4.6.3 Case 3: One Operator, Finite Number of Lines

    Let us now continue with our emergency call center example and consider a situation identical to that of case 1 with one exception. Rather than an infinite number, there is now only a finite number of lines, K, into the switchboard. Furthermore, the analysis will be performed under the assumption that a caller who calls a 911-type number and gets a busy signal becomes discouraged and does not try again. We shall discuss the implications of this assumption at the end of this section.
    The state-transition diagram for case 3 is shown in Figure 4.8. This is a M/M/1 system with finite system capacity equal to K. With respect to our fundamental birth-and-death model,

form4.51a.gif (32326 bytes)

and therefore,

form4.53.gif (10685 bytes)


Note that for p < 1, (4.54) reduces to (4.34), as it should, as K inf.gif (300
bytes). Knowing the steady-state probabilities, we can now obtain expressions for Lbar.gif (304 bytes), Wbar.gif (557 bytes), Lbar.gif (304 bytes)q, and Wbar.gif (557
bytes)q,, one of which is listed in Table 4-1. Aside, however, from the specific form of the results obtained, there are two points which are worth

table4.1.gif (41414 bytes)

remembering about the M/M/1 I system with finite system capacity. Moreover, these points are valid for finite capacity systems in general:

1. Steady-state conditions will be reached in any event, irrespective of the value of p. Because the number of states in the system is finite, there is an upper limit on how long the queue can get. Even when the arrival rate is much larger than the service rate (p >> 1), steady state will be reached: the queue will simply be full most of the time. In such an event a large fraction of the potential facility users will be turned away. That fraction is equal to PK, the probability that the queue is saturated. Note also that for p = 1, Pn = l/(K + 1) for all values of n (i.e., all states are equally likely).

2. The fundamental relationships Lbar.gif (304 bytes) = lamda.gif (291 bytes) Wbar.gif (557 bytes) and Lbar.gif (304 bytes)q = Wbar.gif (557
bytes)q still hold but, as suggested by (4.32), in revised form. Since a fraction & of potential users of the facility are turned away, the actual arrival rate at the queue is equal to A' A A(I - P,,). Thus, the relationships (4. 10) and (4.11) are now revised to the form

form4.55.gif (3967 bytes)

where lamda.gif (291 bytes)' is as defined above. The third fundamental relationship, Wbar.gif (557 bytes)=Wbar.gif (557 bytes)q + 1/mu.gif (300
bytes), still holds.

    In practice it is rather unlikely that an emergency caller who gets a busy signal will refrain from calling the emergency center again. (However, this may be true for nonemergency callers, and it would definitely be true for queueing systems that offer routine types of services that can be obtained with ease at other queueing systems, as well.) If some callers persist in calling the emergency number, the resulting situation is an intermediate one between case 3 and case 1. In fact, in the extreme case when no caller ever becomes discouraged and they all keep trying continually to get a free line, an infinite capacity system will again result. However, during periods of congestion, we now have two queues: a "visible" one consisting of the K callers who have already obtained access to the switchboard, and an "invisible" one consisting of all those trying to obtain such access. In addition, while the former queue is operating on a FCFS basis (because of the existence of the call-ordering "electronic device"), access to the switchboard from the invisible queue is of the SIRO type.
    Finally, it should be clear that for system design purposes, the probability Pk of a full system is often the most important design parameter. For, when Pk, is negligibly small, a finite-capacity system operates, for all practical purposes, like a system with infinite capacity.