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Correlation between arrival and service patterns as a means of queue regulation.
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A major cause of congestion in queuing situations, that is of immoderate waits and lengthening queues, is often the assumed independence of the arrival and service mechanisms. This dissertation is concerned with single server "correlated" models, defined to be such that either the service mechanism is somehow tailored to the arrival pattern, or vice versa. The greatest attention is given to a particular model in which the service time allotted to the nth arrival is Î» Tn , where Î» is a non-time dependent constant and numerically has the value of congestion index, and Tn is the interval between the (n-l)th and the nth arrivals which, it is important to note, could be observed by the server before service is initiated. It is shown that the effect of the correlation mechanism is to reduce congestion under a given level of traffic intensity, as compared with single server systems in which arrivals and service are independent. This result is achieved without inflicting on the service facility the penalty of increased periods of idleness. The particular model is a queuing interpretation of a stochastic-kinematic situation studied by B. W. Conolly in connection with a military tactical analysis. The dissertation is divided into two parts. Part I develops the theory of the main model with particular reference to state probabilities, waiting time, busy period, and output. Some consideration is also give to a related model where service depends on the arrival pattern, and to what is referred to as the "dual" problem in which the arrival mechanism is geared to service capability. Further, the state probabilities at arrival epochs for a conventional M/M/l queue are obtained by employing a simple probabilistic argument. This is needed for Part II. Part II applies the theory to give a practical comparison of the correlation mechanism with the elementary "independent" single server queues M/M/I, M/D/l and D/M/l; and it is shown in detail that the practical result referred to above is achieved. The superiority of the correlation mechanism increases with traffic intensity. State probability, busy period and output comparisons are made only with the M/M/l system. The main conclusions are found to extend also to these processes. It is concluded that, where its application is practicable, a mechanism of correlation can achieve important gains in efficiency.
- Doctoral Dissertations