4.9.3 Nonpreemptive Priorities in a M/M/m System

        In attempting to extend expressions such as (4.107) to the case of queueing systems with many servers, one encounters the same problems that arise in trying to extend the analysis of M/G/1 queueing systems to M/G/m queueing systems. Thus, exact results for multiserver queueing systems with nonpreemptive priorities are available only for the case of negative exponential service times (M/M/m systems). Moreover, the mathematical analysis becomes intractable unless all r priority classes of users have the same mean service time, .
        Under these assumptions-and using the same notation as for the singleserver system-it is not difficult to show (see Problem 4.8) that, as before,

where ₀, is now defined as the expected value of the remaining time until any one of the m servers becomes free following the arrival of a new user (of class k) at the queueing system.¹² (If one or more of the servers are free at  the instant of the new arrival, that remaining time is equal to zero.) From the theory of M/M/m queueing systems and, in particular, from expresion
(4.44), we then have (assuming ₁ + ₂ + . . . + r < m)

₀ = P{all servers are busy}. E[remaining time | all servers are busy]

where P₀ is given by (4.46) and .
        This queueing model-and expression (4.111)-is among the most applicable in urban operations research. Many urban services can be viewed as multiserver systems with Poisson demands of several types and with (nonpreemptive) priority rules that determine the order in which different types of demands will be serviced. In fact, the model places no restrictions on how the types of demand will be defined. Thus, one can, for instance, classify demands according to the region of a city from which they originate or according to the type of service requested or both (e.g., _ij, might be the rate at which j-alarm fires are generated from region i in a city).
        The most restrictive assumption in the model is that service times to all types of demands are assumed to be identical, negative exponential random variables, but even in cases when this is not quite true, expression (4.111) can still be used to obtain some idea of the level of delay experienced by different types of demands for various priority rankings and for different values of m (see also Chapter 5 in [LARS 72b]).

12 As usual for multiserver systems, we define i = _i/ m