisson ; Poisson 1838
Models Approximations
QNets SimNets FNet DNet Models
Accuracy
Phenomenology Value
X
X
Data–Based Prevalent Asymptotic Approximations Models
System (Data)
X X
8
Models
Q F
Value
Data–Based Framework: (Almost) All Models Born Equal
SimNets
D
Data
2
Models
Q F
Value
Data–Based (Asymptotic) Framework: Simulation Mining
SimNets
D
Data
Data-Based
Real-Time
- Creation
- Application
- “Implies” other ServNets
- Virtual Realities
- Validation: Parsimony
- Theory: Reproducibility
- Application: Trust
- Data: Proprietary
3
Models
Q F
Value
Ultimately: Automatic “Discovery, Conformance, Enhancement”
SimNets
D
Data
State-Space Collapse
Multiple Scales (Regimes, UA)
Snapshots
Pathwise Little, ASTA, …
Substitution
Principle
Congestion
Laws
Service
Science
Inference: Patience, Predictors
Performance: G/G/N, QED
Control: Gc , SBR, FJ
Design: - Staffing, Pooling
Predictions
Scope of the Service Industry
Guangzhou Railway Station, Southern China
13
Call Centers, Then Hospitals, Now Internet
Call Centers - U.S. Stat.
◮ $200 – $300 billion annual expenditures
◮ 100,000 – 200,000 call centers
◮ “Window" into the company, for better or worse
◮ Over 3 million agents = 2% – 4% workforce
Healthcare - similar, plus unique challenges:
◮ Cost-figures far more staggering
◮ Risks much higher
◮ ED (initial focus) = hospital-window
◮ Over 3 million nurses
Internet - . . .
14
Call-Center Environment: Service Network
15
Operational Focus
Operational Measures:
◮ Surrogates for overall performance: Financial, Psychological; Clinical
◮ Easiest to quantify, measure, track online, react upon / Research
16
Call-Center Network: Gallery of Models
Agents
(CSRs)
Back-Office
Experts)(Consultants
VIP)Training (
Arrivals(Business Frontier
of the
21th Century)
Redial(Retrial)
Busy
)Rare(
Good
or
Bad
Positive: Repeat Business
Negative: New Complaint
Lost Calls
Abandonment
Agents
Service
Completion
Service Engineering: Multi-Disciplinary Process View
Forecasting
Statistics
New Services Design (R&D)
Operations,
Marketing
Organization Design:
Parallel (Flat)
Sequential (Hierarchical)
Sociology/Psychology,
Operations Research
Human Resource Management
Service Process Design
To Avoid Delay
To Avoid Starvation Skill Based Routing
(SBR) Design
Marketing,
Human Resources,
Operations Research,
MIS
CustomersInterface Design
Computer-TelephonyIntegration - CTIMIS/CS
Marketing
Operations/
Business
Process
Archive
Database
Design
Data Mining:
MIS, Statistics,
Operations
Research,
Marketing
Internet
Chat
Fax
Lost Calls
Service Completion)75% in Banks (
( Waiting Time
Return Time)
Logistics
CustomersSegmentation -CRM
Psychology,
Operations
Research,
Marketing
Expect 3 min
Willing 8 min
Perceive 15 min
Psychological
Process
Archive
Psychology,
Statistics
Training, IncentivesJob Enrichment
Marketing,
Operations Research
Human Factors Engineering
VRU/
IVR
Queue)Invisible (
VIP Queue
(If Required 15 min,
then Waited 8 min)
(If Required 6 min,
then Waited 8 min)
Information Design
Function
Scientific Discipline
Multi-Disciplinary
Index
Call Center Design
(Turnover up to
200% per Year)
(Sweat Shops
of the
21th Century)
Tele-StressPsychology
17
Call-Center Network: Gallery of Models
Agents
(CSRs)
Back-Office
Experts)(Consultants
VIP)Training (
Arrivals(Business Frontier
of the
21th Century)
Redial(Retrial)
Busy
)Rare(
Good
or
Bad
Positive: Repeat Business
Negative: New Complaint
Lost Calls
Abandonment
Agents
Service
Completion
Service Engineering: Multi-Disciplinary Process View
Forecasting
Statistics
New Services Design (R&D)
Operations,
Marketing
Organization Design:
Parallel (Flat)
Sequential (Hierarchical)
Sociology/Psychology,
Operations Research
Human Resource Management
Service Process Design
To Avoid Delay
To Avoid Starvation Skill Based Routing
(SBR) Design
Marketing,
Human Resources,
Operations Research,
MIS
CustomersInterface Design
Computer-TelephonyIntegration - CTIMIS/CS
Marketing
Operations/
Business
Process
Archive
Database
Design
Data Mining:
MIS, Statistics,
Operations
Research,
Marketing
Internet
Chat
Fax
Lost Calls
Service Completion)75% in Banks (
( Waiting Time
Return Time)
Logistics
CustomersSegmentation -CRM
Psychology,
Operations
Research,
Marketing
Expect 3 min
Willing 8 min
Perceive 15 min
Psychological
Process
Archive
Psychology,
Statistics
Training, IncentivesJob Enrichment
Marketing,
Operations Research
Human Factors Engineering
VRU/
IVR
Queue)Invisible (
VIP Queue
(If Required 15 min,
then Waited 8 min)
(If Required 6 min,
then Waited 8 min)
Information Design
Function
Scientific Discipline
Multi-Disciplinary
Index
Call Center Design
(Turnover up to
200% per Year)
(Sweat Shops
of the
21th Century)
Tele-StressPsychology
18
Skills-Based Routing in Call CentersEDA and OR, with I. Gurvich and P. Liberman
Mktg. ⇒
OR ⇒
HRM ⇒
MIS ⇒
19
ER / ED Environment: Service Network
Acute (Internal, Trauma) Walking
Multi-Trauma
20
ED-Environment in Israel
21
Queueing in a “Good" HospitalTong-ren Hospital at 6am, Beijing
22
Emergency-Department Network: Gallery of Models
Imaging
Laboratory
Experts
Interns
Returns (Old or New Problem)
“Lost” Patients
LWBS
Nurses
Statistics,
Human
Resource
Management
(HRM)
New Services Design (R&D)
Operations,
Marketing,
MIS
Organization Design:
Parallel (Flat) = ER
vs. a true ED
Sociology, Psychology,
Operations Research
Service Process
Design
Quality
Efficiency
Customers
Interface Design
Medicine
(High turnovers
Medical-Staff
shortage)
Operations/
Business
Process
Archive
Database
Design
Data Mining:
MIS, Statistics,
Operations
Research,
Marketing
Stretcher
Walking
Service Completion(sent to other department)
( Waiting Time
Active Dashboard )
Patients
Segmentation
Medicine,
Psychology,
Marketing
Psychological
Process
Archive
Human Factors
Engineering
(HFE)
Internal
Queue
Orthopedic
Queue
Arrivals
Function
Scientific Discipline
Multi-Disciplinary
Index
ED-Stress
Psychology
Operations
Research, 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 Design
MIS, HFE,
Operations Research
Psychology,
Statistics
Home
◮ Forecasting, Abandonment = LWBS, SBR ≈ Flow Control
23
ED Patient Flow: The Physicians Viewwith J. Huang, B. Carmeli
✫✪✬✩
❄ ❄ ❄
❅❅❅❅❅❅❘
❈❈❈❈❈❈❲
✠
✒
✁✁✁✁✕
❆❆❆❆❑
❍❍❍❍
❍❍❍❍
❍❍❨✛
❄
✻✻ ✻✻
✲
λ0
1λ0
2λ0
J
· · ·
P 0(j,k)
P (k, l)
d1 d2 dJ
m0
1m0
2m0
J
Triage-Patients
IP-Patients
ExitsS
Arrivals
C1(·) C2(·) C3(·) CK(·)
m1 m2 m3 mK
· · ·
◮ Goal: Adhere to Triage-Constraints, then release In-Process Patients
◮ Model = Multi-class Q with Feedback: Min. convex congestion costs ofIP-Patients, s.t. deadline constraints on Triage-Patients.
24
ED Patient Flow: The Physicians Viewwith J. Huang, B. Carmeli
✫✪✬✩
❄ ❄ ❄
❅❅❅❅❅❅❘
❈❈❈❈❈❈❲
✠
✒
✁✁✁✁✕
❆❆❆❆❑
❍❍❍❍
❍❍❍❍
❍❍❨✛
❄
✻✻ ✻✻
✲
λ0
1λ0
2λ0
J
· · ·
P 0(j,k)
P (k, l)
d1 d2 dJ
m0
1m0
2m0
J
Triage-Patients
IP-Patients
ExitsS
Arrivals
C1(·) C2(·) C3(·) CK(·)
m1 m2 m3 mK
· · ·
◮ Goal: Adhere to Triage-Constraints, then release In-Process Patients
◮ Model = Multi-class Q with Feedback: Min. convex congestion costs ofIP-Patients, s.t. deadline constraints on Triage-Patients.
