4.6 CENTER FOR EMERGENCY CALLS: QUEUEING SYSTEMS OF THE BIRTH-AND-DEATH TYPE
Equipped with the results that we have derived for our fundamental
birthand-death model, we shall now review a problem whose many variations will help
illustrate several of the best-known and most widely used models of queueing systems.
Many cities have by now instituted the use of the telephone number 911
for all types of emergency calls. At the "other end" of the 911 number there is
usually a center for emergency calls employing a number of trained operators. The
sophistication of equipment and operational setup in these centers varies widely from city
to city. Some cities (e.g., New York City) employ quite elaborate schemes for screening
calls and determining their priorities, schemes backed up by special-purpose computers and
communications equipment. Other centers consist of little more than a switchboard and a
number of telephone operators who either process telephone calls themselves (in
cooperation with a number of dispatchers) or transfer calls to the most appropriate city
department (e.g., fire department, emergency medical services department, etc.). It is
interesting to compare, at least in an approximate way, the characteristics of these
centers as a function of different levels of manpower and under various organizational
schemes. Queueing theory offers us a good opportunity to do so.
Throughout the following discussion it will be assumed that the arrival
of calls at a center constitutes a Poisson process (whose mean rate may vary). This
assumption is reasonable, with the possible exception of the occurrence of major incidents
which can be expected to trigger bursts of telephone calls-all reporting the same event
and its repercussions.