Service Capacity Planning

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Chapter 19

• Service Capacity Planning
• -Waiting Lines

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• A common phenomena in service systems is waiting.
  In our daily life, we observe this phenomena
  almost all the time
• in the bank, in the restaurant, during check-out
  in the supermarket,during flight check-in etc.
• The waiting customers need not always be
  people
• jobs waiting to be processed, trucks waiting to
  be loaded, airplanes waiting to land, internet
  requests waiting to be connected etc.
• The time customer spends waiting for service is a
  major determinant of quality.

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Why is there waiting?

• Waiting lines occur naturally because of two
  reasons

1. Customers arrive randomly, not at evenly
placed times nor at predetermined times 2.
Service requirements of the customers are
variable. (Think of a bank for example)

• Because of these two reasons, waiting lines form
  even in underloaded systems.

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Is there a cost of waiting?

• Quantifiable costs
• When the customers are internal (e.g employees
  waiting for making copies), salaries paid to the
  employees
• Cost of the space of waiting (e.g. patient
  waiting room)
• Loss of business (lost profits)
• Hard to quantify costs
• Loss of customer goodwill
• Loss of social welfare (e.g. patients waiting for
  hospital beds)

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Capacity -Waiting Trade-off

• Waiting lines can be reduced by increasing
  capacity
• More service counters
• Adding workers to increase speed

? COST!
Queuing Analysis

• Mathematical analysis of waiting lines
• Goal minimize sum of
• Service capacity costs
• Customer waiting costs
• Economic analysis can be done using either BEA or
  NPV

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Queuing Analysis
Total cost
Customer waiting cost
Capacity cost


Total cost
Cost
Cost of service capacity
Cost of customers waiting
Optimum
Service capacity
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Elements of Queuing System
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Basic System Characteristics

• Population Source
• Infinite source customer arrivals are
  unrestricted
• Finite source number of potential customers is
  limited

• Number of servers (channels)
• Number of service phases
• Arrival and service patterns
• Queue discipline (order of service)

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Modeling the Arrivals
Inter-arrival Time Time between two
consecutive arrivals (a random
quantity) Number Number of arrivals in a unit
time period (also a random quantity) How to
represent this randomness ? ? use common
probability distributions
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Modeling the Arrivals

• Determine the arrival rate (? ) average number
  of arrivals per unit time period
• Represent Number of arrivals with a Poisson
  probability distribution with rate (?).

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Modeling the Arrivals

• Represent Inter-arrival time with an Exponential
  probability distribution with mean (1/?).

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Modeling the Service

• Determine the service rate (?) number of
  customers that can be served per period
• Represent Service time with an Exponential
  probability distribution with mean (1/ ?).

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Important Remarks

• The use of exponential distributions is a
  convention. (it makes mathematical analysis
  simpler)
• You have to justify these distributions used in
  the waiting line models. (by collecting data and
  then checking for a distribution that fits the
  data, via statistical tests such as goodness of
  fit test)
• Poisson arrival is realistic but not exponential
  service

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Queuing Discipline

• First Come First Served (FCFS) is the most
  common, and perhaps the most fair one.
• There are other rules that prioritize customers
• Emergency customers first
• Highest Profit Customers are first
• etc.

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Service Performance Measures

• Average number of customers waiting (in the line
  or in the system)
• Average waiting time of customers
• System Utilization (percent capacity used)
• Probability that an arrival will have to wait

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Queuing Models-Infinite Source

• ? arrival rate, ? service rate (be careful of
  the units)
• c number of channels (it is also sometimes
  represented by M)
• ? utilization
• Lq Average number of customers in queue
• Ls Average number of customers in system
• Wq Average Waiting time in queue
• Ws Average Waiting time in system
• Pn Prob. of n customers in system

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Basic (Steady State) Relationships

• Utilization (should be lt1)
• Average Number in Service
• Average Number in Line (Lq ) model depen.
• Average Number in System
• Average Time Customers Wait in Line
  Wait in System

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Basic (Steady State) Relationships

• One of the most important relationship in
  queueing theory is called Littles Law
• Ls ?Ws
• Lq ?Wq
• Intuitive explanation?

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Two Popular Waiting Line Models

• 1. Model 1 Single Channel
• Model 3 Multiple Channel
• Single phase
• Poisson arrivals
• Exponential service times
• FCFS queue discipline
• No limit on the waiting line length

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Model 1
? Other service measures can be obtained from the
basic relationship formulas.
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Model 3
? Lq formula is complicated. Use Table at the
end of this chapter to read the Lq value.

• ? Use the basic relationship formulas to
  calculate other service measures
• Increasing the number of servers will bring down
  the waiting cost but increase the capacity cost

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Interesting Observations

• Waiting time distribution for an arrival in
  single channel queue joining the queue in steady
  state is exponentially distributed with parameter
  (? - ?).
• Waiting time in the system is less for double
  channel queue with ? and ? than for single
  channel with ? and (?/2).

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Extending Model 1

• ? Other measures from the basic formulas by
  replacing ? with ?eff ?(1- PK) (Why?)
• Note that K includes the person in service also
  in this case not necessary to have ? lt 1.

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Limitations of Waiting Line Models

• Mathematical analysis becomes very complicated or
  intractable for more complex waiting lines. Some
  examples
• Non-Poisson arrivals
• Non-exponential service times
• Complex customer behavior (e.g. customers
  switching between lines, or leaving after some
  time etc)
• multiple phase systems
• ? Simulation can be useful analysis tool