Approximating the Performance of Call Centers with Queues using Loss Models
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Transcript of 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
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)
May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 4
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 5
Building a loss approximation
• Queue with infinite length
• Incoming inputs with infinite patience
Rejected inputs
• No queues
• Rejected if nothing available
May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 6
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 7
• 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
May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 8
• 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.
Building a loss approximation
May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 9
• performance measures of Queueing Systems:– Average Waiting Time (Wq) :
Building a loss approximation
Finding C(s,a) is the key element
May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 10
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 11
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 12
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 13
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.
May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 14
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 15
Simulation experiments
5 Erlangs 5 Erlangs
X = 3 Y = n
X-Y = 7
n from 1 to 10
Experiments with 2 types of demands
May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 16
Simulation experiments
Proportion of Operators for each Type of Demand
2
3
4
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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
May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 17
Simulation experiments
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
May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 18
Simulation experiments
Waiting Probabilities (W.P.) using computed data
0
0,1
0,2
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0,5
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0,8
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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.
May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 19
Simulation experiments
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
May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 20
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 21
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 22
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
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
May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 23
Simulation experiments
0
0,05
0,1
0,15
0,2
0 0,01
0,02
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0,1 0,11
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0,2
Simul Values
Co
mp
Val
ues
Inf X
Inf Y
Sup X
Sup Y
May 11, 2006 Ph. Chevalier, J-C Van den Schrieck, UCL 24
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