Consider a two-server system in which a customer is served first by server 1, then by server 2, and then departs. The service times at server i are exponential random variables with rates μi, i = 1, 2. When you arrive, you find server 1 free and two customers at server 2—customer A in service and customer B waiting in line.

(a) Find PA, the probability that A is still in service when you move over to server 2.

(b) Find PB, the probability that B is still in the system when you move over to server 2.

(c) Find E[T ], where T is the time that you spend in the system.

Hint: Write

where Si is your service time at server i,WA is the amount of time you wait in queue while A is being served, and WB is the amount of time you wait in queue while B is being served.

Transcribed Image Text:

T S+S2+WA+WB