1. Consider an in?nite 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].
2. Suppose that people arrive at a bus stop in accordance with a Poisson process with rate λ. The bus departs at time t. Let X denote the total amount of waiting time of all those who get on the bus at time t. We want to determine Var(X). Let N(t) denote the number of arrivals by time t.
(a) What is E[X|N(t)]?
(b) Argue that Var[X|N(t)]= N(t)t2/12.
(c) What is Var(X)?