4. Suppose that a customer of the M/M/1 system spends the amount of time x > 0 waiting in queue before entering service.

(a) Show that, conditional on the preceding, the number of other customers that were in the system when the customer arrived is distributed as 1 + P , where P is a Poisson random variable with mean λ.

(b) Let W∗

Q denote the amount of time that an M/M/1 customer spends in queue. As a byproduct of your analysis in part (a), show that P{W∗

Q  x} = 1 − λ

μ

if x = 0 1 − λ

μ + λ

μ(1 − e−(μ−λ)x ) if x > 0