5.3.1 Restrictions on Service Time and Queueing Behavior
As we will see, as we proceed from 1 to 2 to N servers, the restrictions we must place on the service-time distribution become more and more severe. With this N = 2 model, we assume that the two servers have the same (general) service-time distribution with mean 1 / . Successive service times must represent independent samples from this distribution, independent of the history of the system, its current state, the identity of the responding unit, or the location of the service request. Thus, unlike the M / G / 1 model (or its modification in the previous section), the N = 2 model cannot explicitly include variations in service times due solely to variation, say, in customer location or server identity. This does not mean that travel time must be ignored in the service time. To the contrary, the mean service time can be set equal to the sum of the estimated mean travel time (estimated prior to the analysis), the on-scene time, and any related off-scene follow-up time. After analysis, mean travel time is a model-computed performance measure; if there is a discrepancy between the estimated and the computed mean travel times, the computed mean travel time should be used to revise the model's mean service time parameter and the analysis should be performed again with the revised value. This iterative procedure, called mean service-time calibration, usually converges very quickly (typically one or two iterations). After calibration, the mean travel time assumed in computing mean service time agrees with that computed by the model. Throughout the following model development, we will assume that such calibration has occurred.
One additional price we pay for a general service-time distribution (rather than, say, an exponential distribution) is that no queue is allowed to form. That is, customers that appear when both units are busy are not serviced by the N = 2 system. We suppose that, in reality, they are handled by some backup system (such as a police department backing up an ambulance service). If we had desired a queue of arbitrary length to be allowed, we would have to restrict further our service-time assumption, requiring the distribution to be negative exponential. This is another example of how an apparently simple modification of a queue's operating behavior can change markedly the ease or difficulty of analyzing the queue.