Customers arrive to a two server system in accordance with a Poisson process with rate . Server 1 is the preferred server, and an arrival

Customers arrive to a two server system in accordance with a Poisson process with rate λ. Server 1 is the preferred server, and an arrival finding server 1 free enters service with 1; an arrival finding 1 busy but 2 free, enters service with 2. Arrivals finding both servers busy do not enter. A customer who is with server 2 at a moment when server 1 becomes free, immediately leaves server 2 and moves over to server 1. After completing a service (with either server) the customer departs. The service times at server i are exponential with rate µ_i , i = 1, 2.

(a) Define states and give the transition diagram.

(b) Find the long run proportion of time the system is in each state.

(c) Find the proportion of all arrivals that enter the system.

(d) Find the average time that an entering customer spends in the system.

(e) Find the proportion of entering customers that complete service with server 2.