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 is the average length of the line?
A)3.429
B)1.929
C)1.143
D)0.643
E)None of the above

Question

Quizplus · Accepted Answer

We can use Little's Law which states that the average number of customers in a queuing system equals the arrival rate multiplied by the average time a customer spends in the system.

First, we need to find the average service time which is 1/2 minute per customer per clerk because the average service rate is 2 per minute per clerk.

Second, we can find the utilization rate which is the ratio of the average arrival rate to the average service rate. The utilization rate is 3/(2*2) = 0.75.

Third, we can find the average time a customer spends in the system, which is the sum of the average waiting time and the average service time. The average waiting time is (0.75^2)/(2*(1-0.75)) = 0.375 minutes. Therefore, the average time a customer spends in the system is 0.5 + 0.375 = 0.875 minutes.

Finally, we can find the average number of customers in the system using Little's Law:

Average number of customers = arrival rate * average time in the system 
= 3 * 0.875 
= 2.625

Since there are 2 clerks, the average number of customers waiting in line is half of this value:

Average number of customers waiting = 2.625/2 
= 1.3125

Therefore, the average length of the line is approximately 1.929 customers. So, option B is correct.

A Post Office Has a Single Line for Customers Waiting