Challenges in Stochastic Modeling of Service Systems ...

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Challenges in StochasticModeling of Service Systems:Illustrations with Call Centers

Pierre L’Ecuyer

Universite de Montreal, Canadaand

Inria–Rennes, France

AEP 10, June 2014

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Simulation Challenges

Want to simulate large complex systems to study their behavior andimprove decision making.

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Examples: Climate change and environment

Need three-dimensional dynamic models for concentrations of greenhousegases CO2, CH4, N2O, freons, etc., aerosols, water vapor, air temperature,water temperature, water currents, surface ice and snow, clouds, etc.Stochastic aspects.

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Simulating a whole human body, or one of its parts.

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Spread of infectious diseases, mutations, resistance.

E.g., virus and bacteria in large populations of humans, animals, andplants. Resistance to antibiotics. Etc.

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Social networks, crowd behavior, evacuations, ...

Fertile ground for agent-based modeling.Behaviors are stochastic and not independent, not easy to model.

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Communication networks (various devices,stochastic demand)

Modeling (or even forecasting) the stochastic demand is difficult.Lots of dependence.

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Transportation networks, logistic and distributionsystems.

Similar problems. Uncertain demands and travel times.

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Stochastic models in finance and economics. Riskanalysis.

Much uncertainty here. Most models are SDEs based on historical data.How to better exploit external knowledge? Dependence between assetprices in a portfolio (beyond correlations) has a large impact on risk.

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Health-care (clinics, hospitals, ambulances, patientevolution)

Stochastic demand, the evolution of each patient involves uncertainty, etc.

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Telephone call centers (or contact centers)

Include sales by telephone, customer service, billing/recovery, publicservices, 911, taxis, pizza order, etc. Employ around 3% of workforce inNorth America. More than agriculture!

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Simulation Challenges

Want to simulate large complex systems to study their behavior andimprove decision making.

I Trustable (valid) stochastic modeling of complex systems.Taking account of various type of external information.

I Simulation-based optimization and control.

I Speed of execution of large simulations.

I Interoperability, interaction between models, reuse, standards, etc.

I Modeling methodology and tools for large and complex systems.Agent-based modeling. Modeling human behavior.

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Big Data

Sometimes huge amounts of data available to build stochastic models.How can we exploit this huge mass of data to build credible models?

How to effectively update the models in real time as new data comes in?

Strong links with data mining, machine learning, Bayesian statistics.

Generally much more complicated than selecting univariate distributionsand estimating their parameters. Model inputs are often multivariatedistributions and stochastic processes, with hard-to-model (but important)dependence between them, and parameters that are themselves stochastic.

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Illustrative Example: A Multiskill Call Center

Different call types. Depends on required skill, language, importance, etc.Agent types (groups). Each has a set of skills to handle certain call types.Service time distribution may depend on pair 〈call type, agent group〉.

λ1 λ2 . . . λK

? ? ?- - -

-

Arrivals

Agent types

Service cdf

AbandonmentsCall routing rules and queues

? ?S1 SJ· · ·

G1,1

?

. . . GK ,1

?

G1,J

?

. . . GK ,J

?

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Baby Example: the N Model

Call Type 1 Call Type 2

Waiting queue

Arrivals

Abandonments

1 2Agent group

Served

Router

Here, group 2 can serve both call types.In large call centers, can have over 100 call types and agent groups.

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Typical call center models

Arrival process is nonstationary and much more complicated than Poisson.Service times are not exponential and not really independent.Abandonments (balking + reneging), retrials, returns, etc.

Skill-based routing: Rules that control in real time the call-to-agent andagent-to-call assignments. Can be complex in general.Static vs dynamic rules. (e.g., using weights).

Agents using fewer skills tend to work faster. Also less expensive.Compromise between single-skill agents (specialists) vs flexible multiskillagents (generalists).

Staffing/scheduling/routing optimization: objective function andconstraints can account for cost of agents, service-level, expected excesswaiting time, average wait, abandonment ratios, agent occupancy ratios,fairness in service levels and in agent occupancies, etc. Various constraintson work schedules.

