Suppose that in the queueing system M/M/2/c, with c = 3, server no. 1 serves only one person at a time, at rate μ1, while server no. 2 can serve one or two persons at a time, from any time instant, at rate 𝜇2 Moreover, when server no. 1 is free, an arriving customer' will go to this server.
(a) Let X(t) be the number of persons in the system at time t. Define a state space such that the stochastic process {X(t),t ≥ 0} is a continuous-time Markov chain.
Remark. Server no. 1 may be free while server no. 2 serves two customers at a time. That is, the two customers finish their service period with server no. 2.
(b) Write the balance equations of the system. Do not solve them.
(c) In terms of the limiting probabilities, what fraction of time does server no. 2 serve two customers at a time, given that she is busy?