4.2 Queueing system with balking We have already noted (cf. Section 4.6) that it is usually difficult to obtain closed-form solutions for queueing systems in which there is user balking or where the rate of service depends on the number of users present. There are exceptions to this.

a. Consider a single-server system with infinite system capacity, Poisson arrivals, and negative exponential service times for which the rates of user

arrivals and of service [cf. (4.58)l are

What is the condition for steady state in this case? Find an expression for .

b. Suppose we now assume that there is no balking (_n= ) but that the service rate depends on the number of users present (_n = c_n. for n = 1, 2, . . .). Find a form for c. such that the expression forP_n becomes identical, as in part (a). To what classical queueing system is this new system equivalent, as far as the service rate is concerned?

c. Find for case (a). Show that, for this case,