1.1. Review by: J K Friend.
Operational Research 13 (1) (1962), 104-106.

The theory of storage embraces the theory of inventories, where input is con- trolled in response to a variable output, and the theory of dams, where output is controlled in response to a variable input. Professor Moran's compact yet powerful monograph is concerned primarily with the development of dam storage models and only one chapter is devoted to inventories, so the title is rather misleading. It does not follow, however, that this book should be ignored by the operational research worker whose day-to-day preoccupations are with inventories rather than dams; in their exploration of a comparatively little-known facet of stochastic process theory, Professor Moran and his colleagues have drawn extensively on the published work on queues and inventories, and there are now signs that the general theory is being enriched by a feedback of ideas in the reverse direction. ... The author is an impressive theoretician and is able to draw on a wide range of mathematical tools in his bid to construct a realistic set of models for dam storage. It is refreshing to be able to report that he keeps his mathematics firmly under control by embedding it in a lucid prose style, so that the reader is not often left in doubt as to the significance of an equation or an assumption to the main trend of the argument.

Professor Moran describes his book as "Dam Theory" being short for "Probability theory of Reservoir Storage," and this title does describe a broad class of problems which are discussed by him. The methods developed are, however, applicable to other classes of problems such as Inventory Storage and the theory of Queues. There are six chapters. In the first chapter we are taken at a brisk canter through characteristic functions and generating functions and the mathematics descriptive of Markoff processes. This is followed by storage problems, Segerdahl's risk theory and inventory control. Chapter 3 concentrates on the mathematical equations concerned with a dam built to store water, the connection of dam and inventory problems and the analogies with queuing theory. This is for time discrete and it is followed by time continuous. The two remaining chapters deal with "Monte Carlo" methods and the programming of storage systems. The author is concerned throughout with mathematical exposition. There is nowhere any suggestion that the mathematical models should be tested against the data of experience, or that such data exists.

One of the fields of operations research which has been studied less extensively in this country than elsewhere is the analysis of the stochastic processes arising in water storage systems. Professor Moran, who has been quite influential in the development of this research, has written an excellent summary. Apparently the publication of the volume was held up for a considerable length of time and some of the more recent work on the theory of dams has not been included. The following example is typical of the type of situation which is analyzed. We examine a water storage system (say a dam or reservoir) whose supply of water is replenished in a random fashion over time. The system is assumed to have a finite capacity so that any excess of water over that capacity is lost if not used immediately. Decisions are made as to the release of water at various moments of time, in order perhaps to provide hydroelectric power, or for the purposes of irrigation. ... Professor Moran remarks on the similarity between the theory of dams and the analysis of inventory policies, both being concerned with storage systems. In fact, Chapter II of this monograph consists of an analysis of several simple inventory models. While the mathematical techniques used in these two fields do occasionally have points of similarity, there is at least one important difference which should be kept in mind.

Professor Moran's book deals with that part of congestion theory applicable to the study of reservoirs and similar storage systems. As the author points out, although his models "are much simpler than those which occur in practice" they do provide background knowledge for those engaged in practical problems. After a general introduction and a chapter on the problem of inventories as stochastic processes, the main theme of the book - the theory of dams in discrete and continuous time - is discussed. These two chapters are followed by a chapter on the solution of these problems by simulation methods and a brief discussion of the programming of storage systems. For such a short book the author has succeeded in including a surprisingly large amount of material and achieved his aim of providing background knowledge. In one respect, however, the author seems to be open to criticism. To include an attack, however justifiable, on the "extreme value" techniques associated in this text with a named author, without including any reference to the work of this author (E J Gumbel) in the bibliography is to do him less than justice.

The book provides a brief account of the work of Professor Moran and his colleagues, notably Gani and Prabhu, on probability problems connected with the study of reservoir storage. The approach will be familiar to those who have read J Gani's paper already published in Series B of this Journal. ... In the main the book is concerned with the mathematical treatment of the operating behaviour of systems in their aspect of stochastic processes, and it will be of more interest to theoreticians than to, say, operational research workers. Although, of course, the type of application concerned is of prime importance in operational research, the emphasis in the latter field is on the design of optimal systems rather than on mathematical analysis of stochastic behaviour.