◮ Solution: In conventional heavy-traffic, asymptotic least-cost s.t. asymptoticcompliance (as in Plambeck, Harrison, Kumar, who applied admission control):
◮ Triage or IP? former, if some deadline is “too" close (least effort)◮ if Triage: Closest deadline (or Van Mieghem’s GLD)◮ if IP: Van Mieghem’s Gcµ, modified for feedback
24
Emergency-Department Network: Flow Control
Imaging
Laboratory
Experts
Interns
Returns (Old or New Problem)
“Lost” Patients
LWBS
Nurses
Statistics,
Human
Resource
Management
(HRM)
New Services Design (R&D)
Operations,
Marketing,
MIS
Organization Design:
Parallel (Flat) = ER
vs. a true ED
Sociology, Psychology,
Operations Research
Service Process
Design
Quality
Efficiency
Customers
Interface Design
Medicine
(High turnovers
Medical-Staff
shortage)
Operations/
Business
Process
Archive
Database
Design
Data Mining:
MIS, Statistics,
Operations
Research,
Marketing
Stretcher
Walking
Service Completion(sent to other department)
( Waiting Time
Active Dashboard )
Patients
Segmentation
Medicine,
Psychology,
Marketing
Psychological
Process
Archive
Human Factors
Engineering
(HFE)
Internal
Queue
Orthopedic
Queue
Arrivals
Function
Scientific Discipline
Multi-Disciplinary
Index
ED-Stress
Psychology
Operations
Research, 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 Design
MIS, HFE,
Operations Research
Psychology,
Statistics
Home
◮ ∗Queueing-Science, w/ Armony, Marmor, Tseytlin, Yom-Tov
◮ ∗Fair ED-to-IW Routing (Patients vs. Staff), w/ Momcilovic, Tseytlin
◮ ∗Triage vs. InProcess/Release (Plambeck et al, van Mieghem) in EDs, w/Carmeli, Huang; Shimkin
◮ ∗Staffing Time-Varying Q’s with Re-Entrant Customers (de Vericourt &Jennings), w/ Yom-Tov
◮ The Offered-Load in Fork-Join Nets (Adlakha & Kulkarni), w/ Kaspi, Zaeid
◮ Synchronization Control of Fork-Join Nets, w/ Atar, Zviran
25
Prerequisite I: Data
Averages Prevalent (and could be useful / interesting).
But I need data at the level of the Individual Transaction:For each service transaction (during a phone-service in a call center,or a patient’s visit in a hospital, or browsing in a website, or . . .), its
operational history = time-stamps of events (events-log files).
26
Interesting Averages: The Human Factor, or
Even “Doctors" Can Manage
Afternoon,
by Case
Morning,
by Hour
Operations Time - Morning (by Hour) vs. Afternoon (by Case):
Ethical?
Even Doctors Can Manage!
0
1
2
3
4
5
6
EEG Orthopedics Surgery Blood Surgery Plastic Surgery Heart/Chest
Surgery
Neuro-Surgery Eyes E.I. Surgery
Department
Ho
urs
AM
PM
27
Beyond Averages: The Human Factor
Histogram of Service-Time in an Israeli Call Center, 1999
January-OctoberJanuary-October
0
2
4
6
8
0 100 200 300 400 500 600 700 800 900
AVG: 185
STD: 238
0
2
4
6
8
?6.83%
November-December
0
2
4
6
8
0 100 200 300 400 500 600 700 800 900
November-December
0
2
4
6
8
0 100 200 300 400 500 600 700 800 900
AVG: 201
STD: 263
5.59%
%
Log-Normal
◮ 6.8% Short-Services:
28
Beyond Averages: The Human Factor
Histogram of Service-Time in an Israeli Call Center, 1999
January-OctoberJanuary-October
0
2
4
6
8
0 100 200 300 400 500 600 700 800 900
AVG: 185
STD: 238
0
2
4
6
8
?6.83%
November-December
0
2
4
6
8
0 100 200 300 400 500 600 700 800 900
November-December
0
2
4
6
8
0 100 200 300 400 500 600 700 800 900
AVG: 201
STD: 263
5.59%
%
Log-Normal
◮ 6.8% Short-Services: Agents’ “Abandon" (improve bonus, rest),(mis)lead by incentives
◮ Distributions must be measured (in seconds = natural scale)
◮ LogNormal service-durations (???, common, more later)
28
Pause for a Commercial:
Pause for a Commercial: The Technion SEE Center
29
Technion SEE = Service Enterprise Engineering
SEELab: Data-repositories for research and teaching
◮ For example:◮ Bank Anonymous: 1 year, 350K calls by 15 agents - in 2000.
Brown, Gans, Sakov, Shen, Zeltyn, Zhao (JASA),paved the way to:
◮ U.S. Bank: 2.5 years, 220M calls, 40M by 1000 agents◮ Israeli Cellular: 2.5 years, 110M calls, 25M calls by 750 agents◮ Israeli Bank: from January 2010, daily-deposit at a SEESafe◮ Home (Rambam) Hospital: 4 years, 1000 beds, ward-level flow◮ 5 EDs: gathered by the late David Sinreich, ED arrivals & LOS
30
Technion SEE = Service Enterprise Engineering
SEELab: Data-repositories for research and teaching
◮ For example:◮ Bank Anonymous: 1 year, 350K calls by 15 agents - in 2000.
Brown, Gans, Sakov, Shen, Zeltyn, Zhao (JASA),paved the way to:
◮ U.S. Bank: 2.5 years, 220M calls, 40M by 1000 agents◮ Israeli Cellular: 2.5 years, 110M calls, 25M calls by 750 agents◮ Israeli Bank: from January 2010, daily-deposit at a SEESafe◮ Home (Rambam) Hospital: 4 years, 1000 beds, ward-level flow◮ 5 EDs: gathered by the late David Sinreich, ED arrivals & LOS
SEEStat: Environment for graphical EDA in real-time
◮ Universal Design, Internet Access, Real-Time Response.
30
Technion SEE = Service Enterprise Engineering
SEELab: Data-repositories for research and teaching
◮ For example:◮ Bank Anonymous: 1 year, 350K calls by 15 agents - in 2000.
Brown, Gans, Sakov, Shen, Zeltyn, Zhao (JASA),paved the way to:
◮ U.S. Bank: 2.5 years, 220M calls, 40M by 1000 agents◮ Israeli Cellular: 2.5 years, 110M calls, 25M calls by 750 agents◮ Israeli Bank: from January 2010, daily-deposit at a SEESafe◮ Home (Rambam) Hospital: 4 years, 1000 beds, ward-level flow◮ 5 EDs: gathered by the late David Sinreich, ED arrivals & LOS
SEEStat: Environment for graphical EDA in real-time
◮ Universal Design, Internet Access, Real-Time Response.
SEEServer: Free for academic use
◮ Register◮ Access U.S. Bank, Bank Anonymous, Home Hospital
30
eg. RFID-Based Data: Mass Casualty Event (MCE)
Drill: Chemical MCE, Rambam Hospital, May 2010
Focus on severely wounded casualties (≈ 40 in drill)Note: 20 observers support real-time control (helps validation)
31
Data Cleaning: MCE with RFID Support
Data-base Company report comment
Asset id order Entry date Exit date Entry date Exit date
4 1 1:14:07 PM 1:14:00 PM
6 1 12:02:02 PM 12:33:10 PM 12:02:00 PM 12:33:00 PM
8 1 11:37:15 AM 12:40:17 PM 11:37:00 AM exit is missing
10 1 12:23:32 PM 12:38:23 PM 12:23:00 PM
12 1 12:12:47 PM 12:35:33 PM 12:35:00 PM entry is missing
15 1 1:07:15 PM 1:07:00 PM
16 1 11:18:19 AM 11:31:04 AM 11:18:00 AM 11:31:00 AM
17 1 1:03:31 PM 1:03:00 PM
18 1 1:07:54 PM 1:07:00 PM
19 1 12:01:58 PM 12:01:00 PM
20 1 11:37:21 AM 12:57:02 PM 11:37:00 AM 12:57:00 PM
21 1 12:01:16 PM 12:37:16 PM 12:01:00 PM
22 1 12:04:31 PM 12:20:40 PMfirst customer is missing
22 2 12:27:37 PM 12:27:00 PM
25 1 12:27:35 PM 1:07:28 PM 12:27:00 PM 1:07:00 PM
27 1 12:06:53 PM 12:06:00 PM
28 1 11:21:34 AM 11:41:06 AM 11:41:00 AM 11:53:00 AMexit time instead of entry time
29 1 12:21:06 PM 12:54:29 PM 12:21:00 PM 12:54:00 PM
31 1 11:40:54 AM 12:30:16 PM 11:40:00 AM 12:30:00 PM
31 2 12:37:57 PM 12:54:51 PM 12:37:00 PM 12:54:00 PM
32 1 11:27:11 AM 12:15:17 PM 11:27:00 AM 12:15:00 PM
33 1 12:05:50 PM 12:13:12 PM 12:05:00 PM 12:15:00 PM wrong exit time
35 1 11:31:48 AM 11:40:50 AM 11:31:00 AM 11:40:00 AM
36 1 12:06:23 PM 12:29:30 PM 12:06:00 PM 12:29:00 PM
37 1 11:31:50 AM 11:48:18 AM 11:31:00 AM 11:48:00 AM
37 2 12:59:21 PM 12:59:00 PM
40 1 12:09:33 PM 12:35:23 PM 12:09:00 PM 12:35:00 PM
- Imagine “Cleaning" 60,000+ customers per day (call centers) !
