The Waiting Time at a Two-Clerk Post Office

Consider a post office with two clerks. Three people, A, B, and C, enter simultaneously. A and B go directly to the clerks, and C waits until either A or B leaves before he begins service. What is the probability that A is still in the post office after the other two have left when

(a) the service time for each clerk is exactly (nonrandom) ten minutes?

(b) the service times are i with probability 1/3 , i = 1, 2, 3?

(c) the service times are exponential with mean 1/μ? 

Solution:

(a) In this scenario, the service time for each clerk is exactly ten minutes. Hence, both A and B will leave the post office after ten minutes, and C will start his service immediately. Therefore, the probability that A is still in the post office after the other two have left is 0.

(b) The service times are i with probability 1/3, i = 1, 2, 3. If A and B receive the shortest service time, i.e., 1 minute, they will leave the post office, and C will start his service immediately. The probability that A is still in the post office after the other two have left is 0.

(c) The service times are exponential with mean 1/μ. In this scenario, it is difficult to find the exact probability that A is still in the post office after the other two have left. However, we can use the theory of queuing systems to estimate the expected waiting time.

Let W1 and W2 be the waiting times of A and B respectively. Then, W1 and W2 are also exponential random variables with mean 1/μ. The expected waiting time can be calculated as E(W1) = 1/μ and E(W2) = 1/μ.

The expected time that A spends in the post office is the sum of the expected service time and the expected waiting time, i.e., E(A) = 1/μ + 1/μ = 2/μ.

The expected amount of time a customer spends in the post office depends on the service time distribution. In the scenario where the service time is non-random and equal to ten minutes, the probability that a customer is still in the post office after the other two have left is 0. In the scenario where the service times are exponential with mean 1/μ, the expected time a customer spends in the post office is 2/μ.