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Examples of common performance measures

Service level: SL(τ) = fraction of calls answered within acceptable waitingtime τ . (May exclude calls that abandon before τ .)May consider its observed value over a fixed time period (randomvariable), or its expectation, or the average in the long run (infinitehorizon), or a tail probability P[SL(τ) ≥ `].

Abandonment ratio: fraction of calls that abandon.

Average waiting time for each call type.

Agent occupancy: fraction of the time where each agent is busy.

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Performance evaluation, single call type

Arrival rate λ, service rate µ, load λ/µ, s servers, waiting time W .

M/M/s queue (Erlang-C). CTMC model.Approx. of P[W > 0], P[W > τ ], and E[W ].

Approximation under quality and efficiency driven (QED) regime:λ→∞ and s →∞ with α = P[W > 0] ∈ (0, 1) fixed.Halfin and Whitt (1981).Square root safety staffing: ds∗ = λ/µ+ β

√λ/µe.

Could make sense for some large call centers.

M/M/s + M queue (Erlang-A).

Approx. of γ = P[abandon], P[W > 0], and α = P[W > τ ].QED(τ): Fix τ , α, and γ > 0.Modified square root rule: ds∗ = (1− γ)λ/µ+ δ

√(1− γ)λ/µe.

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Multiple call types, multiskill agents

Much more difficult.Call routing rules become important and can be complicated.Approximations for service levels are not very good.

Must rely on simulation.

In our lab, we develop ContactCenters, a Java simulation and optimizationsoftware library for contact centers. Also some tools for model estimationfrom data.

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Data on call arrivalsAvailable observations (for each day): X = (X1, . . . ,Xp), arrival countsover (15 or 30 minutes) successive time periods.

020

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0

Quarter of hour

num

ber o

f cal

ls

6am 10am 2pm 5pm 8pm

Ex.: Typical realizations of X for a Monday (15-min periods).Non-stationary. Strong dependence between the Xj ’s.Similar behavior in many other settings: customer arrivals at stores,incoming demands for a product, arrivals at hospital emergency, etc.

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All days, call volumes before and after T = 2 p.m.

0 2000 4000 6000 8000 10000 12000

020

0040

0060

0080

0010

000

# calls arrived this day before T

# ca

lls a

rrive

d th

is da

y af

ter T Mon.

Tue.Wed.Thur.Fri.Sat.Sun.

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Modeling the arrivals

Stationary Poisson process as in Erlang formulas? No.

Poisson process with time-dependent arrival rate λ(t)?Would imply that Var[Xj ] = E[Xj ]. Typically far from true.

True arrival rate depends on several factors that are hard to predict. Wecan view it as stochastic, say

Λj = Bjλj and Xj ∼ Poisson(Λj) over period j , where

B = (B1, . . . ,Bp) = vector of random busyness factors with E[Bj ] = 1,λ = (λ1, . . . , λp) = vector of constant base rates (scaling factors).

Var[Xj ] = E[Var[Xj |Bj ]] + Var[E[Xj |Bj ]] = λj(1 + λjVar[Bj ]).

Dispersion index (DI) and its standardized version (SDI):

DI(Xj) = Var[Xj ]/λj = 1 + λjVar[Bj ] ≥ 1,

SDI(Xj) = (DI[Xj ]− 1)/λj = Var[Bj ].

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Corr[Xj ,Xk ] =Corr[Bj ,Bk ]

[((1 + 1/(Var[Bj ]λj))(1 + 1/(Var[Bk ]λk))]1/2.

We expect Corr[Xj ,Xk ] ≈ Corr[Bj ,Bk ] for “large” periods or heavy traffic,and Corr[Xj ,Xk ] ≈ 0 over very short disjoint periods or light traffic.

In a simulation, we want to generate the Bj ’s, then generate the arrivalsone by one conditional on the piecewise-constant rates Λj .

Another approach (less convenient) is to model and directly generate theXj ’s, then randomize the arrival times.

Modeling the rates is harder because they are not observed!

