We consider a queueing system in which ordinary customers arrive according to a Poisson process with rate λ and are served in a random time having an exponential distribution, with parameter μ, by either of two servers. Furthermore, there is a special customer who, when she arrives in the system, is immediately served by server no. 1, at an exponential rate μS. If an ordinary customer is being served by server no. 1 when the special customer arrives, then this customer is returned to the head of the queue. We suppose that the service times are independent random variables and that the special customer spends an exponential time (independent of the service times), with parameter λS, outside the system between two consecutive visits.
(a) Suppose that if an arbitrary customer is returned to the queue, then he will resume being served as soon as either server becomes available. Define an appropriate state space, and write the balance equations of the system.
(b) Suppose that the system capacity is c = 2, but that if a customer is displaced by the special customer, then she will wait, a few steps behind, until server no. 1 becomes available to resume being served by this server (whether server no. 2 is free or not). Define a state space such that the stochastic process {X(t),t ≥ 0}, where X(t) represents the state of the system at time t, is a continuous-time Markov chain.
(c) Suppose that the system capacity is c = 2 and that, if a customer is displaced by the special customer, then he will go to server no. 2 only if this server is free upon the arrival of the special customer in the system. Otherwise, he will wait, a few steps behind, before server no. 1 becomes available. Let K be the number of times that a given customer, who has started receiving service from server no. 1, will be displaced by the special customer. Calculate P[K = 1] in terms of the limiting probabilities of the system (with an appropriate state space).