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 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 P_k) 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.
    The interesting quantities for this case can be obtained by setting K = m in the expressions obtained for case 4 above (see Table 4-1). However, it is just as easy to work directly with the balance equations and obtain

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).
    It should be clear that the M/M/ queueing system can also be viewed as a special case of M/M/m with no waiting space. In fact, by letting m go to infinity in (4.57) we obtain expression (4.50) for the steady-state probabilities of the M/M/ queueing system.
   

    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!