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Data from an emergency call center (E)Take one call type, Monday to Thursday (similar days), after removingspecial days (holidays, etc.). Other days have different arrival patterns.Day starts at 5 a.m. and is divided into 48 periods of 30 minutes.Mean counts per period, ≈ λj :

10 20 30 400

5

10

15

20

25

Period

Mea

n C

ount

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Emergency call center

10 20 30 40

2

3

4

5

6

7

DI(

Xj)

Period, j

30 min aggregation1 hour aggregation2 hour aggregation4 hour aggregation

10 20 30 40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

(DI(

Xj)−

1)/λ

j

Period, j


DI (left) and SDI (right) as a function of j for different period lengths.

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Corr[Xj ,Xk ] in call center E, for 30 min to 4 hour data aggregations.

10 20 30 40

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Data from a business call center (B)One call type, Tuesday to Friday, after removing special days.Opening hours (8:00 to 19:00) divided into 22 periods of 30 minutes.Monday and Saturday have different patterns. Mean counts per period:

5 10 15 20100

150

200

250

300

350

Period

Mea

n C

ount

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Call center B

5 10 15 20

10

20

30

40

50

60

DI(

Xj)

Period, j


5 10 15 20

0.02

0.025

0.03

0.035

0.04

0.045

0.05

(DI(

Xj)−

1)/λ

j

Period, j


DI (left) and SDI (right) as a function of j for different period lengths.SDI is not as large as for center E, but DI is much larger.

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Corr[Xj ,Xk ] in call center B, for 30 min to 4 hour data aggregations.

5 10 15 20

5

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15

20 0.1

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Data from a public utility call center (U)One call type, data aggregated over 40 15-minute periods per day, from8:00 to 18:00, Monday to Friday, after removing special days.

5 10 15 20 25 30 35 4060

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Period

Mea

n C

ount

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Call center U

5 10 15 20 25 30 355

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30

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40

DI(

Xj)

Period, j

15 min aggregation30 min aggregation1 hour aggregation2 hour aggregation

5 10 15 20 25 30 35

0.03

0.04

0.05

0.06

0.07

0.08

0.09

(DI(

Xj)−

1)/λ

j

Period, j

15 min aggregation30 min aggregation1 hour aggregation2 hour aggregation

DI (left) and SDI (right) as a function of j for different period lengths.

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Corr[Xj ,Xk ] in call center U, for 30 min to 4 hour data aggregations.

10 20 30 40

5

10

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20

25

30

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40

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Rate models

Λj = Bjλj over period j .

Poisson Bj = 1 for all j .PGsingle Bj = B for all j , where B ∼ Gamma(α, α).PGindep Bj ’s are independent, Bj ∼ Gamma(αj , αj).

PG2 Bj = BjB, combines common B and independent Bj ’s.

PG2pow Bj = BjBpj/E[Bpj ].

PGnorta B has gamma marginals Bj and dependence specified bya normal copula (we fit all Spearman correlations).

Bj = G−1j (Φ(Zj)) where Z = (Z1, . . . ,Zp) ∼ N(0,R).

PGnortaR Norma copula with constraints: Corr[Zj ,Zk ] = aρ|j−k| + c .

Difficulty: We want to model the Bj ’s, but they are not observed, only theXj ’s are observed. This makes parameter estimation by maximumlikelihood (ML) much more challenging, because we have no closed formexpression for the likelihood.Moment matching is often possible, but much less robust and reliable.We use Monte Carlo-based methods for ML estimation.

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Example: Likelihood Function for PG2 ModelBi ,j = busyness factor for day i , period j .Bi = busyness factor for day i .

p(X|B, β,α,λ) =

∫ ∞0

. . .

∫ ∞0

I∏i=1

p∏j=1

(λj Bi,j Bi )Xi,j e−λj Bi,j Bi

Xi,j !

ααj

j Bαj−1

i,j e−αj Bi,j

Γ(αj)dBi,j

=I∏

i=1

p∏j=1

∫ ∞0

(λj Bi,j Bi )Xi,j e−λj Bi,j Bi

Xi,j !

αjαj B

αj−1

i,j e−αj Bi,j

Γ(αj)dBi,j

=

[p∏

j=1

αjIαj

Γ(αj)I

]I∏

i=1

p∏j=1

Γ(αj + Xi,j)

Xi,j !

(Biλj)Xi,j

(αj + Biλj)Xi,j+αj

p(X|β,α,λ) =

[p∏

j=1

αjIαj

Γ(αj)I

][I∏

i=1

p∏j=1

Γ(αj + Xi,j)

Xi,j !

