4.8.2 Queueing Systems with No Waiting Space

    One of the few examples of "difficult" queueing systems for which exact results exist are M/G/m systems with no waiting space. Instances in which these models are, at least approximately, applicable in urban operations research are common and usually involve emergency services.

        For example, a few years ago the following situation came under review in one of our major cities. A small fleet of hospital-affiliated ambulances were being used as the primary means of transporting emergency patients to local hospitals. At those times when all ambulances were busy, a back-up fleet of "ambulettes" run by the police department of that city was pressed into service. No queue was allowed to form for service by the hospital-affiliated ambulances. It was generally believed by EMS planners that police ambulettes provided inferior service (a belief that was apparently shared by many city residents). As a result, it was decided to increase the size of the hospitalaffiliated fleet and it was desired to determine the minimum number of active hospital-affiliated ambulances that would be necessary to assure that:

P{a random emergency patient must be served by a police ambulette}leq.gif (53 bytes) f

where f is a threshold probability (0 < f < 1).

        Calls for emergency ambulance service arise in a Poisson fashion in time and the service times for calls (involving a trip to the location of the call, some time spent  there, and, finally, transportation to a hospital) are random variables with "general" pdf's. Therefore, in the presence of a fleet of m hospital-affiliated ambulances, the fraction of calls serviced by police ambulettes is equal to the probability, Pm , that all m hospital ambulances are busy as given by a M/G/m queueing model with no waiting space.

        It has been shown (see, for instance, [GROS 74]) that, quite remarkably, the expressions for the steady-state probabilities for this M/G/m case are identical to those for the corresponding M/M/m system. Thus, as in (4.57), we have

form4.85.gif (18779 bytes)

        To get back to our example, the number of hospital ambulances needed is the smallest m such that Pm leq.gif (53 bytes) f. It is particularly surprising that (4.85) and (4.86) depend only on the mean of the service times.

M/G/inf.gif (300
bytes) queueing system. The M/G/inf.gif (300 bytes) queueing system can now be viewed as a special case of the M/G/m system with no waiting space, by taking the limiting value m rarrow.gif (63
bytes) inf.gif (300 bytes) for the latter system. Thus, we can use directly our last result. Replacing m by inf.gif (300 bytes) in (4.85), we have

form4.87.gif (3450 bytes)

        As one would expect [since (4.85) is identical to (4.57)], this expression is the same as (4.50) for the M/M/inf.gif (300 bytes) system. As with the M/M/inf.gif (300 bytes) system, we also have Lbar.gif (304
bytes) = lambda.gif (179
bytes)/mu.gif (189
bytes), Wbar.gif (557
bytes) = 1/mu.gif (189
bytes), and Lbar.gif (304
bytes)q = Wbar.gif
(557 bytes)q, = 0 in this case.
        It is instructive to derive (4.87) directly because in the process one can also derive the time-dependent probabilities Pn(t) of having n users in the M/G/inf.gif (300 bytes) system at time t assuming that the system was empty at t = 0. In the following we shall use Fs(s) to denote the cdf for the  service times and also use the fact that

formpg225.gif (2968 bytes)

Exercise4A Show that the above relationship holds for any random  variable S that assumes only non-negative values. [Hint: It is easier to work initially with a discrete random variable.]

We define

1. Pn(t) ident.gif (52 bytes) P{at time t there are exactly n users being serviced}.

2. Pn ident.gif (52 bytes) lim trarrow.gif (63 bytes)infty.gif (58 bytes)  Pn(t).

We wish to prove the following:


[Note from (4.88) that, for any given t, P.(t) is a Poisson pmf.]

Proof: Let the queueing system be empty at t = 0 and let

Pn(t | k) ~ P{at time t there are exactly n users being served | exactly k arrivals in [0, t]}

Then, since arrivals occur according to a Poisson process,

form4.89.gif (4265 bytes)

But, as we have seen in Chapter 2, if there are exactly k arrivals from a Poisson process in [0, t], the unordered arrival times are uniformly, independently distributed over [0, t]. So, we now consider a random user arriving in the interval [0, t] (see Figure 4.13). If the random user arrives with x more time left in [0, t], the probability that he is still being serviced at t is [1 - F,(x)]. But the unconditional probability he arrives in [t - x, t - x + dx] is dx/t. Thus, the probability that a random user who arrives any time in [0, tl is still being serviced at time t is

form4.90.gif (9714 bytes)

We now use the fact that the unordered arrival times are independent. Given k arrivals in [0, t], the probability that there are n customers being serviced at t (n leq.gif (53 bytes) k) is

form4.91.gif (3711 bytes)

Substituting (4.90) and (4.91) into (4.89), we obtain (4.88)-after some algebraic manipulation. The steady-state result (4.87) is derived just by letting t rarrow.gif (63
bytes) inf.gif (300 bytes) in (4.88).
        As we have already noted earlier, in order to apply a M/G/inf.gif
(300 bytes) (or M/M/inf.gif (300 bytes)) model, it is not really necessary to have an infinite number of servers. In practice, all that is needed is that the actual number of servers be sufficiently large or that workload be sufficiently small so that queues almost never form. Fire department operations, for example, possess this characteristic. It is highly unlikely that in a major city there will ever be insufficient fire engines or fire ladders to dispatch to a major fire; for if the fire stations in a given area run out of fire companies as a result of one or more multiple alarm fires, then other fire companies are dispatched to the scene, first from neighboring stations and eventually, if necessary, from nearby cities or communities. Assuming then that fire alarms in a region are generated according to a Poisson process, service times for different fire alarms are statistically independent and identically distributed, fire alarm rates and service rates are constant in time, and an infinite number of  fire-fighting units are available, one can use the M/G/inf.gif (300 bytes) model to estimate the distribution of the number of   fire-fighting units which are busy at any one time in an area. Although the validity of some of the assumptions above may be questioned, the results provided by such a model 9 [CHAI 71] have been shown to provide excellent approximations to the true distribution [IGNA 78].

9 The referenced model, due to J. Chaiken, is a modified M/G/inf.gif (300 bytes) model which accounts for the fact that two or more fire units are often simultaneously dispatched to a fire alarm.