The Bulk Service Queue with a General Control Strategy: Theoretical Analysis and a New Computational Procedure

A general framework is developed for analyzing a wide class of vehicle dispatching strategies for bulk arrival, bulk service queues. A simple derivation of the queue length transform for the imbedded Markov chain is provided and a new computational procedure is developed for finding the moments of the queue length distribution. Extensive computational tests are reported which demonstrate that the new procedure is significantly faster and more stable than the standard method referred to in the literature, which requires solving a set of simultaneous linear equations. Formulas for the mean and variance of the length of the queue are provided for the general case of compound Poisson arrivals, random batch capacities, general service times and a general control strategy. We consider the problem of analyzing bulk arrival, bulk service queues in steady state where customers arrive in groups at a point and wait for the next available vehicle. Under the simplest policy, termed here the bulk queue with no control, the vehicle, on arriving, will accept customers up to the capacity of the vehicle and then leave, independent of the number of customers waiting. The available capacity of the vehicle is assumed to be random but independent of the length of the queue or the capacity of all previous vehicles. The time from the departure of one vehicle to the earliest possible departure of the next vehicle is a service period, where successive service periods are assumed to be independently and identically distributed. The end of a service period is referred to as a dispatch instant, since a departure may or may not occur, depending on the control strategy in use and the length of the queue. If the vehicle is simply held until, for example, the length of the queue reaches a specified minimum, then the time during which the departure is delayed is an idle period. Additional constraints are frequently placed on vehicle departures to avoid the possibility of having vehicles depart with uneconomically small loads. In this paper an approach is described for analyzing a very wide class of control strategies, where two of the strategies are of particular interest. The first of these, which has not been dealt with previously in the literature, is a cancellation strategy whereby if the queue is not sufficiently long, the departure is cancelled and any waiting customers must wait an extra service period before they may leave. The instant at which the departure is cancelled is still counted as a dispatch instant even though no departure actually occurred.