]

·I∏

i=1

∫ ∞0

[p∏

j=1

(Biλj)Xi,j

(αj + Biλj)Xi,j+αj

]ββBβ−1

i e−Biβ

Γ(β)dBi .

Want to maximize this. No closed form expression for the last integral.

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How the models match the DI, for Center E

10 20 30 400.5

1

1.5

2

2.5

3

Period

Coe

ffici

ent o

f dis

pers

ion

PoissonPGsinglePGindepPG2PG2powPGnortaData

The DI for the models and data.

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How the models match the correlations, for Center E

10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Period, j

Cor

r(X

− j, X

+ j)

PGsinglePG2PG2powPGnortaPGnortaAR1PGnortaARMData

Sample correlation between past and future demand.

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How the distribution predicted by the model fits thedata out-of-sampleFor each observation i (one day), estimate the model without that day,then for each period j (or block of successive periods) compute interval[Li ,j ,Ui ,j ] such that P[Xi ,j ∈ [Li ,j ,Ui ,j ]] ≈ p (desired coverage) accordingto model, then compute the proportion of days where Xi ,j ∈ [Li ,j ,Ui ,j ] andcompare with p via sum of squares.

RMS Deviation of out-of-sample coverage probability, for call center E:75% target cover 90% target cover

0.5 h 1 h 2 h 4 h 8 h 0.5 h 1 h 2 h 4 h 8 hPoisson 10.7 16.6 23.5 31.3 37.5 8.5 13.8 21.0 30.1 38.7PGsingle 7.2 10.0 12.5 13.5 12.0 5.3 7.8 10.1 11.4 9.0PGindep 1.3 5.3 12.7 21.4 29.9 0.8 4.1 10.2 18.7 29.1PG2 2.1 4.9 8.7 11.4 11.6 1.6 3.8 6.8 9.4 8.8PG2pow 1.5 2.9 4.4 5.1 5.0 1.0 2.0 3.1 3.4 3.0PGnorta 1.3 1.7 1.7 1.7 1.3 0.8 1.1 1.2 1.3 0.8PGnortaR 1.3 2.4 3.4 4.3 4.5 0.9 1.5 2.2 2.7 2.8

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How the models match the DI for Center B

5 10 15 200

2

4

6

8

10

12

14

Period

Coe

ffici

ent o

f dis

pers

ion


The DI for the models and data.

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How the models match the correlations for Center B

5 10 15 200.4

0.5

0.6

0.7

0.8

0.9

1

Period, j

Cor

r(X

− j, X

+ j)


Sample correlation between past and future demand.

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RMS Deviation of out-of-sample coverage probability, for call center B.75% target cover 90% target cover

0.5 h 1 h 2 h 4 h 8 h 0.5 h 1 h 2 h 4 h 8 hPoisson 43.1 50.9 57.5 61.9 66.7 44.7 55.8 64.4 71.3 77.7PGsingle 7.6 7.1 6.1 4.0 2.3 5.8 6.1 5.4 4.0 3.4PGindep 3.1 13.2 27.3 39.3 48.7 2.0 12.1 26.4 41.2 51.9PG2 4.8 4.1 5.1 4.3 2.6 3.0 2.9 3.3 2.9 2.3PG2pow 2.5 3.3 4.1 2.0 0.8 1.7 3.3 3.7 2.8 2.7PGnorta 3.2 3.0 2.7 1.2 1.3 2.0 2.4 2.2 1.7 2.0PGnortaR 3.2 3.1 2.8 1.9 0.5 2.0 2.4 2.3 2.2 2.8

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How the models match the DI for Center U

5 10 15 20 25 30 35 400

2

4

6

8

10

12

Period

Coe

ffici

ent o

f dis

pers

ion


Comparison of the DI for the models and data.

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How the models match the correlations for Center U

5 10 15 20 25 30 35 400.4

0.5

0.6

0.7

0.8

0.9

1

Period, j

Cor

r(X

− j, X

+ j)


Comparison of sample correlation between past and future demand.

