Approximating the Performance of Call
Centers with Queues using Loss Models
Ph. Chevalier, J-Chr. Van den Schrieck
Université catholique de Louvain
Observation
•High correlation between performance of
configurations in loss system and in systems with
queues
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
2
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)
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
3
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
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
4
Building a loss approximation
• Queue with
infinite length
• No queues
• Rejected if
nothing
available
• Incoming
inputs with
infinite
patience
Rejected inputs
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
5
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
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
6
Building a loss approximation
• multiple skill example
Type X-Calls
Type Y-Calls
X
Type Z-Calls
Y
Z
X-Y
X-Y-Z
Lost calls
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
7
Building a loss approximation
• 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.
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
8
Building a loss approximation
• performance measures of Queueing
Systems:
– Average Waiting Time (Wq) :
Finding C(s,a) is the key element
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
9
Erlang formulas
• Link between Erlang B and Erlang C:
Where B(s,a) is the Erlang B formula with parameters « s » and « a » :
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
10
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
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
11
Approximations
• Hayward Loss:
Where:
• ν is the overflow rate
• z is the peakedness of the incoming flow,
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
12
Approximations
• Idea: virtually allocate operators to the different
flows i.o. to make separated systems.
Sx
Sx
+
Sxy’
Sy
Sxy
Sxy
Sy
+
Sxy’’
Operators: allocated according to
their utilization by the different
flows.
May 11, 2006
Sx
Ph. Chevalier, J-C Van den Schrieck, UCL
Sy
13
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
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
14
Simulation experiments
Experiments with 2 types of
demands
5 Erlangs
5 Erlangs
X=3
Y=n
n from 1 to 10
X-Y = 7
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
15
Simulation experiments
Proportion of Operators for each Type of Demand
12
Loss System (simulated)
11
10
9
8
Operators to X-flow
7
Operators to Y-flow
6
5
4
3
2
2
4
6
8
10
12
Queueing System (sim ulated)
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
16
Simulation experiments
Waiting Probabilities (W.P.) using simulation data
0,9
Computed W.P.
0,8
0,7
0,6
0,5
W.P. X
0,4
W.P. Y
0,3
0,2
0,1
0
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
Simulated W.P.
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
17
Simulation experiments
Waiting Probabilities (W.P.) using computed data
0,9
0,8
Computed W.P.
0,7
0,6
0,5
W.P. X
0,4
W.P. Y.
0,3
0,2
0,1
0
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
Simulated W.P.
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
18
Simulation experiments
Accuracy of the Approximation compared with the
Simulations
0,025
Waiting Probability
Waiting Probability X
0,02
Waiting Probability Y
0,015
0,01
0,005
0
May 11, 2006
General Waiting
Probability
N = Sim
B = Sim
N = Sim
B = Comp
N = Comp
B = Comp
Ph. Chevalier, J-C Van den Schrieck, UCL
19
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
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
20
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.
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
21
Simulation experiments
Bounds for Average Waiting Time
Computed Waiting Time
2,5
2
Inf Bound for X
1,5
Inf Bound for Y
Sup Bound for X
1
Sub Bound for Y
0,5
0
0
0,5
1
1,5
2
2,5
Simulated Waiting Time
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
22
Simulation experiments
Comp Values
0,2
0,15
0,1
Inf X
Inf Y
Sup X
Sup Y
0,05
0
0 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,1 0,1 0,1 0,1 0,1 0,1 0,1 0,1 0,1 0,1 0,2
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
Simul Values
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
23
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
May 11, 2006
Ph. Chevalier, J-C Van den Schrieck, UCL
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