4.6.4 Case 4: Operator, Finite Number of Lines Case
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,
P_{k};
and (2) some other levelofservice 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). 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,
P_{m}. (i.e.,
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
Section 4.8). 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 servicetime distribution, that is, for M/G/m queueing systems with no waiting space!
