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Capacity Management in Call Centers

Basic Models and Links to Current Research

from a review article authored with

Ger Koole and Avishai Mandelbaum

Outline:

• Tutorial

– background on how calls are handled

– basic model of call-center operations

• Research overview

– system primitives

– basic model

– routing and networking

– further directions for research

1

Tutorial, background...

Typical call-center environments:

Top figure from Larreche et al. (1997).

2


Representative infrastructure for asingle location:

customers PABX

agents

IVR/VRU

ACD

CTIserverPSTN customer

data servertrunk lines

3


Queueing model associated with asingle location:

retrials

arrivals

abandon

queue

busy

lost calls

retrials

lost calls returns

N = 3 CSR-servers

5 = (k – N) places in queue

w = 5 work stations

k = 8 trunk lines (not visible)

Call-center hardware Queueing model parameters

4

Tutorial, basic model...

Traditional model used in capacitymanagement:

• Work arriving over time

• Short-term capacity requirements

• Medium-term shift scheduling and rostering

• Long-term hiring and training

5


Number of calls arriving over severaltime horizons:

…each month of the year …each day of the month

…each hour of the day …each minute of the hour

Number of calls arriving…

6


Work modelled as arriving in discretetime buckets:

0

5

10

15

20

25

30

8 9 10 11 12 1 2 3 4 5

½-hour interval i

Ri = λi/µ = work arriving during i

7


Waiting-time standards drivecapacity requirements:

• ASA = average speed of answer

E[wait]

• TSF = telephone service factor (or service level)

P{wait ≤ T}

• Fraction of calls abandoned before served

P{abandon}

• Determining capacity requirements for interval i:

e.g. Ni = min{N | E[wait] ≤ ASA∗}

8


A report with 1/2-hour intervals:

9


Performance estimate usesM/M/N/∞ model:

x xx

xxx

x• no blocking, abandonment, or retrials

• fixed arrival and service rates

• exponential interarrival and service times

• measures of stationary performance

10


Estimates require only λi, µ, N :

• P{wait > 0}

= 1−∑N−1

m=0 Rim/m!∑N−1

m=0 Rim/m! + (Ri

N/N !) / (1− Ri/N)

(Erlang-C formula)

• P{wait ≤ T}

= 1 − P{wait > 0} · P{wait > T |wait > 0}

= 1 − P{wait > 0} · e−(Nµ−λi)

• E[wait]

= P{wait > 0} · E[wait|wait > 0]

= P{wait > 0} · (Nµ − λi)−1

11


Example with 30-second ASA:

0

5

10

15

20

25

30

8 9 10 11 12 1 2 3 4 5

½-hour interval i

offered load idleness for ASA

Ni = number of CSRs working in i

12


Determining staffing requirementsfor a day or week:

• Data

Ni = num. CSRs required for interval i

cj = cost of putting a CSR on schedule j

aij ={

1 if sched. j take calls during int. i,0 otherwise.

• Decision variables

xj = num. CSRs to work on schedule j

• Math program to choose schedules

min{c′x |Ax ≥ N ;x ≥ 0;x integer}

13


Example with 4-hour shifts on the1/2-hour:

total Ni1 13 121 1 21 211 1 1 21 191 1 1 1 21 191 1 1 1 1 24 201 1 1 1 1 1 24 211 1 1 1 1 1 1 28 201 1 1 1 1 1 1 1 28 19

1 1 1 1 1 1 1 1 17 171 1 1 1 1 1 1 1 14 13

1 1 1 1 1 1 1 1 14 131 1 1 1 1 1 1 1 19 19

1 1 1 1 1 1 1 1 19 191 1 1 1 1 1 1 19 17

1 1 1 1 1 1 15 111 1 1 1 1 15 15

1 1 1 1 13 131 1 1 8 8

1 1 8 8

8:00-8:298:30-8:599:00-9:299:30-9:5910:00-10:2910:30-10:5911:00-11:2911:30-11:5912:00-12:2912:30-12:591:00-1:291:30-1:592:00-2:292:30-2:593:00-3:293:30-3:594:00-4:294:30-4:595:00-5:295:30-5:59 1 3 3

Shifts = xj8 8:30 9 9:30 10 10:30 11 11:30 12 12:30 1 1:30 2

43 13 8 0 0 3 0 4 0 2 5 2E-10 5 3

14


Excess capacity resulting from thescheduling IP:

0

5

10

15

20

25

30

8 9 10 11 12 1 2 3 4 5

½-hour interval i

head

coun

t

offered load idleness for ASA slack from shifts

15


More on the scheduling process:

• Scheduling IP recommends schedules

• Rostering assigns CSRs to schedules

(shift-bidding)

• Scheduling IP also recommends a number of

CSRs, 1′x

• Target headcount 1′x typically inflated to

accommodate “shrinkage”

• Target headcount 1′x can be used in a

longer-term hiring problem

16


Longer-term hiring and training:

• nt = 1′xt

target headcount for scheduling period t

• τ

lead time required to hire and train a new CSR

• Then in scheduling period t hirent+τ −t+τ−1∑

j=t

yj(1− β)τ−(j−t)

+

where

nt+τ = target headcount for hiring today

yj = CSRs that start working in period j

β = 1-period attrition rate

17


ACD reporting system drivescapacity planning:

18

Research, model primitives...

