6. 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 ??, and will enter the system if
either server is free. The service time of a type 1 customer is
exponential with rate ??. Type 2 customers arrive according to a
Poisson process having rate ??. 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 ??. 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.
7. In terms of the solution of the balance equations, find
(a) the average amount of time an entering customer spends in the
system;
(b) the fraction of served customers that are type 1.
6.
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