9.2.4 The problem is to model a queueing system having finite capacity. We assume arrivals according to a Poisson process of rate , with independent exponentially distributed service times having mean 1=, 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 D 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?