Approximating the Performance of Call Centers with Queues using Loss Models

Approximating the Performance of Call Centers with Queues using Loss

Models

Ph. Chevalier, J-Chr. Van den Schrieck Université catholique de Louvain

May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 2

Observation

•High correlation between performance of configurations in loss system and in systems with queues


Loss models are easier than queueing models

• Smaller state space.• Easier approximation methods for

loss systems than for queueing systems.(e.g. Hayward, Equivalent Random

Method)


Main assumptions

• Multi skill service centers (multiple independant demands)

• Poisson arrivals• Exponential service times• One infinite queue / type of demand• Processing times identical for all

type


Building a loss approximation

• Queue with infinite length

• Incoming inputs with infinite patience

Rejected inputs

• No queues

• Rejected if nothing available



• Server configuration– Use identical configuration in loss

system• Routing of arriving calls

– Can be applied to loss systems• Scheduling of waiting calls

– No equivalence in loss systems– Difficult to approximate systems with

other rules than FCFS


• multiple skill example

Lost calls

Type Z-Calls

Z

Type X-Calls Type Y-Calls

X Y

X-Y

X-Y-Z



• performance measures of Queueing Systems:– Probability of Waiting:

Erlang C formula (M/M/s system):

With• « a » = λ / μ, the incoming load (in Erlangs).• « s » the number of servers.



• performance measures of Queueing Systems:– Average Waiting Time (Wq) :


Finding C(s,a) is the key element


Erlang formulas

• Link between Erlang B and Erlang C:

Where B(s,a) is the Erlang B formula with parameters « s » and « a » :


Approximations

• We try to extend the Erlang formulas to multi-skill settings– Incoming load « a »: easily determined– B(s,a) : Hayward approximation– Number of operators « s » : allocation

based on loss system


Approximations

• Hayward Loss:

Where:• ν is the overflow rate• z is the peakedness of the incoming flow,


Approximations

• Idea: virtually allocate operators to the different flows i.o. to make separated systems.

Sx Sy

Sxy

Sx Sy

SxySxy’ Sxy’’+ +

Sx Sy

Operators: allocated according to their utilization by the

different flows.


Simulation experiments

• Description– Comparison of systems with loss and

of systems with queues. Both types receive identical incoming data.

– Comparison with analytically obtained information.

• analysis of results



5 Erlangs 5 Erlangs

X = 3 Y = n

X-Y = 7

n from 1 to 10

Experiments with 2 types of demands



Proportion of Operators for each Type of Demand

2

3

4

5

6

7

8

9

10

11

12

2 4 6 8 10 12

Queueing System (simulated)

Lo

ss S

yste

m (

sim

ula

ted

)

Operators to X-f low

Operators to Y-f low



Waiting Probabilities (W.P.) using simulation data

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Simulated W.P.

Co

mp

ute

d W

.P.

W.P. X

W.P. Y



Waiting Probabilities (W.P.) using computed data

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Simulated W.P.

Co

mp

ute

d W

.P.

W.P. X

W.P. Y.



Accuracy of the Approximation compared with the Simulations

0

0,005

0,01

0,015

0,02

0,025

Wai

tin

g P

rob

abili

ty

Waiting Probability X

Waiting Probability Y

General WaitingProbability

N = SimB = Sim

N = SimB = Comp

N = CompB = Comp


Average Waiting Time

• The interaction between the different types of demand is a little harder to analyze for the average waiting time.

– Once in queue the FCFS rule will tend to equalize waiting times

– Each type can have very different capacity dedicated

=> One virtual queue, identical waiting times for all types

=> Independent queues for each type, different waiting times


Average Waiting Time

• We derivate two bounds on the waiting time:

1. A lower bound: consider one queue ; all operators are available for all calls from queue.

2. An upper bound: consider two queues ; operators answer only one type of call from queue.



Bounds for Average Waiting Time

0

0,5

1

1,5

2

2,5

0 0,5 1 1,5 2 2,5

Simulated Waiting Time

Co

mp

ute

d W

aiti

ng

Tim

e

Inf Bound for X

Inf Bound for Y

Sup Bound for X

Sub Bound for Y



0

0,05

0,1

0,15

0,2

0 0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,1 0,11

0,12

0,13

0,14

0,15

0,16

0,17

0,18

0,19

0,2

Simul Values

Co

mp

Val

ues

Inf X

Inf Y

Sup X

Sup Y


Limits and further research

• Service time distribution : extend simulations to systems with service time distributions different from exponential

• Approximate other performance measures

• Extention to systems with impatient customers / limited size queue

Documents

Approximating the Performance of Call Centers with Queues using Loss Models