# 13.7.1 Exercise: Queues as an Effect of Random Load Fluctuations (1)

### Intended learning outcomes: Answer a number of questions using the relevant formulas in queuing theory.

Answer the following questions using the relevant formulas in queuing theory (refer to Figure 13.2.2.4):

a.    How many parallel workstations are needed to have an expected wait time of less than 10 hours, if capacity utilization is 0.95, the mean of the operation time is 2 hours, and the coefficient of variation of the operation time is 1?

b.    The capacity is 10 hours. How much does the expected wait time increase if load rises from 4 to 8 hours?

c.    How is the expected wait time affected when the coefficient of variation increases from 1 to 2?

Solutions:

a.    s = 0.95 / (1 – 0.95)  *  (1 + (1 * 1)) / 2  *  2 / 10) = 3.8. Thus, with four workstations, the expected wait time will be 9.5 hours.

b.    Capacity utilization increases from 4/10 to 8/10. Therefore, the respective factor in the formula for the expected wait time increases from 0.4 / (1 – 0.4) = 2/3 to 0.8 / (1 – 0.8) = 4. The new factor is 4 / (2/3) = 6 times greater than the old factor. Thus, the expected wait time increases by a factor of 6.

c.    The respective factor in the formula for the expected wait time increases from (1 + (1 * 1)) / 2 = 1 to (1 + (2 * 2)) / 2 = 2.5. Thus, the expected wait time increases by the factor 2.5.