This small monograph is a handy compendium of recent work on that part of the Theory of Queues having particular relevance to storage problems. The author subdivides his subject-matter into two parts - 'Inventory Storage' and 'Reservoir Storage' (the author's preference - and also the reviewer's - although he chooses to call it 'Probability Theory of Dams'). The main difference is one of emphasis : in inventory problems it is typically easier to control supply (e.g. entry of goods into a warehouse) than demand (e.g. sale of goods), while in reservoir problems the positions of supply and demand are, on the whole, reversed. It is a convenient, though by no means a rigid, distinction. Also the aims of operating policy are usually different in the two types of problem. ... Despite some unevenness in the relative amounts of detail accorded to different topics, and the lack of numerical illustration, this book should be a worth-while purchase for persons wishing to possess a good knowledge of one of the most fruitful fields of statistical research in recent years. Provided they are not intimidated by some of the names - such as Riemann-Stieltjes, Laplace- Stieltjes, Bachelier-Wiener - appearing in Chapter 1, they will find the mathematics in the rest of the book well under control, and serving its proper purpose of clarifying the argument.

This is a book about dams. Professor Moran is at the Australian National University at Canberra, and I imagine that dams have great practical interest there. For many years he has been interested in estimating the probability that a dam will go dry or that it will overflow. He is also interested in how one finds a program of releasing water from a dam in such a way as to optimize the operations of a hydro- electric plant. ... The analogies between dams and queues or inventories are not pursued beyond the third chapter, in which it is merely mentioned. If these analogies are indeed valid they deserve more treatment. Without this treatment the title is misleading, for we find we are storing only water.

After introducing some necessary (mainly mathematical) definitions, notations, ideas and techniques the book consists largely of a development of the distribution theory arising from the various models (of stochastic process type) which have been proposed for dealing with inventory and dam theory. It closes with two short chapters: one on Monte Carlo methods (of direct simulation type) and the other on the best or good Rules of release (of stock or water, as the case may be) by means of such examples as that where several dams in series, with electric generators of different efficiencies, have to supply a given amount of power with minimal expected loss due to overflow. The Rule of Release problem is shown in the examples to reduce to one of Linear Programming or Calculus of Variations, and methods of solution of these are discussed or referred to. ... In short, this book is an admirably clear introduction to the probability theory of inventory and dam processes, assuming only a knowledge of elementary statistical and stochastic process theory and a corresponding level of mathematics.

The 'storage' considered in this monograph is chiefly that of water in a dam, though a short discussion is given of the related problem of goods inventories. The effect of random variability of supply as well as of demand is discussed and the relation to Markov processes is indicated, though no detailed analysis is presented. The majority of the space is devoted to a discussion of the single-dam operation, using discrete-time periods and, later, considering time as a continuous variable. A few of the many possible criteria of optimal operation are mentioned, though no unified picture of the decision problems, which face the operating head, is presented. There is a short outline of the use of 'Monte Carlo' and machine-simulation techniques in such studies.

The author, a pioneer in the application of probability theory to problems of water storage in dams, presents a concise and lucid account of the present status of the subject. In the first two chapters the mathematical preliminaries (including Markov processes) and inventory problems are briefly summarized. The core of the book is in the remaining four chapters: Dams - Discrete Time; Dams - Continuous Time; Monte Carlo and Other Statistical Methods; The Programming of Storage Systems. A bibliography consisting chiefly of post-1950 items bears witness to the novelty of the topic. This book can be recommended not only to the designer concerned with problems of dam construction but to any one interested in applications of recently developed techniques in probability theory.

The author gives a useful connected account of some probability problems that arise in theories of inventories and of dams. Inventories: brief accounts are given of Hammersley's storage model and Segerdahl's theory of insurance risk, followed by a more detailed comparison of two replacement policies based on work of Pitt and Gani, and reference to more complex models. Dams: the account is based on the author's pioneer work on finite and infinite dam models, and developments by Gani, Kendall and other speakers at a Royal Statistical Society symposium, with mention of further work not published at the time of writing (1958). The author draws attention to the usefulness, but also to the inherent limitations, of these storage models. He concludes with chapters on Monte Carlo, and other, statistical methods, and on the programming of storage systems.