- “Psychology" of Data Trust and Transfer (e.g. 2 years till transfer)32
Event-Logs in a Call Center (Bank Anonymous)
A Data Sample (Excel worksheet) vru+line call_id customer_id priority type date vru_entry vru_exit vru_time q_start q_exit q_time outcome ser_start ser_exit ser_time server
AA0101 44749 27644400 2 PS 990901 11:45:33 11:45:39 6 11:45:39 11:46:58 79 AGENT 11:46:57 11:51:00 243 DORIT
AA0101 44750 12887816 1 PS 990905 14:49:00 14:49:06 6 14:49:06 14:53:00 234 AGENT 14:52:59 14:54:29 90 ROTH
AA0101 44967 58660291 2 PS 990905 14:58:42 14:58:48 6 14:58:48 15:02:31 223 AGENT 15:02:31 15:04:10 99 ROTH
AA0101 44968 0 0 NW 990905 15:10:17 15:10:26 9 15:10:26 15:13:19 173 HANG 00:00:00 00:00:00 0 NO_SERVER
AA0101 44969 63193346 2 PS 990905 15:22:07 15:22:13 6 15:22:13 15:23:21 68 AGENT 15:23:20 15:25:25 125 STEREN
AA0101 44970 0 0 NW 990905 15:31:33 15:31:47 14 00:00:00 00:00:00 0 AGENT 15:31:45 15:34:16 151 STEREN
AA0101 44971 41630443 2 PS 990905 15:37:29 15:37:34 5 15:37:34 15:38:20 46 AGENT 15:38:18 15:40:56 158 TOVA
AA0101 44972 64185333 2 PS 990905 15:44:32 15:44:37 5 15:44:37 15:47:57 200 AGENT 15:47:56 15:49:02 66 TOVA
AA0101 44973 3.06E+08 1 PS 990905 15:53:05 15:53:11 6 15:53:11 15:56:39 208 AGENT 15:56:38 15:56:47 9 MORIAH
AA0101 44974 74780917 2 NE 990905 15:59:34 15:59:40 6 15:59:40 16:02:33 173 AGENT 16:02:33 16:26:04 1411 ELI
AA0101 44975 55920755 2 PS 990905 16:07:46 16:07:51 5 16:07:51 16:08:01 10 HANG 00:00:00 00:00:00 0 NO_SERVER
AA0101 44976 0 0 NW 990905 16:11:38 16:11:48 10 16:11:48 16:11:50 2 HANG 00:00:00 00:00:00 0 NO_SERVER
AA0101 44977 33689787 2 PS 990905 16:14:27 16:14:33 6 16:14:33 16:14:54 21 HANG 00:00:00 00:00:00 0 NO_SERVER
AA0101 44978 23817067 2 PS 990905 16:19:11 16:19:17 6 16:19:17 16:19:39 22 AGENT 16:19:38 16:21:57 139 TOVA
AA0101 44764 0 0 PS 990901 15:03:26 15:03:36 10 00:00:00 00:00:00 0 AGENT 15:03:35 15:06:36 181 ZOHARI
AA0101 44765 25219700 2 PS 990901 15:14:46 15:14:51 5 15:14:51 15:15:10 19 AGENT 15:15:09 15:17:00 111 SHARON
AA0101 44766 0 0 PS 990901 15:25:48 15:26:00 12 00:00:00 00:00:00 0 AGENT 15:25:59 15:28:15 136 ANAT
AA0101 44767 58859752 2 PS 990901 15:34:57 15:35:03 6 15:35:03 15:35:14 11 AGENT 15:35:13 15:35:15 2 MORIAH
AA0101 44768 0 0 PS 990901 15:46:30 15:46:39 9 00:00:00 00:00:00 0 AGENT 15:46:38 15:51:51 313 ANAT
AA0101 44769 78191137 2 PS 990901 15:56:03 15:56:09 6 15:56:09 15:56:28 19 AGENT 15:56:28 15:59:02 154 MORIAH
AA0101 44770 0 0 PS 990901 16:14:31 16:14:46 15 00:00:00 00:00:00 0 AGENT 16:14:44 16:16:02 78 BENSION
AA0101 44771 0 0 PS 990901 16:38:59 16:39:12 13 00:00:00 00:00:00 0 AGENT 16:39:11 16:43:35 264 VICKY
AA0101 44772 0 0 PS 990901 16:51:40 16:51:50 10 00:00:00 00:00:00 0 AGENT 16:51:49 16:53:52 123 ANAT
AA0101 44773 0 0 PS 990901 17:02:19 17:02:28 9 00:00:00 00:00:00 0 AGENT 17:02:28 17:07:42 314 VICKY
AA0101 44774 32387482 1 PS 990901 17:18:18 17:18:24 6 17:18:24 17:19:01 37 AGENT 17:19:00 17:19:35 35 VICKY
AA0101 44775 0 0 PS 990901 17:38:53 17:39:05 12 00:00:00 00:00:00 0 AGENT 17:39:04 17:40:43 99 TOVA
AA0101 44776 0 0 PS 990901 17:52:59 17:53:09 10 00:00:00 00:00:00 0 AGENT 17:53:08 17:53:09 1 NO_SERVER
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VRURetail
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NonBusiness LineSubanco
NonBusiness LinePriority Service
NonBusiness Line NonCC ServiceCCO
NonCC ServiceQuick&Reilly
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Overnight ClosedEBO
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TrunkOnline Banking
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Trunk
Trunk
Trunk
Trunk
InternalTotal
OutgoingTotal
NormalTermination
NormalTermination
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OtherUnhandled
Transfer
Transfer
1 Day in a Call Center
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USBank
April 2, 2001
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NonCC ServiceQuick&Reilly
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InternalTotal
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NormalTermination
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1 Day in a Call Center
- Customers
USBank
April 2, 2001
ILDUBank
January 24, 2010
from 06:00 to 23:59:59
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January 24, 2010
from 06:00 to 23:59:59
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ED
Pediatric
Neurology Nephrology Derrmatology
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Cardiology
Rheumatology
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Psychiatry Ophthalmology
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1 Day in a Hospital
- Patients
Rambam
August, 2004
Released (Home)
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Transfer
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serveng
lectures
references
predictionmay1809.pdf
homeworks
hw1_amusement_park_mgt.pdf
recitations
hazard_phase_type.pdf
course2011winter
icpr_service_engineering.pdfon_cc_data_frommsom.pdf pivot_table.pdf
thesis_polyna.pdf
proposal_polina.pdf
moss.pdf
files
nurse_proj_psu_tech.pdf
national_cranberry_1.pdf
hw9par_sol_2012w.pdf
syllabus8.pdfexams
statanalysis.pdf
agent-heterogeneity-brown-book-final.pdf
1_1_qed_lecture_introduction_2011w.pdf
course2004
retail.pdf hall_6.pdf
jasa_callcenter.pdf dantzig.pdf
bacceli.pdf
avishai-class-april-2003.ppt
hw7_2011w_par_sol_part2.pdf
ibmrambam technion srii presentation final.ppt
dimensioning.pdf
palmannoyance.pdf
yair_sergurywaitingtimeanalysis.pdf
nejmsb1002320.pdf
qed_qs_scientific_generic_caschina.pdf
web_summary13_2011s.pdf
4callcenters_coverpage.pdf
documentation.pdf rec9part2.pdf
web_summary9_2010w.pdf
qed_qs_scientific_wharton_stat_seminar.pdf
l2_measurements_class_2009s.pdf
ccreview.pdfvdesign_ecompanion.pdf
wharton_lecture_avi_printout.pdflarson_atoa.pdf
proposal_michael.pdf
stanford_seminar_avi_printout.pdf
solver_guide.pdf
fitz.pdf
rec10_part2.pdf rec5.pdf
paxson.pdf
connect_to_see_terminal.pdf
project_ug_simulationinterface.pdf revenue_manage.pdf
hw7_2009s_forecast.xls
syllabus2.pdf
infocom04.pdf
hall_transportation.pdf
whitt_handout_chapter1.pdf
fromprojecttoprocess.pdf
nytimes.2007-06-23.pdf ds_pert_lec1.pdf syllabus6.pdf hw9_2011w.pdf hw7_2011w.pdf
mmng_constraint_supp_or.pdf
awam-april_2011.pdfwhitt_scaling_slides.pdfmmng_seminar.pdf
syllabus4.pdf
syllabus12_2012w.pdf
syllabus5.pdf
mm1_strong_apprximations.pdf seestat_tutorial.ppt
kaplan_porter_2011-9_how-to-solve-the-cost-crisis-in-health-care_hbr.pdf
rec9_part2.pdf
callcenterdata
bocaraton.ps
gccreport 20-04-07 uk version.pdf
hall_manpower_planning.pdf
bi18.pdf
syllabus12.pdf fluidview_2010w_short_version.pdf hw7_2009s_par_sol_part2.pdf
moed_b_2007w_solution.pdf
newell.pdf
rec11.pdf
moedb_2009w.pdf
rec3.pdf
hw10_par_sol_2011s.pdf
thirdlevel support ijtm1_cs.pdf
hw4_full.pdf
moedb_2010s_sol.pdf
chandra.pdf
gurvich_seminar.pdf
ed_sinreich_marmor.ppt regimes_supplement.pdf
web_summary10_2011s.pdf hw6par_sol_2009s.xls
exam_2005s_sol.pdf
servicefull.pdf
witor09_empirical_adventures_sep2009.pdf
web_summary12_2011s.pdf
heb_summary_yulia.pdf
hw9par_sol_2011s.pdf
seminar_yulia_9_2.pdf
