Question: Two queues in series. Consider a two-station queueing network in which arrivals only occur at the first server and do so at rate 2. If

Two queues in series. Consider a two-station queueing network in which arrivals only occur at the first server and do so at rate 2. If a customer finds server 1 free he enters the system; otherwise he goes away. When a customer is done at the first server he moves on to the second server if it is free and leaves the system if it is not. Suppose that server 1 serves at rate 4 while server 2 serves at rate 2. Formulate a Markov chain model for this system with state space f0; 1; 2; 12g where the state indicates the servers who are busy. In the long run

(a) what proportion of customers enter the system?

(b) What proportion of the customers visit server 2?

(c) Compute ExT0 for x D 1; 2; 12 and use this to determine the expected duration of the busy period E1T0.

(d) Compute E1T0 from .0/ by using our result for alternating renewal processes.

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