1. Example 14.3 Queuing

2. 14.114.1 | 14.2 | 14.4 | 14.5 | 14.6 | 14.7 |14.8 |14.914.214.414.514.614.714.814.9 Background Information n County Bank has several branch locations n At one of these locations, customers arrive at a Poisson rate of 150 per hour. n The branch employs 6 tellers. n Each teller takes, on average, 2 minutes to serve a customer, and service times are exponentially distributed. n Also, all tellers perform all takes, so that customers can go to any of the 6 tellers.

3. 14.114.1 | 14.2 | 14.4 | 14.5 | 14.6 | 14.7 |14.8 |14.914.214.414.514.614.714.814.9 Background Information -- continued n Customers who arrive and find all 6 servers busy join a single queue and are then served in FCFS fashion. n As a first step, the bank manager wants to develop a queuing model of the current system. n Then he wants to find the “best” number of tellers, given that tellers are paid $8 per hour.

4. 14.114.1 | 14.2 | 14.4 | 14.5 | 14.6 | 14.7 |14.8 |14.914.214.414.514.614.714.814.9 MMS_TEMPLATE.XLS n 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. n However, the details are fairly complex and will not be given here. n Instead, we provide a template in this file for performing the calculations. n The template can be seen on the next slide.

5. 14.114.1 | 14.2 | 14.4 | 14.5 | 14.6 | 14.7 |14.8 |14.914.214.414.514.614.714.814.9

6. 14.114.1 | 14.2 | 14.4 | 14.5 | 14.6 | 14.7 |14.8 |14.914.214.414.514.614.714.814.9 Solution n All you need to do is enter the inputs in cells B4 through B7 and then click on the button. n This button runs a macro that calculates all of the necessary outputs and places them in the appropriate cells. n 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. 14.114.1 | 14.2 | 14.4 | 14.5 | 14.6 | 14.7 |14.8 |14.914.214.414.514.614.714.814.9 Solution -- continued n We can find the expected fraction of time each teller is busy as L serv /s. n Then the expected fraction of time each teller is busy is L serv /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. n 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. n We now turn to the economic analysis.

8. 14.114.1 | 14.2 | 14.4 | 14.5 | 14.6 | 14.7 |14.8 |14.914.214.414.514.614.714.814.9 Economic Analysis n There is a cost and benefit from adding a teller. n The cost is the wage rate paid to the extra teller, $8 per hour. n The benefit is that customers wait less time in the bank. n The problem is evaluating the cost of waiting in line. n 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. 14.114.1 | 14.2 | 14.4 | 14.5 | 14.6 | 14.7 |14.8 |14.914.214.414.514.614.714.814.9 Economic Analysis -- continued n In any case, the key to the trade-off is assessing a unit cost, c Q, per customer per hour of waiting in the queue. n If the manager can assess this unit cost, then the total expected cost per hour and each waits waiting is c Q W Q. n Then we can trade off this waiting cost against the cost of hiring extra tellers.

10. 14.114.1 | 14.2 | 14.4 | 14.5 | 14.6 | 14.7 |14.8 |14.914.214.414.514.614.714.814.9 MMS_OPT_TEMPLATE.XLS n We provide another template in this file that helps solve the problem. n 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. n You should not enter the numbers of servers as an input. n Instead, the macro calculates selected summary measures of the system for several choices of the number of servers.

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12. 14.114.1 | 14.2 | 14.4 | 14.5 | 14.6 | 14.7 |14.8 |14.914.214.414.514.614.714.814.9 Solution -- continued n Specifically, it begins by using the smallest umber of servers required to keep the system stable. n This procedure requires a value for c Q in cell B8. n Because this value is probably very difficult for a bank manager to assess, we can instead use an indirect approach. n We will find ranges for c Q where a specific number of servers is economically optimal.

13. 14.114.1 | 14.2 | 14.4 | 14.5 | 14.6 | 14.7 |14.8 |14.914.214.414.514.614.714.814.9 Solution -- continued n The output form the template can be seen on the next slide. n The results imply that it is best to use 6 tellers when c Q < $3.76. n Otherwise, if c Q < $15.24, it is best to use 7 tellers. n Finally, for c Q between $15.24 and $20, it is best to use 8 tellers.

14. 14.114.1 | 14.2 | 14.4 | 14.5 | 14.6 | 14.7 |14.8 |14.914.214.414.514.614.714.814.9