bpr_2003.pdf staffing_italian_us_2011w.xlsdesignofqueues.pdf thepsychologyofwaitinglines.pdf
erlangabc.pdf
studentsevaluations.pdf hw8_2011s.pdf
hw9_2012w.pdf
edie.pdf
syllabus9.pdf
ds_pert_lec2.pdf
zohar_seminar.ppt
hist-normal.pdf
vandergraft.pdf
rec13.pdf
desighn_ivr.pdf
cohen_aircraft_failures.pdf
sbr.pdf
nano.pdf
hw2_2011w.pdfseminar_presentation_boaz_estimating
er load.ppt
project_ug_meirav.pdf see_report_output_june_2011.pdf
see_report_2010.pdf
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serveng
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hall_flows_hospitals_chapter1text.pdf
references
4callcenterscourse2004 homeworks
icpr_service_engineering.pdf keycorp.pdf
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revenue_manage.pdfhw1_amusement_park_mgt.pdf
recitations
setup.exe
garnett.pdf abandon02.pdf erlang_a.pdf ccbib.pdf
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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
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setup.exe
garnett.pdf abandon02.pdf erlang_a.pdf ccbib.pdf
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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
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rec10_part2.pdf
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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
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USBank April 2, 2001
Zoom Out: Call Center Network Inter-Queues
Daily
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ILBank
May 2, 2008
ILDUBank
January 24, 2010
Agent #043
Shift: 16:00 - 23:45
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Israeli Telecom
February 10, 2008
N -Design
L -Design
Agent Pool
Customer Class
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1253823 843343 1390109630 4454 1240792 3793 49 735
1433 193
Goal: Data-Based Real-Time Simulation (SimNet)
Private Prepaid Language 3 Language 2 Private Language 2 Prepaid Language 2 Prepaid Low Priority Internet Surfing Internet
Private VIP Private Business Business VIP Business Customer Retention Private Customer Retention Overseas
What is lacking?
1. Dynamics
2. Durations
3. Protocols
Israeli Telecom
February 10, 2008
8 AM - 9 AM
USBank
April 2, 2001
8 AM - 9 AM
9 AM - 10 AM
10 AM - 11 AM
11 AM - 12 PM
Dynamics: Time-Varying Arrival-Rates
2 Daily Peaks
CC: Dec. 1995, (USA, 700 Helpdesks) CC: May 1959 (England)
Dec 1995!
(Help Desk Institute)
Arrival
Time
Time
24 hrs
% Arrivals
Q-Science
May 1959!
Arrival
Rate
Time
24 hrs
CC: Nov. 1999 (Israel) ED: Jan.–July 2007 (Israel)
Daily
0.00
2.50
5.00
7.50
10.00
12.50
15.00
17.50
20.00
22.50
25.00
27.50
30.00
0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00
Avera
ge n
um
be
r of
cases
Time (Resolution 60 min.)
HomeHospital Patients Arrivals to Emergency DepartmentTotal for January2007 February2007 March2007 April2007 May2007 June2007 July2007,Sundays
54
Durations: Phone Calls (2 Surprises)
Israeli Call Center, Nov–Dec, 1999
Log(Service Times)
0 2 4 6 8
0.0
0.1
0.2
0.3
0.4
Log(Service Time)
Pro
port
ion
LogNormal QQPlot
Log-normal
Serv
ice t
ime
0 1000 2000 3000
01000
2000
3000
◮ Practically Important: (mean, std)(log) characterization
◮ Theoretically Intriguing: Why LogNormal ? Naturally multiplicative
but, in fact, also Infinitely-Divisible (Generalized Gamma-Convolutions)
56
Durations: Answering Machine
Israeli Bank: IVR/VRU Only, May 2008
IVR_only
May 2008, Week days
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
00:00 00:30 01:00 01:30 02:00 02:30 03:00 03:30 04:00 04:30
Time(mm:ss) (Resolution 1 sec.)
Rela
tive f
req
uen
cie
s %
mean=99st.dev.=101
Tree-Network Tomography:Reconstruct topologyfrom root and leaves
Mixture: 7 LogNormals
0.00
0.10
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
Rela
tiv
e f
req
uen
cie
s %
Time(mm:ss) (Resolution 1 sec.)
Fitting Mixtures of Distributions for VRU only timeMay 2008, Week days
Time(mm:ss) (Resolution 1 sec.)Empirical Total Lognormal Lognormal Lognormal
57
Durations: Waiting Times in a Call Center
⇒ Protocols
Exponential in Heavy-Traffic (min.)Small Israeli Bank
0.00
2.50
5.00
7.50
10.00
12.50
15.00
17.50
20.00
22.50
25.00
27.50
30.00
0.00 100.00 200.00 300.00 400.00 500.00 600.00
Rela
tive f
requencie
s %
Time(seconds)( resolution 30)
AnonymousBank Handled Wait Time, TOTALTotal for January1999 February1999 March1999 April1999 May1999 June1999 July1999 August1999 September1999 October1999
November1999 December1999,Weekdays
Empirical Exponential (scale=106.67)
Mean = 106 SD = 109
Routing via Thresholds (sec.)Large U.S. Bank
0.00
2.50
5.00
7.50
10.00
12.50
15.00
17.50
20.00
2.00 7.00 12.00 17.00 22.00 27.00 32.00 37.00
Rela
tive f
requencie
s %
Time(seconds)( resolution 1)
USBank Wait time(waiting), RetailMay 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
Rela
tive f
requencie
s %
Time(seconds)( resolution 1)
ILBank Wait time (all)August 2007, Weekdays
58
LogNormal & Beyond: Length-of-Stay in a Hospital
Israeli Hospital, in Days: LN
0 2 4 6 8 10 13 16 19 22 25 28 31 34 37 40 43 46 49
59
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
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 ?◮ Hourly Scale: Staffing,. . .
◮ Daily: Flow / Bed Control,. . .
Arrivals, Departures, # Patients in Ward A, by Hour
59
Protocols + Psychology
Patient Customers, Announcements, Priority Upgrades
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0 60 120 180 240 300 360 420 480 540 600 660 720 780 840 900
Hazard
Function
Time (seconds)
USBank December 2002, Week days, Quick&Reilly
time willing to wait (hazard rate)
time willing to wait (Pspline)
virtual wait (hazard rate)
virtual wait (Pspline)
61
Dynamics: Parsimonious Models (Congestion Laws)
3 Queue-Lengths at 30 sec. resolution (ILBank, 10/6/2007)
ILBank Customers in queue (average)
10.06.2007
0.00
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
of
cases
Medium priority Low priority Unidentified
Queue “Shape"
ILBank Customers in queue (average)
10.06.2007
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
06:00 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00
Time (Resolution 30 sec.)
Perc
enta
ge
Medium priority Low priority Unidentified
◮ Area normalized to 100%
◮ State-Space Collapse
55
Little’s Law: Call Center & Emergency Department
Time-Gap: # in System lags behind Piecewise-Little (L = λ×W )
USBank Customers in queue(average), Telesales
10.10.2001
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
180.00
200.00
07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00
Time (Resolution 30 min.)
Nu
mb
er
of ca
se
s
Customers in queue(average) Little's law
HomeHospital Average patients in ED
February 2004, Wednesdays
0.00
10.00
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
rag
e n
um
be
r o
f ca
se
s
Average patients in ED Lambda*E(S) smoothed
62
Little’s Law: Call Center & Emergency Department
Time-Gap: # in System lags behind Piecewise-Little (L = λ×W )
USBank Customers in queue(average), Telesales
10.10.2001
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
180.00
200.00
07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00
Time (Resolution 30 min.)
