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 254-1.gif (10696 bytes)

What is the condition for steady state in this case? Find an expression for rho.gif (189 bytes).

b. Suppose we now assume that there is no balking (lambda.gif (179 bytes)n= lambda.gif (179 bytes)) but that the service rate depends on the number of users present (mu.gif (189 bytes)n = cn.mu.gif (189 bytes) for n = 1, 2, . . .). Find a form for c. such that the expression forPn 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 Lbar.gif (304 bytes) for case (a). Show that, for this case,

254-2.gif (2157 bytes)