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THE BEHAVIOR OF A SINGLE CHANNEL QUEUE WITH PERIODIC EXPONENTIALLY DISTRIBUTED INTERARRIVAL TIMES AND SERVICE TIMES
CENTER FOR NAVAL ANALYSES WASHINGTON D C OPERATIONS EVALUATION GROUP
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The operation of the Naval Communication System was studied from a mathematical model which simulates the behavior of the system. This model was used to predict possible improvements in the operation of the system. Since messages passing through the Naval Communications System generally form queues or waiting lines, it is natural to construct the mathematical model using queueing theory. A mathematical model is developed for a simple queueing system, a single channel queue with exponential and periodically varying interarrival and service time distributions. In future work this model will be generalized to include more complex queueing systems, such as queues with priorities and networks of queues. Two methods are applied for solving equations which describe the queueing systems a generating function met od, and a matrix method. The generating function method produce not only approximate expressions for the mean and variance of the number of messages in the system as a function of time but also conditions for queue stability. The matrix method produces expressions for the probability that there are n messages in the system at time t. Both of these methods are applicable to more complex queueing systems. Other results include an expression relating the mean messages delay time to the number of messages in the system and to the expected number of arrivals, and four measures of effectiveness for queueing systems. Author
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