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. and therefore,
Note that for p < 1, (4.54) reduces to (4.34), as it should, as K . Knowing the steady-state probabilities, we can now obtain expressions for , , q, and 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 remembering about the M/M/1 I system with finite system capacity. Moreover, these points are valid for finite capacity systems in general:
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. |