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RMS Deviation of out-of-sample coverage probability, for call center U.75% target cover 90% target cover

1/4 h 1/2 h 1 h 2 h 4 h 1/4 h 1/2 h 1 h 2 h 4 hPoisson 38.9 47.1 53.9 59.6 64.6 39.2 50.9 59.7 67.5 74.4PGsingle 8.6 8.0 6.9 4.0 1.7 7.3 7.0 5.4 3.1 1.4PGindep 4.5 10.5 24.6 36.5 46.3 1.8 8.4 22.3 37.8 51.2PG2 4.4 3.4 3.8 3.3 2.2 2.0 3.0 3.5 2.5 1.7PG2pow 4.0 2.3 2.4 2.7 2.0 1.5 1.7 1.6 1.1 1.1PGnorta 4.4 4.1 3.9 3.4 2.7 1.8 2.2 2.4 2.3 2.4PGnortaR 4.4 4.0 4.2 4.0 3.2 1.8 2.3 2.5 2.7 2.2

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More on arrival process modeling

Modeling the arrival rates over successive days.Dependence between the days.Seasonal effects (day of the week, period of the year).Special days (holidays, special events, etc.).External effects (weather, marketing campaigns, etc.).

Dependence between call types: the arrival rate should in fact be amultivariate process. Modeling via copulas.

Arrival bursts in emergency call center.

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Modeling the service times

In call center U, the available data for service times is the number of callsof each type handled by each agent on each day, and the average durationof these calls. From this, we can estimate the mean and variance of aservice times and match those to the mean and variance of a distributionsuch as lognormal or gamma.

Service times are usually not exponential.

Common assumption: the distribution depends only on the call type.

But on closer examination, we find that it depends on the individualagent, on the number of call types that the agent is handling, and maychange with time (learning effect, motivation and mood of agent, etc.).

This is an important fact to consider when making work schedules!

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Average service time per agent for one call type, incenter U (more than 1000 agents)

+++

++++++++++++

+

++++++++ +++++ +++ ++ +

+

+++ ++ + ++ ++++

+++

++ ++++ + ++ + +++ + ++ +++ +++ + + + ++++ +++++ ++ ++ + + ++ ++ + ++ +++++ + + + +++ ++ ++ ++ +++ +++ +++ + +++ ++ ++ ++++ ++++ ++ + +++ +++ +++++ ++ ++ ++ ++ +++ + ++ ++ ++ ++ ++ ++

+

+ ++ ++++ ++++ ++ ++++ ++ +++++ + +

+

++++ ++ + +++ +++ ++ ++++ ++ + + ++ ++++ +++ ++++ +

+

+ + ++ +++ + ++ +++ + ++ ++ ++ ++++ ++++ ++ +++++ ++ +++ ++ ++++ ++ ++++ ++++ + ++ ++ +++ +++ +

++

+ ++ ++++ ++ ++ ++ +++ +++ +++++ +++ +++ ++++ + ++ +++ + +++++ +++ ++ ++ + +++ ++++ + ++ ++++ ++++ ++ +++ + +++ ++ + ++

+

++ + + +++ ++ + ++ +++++ +++ + +++ +++ +++ ++++ ++ ++ ++++ +++ +++ +++ + ++ +++ ++++ + +++ +++ +++ ++ ++ ++++ ++ +++ +++ + +++ ++ ++ ++++++ ++ ++ +++ + ++++ ++ +

+

+ ++++ + ++++ + +++ +++ + +++ ++ +++ ++ +++++ + ++ +++++ +++ ++ + ++ ++ ++ +++ + ++ ++++ ++++ +++ ++ ++++ +++ ++ ++

+

++++++ ++ +++ + +++ ++ +++ ++++ ++ ++ ++ ++ + +++ +++ ++++ + ++++ ++++ + ++ +++ ++ ++ ++ ++ +++ +++ ++ ++++ +++++ ++ + ++ ++ + +++ +++ + ++ +++++ + +++ + + ++ ++ ++

+

++++ +++ ++ +++++ + ++++++ + + ++ +++++ ++ ++ ++ ++ +++ + +++ ++ +++ + ++++++ ++ ++ ++ ++++ +++ +++ +++++++ + ++ ++ + + +++ ++ +++ +++ +++ + +++ ++ ++++++ + ++ ++ + +++ + +++ +++++++ ++++ ++ ++ +++++ + + +++ +++++ + + ++ +++++ +++ +++ ++ ++ +++ + +++++ ++ ++ ++ + ++++ + + +++ ++ +++ + + ++ + +++ +++ ++ + ++++ ++++++ ++++++ +

