The Exponential Distribution

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Example 14.1

• The Exponential Distribution

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Background Information

• A bank manager would like to use an analytical
  queuing model to study the congestion at the
  banks automatic teller machines (ATMs).
• A simple model of this system requires that the
  interarrival times and service times are
  exponentially distributed.
• During a period of time when business is fairly
  steady, several employees use stopwatches to
  gather data on interarrival times and service
  times.

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Background Information -- continued

• These are listed in the table on the next slide.
• The bank manager wants to know, based on these
  data, whether it is reasonable to assume
  exponentially distributed interarrival times and
  service times.
• In each case he also wants to know the
  appropriate value of ?.

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EXPONFIT.XLS

• To see whether these times are consistent with
  the exponential distribution, we plot histograms
  of the interarrival times and the service times.
• The histograms appear on the next slides.
• This file contains the model.

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Solution

• The histogram of the interarrival times appears
  to be quite consistent with the exponential
  density shown.
• Its highest bar is at the left, and the remaining
  bars fall off gradually from left to right.
• On the other hand, the histogram of the service
  times is not shaped like the exponential density.
• Its highest bar is not the one farthest to the
  left but instead corresponds to the category 15
  to 30 seconds.

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Solution -- continued

• There is some minimum time required to process
  any customer, regardless of the task, so that the
  most likely times are not close to 0.
• Therefore, the exponential assumption for
  interarrival times is reasonable, but it is
  questionable for service times.
• In either case, if the manager decides to accept
  the exponential assumption, the parameter ? is
  the rate of arrivals and is estimated by the
  reciprocal of the average of the observed times.

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Solution -- continued

• For interarrival times, this estimate of ? is the
  reciprocal of the average in cell C39 1/25.3
  0.0395 that is, 1 arrival every 25.3 seconds.
• For service times, the estimated ? is the
  reciprocal of the average in cell G39 1/22.3
  0.0448 that is, 1 service every 22.3 seconds.