Queuing

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Example 14.4

• Queuing

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Background Information

• Which system has the better steady-state
  characteristics such as LSys, WSys, LQ, and WQ a
  single-server system where the single server can
  serve 30 customers per hour, or a 5-server system
  where each of the servers can serve 6 customers
  per hour?
• For each system we will, assume that customers
  arrive according to a Poisson process at rate 25
  hour.

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Solution

• First, note that the two models are comparable in
  the sense the ?fast s?slow because ?fast30,
  s5, ?slow6.
• Equivalently, the server utilization is 5/6 for
  each.
• The spreadsheets on the next two slides answer
  our question.
• As you can see, the comparison is not entirely
  clear-cut.

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Solution -- continued

• The M/M/1 system has a smaller LSys but a larger
  LQ. Similarly, it has a smaller WSys but a
  larger WQ.
• Finally, the M/M/1 system is worse in the sense
  that is has a smaller percentage of customers who
  experience no waiting in line and a larger
  percentage who must wait in lone at least 0.25
  hour.
• The basic conclusion is that if you hate to wait
  in a queue, you should prefer the system with
  multiple slow servers.

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Solution -- continued

• However, once it is your turn to be served, you
  will clearly prefer the system with the single
  fast server.
• In this latter system you will spend less total
  time in the system, but more of it will be spent
  waiting in line. Take your choice!