Let X(t) and Y(t) be independent, Poisson random variables with means λt and µt, respectively. Show that the generating function z(s) for the difference

Apply this result to the following problem. Travelers and taxis arrive independently at a service point at random rates λ and µ, respectively. Let Z(t) denote the queue size at time t, a negative value representing a line of taxis and a positive value representing a queue of travelers. Beginning with Z(O)=O, show that the distribution of Z(t) is given by the difference between independent Poisson variables of means λt and µt.