Nu
mb
er
of ca
se
s
Customers in queue(average) Little's law
HomeHospital Average patients in ED
February 2004, Wednesdays
0.00
10.00
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
rag
e n
um
be
r o
f ca
se
s
Average patients in ED Lambda*E(S) smoothed
⇒ Time-Varying Little’s Law
◮ Berstemas & Mourtzinou;
◮ Fralix, Riano, Serfozo; . . .
62
Prerequisite II: Models (FNets)
“Laws of Large Numbers" capture Predictable Variability
Deterministic Models: Scale Averages-out Stochastic Individualism
70
The Basic Service-Network Model: Erlang-R
w/ G. Yom-Tov Needy
(st-servers)
rate-
Content
(Delay)
rate -
1-p
p
1
2
Arrivals
Poiss( t)Patient discharge
Erlang-R (IE: Repairman Problem 50’s; CS: Central-Server 60’s) =
2-station “Jackson" Network = (M/M/S, M/M/∞) :
◮ λt – Time-Varying Arrival rate
◮ St – Number of Servers (Nurses / Physicians)
◮ µ – Service rate (E [Service] = 1µ
)
◮ p – ReEntrant (Feedback) fraction
◮ δ – Content-to-Needy rate (E [Content] = 1δ)
71
Fluid Model ↔ (Time-Varying) Erlang-R System
Needy
(st-servers)
rate-
Content
(Delay)
rate -
1-p
p
1
2
Arrivals
Poiss( t)Patient discharge
FNet of a 2-station “Jackson" Network:
d
dtq1
t = λt − µ ·(
q1t ∧ st
)
+ δ · q2t ,
d
dtq2
t = p · µ ·(
q1t ∧ st
)
− δ · q2t .
(1)
72
Erlang-R: Fitting a Simple Model to a Complex Reality
Chemical MCE Drill (Israel, May 2010)
Arrivals & Departures (RFID) Erlang-R (Fluid, Diffusion)
0
10
20
30
40
50
60
11:02 11:16 11:31 11:45 12:00 12:14 12:28 12:43 12:57 13:12 13:26
To
ta
l N
um
be
r o
f P
atie
nts
Time
Cumulative Arrivals
Cumulative Departures
15
20
25
30
erofMCEPatientsinED
Actual Q(t)
Fluid Q(t)
Lower Envelope Q(t) (Theoretical)
Upper Envelope Q(t) (Theoretical)
Fluid Q1
0
5
10
11:02 11:16 11:31 11:45 12:00 12:14 12:28 12:43 12:57 13:12 13:26
Numbe
Time
◮ Recurrent/Repeated services in MCE Events: eg. Injection every 15 minutes
73
Erlang-R: Fitting a Simple Model to a Complex Reality
Chemical MCE Drill (Israel, May 2010)
Arrivals & Departures (RFID) Erlang-R (Fluid, Diffusion)
0
10
20
30
40
50
60
11:02 11:16 11:31 11:45 12:00 12:14 12:28 12:43 12:57 13:12 13:26
To
ta
l N
um
be
r o
f P
atie
nts
Time
Cumulative Arrivals
Cumulative Departures
15
20
25
30
erofMCEPatientsinED
Actual Q(t)
Fluid Q(t)
Lower Envelope Q(t) (Theoretical)
Upper Envelope Q(t) (Theoretical)
Fluid Q1
0
5
10
11:02 11:16 11:31 11:45 12:00 12:14 12:28 12:43 12:57 13:12 13:26
Numbe
Time
◮ Recurrent/Repeated services in MCE Events: eg. Injection every 15 minutes
◮ Fluid (Sample-path) Modeling, via Functional Strong Laws of Large Numbers
◮ Stochastic Modeling, via Functional Central Limit Theorems
◮ ED in MCE: Confidence-interval, usefully narrow for Control◮ ED in normal (time-varying) conditions: Personnel Staffing73
An Asymptotic Framework: Erlang-R in the ED
System = Emergency Department (eg. Rambam Hospital)
◮ SimNet = Customized ED-Simulator (Marmor & Sinreich)
◮ QNet = Erlang-R (time-varying 2-station Jackson; w/ Yom-Tov)
◮ FNets = 2-dim dynamical system (Massey & Whitt)
◮ DNets = 2-dim Markovian Service Net (w/ Massey and Reiman)
74
An Asymptotic Framework: Erlang-R in the ED
System = Emergency Department (eg. Rambam Hospital)
◮ SimNet = Customized ED-Simulator (Marmor & Sinreich)
◮ QNet = Erlang-R (time-varying 2-station Jackson; w/ Yom-Tov)
◮ FNets = 2-dim dynamical system (Massey & Whitt)
◮ DNets = 2-dim Markovian Service Net (w/ Massey and Reiman)
Asymptotic Framework
◮ Data and Measurements
◮ Fit a simple model (time-varying Erlang-R) to a complex reality(ED Physicians)
◮ Develop FNets (Offered-Load of Physicians) and (relevant) DNet(if needed)
◮ Use FNet / DNet for Design (√
-Staffing), Analysis, . . .
◮ Simulate reality (ED with√
-staffing of Physicians)
◮ Validation: stable performance, confidence intervals, . . .
74
Case Study: Emergency Ward Staffing
Many-Server (↑ ∞) Approximations for Small Systems (1-7)
◮ Staffing resolution: 1 hour◮ Lower bound: 1 doctor per type◮ Flexible (time-varying square-root) staffing: Yunan’s Lecture◮ Rounding effects ⇒ Not all performance levels achievable
90 Servers 1-7 Doctors
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P(W
>0)
0
0.1
0.2
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100105110115
Time [Hour]
beta=0.1 beta=0.3 beta=0.5 beta=0.7 beta=1 beta=1.5
0.3
0.4
0.5
0.6
0.7
0.8
0.9
P(W
>0)
0
0.1
0.2
0 1000 2000 3000 4000 5000 6000 7000
Time
Beta=0.1 Beta=0.5 Beta=1 Beta=1.5
75
Protocols: Staffing (N) vs. Offered-Load (R = λ× E(S))
IL Telecom; June-September, 2004; w/ Nardi, Plonski, Zeltyn
2205 half-hour intervals (13 summer weeks, week-days)63
Number of patients in ED
0
250
500
750
1000
1250
1500
1750
2000
2250
2500
0 10 20 30 40 50 60 70 80 90 100
Fre
que
ncie
s
Number of patients (resolution 1)
HomeHospital Time by ED Internal and Surgery state (sec.)January 2004-October 2007, all days
0
20
40
60
80
100
120
140
160
180
200
220
240
260
0 10 20 30 40 50 60 70 80 90 100
Fre
quencie
s
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 c
ases
Time (Resolution 60 min.)
