20. There are two types of customers. Type 1 and 2 customers arrive in accordance with independent Poisson processes with respective rate λ1 and λ2. There are two servers. A type 1 arrival will enter service with server 1 if that server is free; if server 1 is busy and server 2 is free, then the type 1 arrival will enter service with server 2. If both servers are busy, then the type 1 arrival will go away. A type 2 customer can only be served by server 2; if server 2 is free when a type 2 customer arrives, then the customer enters service with that server. If server 2 is busy when a type 2 arrives, then that customer goes away. Once a customer is served by either server, he departs the system. Service times at server i are exponential with rate μi, i = 1, 2.

Suppose we want to find the average number of customers in the system.

(a) Define states.

(b) Give the balance equations. Do not attempt to solve them.

In terms of the long-run probabilities, what is

(c) the average number of customers in the system?

(d) the average time a customer spends in the system?