A queueing system having two servers and no queue

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.

Answer

There are two types of customers. Type i 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 service 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 customer 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 the states. State definition = (type of customer on server 1, type of customer on server 2) State space = {(0,0), (1,0), (0,1), (0,2), (1,1), (1,2)}