HomeHospital Number of Patients in Emergency Department Internal and Surgery,January 2004-October 2007, all days
0
250
500
750
1000
1250
1500
1750
2000
2250
2500
0 10 20 30 40 50 60 70 80 90 100
Fre
que
ncie
s
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
Time by ED Internal and Surgery state (sec.) Statistics
N 120960000
N(average per day) 86400
Mean 34.1
Standard Deviation 16.34
Variance 266.87
Median 32
Minimum 0
Maximum 99
Skewness 0.524
Kurtosis -0.44452
Standard Error Mean 0.00149
Interquartile Range 25
Mean Absolute Deviation 13.7
Median Absolute Deviation(MAD) 12
Coefficient of Variation (CV) (%) 47.9
L-moment 2 (half of Gini's Mean Difference) 9.25
L-Skewness 0.121
L-Kurtosis 0.0561
Coefficient of L-variation (L-CV)(%) (Gini's Coefficient) 27.12
Parameter Estimates
Components Mixing Proportions (%) Location Scale Shape Mean Standard Deviation
1. Normal 21.45 17.59 5.39 17.59 5.39
2. Gamma 40.19 7.00 4.55 31.83 14.99
3. Weibull 38.36 49.28 3.61 44.42 13.679
Goodness-of-Fit Tests
Tests Statistic DF p Value
Residuals Std 0.011
Kolmogorov-Smirnov 0.028 <.0001
Cramer-von Mises 14953.04 <.0001
Andersen-Darling 96156.09 <.0001
Chi-Square 460741.3 94 <.0001
0
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
15 20 25 30 35 40 45 50
Fre
quen
cie
s
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)
Time by ED Internal state (sec.), [13:00-23:00) Statistics
N 1656000
N(average per day) 36000
Mean 33.22
Standard Deviation 5.756
Variance 33.13
Median 33
Minimum 16
Maximum 54
Skewness 0.282
Kurtosis -0.31791
Standard Error Mean 0.00447
Interquartile Range 8
Mean Absolute Deviation 4.712
Median Absolute Deviation(MAD) 4
Coefficient of Variation(CV) (%) 17.33
L-moment 2 (half of Gini's Mean Difference) 3.263
L-Skewness 0.0601
L-Kurtosis 0.0924
Coefficient of L-variation(L-CV)(%) (Gini's Coefficient) 9.82
Parameters for Normal Distribution
Parameter Estimate
mu 33.22
sigma 5.76
mean 33.22
std 5.756
Goodness-of-Fit Tests for Normal Distribution
Test Statistic DF p Value
Residuals Std 0.025
Kolmogorov-Smirnov 0.069 <.0001
Cramer-von Mises 1068.72 <.0001
Anderson-Darling 6193.27 <.0001
Chi-Square >1000 34 <.0001
Internal ED, non-holiday Mondays, 13:00-23:00
Our EDA “implies" (w/ Armony, Marmor, Tseytlin, Yom-Tov):
◮ Israeli ED censusd= M/M/∞: “Secret" behind
√-Staffing
68
Internal ED, non-holiday Mondays, 13:00-23:00
Our EDA “implies" (w/ Armony, Marmor, Tseytlin, Yom-Tov):
◮ Israeli ED censusd= M/M/∞: “Secret" behind
√-Staffing
◮ EDd= Reversible Birth-Death process, hence conditioning on
finite capacity, say B, gives rise to M/M/B/B (Erlang-B)
◮ U.S. ED censusd= M/M/B/B, which can be (has been) used to
analyze Ambulance Diversion (ED Blocking)
68
Internal ED, non-holiday Mondays, 13:00-23:00
Our EDA “implies" (w/ Armony, Marmor, Tseytlin, Yom-Tov):
◮ Israeli ED censusd= M/M/∞: “Secret" behind
√-Staffing
◮ EDd= Reversible Birth-Death process, hence conditioning on
finite capacity, say B, gives rise to M/M/B/B (Erlang-B)
◮ U.S. ED censusd= M/M/B/B, which can be (has been) used to
analyze Ambulance Diversion (ED Blocking)
◮ Puzzle (getting ahead): Is the ED an Erlang-A system withµ = θ? then the “effective number of servers" can be, perhaps,deduced via those LWBS = Left Without Being Seen (or LAMA)
Simple models at the service of complex realities
68
Prerequisite II: Models (DNets, QED Q’s)
Traditional Queueing Theory predicts that Service-Quality andServers’ Efficiency must be traded off against each other.
For example, M/M/1 (single-server queue): 91% server’s utilizationgoes with
Congestion Index =E [Wait ]
E [Service]= 10,
and only 9% of the customers are served immediately upon arrival.
77
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:
◮ Call Centers: Wait “seconds" for minutes service;◮ Transportation: Search “minutes" for hours parking;◮ Hospitals: Wait “hours" in ED for days hospitalization in IW’s.
77
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:
◮ Call Centers: Wait “seconds" for minutes service;◮ Transportation: Search “minutes" for hours parking;◮ Hospitals: Wait “hours" in ED for days hospitalization in IW’s.
Moreover, a significant fraction not delayed in queue: e.g. in well-run
◮ CCs: 50% served “immediately" & 90% utilization ⇒ QED
◮ EDs + IWs: ?
77
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:
◮ Call Centers: Wait “seconds" for minutes service;◮ Transportation: Search “minutes" for hours parking;◮ Hospitals: Wait “hours" in ED for days hospitalization in IW’s.
Moreover, a significant fraction not delayed in queue: e.g. in well-run
◮ CCs: 50% served “immediately" & 90% utilization ⇒ QED
◮ EDs + IWs: ? Multiple scales! IW-“Beds" (10’s) are QED whileIW-Doctors (1‘s) are in conventional heavy-traffic (hours wait forminutes service), hence the bottlenecks77
The Basic Staffing Model: Erlang-A (M/M/N + M)
agents
arrivals
abandonment
1
2
n
…
queue
Erlang-A (Palm 1940’s) = Birth & Death Q, with parameters:
◮ λ – Arrival rate (Poisson)
◮ µ – Service rate (Exponential; E [S] = 1µ )
◮ θ – Patience rate (Exponential, E [Patience] = 1θ )
◮ N – Number of Servers (Agents).
78
Erlang-A: Practical Relevance?
Experience:
◮ Arrival process not pure Poisson (time-varying, σ2 too large)
◮ Service times not Exponential (typically close to LogNormal)
◮ Patience times not Exponential (various patterns observed).
79
Erlang-A: Practical Relevance?
Experience:
◮ Arrival process not pure Poisson (time-varying, σ2 too large)
◮ Service times not Exponential (typically close to LogNormal)
◮ Patience times not Exponential (various patterns observed).
◮ Building Blocks need not be independent (eg. long waitassociated with long service; w/ M. Reich and Y. Ritov)
◮ Customers and Servers not homogeneous (classes, skills)
◮ Customers return for service (after busy, abandonment;dependently; P. Khudiakov, M. Gorfine, P. Feigin)
◮ · · · , and more.
79
Erlang-A: Practical Relevance?
Experience:
◮ Arrival process not pure Poisson (time-varying, σ2 too large)
◮ Service times not Exponential (typically close to LogNormal)
◮ Patience times not Exponential (various patterns observed).
◮ Building Blocks need not be independent (eg. long waitassociated with long service; w/ M. Reich and Y. Ritov)
◮ Customers and Servers not homogeneous (classes, skills)
◮ Customers return for service (after busy, abandonment;dependently; P. Khudiakov, M. Gorfine, P. Feigin)
◮ · · · , and more.
Question: Is Erlang-A Relevant?
YES ! Fitting a Simple Model to a Complex Reality, bothTheoretically and Practically
79
Asymptotic Landscape: 9 Operational Regimes, and then some
Erlang-A, w/ I. Gurvich & J. HuangErlang-A Conventional scaling Many-Server scaling NDS scaling
µ & θ fixed Sub Critical Over QD QED ED Sub Critical OverOffered load 1
1+δ1− β
√n
11−γ
11+δ
1− β√
n
11−γ
11+δ 1− β
n
11−γper server
Arrival rate λ µ1+δ
µ− β√
nµ µ
1−γnµ1+δ
nµ− βµ√
n nµ1−γ
nµ1+δ
nµ− βµ nµ1−γ
# servers 1 n nTime-scale n 1 n
Impatience rate θ/n θ θ/n
Staffing level λµ(1 + δ) λ
µ(1 + β
√n) λ
µ(1− γ) λ
µ(1 + δ) λ
µ+ β
√
λµ
λµ(1− γ) λ
µ(1 + δ) λ
µ+ β λ
µ(1− γ)
Utilization 11+δ
1−√
θµ
h(β)√
n1 1
1+δ1−
√
θµ
h(β)√
n1 1
1+δ1−
√
θµ
h(β)n
1
E(Q) 1δ(1+δ)
√ng(β) nµγ
θ(1−γ)1δn
√ng(β)α nµγ
θ(1−γ)o(1) ng(β) n2µγ
θ(1−γ)
P(Ab) 1n
1δ
θµ
θ√
nµg(β) γ 1
n
(1+δ)δ
θµn
θ√
nµg(β)α γ o( 1
n2 )θ
nµg(β) γ
P(Wq > 0) 11+δ
≈ 1 n α ∈ (0,1) ≈ 1 ≈ 0 ≈ 1
P(Wq > T ) 11+δ
e−
δ1+δ
µT 1 +O( 1√
n) 1 +O( 1
n) ≈ 0 f(T ) ≈ 0 Φ(β+
√
θµT )
Φ(β)1 +O( 1
n)
CongestionEWq
ES1δ
√ng(β) nµγ/θ 1
n
(1+δ)δ
nα√
ng(β) µγ
θo( 1
n) g(β) nµγ/θ
◮ Conventional: Ward & Glynn (03, G/G/1 + G)
◮ Many-Server:◮ QED: Halfin-Whitt (81), Garnett-M-Reiman (02)◮ ED: Whitt (04)◮ NDS: Atar (12)
91
Asymptotic Landscape: 9 Operational Regimes, and then some
Erlang-A, w/ I. Gurvich & J. HuangErlang-A Conventional scaling Many-Server scaling NDS scaling
µ & θ fixed Sub Critical Over QD QED ED Sub Critical OverOffered load 1
1+δ1− β
√n
11−γ
11+δ
1− β√
n
11−γ
11+δ 1− β
n
11−γper server
Arrival rate λ µ1+δ
µ− β√
nµ µ
1−γnµ1+δ
nµ− βµ√
n nµ1−γ
nµ1+δ
nµ− βµ nµ1−γ
# servers 1 n nTime-scale n 1 n
Impatience rate θ/n θ θ/n
Staffing level λµ(1 + δ) λ
µ(1 + β
√n) λ
µ(1− γ) λ
µ(1 + δ) λ
µ+ β
√
λµ
λµ(1− γ) λ
µ(1 + δ) λ
µ+ β λ
µ(1− γ)
Utilization 11+δ
1−√
θµ
h(β)√
n1 1
1+δ1−
√
θµ
h(β)√
n1 1
1+δ1−
√
θµ
h(β)n
1
E(Q) 1δ(1+δ)
√ng(β) nµγ
θ(1−γ)1δn
√ng(β)α nµγ
θ(1−γ)o(1) ng(β) n2µγ
θ(1−γ)
P(Ab) 1n
1δ
θµ
θ√
nµg(β) γ 1
n
(1+δ)δ
θµn
θ√
nµg(β)α γ o( 1
n2 )θ
nµg(β) γ
P(Wq > 0) 11+δ
≈ 1 n α ∈ (0,1) ≈ 1 ≈ 0 ≈ 1
P(Wq > T ) 11+δ
e−
δ1+δ
µT 1 +O( 1√
n) 1 +O( 1
n) ≈ 0 f(T ) ≈ 0 Φ(β+
√
θµT )
Φ(β)1 +O( 1
n)
CongestionEWq
ES1δ
√ng(β) nµγ/θ 1
n
(1+δ)δ
nα√
ng(β) µγ
θo( 1
n) g(β) nµγ/θ
◮ Conventional: Ward & Glynn (03, G/G/1 + G)
◮ Many-Server:◮ QED: Halfin-Whitt (81), Garnett-M-Reiman (02)◮ ED: Whitt (04)◮ NDS: Atar (12)
◮ “Missing": ED+QED; Hazard-rate scaling (M/M/N+G); Time-Varying,Non-Parametric; Moderate- and Large-Deviation; Networks; Control
91
Universal Approximations: Erlang-A (M/M/N + M)
agents
arrivals
abandonment
1
2
n
…
queuew/ I. Gurvich & J. Huang
92
Universal Approximations: Erlang-A (M/M/N + M)
agents
arrivals
abandonment
1
2
n
…
queuew/ I. Gurvich & J. Huang
◮ QNet: Birth & Death Queue, with B - D rates
F (q) = λ− µ · (q ∧ n)− θ · (q − n)+, q = 0, 1, . . .