+

+ ++++ + +++++ ++ ++++

+++

++ ++ ++++ +++++ ++ +++++ + ++++ +++++++ + +++ ++++++

+

++++++++

++

++++++++++ +++++++

+++

+

+

+

++

+

+

+

+

+++

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0Temps de service moyen par agent pour le type d'appel R_Facture_F

Nombre d'appels

Tem

ps d

e se

rvic

e

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Another call type

++

++

+

+++++++++

+ +++ ++ ++++++ + + +++ +++ +++ ++++ ++ ++ ++ ++ ++ + + +++ + ++ ++ + + ++ ++ +++ ++ ++ ++++ ++++ +++ ++ +++ +++ ++++ + +++ +++ + ++ + ++ +++ +++ ++++ ++ ++ ++ ++++ +++ ++++ + +++ ++ + + +++ ++ +++ ++++ ++ +++ + ++ + +++

+++

+ ++ + ++ +++ +++ + ++ ++ +++ ++ + + +++ ++ + + +++ + +++ ++++ + +++ ++ ++ ++ +++ + ++ ++ ++ ++++ + ++ + ++ ++++ + ++++ + + ++ + ++ ++ + +++ ++

+

+ +++ ++ ++ ++ ++ ++ ++ +++++ +++ +++++ +++

+

+ +

+

+++++ ++++++

+++

+

+++

+

+

+

0 500 1000 1500

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0

Temps de service moyen par agent pour le type d'appel R_Coupure_F

Nombre d'appels

Tem

ps d

e se

rvic

e

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Another call type

+

+++ +++++ ++++++ ++++++++

+

++++ +++ + ++ +++ +++++ ++ ++ ++ +++ ++++ ++++ ++++ +++ +++++++++ ++ +++++++ ++ ++ +++ +++ ++ ++ ++ ++++ + +++ +++ ++++ ++ ++++ ++++ ++++ +++++ +++ +++ ++++++ ++ ++++ ++++ +++++ ++ +++++ ++ +++ +++++ +++ ++++ +++ +++++ +++ ++ +++++ ++ ++ ++++++++++ +++++ +++ +++ + +++++ +++ + +++ ++ ++++ +++++++++

+

+ ++++++++++ ++++++ + +++ + ++++ +++ ++ + +++ +++++ ++++++ ++++ ++ +++++++++++ +++ ++ +++ ++ +++++ ++++ ++

+

+ ++ ++ +++ ++ +++++ ++ ++++ ++

+

+ +++ ++ +++++++ + +++ ++++ ++++ ++ ++ ++ +++++ ++++++++++ +++ ++++ ++ + +++ ++++++ ++

+

+ ++++ +++ ++++ ++++++ ++ + ++++++ +++++ + +++++++ ++ ++++ +++ + ++++++++ ++ ++++++ + +++++ ++ ++ +++ ++ +++ ++ ++++ +++ +++ ++ +

+

+++ +++++ ++ +++ +++++++ ++++++++++++ ++ ++++ +++ ++ +++

+

+++++++++++ +++ ++ ++++++++ +++++ +++

+

++++ ++ ++ +++++++ + +++

+

++ +++++ +++ +++ +++

+

+++++++++++++ +++++ +++ +++++ +++ +++++ ++++ +++ + ++ ++

+

++++ ++++++++++++

++

+++ ++ ++++

+

++++++++

+

++++ +++ ++++++++++

+

+++

+

+++

+

++++++

+

+

+

++

++

+++

+++

++++

+

+

+++

+

++

+

+

+

+

+

++

++++++++++++

++

+

+

+++++

+

++++++++ ++++

+

+

+

++++ ++

+

++

0 1000 2000 3000 4000 5000 6000

020

040

060

080

010

00

Temps de service moyen par agent pour le type d'appel R_Panne_F

Nombre d'appels

Tem

ps d

e se

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e

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Four different agents, same call typeAll have handled more than 1000 calls. Daily averages:

