Consider a queueing system with two classes of customers. Type A customer arrivals are Poisson with rate λA and type B customer arrivals are Poisson with rate λB. The service time for type A customers is exponential with rate μA and that for type B is exponential with rate μB. Each service time is independent of all other service times and of all arrival epochs.
(a) First assume there are infinitely many identical servers, and each new arrival immediately enters an idle server and begins service. Let the state of the system be (i, j), where i and j are the numbers of type A and B customers respectively in service. Draw a graph of the state transitions for i ≤ 2, j ≤ 2. Find the steady-state PMF, {p(i, j); i, j ≥ 0}, for the Markov process. Hint: Note that the type A and type B customers do not interact.
(b) Assume for the rest of the exercise that there is some finite number m of servers. Customers who arrive when all servers are occupied are turned away. Find the steady- state PMF, {p(i, j); i, j ≥ 0, i + j ≤ m}, in terms of p(0, 0) for this Markov process. Hint:
Combine (a) with the result of Exercise 7.25.
(c) Let Qn be the probability that there are n customers in service at some given time in steady state. Show that Qn = p(0, 0)ρn/n! for 0 ≤ n ≤ m, where ρ = ρA + ρB, ρA = λA/μA, and ρB = λB/μB. Solve for p(0, 0).
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.