92
Universal Approximations: Erlang-A (M/M/N + M)
agents
arrivals
abandonment
1
2
n
…
queuew/ I. Gurvich & J. Huang
◮ QNet: Birth & Death Queue, with B - D rates
F (q) = λ− µ · (q ∧ n)− θ · (q − n)+, q = 0, 1, . . .
◮ FNet: Dynamical (Deterministic) System – ODE
dxt = F (xt)dt , t ≥ 0
92
Universal Approximations: Erlang-A (M/M/N + M)
agents
arrivals
abandonment
1
2
n
…
queuew/ I. Gurvich & J. Huang
◮ QNet: Birth & Death Queue, with B - D rates
F (q) = λ− µ · (q ∧ n)− θ · (q − n)+, q = 0, 1, . . .
◮ FNet: Dynamical (Deterministic) System – ODE
dxt = F (xt)dt , t ≥ 0
◮ DNet: Universal (Stochastic) Approximation – SDE
dYt = F (Yt)dt +√
2λ dBt , t ≥ 0
92
Universal Approximations: Erlang-A (M/M/N + M)
agents
arrivals
abandonment
1
2
n
…
queuew/ I. Gurvich & J. Huang
◮ QNet: Birth & Death Queue, with B - D rates
F (q) = λ− µ · (q ∧ n)− θ · (q − n)+, q = 0, 1, . . .
◮ FNet: Dynamical (Deterministic) System – ODE
dxt = F (xt)dt , t ≥ 0
◮ DNet: Universal (Stochastic) Approximation – SDE
dYt = F (Yt)dt +√
2λ dBt , t ≥ 0
eg. µ = θ : x = λ− µ · x , Y = OU process
Accuracy increases as λ ↑ ∞ (no additional assumptions)92
Value of Universal Approximation
◮ Tractable - closed-form stable expressions
◮ Accurate - more than heavy traffic limits
◮ Robust - all many-server regimes, and beyond, with hardly anyassumptions
◮ Value
◮ Performance Analysis◮ Optimization (Staffing)◮ Inference (w/ G. Pang)◮ Simulation (w/ J. Blanchet)
◮ Limitation: Steady-State (but working on it)
Why does it work so well?
Coupling “Busy" + “Idle" Excursions of B&D and the correspondingDiffusion (durations order 1√
λ)
93
Universal Diffusion: Tractability
◮ Density function of Y (∞)− n:
π(x) =
õ
√λ
φ(√µ(x/
√λ+β/µ))
Φ(β/√µ) p(β, µ, θ), if x ≤ 0,
√θ√λ
φ(√θ(x/
√λ+β/θ))
1−Φ(β/√θ)
(1− p(β, µ, θ)), if x > 0,
Here β := (nµ− λ)/√λ
and
p(β, µ, θ) =
[
1 +
√
µ
θ
φ(β/√µ)
Φ(β/√µ)
1− Φ(β/√θ)
φ(β/√θ)
]−1
.
94
Universal Approximation: Accuracy
◮ ∆λ is the “balancing" state, obtained by solving
λ = µ(n ∧∆λ) + θ(∆λ − n)+.
Solution: ∆λ = λµ −
(
λµ − n
)+(
1− µθ
)
.
Specifically: QD = λµ ; ED = n + 1
θ (λ− nµ); QED = n +O(√λ))
◮ Centered processes (excursions):
Qλ(·) = Q(·)−∆λ, Yλ(·) = Y (·)−∆λ.
TheoremFor f bounded by an m-degree polynomial (m ≥ 0):
Ef (Qλ(∞))− Ef (Yλ(∞)) = O(√λ
m−1).
◮ Accuracy: higher than heavy-traffic limits
95
Universal Approximation: Why 2λ?
◮ Semi-martingale representation of the B&D process:Fluid + Martingale
◮ Predictable quadratic variation:
∫ t
0
[λ+ µ(Qs ∧ n) + θ(Qs − n)+]ds
◮ In steady-state, arrival rate ≡ departure rate:
λ = E[µ(Qs ∧ n) + θ(Qs − n)+]
◮ Expectation of the predictable quadratic variation:
E
∫ t
0
[λ+ µ(Qs ∧ n) + θ(Qs − n)+]ds = 2λt
◮ dMartingalet ≈√
2λ · dBrowniant
96
Reconciling Steady-State and Time-Varying Models
◮ Challenge: Accommodate time-varying demand (routine)◮ Prerequisite: Flexible Capacity
◮ As in Call Centers and to a degree in Healthcare,◮ In contrast to rigid (fixed) staffing level during a shift: doomed to
alternate between overloading and underloading
97
Reconciling Steady-State and Time-Varying Models
◮ Challenge: Accommodate time-varying demand (routine)◮ Prerequisite: Flexible Capacity
◮ As in Call Centers and to a degree in Healthcare,◮ In contrast to rigid (fixed) staffing level during a shift: doomed to
alternate between overloading and underloading
◮ Idea/Goal: In the face of time-varying demand, designtime-varying staffing which accommodates demand such thatperformance is stable over time
◮ Solution: In fact, a time-varying system with Steady-Stateperformance, at all times, via (Modified) Offered-Load(Square-Root) Staffing.
97
Reconciling Steady-State and Time-Varying Models
◮ Challenge: Accommodate time-varying demand (routine)◮ Prerequisite: Flexible Capacity
◮ As in Call Centers and to a degree in Healthcare,◮ In contrast to rigid (fixed) staffing level during a shift: doomed to
alternate between overloading and underloading
◮ Idea/Goal: In the face of time-varying demand, designtime-varying staffing which accommodates demand such thatperformance is stable over time
◮ Solution: In fact, a time-varying system with Steady-Stateperformance, at all times, via (Modified) Offered-Load(Square-Root) Staffing.
◮ History:◮ Jennings, M., Reiman, Whitt (1996): Emergence of the
phenomenon, via infinite-server heuristics◮ Feldman, M., Massey, Whitt (2008): Stabilize delay probability via
QED staffing (justified theoretically only for Erlang-A with µ = θ)◮ Liu and Whitt (ongoing): Stabilize abandonment probability by ED
staffing, via a corresponding network, theoretically and empirically◮ Huang, Gurvich, M. (ongoing): QED theory
97
The Offered-Load R(t), t ≥ 0 (R(t)↔ R)
Empirically (in SEEStat):◮ Process: L(·) = Least number of servers that guarantees no delay.◮ Offered-Load Function R(·) = E [L(·)]
98
The Offered-Load R(t), t ≥ 0 (R(t)↔ R)
Empirically (in SEEStat):◮ Process: L(·) = Least number of servers that guarantees no delay.◮ Offered-Load Function R(·) = E [L(·)]
ILTelecom , Private
12.05.2004
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
45.00
50.00
55.00
60.00
07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00
Time (Resolution 30 min.)
