Consider a queueing system having two servers and no queue. There are two types of customers. Type 1 customers arrive according to a Poisson process having rate λ1, and will enter the system if either server is free. The service time of a type 1 customer is exponential with rate μ1. Type 2 customers arrive according to a Poisson process having rate λ2. A type 2 customer requires the simultaneous use of both servers; hence, a type 2 arrival will only enter the system if both servers are free. The time that it takes (the two servers) to serve a type 2 customer is exponential with rate μ2. Once a service is completed on a customer, that customer departs the system.

(a) Define states to analyze the preceding model.

(b) Give the balance equations.

In terms of the solution of the balance equations, find

(c) the average amount of time an entering customer spends in the system;

(d) the fraction of served customers that are type 1.