Queuing

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Example 14.5

• Queuing

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Background Information

• Over a period of time, the County Bank branch
  office from Example 14.3 has been experiencing a
  steady increase in the customer arrival rate.
• This rate has increases from the previous value
  of 150 customers per hour to 160, then to 170,
  and it is still increasing.
• During this time, the number of tellers has
  remained constant at 6, and the mean service time
  per teller has remained constant at 2 minutes.
  The bank manager has seen an obvious increase in
  back congestion.
• Is this reinforced by the M / M / s model? What
  will happen if the arrival rate continues to
  increase?

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Solution

• Because s? has stayed constant at value 6(30)
  180, the server utilization, ?/(s?), has climbed
  from 150/1800.833 to 160/1800.889 to
  170/1800.944 and is still climbing.
• We know that ? must stay below 180 or the system
  will become unstable, but what about values of ?
  slightly below 180?
• We recalculated the spreadsheet from the previous
  example and obtained the results in the table on
  the next slide.

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Solution -- continued

• Although each column of this table represents a
  stable system, the congestion is becoming
  unbearable.
• When ? 178, the expected line length is over 80
  customers, and a typical customer must wait about
  a half hour in line. Things are twice as bad when
  ?179.
• The conclusion should be clear to the bank
  manager.

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Solution -- continued

• Something must be done to alleviate the
  congestion probably extra tellers and the
  bank will no doubt take such measures if it wants
  to stay in business.
• However, the point of the example is that systems
  moving toward the borderline of stability can
  become extremely congested.
• As the results in the table indicate, there is a
  huge difference between a system with a server
  utilization of 0.9 and one with a server
  utilization f 0.99!