Question

2. [3] Suppose you arrive to a service system with three parallel servers. All servers are busy, and there is no customer ahead of you in the queue. As soon as one of the servers is free, you will be served by that server. Service times at each server are exponentially distributed with rates 1, 2, and us. What is the expected time you will spend in the system?

Answer 1

(a) If you arrive to find all three servers busy and no one waiting, find the expected time until you depart the system. Solution: Let T, denote this time, and W and S be our waiting and service times, respectively. The time until any customer in service leaves is exponentially distributed with rate μ1 + μ2 + μ3. Thus For i 1,2,3, let Ai be the event in which the customer at server i finishes first. This occurs with probability μ/(U1 + μ2 + μ3). conditional on Ai occurring, our service time is exponentially distributed with rate μ due to the memoryless property. Thus, we have χι i- 1 which yields (b) If you arrive to find all three servers busy and one person waiting, find the expected time until you depart the system. Solution: Let T2 denote the above time, and V the time until the other waiting customer enters service. Since V is again exponentially distributed with rate μι + μ2 + μ3, and the time we spend in the system thereafter is Ti, we have