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Some Aspects of the Mathematical Theory of Storage SOME ASPECTS OF THE KITHEM1TICAL THEORY OF STORAGE BY A. M. HASOFER, B.E.E., B.Sc., submitted in fulfilment of the requirements for the Degree of Doctor of Philosophy UNIVERSITY OF TASMANIA HOBART July, 1964 Except as stated herein, this thesis contains no material which has been accepted for the award of any other degree or diploma in any University, and, to the best of the author's knowledge and belief, contains no copy or paraphrase of material previously published or written by another person, except when due reference is made in the text of the thesis. \Ao Signed: (A. M. Hasofer) PREFACE The research work embodied in this thesis was done during the years 1961 — . 1964 at the Department of Mathematics, the University of Tasmania. The following parts of the thesis are original: Chapter 1, section 7; Chapter 2, sections 2, 7 and 11; Chapter 3, section 1, Theorems 3., and 7.6 of section 4, sections 5 to ii; Chapter 4 most of sections 2 to 6; Chap- ter 5, most of sections 4 and 5; the whole of Chapters 6 and 7. Most of the new. results have been published and may be found in papers 02j, L33j, 1r. 34j , and [35]. I would like to take this opportunity to thank my supervisor, Professor L. S. Goddard, who has untiringly given me much of his time as well as valuable advice throughout the preparation of this thesis. I would also like to thank Professor E. J. G. Pitman, who introduced me to Mathematical Statistics and who taught me the importance of rigour in Probability Theory. This thesis is imbued throughout with the spirit of his teaching. Finally I would like to thank my wife Atara, who has cheer- fully put up for the last three years with all the sleepless nights and fits of temper incident to the writing of the thesis and has unceasingly helped me with typing, checking and duplicat- ing. SUMMARY In this thesis, a storage model with infinite capacity, additive stochastic input and unit release per unit time is investigated. Th content of the store in the deterministic case is defined as the unique solution of an integral equation. Properties of non-negative additive stochastic processes are obtained. These properties are used to study the distribution of the time of first emptiness when the input is stationary, and the distribution of the content. Applications to dams and queues with specific input laws are given. In particular, the waiting time for the queues M/M/1 and M/G/1, and the content of the dam with Gamma input are studied in detail. The dam with Inverse Gaussian input is introduced and its transient solution obtained explicitly. Finally, in the case of a Compound Poisson input, the con- tinuity and differentiability of the distribution of the content are investigated. A non-stationary Compound Poisson input is considered, and it is shown that the probability of the store being empty and the Laplace transform of the content can be expanded in a power series. When the parameter of the input is periodic, it is shown that all terms of the series expansion are asymptotically periodic, and explicit expressions for the leading terms are obtained. (i) CONTENTS INTRODUCTION Page 1. General alitline of the thesis. vi 2. A survey of recent literature on storage. xiv CHAPTER 1 A DETERMINISTIC INVESTIGATION OF THE STORAGE MODEL WITH INFINITE CAPACITY 1. Definition and elementary properties of the input and output functions. 2. The case of an input function which is a step function. 5 3. The case of an input function which has a continuous derivative. 9 4. The function V (-0 and its use in the definition of (i) 12 5. The formal definition of the content of the store. 14 6. Various interpretations of the model. 22 CHAPTER 2 THE STRUCTURE OF NON-NEGATIVE ADDITIVE PROCESSES 1. Definitions and elementary properties. 25 2. Complete monotonicity ofQ.:)N,t,,S) in A . 28 3. The relationship between N4( -30 and the deriv- atives of <(t,(-). 4. The Poisson Process as the simplest type of additive process. 33 5. The Compound Poisson process. 36 6. The input process of the queue M/M/1. 39 7. Bunched arrivals. 4n 8. Processes Where the sample functions are not a.s. step functions With isolated discontinuities. 4"4 9. On the derivative of the sample functions of non- negative additive processes. 44 in. The Gamma process 46 11. The Inverse Gaussian process. 48 12. The non-homogeneous additive process. 50 CHAPTER 3 THE TIME OF FIRST EMPTINESS AND ITS DISTRIBUTION 1. Definition, measurability and elementary properties. 2. The L.S. transform of 73( ,;0 in the case of a hom- ogeneous input process. 56 3. Some properties of ()(p) and the corresponding properties of . 4. The uniqueness of the solution of lc . y(). 67 5. Some examples. 