0 100 200 300 400

020

040

060

080

010

00Comparatif d'agents sur le type R_Facture_F

Index du jour

Tem

ps d

e se

rvic

e

●

●

●●

●

●

●

●●

●●

●●

●

●

●

●●●

●

●

●

●

●●

●● ●

● ●●●●●●

●

●●

●●

●●

●

●●●

●

●

●

●

●

●●

●

●●

●

●●

●

●●

●

●

●

●

AY2915 (1419)

BA9537 (1588)

BM5697 (1259)

CP4704 (1185)

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Same agent, 4 call types, weekly averages

0 10 20 30 40 50 60

5010

015

020

025

030

035

0

Temps de service hebdomadaire moyen de l'agent par type d'appel

Index de la semaine

Tem

ps d

e se

rvic

e ●

●

●●

●

● ●

●●

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●

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●

●● ●

●

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● ●●

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●

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●

● ●

●

●

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●

●

●

●

● ● ● ●

●

●

●

C_PANNE_F

R_ED_F

R_EDBloque_F

R_Panne_F

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Same agent, 6 call types

0 10 20 30 40 50 60

100

200

300

400

500

600


Index de la semaine

Tem

ps d

e se

rvic

e

●

●

C_PANNE_F

R_AideTech_F

R_Budget_F

R_Facture_F

R_MVE_F

R_Panne_F

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Same agent, 8 call types

0 10 20 30 40 50 60

020

040

060

080

0


Index de la semaine

Tem

ps d

e se

rvic

e

●

●

●

●

●

●

C_PANNE_F

R_Budget_F

R_Facture_A

R_Facture_F

R_MVE_F

R_Panne_F

R_RetProact_F

R_TransfSpec1_F

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Modeling the evolution of service time averages

For given agent and call type, day i :

Mi = βdi + αi + Γwi + εi ,

where di = type of day i , wi = week of day i , and Γi is a random effectthat may follow, e.g., an AR process:

Γw = ρΓw−1 + ψw .

The εi and ψw are residuals (noise).This gives much better predictions than just taking the overall average foreach agent.

There may also be common effects across agents.

Would like to model the evolution of all distribution parameters.

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Performance measures and optimization

For a given staffing and routing strategy, the SL on a given day (or givenperiod) is a random variable. We may be interested in its distribution.

What if we pay a penalty iff the SL is below a given number today?

After solving some work-schedule optimization problem in some callcenter, we re-simulated with our best feasible solution for 10000 days, andcomputed the empirical distribution of the SL.

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SL0.85 0.87 0.89 0.91 0.93 0.95

Frequency

0

500

1000

1500 mean = 0.906

Distribution of Global SL

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SL0.2 0.4 0.6 1

Frequency

0

250

500

750

1000

1250

1500mean = 0.821

0.80

Distribution of SL Period: 7

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SL0.2 0.4 0.6 1

Frequency

0

250

500

750

1000

1250

1500

mean = 0.806

0.80


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SL0.5 0.6 0.7 0.9 1

Frequency

0

250

500

750

1000

1250

1500

1750

2000mean = 0.915

0.80


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SL0.5 0.6 0.7 0.9 1

Frequency

0

1250

2500

3750 mean = 0.965

0.80


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SL0.65 0.75 0.85 0.95

Frequency

0

500

1000

1500

2000

mean = 0.832

0.80

Distribution of SL for Call Type 3

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SL0.7 0.9 1

Frequency

0

250

500

750

1000

1250

1500

1750

2000

2250 mean = 0.937

0.80

Distribution of SL for Call Type 17

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SL0 0.25 0.5 0.75 1

Frequency (×103)

0

1

2

3

4

5

6

7

8

mean = 0.894

0.80

Distribution of SL for Call Type 2 in Period 20

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SL0 0.25 0.5 0.75 1

Frequency (×103)

0

1

2

3

4

5

6

7

mean = 0.911

0.80

Distribution of SL for Call Type 12 in Period 30

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Example of scheduling optimization problemSuppose the routing rules are fixed.

Several call types, several agent types, several time periods.

A shift type specifies the time when the agent starts working, when he/shefinishes, and all the lunch and coffee breaks.cs,q = cost of an agent of type s having shift type q.

