4.6.4 Case 4: Operator, Finite Number of LinesCase 4 involves a finite number, m, of operators and a finite number, K, of lines. In many ways, this is probably the most general and appropriate model for a simple emergency call center. The "design parameters" then involve the determination of the right number of operators and lines so that a combination of objectives will be achieved. These objectives might be in the form of specifications, for instance, of an upper limit on (1) the probability that a caller receives a busy signal, Pk; and (2) some other level-of-service indicator, such as the expected system occupancy time, , for a random accepted caller. An interesting analysis of this type has been reported for the 911 emergency call center in New York City [LARS 12a] (see also Chapter 8).
A state-transition diagram for a M/M/m system with a finite system capacity is shown in Figure 4.9, and expressions for some quantities related to this case are listed in Table 4-1. The algebra involved gets quite tedious for the general case, but numerical applications are straightforward and can be performed easily with a hand calculator.
It should finally be emphasized that, unless otherwise stated, the implicit assumption in finite-system-capacity queueing models is that prospective users who find a full system (with probability Pk) are permanently lost to the system.
Special case: K= m (Erlang's loss
formula). A special
instance of the case 4 system is when the capacity of the system, K,
is equal to the
number of servers, m. Such would be the case if there were
one telephone line for
each operator. Obviously, in such a system, there is no waiting
space at all and users
who, on arrival, find all servers busy are simply turned away.
Historically, this was one
of the first queueing systems ever to be investigated in depth. This
was done by A. K.
Erlang of Denmark (generally considered to be the "father"
of queueing theory)
during the first decade of this century.
In particular, the probability of a full system,
the probability that an arriving user will find a full system and be
turned away) is
widely known as Erlang's loss formula and has been
extensively tabulated for
different values of the ratio / (which is not restricted as to
magnitude) and of the
number of servers m. Erlang's loss formula has also been
used widely in
applications of queueing theory to urban service systems (see also
We shall also anticipate here an interesting result that will be presented later in this chapter. It turns out that (4.57) holds for any service-time distribution, that is, for M/G/m queueing systems with no waiting space!