Example 14.5

1. Example 14.5 Queuing

2. 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?

3. Solution • Because s has stayedconstant at value 6(30) = 180, the server utilization, /(s), has climbed from 150/180=0.833 to 160/180=0.889 to 170/180=0.944 – 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.

5. 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.

6. 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!