The arrival process:

• Time-inhomogeneous process

– Andrews and Cunningham (1995)

– Massey et al. (1996)

– Brown et al. (2002)

– Henderson (2003)

– Avramidis et al. (2004)

– Huang and Shen (2004)

• Uncertain rates

– Whitt (1999)

– Jongbloed and Koole (2001)



• Correlation among rates

– Andrews and Cunningham (1995)

– Whitt (1999)



19


Service times:

• Exponential distribution

– Kort (1983)

– Harris et al. (1987)

• Gamma distribution

– Chlebus (1997)

• Lognormal distribution

– Bolotin (1994)

– Mandelbaum et al. (2001)


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Abandonment:

• Earlier work on impatience and abandonment

– Palm (1943)

– Kort (1983)

– Roberts (1979)

• Abandonment behavior in a call center

– Mandelbaum et al. (2001)

– Zohar et al. (2002)


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Research, basic model...

System performance and limits:

• Many-server limits for M/M/N model

– Halfin and Whitt (1982)

– Borst et al. (2004)

• Systems with abandonment

– Baccelli and Hebuterne (1981)

– Brandt and Brandt (1999a,b)

– Garnett et al. (2002)

– Mandelbaum and Zeltyn (2004)

– Whitt (2004a,b,c,d)

• Also blocking, retrials, dependent λ...

– Jagerman (1974)

– Massey and Wallace (2002)

– Whitt (2002,2003)

– Aguir et al. (2004)

22


Staffing for a more complex arrivalprocess:

• Time-varying rates – numerical methods

– Yoo (1996)

– Ingolfsson et al. (2002)

• Time-varying rates – approximate methods

– PSA – Green and Kolesar (1991)

– fluids – Mandelbaum et al. (1995...2000)

– Mt/G/∞ systems

Eick et al. (1993a,b)

Jennings et al. (1996)

Massey and Whitt (1997)

Whitt (1999)

• Uncertain arrival rates (and absenteeism)

– Ross (2001)

– Chen and Henderson (2001)

– Jongbloed and Koole (2001)

– Whitt (2004e)

– Harrison and Zeevi (2005)

23


Higher-level problems:

• Solutions to scheduling and rostering problems

– break placement: Segal (1974)

– column restriction

Henderson and Berry (1976)

– alternative formulations

Aykin (1996), Brusco and Jacobs (2000)

– rostering: Thompson (1997)

• Joint staffing and scheduling

– Yoo (1996), Ingolfsson at al. (2000,2002)

– Atlason et al. (2002)

– Koole and van der Sluis (2003)

• Long-term hiring and training

– Aksin (2002)

– Gans and Zhou (2002)

24

Research, routing and networking...

Skills-based routing:

• The general routing problem

1 2 3 4 5

1 2 3 4 5 6types of calls

pools of CSRs

feasible routings

• Special routing structures

“I” “N” “X” “W” “M”“V”

1 2 2 2 2 11 1 1 1 3 2

1 2 2 2 11 1 1 1 32

25


Non-scaled analysis:

• Exact analysis of special designs

– Bhulai and Koole (2000)

– Gans and Zhou (2003,2004)

• Heuristic policies for special designs

– Brandt and Brandt (1997)

– Stanford and Grassmann (2000)

– Shumsky (2004)

– Stolletz and Helber (2004)

– Chevalier et al. (2004)

• Heuristic policies for more general designs

– Koole and Talim (2000)

– Borst and Seri (2000)

– Wallace and Whitt (2004)

26


Asymptotic analysis:

• “ED” regime

– batching policies –Harrison and Lopez (1999)

– thresholds, N-design – Bell and Williams (2001)

– Gcµ rules – Mandelbaum and Stolyar (2003)

• “QED” regime

– Armony and Maglaras (2004a,b)

– Armony and Mandelbaum (2004)


– Atar et al. (2004)

• Other scalings


– Bassamboo, Harrison and Zeevi (2004)

27


Networking:

• Network structures

network ACD

network

call centers

load balancing overflow load balancingand overflow

• Some recent telecom research

– Borst et al. (1996)

– Kogan et al. (1997)

– Servi and Humair (1999)

• SBR methods may also apply

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On the horizon...

Examples of further directions forresearch:

• A broader view of the service process

– Srinivasan and Talim (2004)

– Aksin and Harker (2001,2003)

– de Vericourt and Zhou (2003)

• Better understanding of customer and CSR

behavior

– Carmon and Kahneman (2002)

– Shumsky and Pinker (2003)

• Multiple levels of equilibria

– Whitt (2003)

– Armony and Maglaras (2004a,b)

• Customer relationship management by the

numbers

– Ariely et al. (2002)

– Gunes and Aksin (2003)

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