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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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May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 3
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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Building a loss approximation
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
Building a loss approximation
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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.
Building a loss approximation
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performance measures of Queueing
Systems:
Average Waiting Time (Wq) :
Building a loss approximation
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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Simulation experiments
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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Simulation experiments
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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Simulation experiments
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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Simulation experiments
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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Simulation experiments
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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Simulation experiments
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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Simulation experiments
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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