Queuing

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Example 14.3

• Queuing

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Background Information

• County Bank has several branch locations
• At one of these locations, customers arrive at a
  Poisson rate of 150 per hour.
• The branch employs 6 tellers.
• Each teller takes, on average, 2 minutes to serve
  a customer, and service times are exponentially
  distributed.
• Also, all tellers perform all takes, so that
  customers can go to any of the 6 tellers.

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Background Information -- continued

• Customers who arrive and find all 6 servers busy
  join a single queue and are then served in FCFS
  fashion.
• As a first step, the bank manager wants to
  develop a queuing model of the current system.
• Then he wants to find the best number of
  tellers, given that tellers are paid 8 per hour.

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MMS_TEMPLATE.XLS

• As in the M / M / 1 system, there are formulas
  for the steady state probabilities, and these can
  be used to find summary measures such as L and W.
  
• However, the details are fairly complex and will
  not be given here.
• Instead, we provide a template in this file for
  performing the calculations.
• The template can be seen on the next slide.

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Solution

• All you need to do is enter the inputs in cells
  B4 through B7 and then click on the button.
• This button runs a macro that calculates all of
  the necessary outputs and places them in the
  appropriate cells.
• From the template, we see that when there are 6
  tellers and the server utilization is 0.833, the
  expected number of customers in the system is
  7.94 and the expected time a typical customer
  spends in the system if 0.053 hour.

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Solution -- continued

• We can find the expected fraction of time each
  teller is busy as Lserv/s.
• Then the expected fraction of time each teller is
  busy is Lserv/s 5/6 0.833. If this number
  doesnt ring a bell, it should it is the server
  utilization in cell B13. This is no coincidence.
• The server utilization in an M / M / 1 system,
  calculated as the arrival rate divided by the
  maximum service rate, is always the expected
  fraction of time a typical server is busy.
• We now turn to the economic analysis.

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Economic Analysis

• There is a cost and benefit from adding a teller.
• The cost is the wage rate paid to the extra
  teller, 8 per hour.
• The benefit is that customers wait less time in
  the bank.
• The problem is evaluating the cost of waiting in
  line.
• This is not an out-of-pocket cost for the bank,
  but the back realizes that it is an indirect cost
  in that customers who experience long waits might
  take their business elsewhere.

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Economic Analysis -- continued

• In any case, the key to the trade-off is
  assessing a unit cost, cQ, per customer per hour
  of waiting in the queue.
• If the manager can assess this unit cost, then
  the total expected cost per hour and each waits
  waiting is cQ?WQ.
• Then we can trade off this waiting cost against
  the cost of hiring extra tellers.

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MMS_OPT_TEMPLATE.XLS

• We provide another template in this file that
  helps solve the problem.
• You now need to provide the arrival rate, the
  service rate per server, the wage rate per
  server, and the unit waiting cost per customer
  per unit time in line.
• You should not enter the numbers of servers as an
  input.
• Instead, the macro calculates selected summary
  measures of the system for several choices of the
  number of servers.

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Solution -- continued

• Specifically, it begins by using the smallest
  umber of servers required to keep the system
  stable.
• This procedure requires a value for cQ in cell
  B8.
• Because this value is probably very difficult for
  a bank manager to assess, we can instead use an
  indirect approach.
• We will find ranges for cQ where a specific
  number of servers is economically optimal.

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Solution -- continued

• The output form the template can be seen on the
  next slide.
• The results imply that it is best to use 6
  tellers when cQ lt 3.76.
• Otherwise, if cQ lt 15.24, it is best to use 7
  tellers.
• Finally, for cQ between 15.24 and 20, it is
  best to use 8 tellers.

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