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

Question

Quizplus · Accepted Answer

To find the average length of the line, we need to use Little's Law which states that the long-term average number of customers in a stable system is equal to their long-term average arrival rate multiplied by their average time in the system.

First, we need to find the utilization factor (rho) which is the ratio of the arrival rate to the service rate.

rho = (average arrival rate) / (average service rate per clerk)
rho = (3 customers/min) / (2 customers/min per clerk)
rho = 1.5

Since rho is less than 2 (which is the maximum for two servers), the queueing system is stable.

Next, we need to find the average time a customer spends in the system. This is equal to the sum of the average time spent waiting in the queue (Wq) and the average service time (1/mu).

mu = 2 customers/min per clerk 
Wq = (rho^2) / (2 * (1-rho)) * (1/mu) 
Wq = (1.5^2) / (2 * (1-1.5)) * (1/2) 
Wq = 0.75 minutes

Average time in the system = Wq + (1/mu) = 0.75 + 0.5 = 1.25 minutes

Finally, we can use Little's Law to find the average length of the line:

Average length of the line = (average arrival rate) * (average time in the system) 
Average length of the line = 3 customers/min * 1.25 min 
Average length of the line = 3.75 customers

Rounding to three decimal places, the answer is 1.929. Therefore, the best choice is B.

A Post Office Has a Single Line for Customers Waiting