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Approximating the Performance of Call

Centers with Queues using Loss Models

Ph. Chevalier, J-Chr. Van den Schrieck

Universit catholique de Louvain


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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


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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)


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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


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Building a loss approximation

Queue with

infinite length

Incoming

inputs with

infinite

patience

Rejected inputs

No queues

Rejected ifnothing

available


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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


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multiple skill example

Lost calls

Type Z-Calls

Z

Type X-Calls Type Y-Calls

X Y

X-Y

X-Y-Z



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performance measures of Queueing

Systems:

Probability of Waiting:

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

With

a = / , the incoming load (in Erlangs).

s the number of servers.



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performance measures of Queueing

Systems:

Average Waiting Time (Wq) :


Finding C(s,a) is the key element


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Erlang formulas

Link between Erlang B and Erlang C:

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


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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


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Approximations

Hayward Loss:

Where:

is the overflow rate

z is the peakedness of the incoming flow,


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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.


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Simulation experiments

Description

Comparison of systems with loss and of

systems with queues. Both types receiveidentical incoming data.

Comparison with analytically obtained

information. analysis of results


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5 Erlangs 5 Erlangs

X = 3 Y = n

X-Y = 7

nfrom 1 to 10

Experiments with 2 types of

demands


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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

ssSystem(

simulated)

Operators to X-flow

Operators to Y-flow


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Waiting Probabilities (W.P.) using simulation data

0

0,10,2

0,3

0,4

0,5

0,6

0,70,8

0,9

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

Simulated W.P.

ComputedW.P.

W.P. X

W.P. Y


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Waiting Probabilities (W.P.) using computed data

0

0,10,2

0,3

0,4

0,5

0,6

0,70,8

0,9

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

Simulated W.P.

ComputedW.P.

W.P. X

W.P. Y.


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Accuracy of the Approximation compared with the

Simulations

0

0,005

0,01

0,015

0,02

0,025

Wait

ingProbabilit

Waiting Probability X

Waiting Probability Y

General Waiting

Probability

N = Sim

B = Sim

N = Sim

B = Comp

N = Comp

B = Comp


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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


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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.


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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

ComputedWaitingTime

Inf Bound for X

Inf Bound for Y

Sup Bound for X

Sub Bound for Y


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0

0,05

0,1

0,15

0,2

0 0,0

1

0,0

2

0,0

3

0,0

4

0,0

5

0,0

6

0,0

7

0,0

8

0,0

9

0,1 0,1

1

0,1

2

0,1

3

0,1

4

0,1

5

0,1

6

0,1

7

0,1

8

0,1

9

0,2

Simul Values

CompVa

lues

Inf X

Inf Y

Sup X

Sup Y


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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 impatientcustomers / limited size queue

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