2.4. The problem is to model a queueing system having finite capacity.

We assume arrivals according to a Poisson process of rate A, independent exponentially distributed service times having mean 1/μ, a single server, and a finite system capacity N. By this we mean that if an arriving customer finds that there are already N customers in the system, then that customer does not enter the system and is lost.

Let X(t) be the number of customers in the system at time t. Suppose that N = 3 (2 waiting, 1 being served).

(a) Specify the birth and death parameters for X(t).

(b) In the long run, what fraction of time is the system idle?

(c) In the long run, what fraction of customers are lost?