A post office has a single line for customers waiting for the next available postal clerk.There are two postal clerks who work at the same rate.The arrival rate of customers follows a Poisson distribution,while the service time follows an exponential distribution.The average arrival rate is three per minute and the average service rate is two per minute for each of the two clerks.What proportion of time are both clerks idle?
A)0.643
B)0.250
C)0.750
D)0.143
E)None of the above

Question

Quizplus · Accepted Answer

Since the average arrival rate is 3 per minute and there are two clerks, the average service rate is 4 per minute.
The utilization factor(u) = Average service rate / Average arrival rate = 4/3
The probability that both clerks are idle at any given moment is equal to (1-u)^2 = (1-4/3)^2 = 1/9.
Therefore, the proportion of time that both clerks are idle is 1/9 * 2 = 0.222 or 0.222/0.888 = 0.25 (since the system is in equilibrium, the probability that the system is idle is equal to the sum of the probabilities of each state in which both clerks are idle.)

A Post Office Has a Single Line for Customers Waiting