The decision variables x and z are:(i) xs,q = number of agents of type i having shift type q;(ii) z`,s,j = number of agents of type ` that work as type-s agents inperiod j , with Ss ⊂ S` (they use only part of their skills).

This determines indirectly the staffing vector y, where ys,j = num. agentsof type s in period j , and aj ,q = 1 iff shift q covers period j :

ys,j =∑q

aj ,qxs,q +∑l∈S+

s

zl ,s,j −∑l∈S−s

zs,l ,j for all s, j .

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Scheduling Optimization Problemx = vector of shifts; c = their costs; y = staffing vector;(Long-run) service level for type k in period j (depends on entire vector y):

gk,j(y) =E[num. calls type k in period j answered within time limit]

E[num. calls type k in period j , ans., or abandon. after limit].

(P0) : [Scheduling problem]

min ctx =∑I

s=1

∑Qq=1 cs,qxs,q

subject to Ax + Bz = y,gk,j(y) ≥ lk,j for all k , j ,gj(y) ≥ lj for all j ,gk(y) ≥ lk for all k,g(y) ≥ l ,x ≥ 0, z ≥ 0, y ≥ 0, and integer.

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Sample-path optimization via simulation

We simulate n independent operating days of the center, to estimate thefunctions g .

Let ω represent the source of randomness, i.e., the sequence ofindependent uniform r.v.’s underlying the entire simulation (n runs).

The empirical SL’s over the n simulation runs are:gn,k,j(y, ω) for call type k in period j ;gn,j(y, ω) aggregated over period j ;gn,k(y, ω) aggregated for call type k ;gn(y, ω) aggregated overall.

For a fixed ω, these are deterministic functions of y.

We replace the (unknown) functions g(·) by g(·, ω) and optimize.

To compute them at different values of y, we use simulation withwell-synchronized common random numbers. Discuss.

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Empirical (sample) scheduling optimization problem

(SP0n) : [Sample scheduling problem]

min ctx =∑

s

∑Qq=1 cs,qxs,q

subject to Ax + Bz = y,gn,k,j(y) ≥ lk,j for all k , j ,gn,j(y) ≥ lj for all j ,gn,k(y) ≥ lk for all k ,gn(y) ≥ l ,x ≥ 0, z ≥ 0, and integer.

Theorem: When n→∞, the optimal solution of SP0n converges w.p.1 tothat of P0. Moreover, if a standard large deviation principle holds for g(which is typical), the probability that the two solutions differ converges to0 exponentially with n. [Adaptation of Vogel 1994, for example.]

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Solving the sample optimization problem

Integer programming with cutting planes.[Atlason, Epelman, and Henderson, 2004; Cezik and L’Ecuyer 2005]Replace the nonlinear constraints in SP0n by a set of linear constraints.This gives an integer program (IP).

We start with a relaxation of the IP problem (fewer constraints).Then, at each step, use simulation to compute the service levels in SP0nfor the optimal solution y of the current IP.For each SL constraint that is not satisfied, add a cut based on estimatedsubgradient.Stop when all SL constraints of SP0n are satisfied.

In practice, for large problems, we solve the IP as an LP and round thesolution (at each step, to be able to simulate). We select a roundingthreshold δ (usually around 0.5 or 0.6). Heuristic!

Phase II: run longer simulation to perform a local adjustment to the finalsolution, using heuristics (add, remove, switch).

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Other objectives and constraints(alternative formulations)

Chance constraints: Replace long-term average gk,j(y) by a tail probabilityof the service level, e.g.:

P[SLk,j(τ) ≥ lk,j ] ≥ αk,j for all k , j .

Optimizing call routing rules.

Replace constraints by penalties.

Etc.

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Conclusion

Simulation can be useful only to the extent that we can trust the model.

We can do more and more runs and compute arbitrarily tight confidenceintervals on certain unknown quantities, but this can be meaningless if themodel is not accurate enough.

Huge masses of data are becoming available (currently) at a rate neverseen before and that increases exponentially. Exploiting this data to buildcredible and valid stochastic models of complex systems is in my opinionthe biggest challenge that we now face for simulation.

Challenges in Stochastic Modeling of Service Systems ...

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