Question

Consider a post office that is run by two clerks. Suppose that when Mr. Simon enters the systems he discovers Mr. Garfunkel is being served by clerk 1 and Mrs. Oates by clerk 2. Suppose that Mr. Simon is told that his service will begin as soon as either Garfunkel or Oates leaves. If clerk i serves at an exponential rate ?i, i=1,2, what is the probability that, of the 3 customers, Mr. Simon is the last one to leave the post office?

Answer 1

Solution:

Let X be the time when the first person leaves the system,

Y be the total waiting that remaining person and

Z be total waiting that Mr. Simon has.

Now the probability that Mr. Simon is not the last person to leave the post office is

According to the memoryless propoerty, if X is exponentially distributed, then

Substituting this into above X = x, Y = z, we get

Since Y and Z are both exponentially distributed random variables with parameter ?, therefore

P (Y $\geq$ Z) = 1/2