Low, Siew Nghee
This thesis extends the application of waiting line theory to situations where both arrival rate and service rate distributions are arbitrary or non-random. It does so only for single channel, single phase, steady state, infinite queues with no feed-back. Previous work by A.K. Erlang had shown that queuing characteristics could be predicted for one case of an arbitrary service rate distribution, the constant service time. Also, F. Pollaczek had shown that, where arrival rates are random, queue lengths and waiting times were independent of the form of the service rate distribution, being functions of the coefficient of variance squared. But all of the works assumed random arrivals around a stable mean arrival rate and, except for the constant service time case, most applications were limited to cases where both arrival and service rates were random. This restriction has limited applications severely and has required that most analysis of queuing characteristics be done by simulation. This study develops and proves by inference the hypothesis that system length is dependent on these factors only: the square of the coefficient of variance of the interarrival time distribution, C²a, the square of the coefficient of variance of the service time distribution, C²s, and the ratio of mean arrival rate to mean service rate, p. Through a combination of calculation and simulation a set of curves has been developed covering values, C²a from 0 to 6, C²s from 0 to 6 and of p from 0.1 to 0.9. These curves permit the prediction of system length, and then of average queue length and waiting time, for any case where only the mean and variance of the arrival and service time distributions are known, even though nothing is known about the form of the distributions. In the usage of the set of graphs (figures 10-29), the following steps are all that is required to obtain the necessary characteristics: a) Calculate the average interarrival time, [formula omitted] (Total time of observation/Total number of Arrivals). b) Calculate the variance for interarrival times, [formula omitted] Total number of arrivals. c) Calculate the fractional coefficient of variance squared for interarrival time distribution, [formula omitted]. d) Calculate the average service time, [formula omitted] (Total time service facility is in operation/ Total number serviced). e) Calculate the variance for service times, [formula omitted] Total number serviced. f) Calculate the fractional coefficient of variance squared for service time distribution, [formula omitted]. g) Calculate the utilization factor, p = (Average service time/Average interarrival time). h) With the values p, C²a, and C²s, read from the set of graphs (figures 10-29) the verticle axis, L. i)Compute Lq, W and Wq. / Business, Sauder School of / Graduate
This thesis examines methods for predicting queue length of single server queues in order to evaluate how the practitioner may achieve greatest accuracy. Because accuracy is dependent on the correct estimation of the rate parameters of the population distributions and the choice of the appropriate method of prediction, the effects of errors in both of these are examined. A computer simulation model written in GPSS/360 is used to create a real world from which data is drawn and where long run performance represents the correct solution. For four values of rho nine simulations are run, each with a unique combination of inter-arrival and service time distributions. In each of the 36 runs 10,000 arrivals are generated from which two samples of size 36 and 100 are taken and from which the generated queue statistics form the standard. A statistical analysis is used to detect samples taken from exponential distributions. The lack of a suitable test for small samples led to the development of a test based on the correlation coefficient of the sample times and pre-computed standard data. Estimates for queue length are found with classical queueing formulae and solution methods suggested by Marshall. These predictions are done without prior knowledge of rate parameters and queue type which are estimated from the samples. Then the estimated solutions are compared to the real world solution derived from the simulation. Estimation error for each method is measured and conclusions are drawn as to their accuracy in predicting queue length. It is found that accurate queue length estimation is possible using methods that can be applied without a great deal of prior mathematical knowledge. The classical formulae are accurate only when applied to queues with exponential inter-arrival times and are found to overestimate when applied to other queue types. The Increasing Failure Rate (IFR) bounds on queue length provide a satisfactory method of estimation for the general class of queues. / Business, Sauder School of / Graduate
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. / Science, Faculty of / Mathematics, Department of / Graduate
Deveault, Andrée L.
No description available.
Leontas, Angela Zoi
01 January 2006
The thesis introduces the theory of queueing systems and demonstrates its applicability to real life problems. It discusses (1) Markovian property and measures of effectiveness with exponential interarrival and service times; (2) Erlang service times, and a single server; (3) different goodness-of-fit tests that can be used to determine whether the exponential distribution is appropriate for a given set of data. A single server queueing system with exponential interarrival times and Erlang service times is simulated using Visual Basic for Applications (VBA).
Many queueing systems have an arrival process that can be modeled by a Markov-modulated Poisson process. The Markov-modulated Poisson process (MMPP) is a doubly stochastic Poisson process in which the arrival rate varies according to a finite state irreducible Markov process. In many applications of MMPPs, the point process is constructed by superpositions or similar constructions, which lead to modulating Markov processes with a large state space. Since this limits the feasibility of numerical computations, a useful problem is to approximate an MMPP represented by a large Markov process by one with fewer states. We focus our attention in particular, to approximating a simple but useful special case of the MMPP, namely the Birth and Death Modulated Poisson process. In the validation stage, the quality of the approximation is examined in relation to the MMPP/G/1 queue.
Cheung Ka Wo. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 61-63). / Abstracts in English and Chinese. / Chapter 1. --- Introduction --- p.1 / Chapter 2. --- A review of Queueing theory --- p.7 / Chapter 2.1 --- Introduction to Queueing theory --- p.7 / Chapter 2.2 --- The M/M/l Queue --- p.13 / Chapter 2.3 --- The M/M/c Queue --- p.15 / Chapter 3. --- Model --- p.18 / Chapter 3.1 --- Hierarchy --- p.18 / Chapter 3.2 --- Random-assigning Model and M/M/l queues --- p.23 / Chapter 3.3 --- Idle-assigning Model and M/M/c queues --- p.26 / Chapter 3.4 --- Total Cost --- p.29 / Chapter 4. --- Optimization --- p.36 / Chapter 4.1 --- Minimum Total Cost --- p.36 / Chapter 4.2 --- The Optimal Number of Workers --- p.42 / Chapter 4.3 --- General Optimization --- p.45 / Chapter 5. --- Routing --- p.52 / Chapter 6. --- Conclusion --- p.56 / Appendix --- p.58 / Detail of computational results given in Section 4.2 --- p.58 / Reference --- p.61
Rumsewicz, Michael P.
(has links) (PDF)
Includes summary. Bibliography: leaves 108-112.
Northcote, Bruce S. (Bruce Stephen)
Bibliography: leaves 119-125. / viii, 125 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1995?
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Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.
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