Assume that (An : n0) is a sequence of random variables with A0=0 in which An represents the arrival timeof customer n. The system evolves in discrete time. Suppose also that Vn is the service time for customer n, and let n=AnAn1 for all n1 (i.e. n is the interarrival time between the n-th customer and the (n1)th customer).

For simplicity, let us focus on the special case in which n=1 for all n1 and the Vn's are iid, positive, and integer-valued with P(Vn=1)>0.

Assume that this system has infinitely many servers-so nobody waits, when a customer joins, a server starts working on the customer's job. Denote by Xn the amount of additional time that it would take the system to "drain" to the empty state if no new jobs were to join the system after time n.

a.)Argue that Xn+1 = max (Xn1, Vn+1) for all n0.

b.)Is X = (Xn : n0) a Markov chain? Why or why not?

c.)Compute P0(X34) in terms of the distribution of the Vi's. Remark: Recall from class that the notation Px(A) is shorthand for P(A|X0=x) for all events A.