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UBC Theses and Dissertations

Closure methods for the single-server retrial queue Sabo, David Warren


This work focuses on the development and evaluation of so-called "closure methods" for solving the equations governing the time-dependent behaviour of single-server retrial queues. These methods involve assuming that particular known algebraic relationships between various characteristics of the corresponding steady-state queue also apply approximately when the queue is not at steady-state. The objective is to replace a problem requiring the solution of dozens or hundreds of simultaneous linear differential equations with a system of a few differential equations that has a solution that approximates those queue characteristics of immediate interest. The viability of such closure methods is assessed by examining the results of a series of test calculations. The methods described in this thesis apply to a retrial queue in which inter-arrival times for new customers, inter-retrial times, and service times are all assumed to be exponentially distributed. The steady-state solution for such a queue is described in some detail. A survey of the literature indicates that the description of this steady-state retrial queue has become quite sophisticated, whereas only very tentative steps have been taken in the study of the time-dependent behaviour of such queues. On the other hand, the time-dependent behaviour of the simple M/M/s queues have been studied to a much greater extent. The apparent value of closure methods in computing approximations to various basic time-dependent M/M/s queue characteristics motivated this examination of the extension of such methods to the single-server retrial queue. After discussing the basic approach to be used in devising and testing prospective closure methods for the single-server retrial queue, a variety of such methods is presented, with each being tested in considerable detail. It is found that three of the methods devised give results of comparable or better accuracy than those closure methods for the simple M/M/s queues which motivated this study. All recommended closure methods developed here involve systems of either two or three differential equations and permit the calculation of good approximations to four of the characteristics of greatest interest for non-stationary queues: the probability that the server is idle, the mean queue length, the variance of the queue length, and the conditional mean number of customers in the system given that the server is idle. Each of the methods presented is tested for queues with constant mean arrival, retrial and service rates, as well as for queues in which arrival and retrial rates vary sinusoidally with time.

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