Question

(4) In a shop there are two cashiers (A and B) with a single queue for them. Customers arrive at the queue as a Poisson process with rate λ, and wait for the first available cashier. If both cashiers are available, they pick one equally likely. Each cashier finishes with a customer after an exponential waiting time, with parameters µa and µb for cashier A and B, respectively. Assume that λ < µa+µb. (a) Formulate a Markov chain model with state space S = {0,a,b,2,3,...}, where a and b mean that only cashier A or only B is busy, and the numbers mean the number of customers in the system. Give all the transition rates. [5 marks] (b) Write down the detailed balance equations. [5 marks] (c) Find the stationary probabilities of the process. [5 marks] (d) Calculate the mean queue length in the stationary state (not counting people being served)? [5 marks]

Answer 1