Consider an infinite server queuing system in which customers arrive in accordance with a Poisson process with rate λ, and where the service distribution is exponential with rate μ. Let X(t) denote the number of customers in the system at time t. Find

(a) E[X(t + s)|X(s) = n];

(b) Var[X(t + s)|X(s) = n].

Hint: Divide the customers in the system at time t + s into two groups, one consisting of “old” customers and the other of “new” customers.

(c) Consider an infinite server queuing system in which customers arrive according to a Poisson process with rate λ, and where the service times are all exponential random variables with rate μ. If there is currently a single customer in the system, find the probability that the system becomes empty when that customer departs.