Example 14.3

1. Example 14.3 Queuing

2. 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.

3. 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.

4. 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.

6. 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.

7. 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 doesn’t 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.

8. 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.

9. 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.

10. 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.

12. 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.

13. 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 < $3.76. • Otherwise, if cQ < $15.24, it is best to use 7 tellers. • Finally, for cQ between $15.24 and $20, it is best to use 8 tellers.