18. Arrivals to a three-server system are according to a Poisson process with rate . Arrivals finding

18. Arrivals to a three-server system are according to a Poisson process with rate λ.

Arrivals finding server 1 free enter service with 1. Arrivals finding 1 busy but 2 free enter service with 2. Arrivals finding both 1 and 2 busy do not join the system. After completion of service at either 1 or 2 the customer will then either go to server 3 if 3 is free or depart the system if 3 is busy. After service at 3 customers depart the system. The service times at i are exponential with rate μi , i = 1, 2, 3.

(a) Define states to analyze the above system.

(b) Give the balance equations.

(c) In terms of the solution of the balance equations, what is the average time that an entering customer spends in the system?

(d) Find the probability that a customer who arrives when the system is empty is served by server 3.