75 6. The inversion of Clp4j ) when the input has a density function. 78 7. The inversion of (13 1 ,0 in the case of a compound Poisson input. 81 8. The case of a discrete input. 86 ( iii) 9. Some explicit expressions for the time of first emptiness. 87 10. The general solution of Kendall's integral equation. 91. 11. The distribution of the busy period. 95 CHAPTER 4 THE DISTRIBUTION OF THE CONTENT 1. The fundamental equation for the Laplace transform of the content distribution. 99 2. The inversion of the fundamental formula. 102 3. The calculation of W(t,o) in the stationary case. 107 4. The asymptotic beha-iour of the content in the stationary case. 113 5. The inversion of the Pollaczek-Khintchine formula. 122 6. The asymptotic behaviour of W -)c) 126 CHAPTER 5 SOME EXMFLES 1. The content distribution for various initial contents. 170 2. The waiting time for the queue M/M/1 131 3. The content of the dam with simple Poisson input. 174 4. The content of the dam with Gamma input. 176 5. The content of the dam with Inverse Gaussian input. 137 CHAFTER 6 THE CONTINUITY AND DIFFERENTIABILITY OF W(t, 1. Preliminary remarks. 142 2. Canditions for the continuity and differentiability of 144 . CHAPTER 7 THE CASE OF A NON-STATIONARY COMPOUND POISSON INPUT 1, Introduction. 151 2. Remarks on the Poisson process with periodic par- ameter. 3. An outline of the approach. 155 4. Some analytical properties of the functions R 157 5. Derivation of a power series for W(t /O) . 163 6. Some theorems on Laplace transforms. 169 7, The Laplace transforms of the coefficients in the power series for W(t,o) 171 8. The asymptotic behaviour of the F= (t) when T,(t) is periodic. 175 9. The asymptotic behaviour of the Laplace transform of the content distribution. 182 10. Explicit expressions for the leading terms in the case of a simple harmonic input, 185 REFERENCES 189 INTRODUCTION AND SURVEY OF THE LITERATURE 1. General outline of the thesis The theory of storage has attracted much attention in recent years. Although the impetus was first given by economic problems of inventory and provisioning and engineering problems in dam design, it soon appeared that storage models had an intrinsic mathematical interest. Storage models with stochastic input are analogous to models in queueing and renewal theory, and provide interesting examples of Markov processes having unusual properties. In this thesis, the following model is investigated z an input (t ) which is a stochastic process with independent increments, is fed into a store, over an interval of time t The output from the store is of one unit per unit time, except - when the store is empty. The two main processes investigated are the time of first emptiness and the content of the store at any time t This abstract model contains, as special cases, the follow- ing models which have been extensively studied (a) the single-server queue with Poisson arrivals and exponential service times (M/M/1). (b) the single-server queue with Poisson arrivals and general service times (M/G/1). (c) the single-server queue with bulk arrivals at points of time which follow a Poisson distribution and either exponential or general service times. (d) the infinite dam with Poisson input and constant- rate release. (e) the infinite dam with constant inputs at equidistant points of time and releases following the negative expon- ential distribution. (f) the infinite dam with a Gamma-distributed input, and constant-rate release. The advantage of using a single abstract model for these various situations is, of course, that of being able to use a unified technique to obtain results which have been established previously by widely varying methods, and then only for special cases. By using the general method, we are also able to obtain new results not previously published. Another feature of this thesis is the emphasis on contin- uous parameter methods. Many of the important results in the field under investigation have been obtained by limiting methods e.g. Moran [521], Gani and Prabhu [29]. However, it seems simpler to study the continuous-time model directly, and it turns out in fact that the required results can be obtained just as easily in this way as with limiting methods. It should be emphasised that only results relating to the waiting time can be obtained by our technique, when it is applied to queueing models. The queue length cannot be studied by this method. The main contents of this thesis are as follows: Chapter 1 deals with the deterministic version of the mOdel under investigation. It is interesting to note that the problem of the definition of the content of a store with completely general inputs and outputs, though a natural one, has not been given attention until quite recently.
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