1. Example 14.5 Queuing

2. 14.114.1 | 14.2 | 14.3 | 14.4 | 14.6 | 14.7 |14.8 |14.914.214.314.414.614.714.814.9 Background Information n 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. n This rate has increases from the previous value of 150 customers per hour to 160, then to 170, and it is still increasing. n 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. n Is this reinforced by the M / M / s model? What will happen if the arrival rate continues to increase?

3. 14.114.1 | 14.2 | 14.3 | 14.4 | 14.6 | 14.7 |14.8 |14.914.214.314.414.614.714.814.9 Solution n Because s  has stayed constant 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. n We know that must stay below 180 or the system will become unstable, but what about values of slightly below 180? n We recalculated the spreadsheet from the previous example and obtained the results in the table on the next slide.

4. 14.114.1 | 14.2 | 14.3 | 14.4 | 14.6 | 14.7 |14.8 |14.914.214.314.414.614.714.814.9 Effects of Increasing Arrival Rate Customer Arrival Rate ( ) 150160170175178179 Utilization0.8330.8890.9440.9720.9890.994 L sys 7.9411.0420.1438.1892.21182.22 Lq2.945.7114.4732.3586.28176.25 W sys 3.184.147.1113.0931.0861.08 WQWQ 1.182.145.1111.0929.0859.08

5. 14.114.1 | 14.2 | 14.3 | 14.4 | 14.6 | 14.7 |14.8 |14.914.214.314.414.614.714.814.9 Solution -- continued n Although each column of this table represents a stable system, the congestion is becoming unbearable. n 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. n The conclusion should be clear to the bank manager.

6. 14.114.1 | 14.2 | 14.3 | 14.4 | 14.6 | 14.7 |14.8 |14.914.214.314.414.614.714.814.9 Solution -- continued n 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. n However, the point of the example is that systems moving toward the borderline of stability can become extremely congested. n 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!