Approximating the Performance of Call Centers with Queues using Loss Models

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Approximating the Performance of Call Centers
with Queues using Loss Models

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

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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

• No queues
• Rejected if nothing available

Rejected inputs
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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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Building a loss approximation

• multiple skill example

Type Z-Calls
Z
X-Y-Z
Lost calls
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Building a loss approximation

• performance measures of Queueing Systems
• Probability of Waiting
• Erlang C formula (M/M/s system)
• With
•  a  ? / µ, the incoming load (in Erlangs).
•  s  the number of servers.

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Building a loss approximation

• 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
Sx
Sy


Sxy
Sxy
Sxy
Sxy
Operators allocated according to their
utilization by the different flows.
Sx
Sy
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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

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Simulation experiments
Experiments with 2 types of demands
n from 1 to 10
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Simulation experiments
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Simulation experiments
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Simulation experiments
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Simulation experiments
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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

gt One virtual queue, identical waiting times for
all types
gt Independent queues for each type, different
waiting times
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Average Waiting Time

• We derivate two bounds on the waiting time
• A lower bound consider one queue all operators
  are available for all calls from queue.
• An upper bound consider two queues operators
  answer only one type of call from queue.

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Simulation experiments
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Simulation experiments
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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 impatient customers /
  limited size queue