Nu
mb
er
of ca
se
s
Offered load Abandons Av_agents_in_system
98
Time-Varying Arrival Rates
Square-Root Staffing:N(t) = R(t) + β
√
R(t) , −∞ < β <∞.
R(t) is the Offered-Load at time t ( R(t) 6= λ(t)× E[S] )
Arrivals, Offered-Load and Staffing
0
50
100
150
200
2500 1 2 4 5 6 7 8
10
11
12
13
14
16
17
18
19
20
22
23
0
500
1000
1500
2000
Arr
iva
ls p
er
ho
ur
beta 1.2 beta 0 beta -1.2 Offered Load Arrivals
QDQED
ED
99
Time-Stable Performance of Time-Varying Systems
Delay Probability = as in the Stationary Erlang-A / R
Delay Probability
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 4 5 6 7 8
10
11
12
13
14
16
17
18
19
20
22
23
beta 2 beta 1.6 beta 1.2 beta 0.8 beta 0.4 beta 0
beta -0.4 beta -0.8 beta -1.2 beta -1.6 beta -2
100
Time-Stable Performance of Time-Varying Systems
Waiting Time, Given Waiting:Empirical vs. Theoretical Distribution
Waiting Time given Wait > 0:
beta = 1.2 QD ( 0.1)
0
0.05
0.1
0.15
0.2
0.25
0.0
00
0.0
02
0.0
04
0.0
06
0.0
08
0.0
10
0.0
12
0.0
14
0.0
16
0.0
18
0.0
20
ho
urs
Simulated Theoretical (N=191)
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).
101
Calculating the Offered-Load R(t), Theoretically
◮ Offered-Load Process: L(·) = Least number of servers thatguarantees no delay.
◮ Offered-Load Function R(t) = E [L(t)], t ≥ 0.
Think Mt/G/N?t + G vs. Mt/G/∞: Ample-Servers.
102
Calculating the Offered-Load R(t), Theoretically
◮ Offered-Load Process: L(·) = Least number of servers thatguarantees no delay.
◮ Offered-Load Function R(t) = E [L(t)], t ≥ 0.
Think Mt/G/N?t + G vs. Mt/G/∞: Ample-Servers.
Four (all useful) representations, capturing “workload before t":
R(t) = E [L(t)] =
∫ t
−∞
λ(u) · P(S > t − u)du = E
[
A(t)− A(t − S)
]
=
= E
[∫ t
t−S
λ(u)du
]
= E [λ(t − Se)] · E [S] ≈ ... .
◮ {A(t), t ≥ 0} Arrival-Process, rate λ(·);◮ S (Se) generic Service-Time (Residual Service-Time).
102
Calculating the Offered-Load R(t), Theoretically
◮ Offered-Load Process: L(·) = Least number of servers thatguarantees no delay.
◮ Offered-Load Function R(t) = E [L(t)], t ≥ 0.
Think Mt/G/N?t + G vs. Mt/G/∞: Ample-Servers.
Four (all useful) representations, capturing “workload before t":
R(t) = E [L(t)] =
∫ t
−∞
λ(u) · P(S > t − u)du = E
[
A(t)− A(t − S)
]
=
= E
[∫ t
t−S
λ(u)du
]
= E [λ(t − Se)] · E [S] ≈ ... .
◮ {A(t), t ≥ 0} Arrival-Process, rate λ(·);◮ S (Se) generic Service-Time (Residual Service-Time).
◮ Relating L, λ,S (“W”): Time-Varying Little’s Formula.Stationary models: λ(t) ≡ λ then R(t) ≡ λ× E[S].
102
Calculating the Offered-Load R(t), Theoretically
◮ Offered-Load Process: L(·) = Least number of servers thatguarantees no delay.
◮ Offered-Load Function R(t) = E [L(t)], t ≥ 0.
Think Mt/G/N?t + G vs. Mt/G/∞: Ample-Servers.
Four (all useful) representations, capturing “workload before t":
R(t) = E [L(t)] =
∫ t
−∞
λ(u) · P(S > t − u)du = E
[
A(t)− A(t − S)
]
=
= E
[∫ t
t−S
λ(u)du
]
= E [λ(t − Se)] · E [S] ≈ ... .
◮ {A(t), t ≥ 0} Arrival-Process, rate λ(·);◮ S (Se) generic Service-Time (Residual Service-Time).
◮ Relating L, λ,S (“W”): Time-Varying Little’s Formula.Stationary models: λ(t) ≡ λ then R(t) ≡ λ× E[S].
QED-c: Nt = Rt + βRct , 1/2 ≤ c < 1; (c = 1 separate analysis).
102
Extending the Notion of the “Offered-Load"
◮ Business (Banking Call-Center): Offered Revenues
◮ Healthcare (Maternity Wards): Fetus in stress
◮ 2 patients (Mother + Child) = high operational and cognitive load◮ Fetus dies ⇒ emotional load dominates
◮ ⇒◮ Offered Operational Load
◮ Offered Cognitive Load
◮ Offered Emotional Load
◮ ⇒ Fair Division of Load (Routing) between 2 Maternity Wards:One attending to complications before birth, the other tocomplications after birth, and both share normal birth
106
ServNets: Data-Based Online Automatic Creation
w/ V. Trofimov, E. Nadjharov, I. Gavako = Technion SEELab
◮ ServNets = QNets, SimNets, FNets, DNets
◮ SimNets of Service Systems = Virtual Realities
◮ SimNets also of QNEts, FNets, DNets
eg. ED MD (Physics): Where are the Differential Equations?
112
ServNets: Data-Based Online Automatic Creation
w/ V. Trofimov, E. Nadjharov, I. Gavako = Technion SEELab
◮ ServNets = QNets, SimNets, FNets, DNets
◮ SimNets of Service Systems = Virtual Realities
◮ SimNets also of QNEts, FNets, DNets
eg. ED MD (Physics): Where are the Differential Equations?
◮ Ultimately, Research Labs will become necessary (hence mustbe funded!): offering universal access to data and ServNets, andtheir analysis
112
ServNets: Data-Based Online Automatic Creation
w/ V. Trofimov, E. Nadjharov, I. Gavako = Technion SEELab
◮ ServNets = QNets, SimNets, FNets, DNets
◮ SimNets of Service Systems = Virtual Realities
◮ SimNets also of QNEts, FNets, DNets
eg. ED MD (Physics): Where are the Differential Equations?
◮ Ultimately, Research Labs will become necessary (hence mustbe funded!): offering universal access to data and ServNets, andtheir analysis
◮ Data-based Research: Tradition in Physics, Chemistry, Biology;Psychology (now also in Transportation (Science) and(Behavioral) Economics)
◮ Why not in Service Science / Engineering / Management ?
112
ServNets: Data-Based Online Automatic Creation
w/ V. Trofimov, E. Nadjharov, I. Gavako = Technion SEELab
◮ ServNets = QNets, SimNets, FNets, DNets
◮ SimNets of Service Systems = Virtual Realities
◮ SimNets also of QNEts, FNets, DNets
eg. ED MD (Physics): Where are the Differential Equations?
◮ Ultimately, Research Labs will become necessary (hence mustbe funded!): offering universal access to data and ServNets, andtheir analysis
◮ Data-based Research: Tradition in Physics, Chemistry, Biology;Psychology (now also in Transportation (Science) and(Behavioral) Economics)
◮ Why not in Service Science / Engineering / Management ?
◮ Moreover, address the Reproducibility and ProprietaryCrisis in Scientific Research
112
Data-Based Creation ServNets: some Technicalities
◮ ServNets = QNets, SimNets, FNets, DNets
◮ Graph Layout: Adapted from but significantly extends Graphviz(AT&T, 90’s); eg. edge-width, which must be restricted topoly-lines, since there are “no parallel Bezier (Cubic) curves(Bn(p) = EpF [B(n, p)], 0 ≤ p ≤ 1)
◮ Algorithm: Dot Layout (but with cycles), based on Sugiyama,Tagawa, Toda (’81): “Visual Understanding of HierarchicalSystem Structures"
113
Data-Based Creation ServNets: some Technicalities
◮ ServNets = QNets, SimNets, FNets, DNets
◮ Graph Layout: Adapted from but significantly extends Graphviz(AT&T, 90’s); eg. edge-width, which must be restricted topoly-lines, since there are “no parallel Bezier (Cubic) curves(Bn(p) = EpF [B(n, p)], 0 ≤ p ≤ 1)
◮ Algorithm: Dot Layout (but with cycles), based on Sugiyama,Tagawa, Toda (’81): “Visual Understanding of HierarchicalSystem Structures"
◮ Draws data directly from SEELab data-bases:◮ Relational DBs (Large! eg. USBank Full Binary = 37GB, Summary
Tables = 7GB)◮ Structure: Sequence of events/states, which (due to size)
partitioned (yet integrated) into days (eg. call centers) or months(eg. hospitals)
◮ Differs from industry DBs (in call centers, hospitals, websites)
113