2.1. Review by: Motoo Kimura.
The Quarterly Review of Biology 39 (1) (1964), 86.

In this book, the mathematical theory of population genetics in the strict sense is presented as a branch of applied mathematics. ... The book may not be intelligible unless the reader has considerable mathematical background. It will be most useful for mathematicians, statisticians, and physicists who want to learn the mathematical theory of population genetics. In this respect, the book will make a good contribution towards promoting the development of this branch of science.

This monograph is in the author's words, "an attempt to give a systematic account of the mathematical aspects of the genetics of natural populations." The treatment is mathematical and "there is no extensive discussion of the application of the theory in practice, or of its significance for evolutionary theory." ... The subject matter is treated quite rigorously as a branch of applied mathematics but the mathematics used in the construction of the models is not developed in the text. 'A priori' familiarity with stochastic processes, and in particular Markov processes, as well as a general knowledge of population genetics is required.

The scope of this monograph is described as follows by the author: "The treatment is mathematical and there is no extensive discussion of the theory in practice or of its significance for evolutionary theory." There is indeed more rigorous treatment than usual of certain mathematical problems that have arisen in its field but others of equal or greater importance are touched on only slightly if at all. A valuable feature is the abundance of references to earlier work, but this is not so complete that new contributions can always be distinguished from earlier work presented without citation. ... It should be a useful book on the topics which it includes.

This is a systematic development of the deterministic and stochastic theory of population genetics written with the emphasis on rigorous and detailed mathematics. It is not clear to whom the book is addressed. A good working knowledge of the elementary ideas of theoretical genetics and its terminology is assumed; the reader's vocabulary must contain words like allopolyploidy, epistasis, pleotropic, monoecious, autosomal, etc (in their technical meanings). Simple ideas of linkage, cross-over, recombination fraction and so on must be familiar. On the other hand the book is basically a text in applied probability; the reader must be able to follow the matrix, partial differential and integral equation methods of stochastic process theory. The more complicated genetical schemes are explained as they arise. There is no direct reference to numerical applications, let alone comparison with actual population records (which are in any event sporadic and largely inadequate). Instead there are many brief references to such applied literature as exists. This leads to slightly oversimplified statements. For example, the untutored reader might imagine from the discussion here that the Canadian lynx data was an example of a simple sinusoidal regression with an added random error. Whilst Lynx Canadensis has undoubtedly enjoyed a marked 9 3/4-year periodicity in population size over the last century, and less certainly over the previous one, this does not rule out a damped oscillation as the author implies; it merely restricts the range of parameters. The author, though, does give a careful underlining of the mathematical limitation of the models considered, for the benefit of non-mathematicians.

In this excellent book Professor Moran presents "a systematic account of the mathematical aspects of the genetics of natural populations" - a study in applied mathematics with little reference to applications. In this subject old theories are seldom completely superseded by new: all contribute to current thought. In each topic the author presents each theory and comments on the breadth of theoretical results to which each leads. ... One feature of the subject is the wide variety of unsolved problems, commented on in the text. I hope that many mathematicians see this book, for I am sure the author will arouse their interest, to their - and the subject's - advantage.

3.1. Review by: F N David.
Biometrika 51 (3/4) (1964), 532.

The authors have collected together theorems connected with the geometry of points and lines in one, two and three dimensions, with a valuable emphasis on the two-dimensional case. This collection can have been no small task since the English mathematicians of the late nineteenth century published their researches in many places and the contribution of any one person, with the exception of Morgan Crofton, was often not very large. The authors divide their work into the distribution of points in Euclidean space, random lines in two and three dimensions, random planes and random rotations, and coverage problems. ... The monograph can be recommended.

Geometrical Probability is a fascinating but of late neglected branch of mathematics; it has been developed by Crofton, Minkowski, Lebesgue, and Steinhaus, among others. The content of the field is as rich as these names promise. The monograph under review is, to my knowledge, the only recent survey of the field, and is therefore a valuable contribution to the literature. Kendall and Moran describe a great variety of topics, with applications in disciplines ranging from astronomy through forestry to molecular physics. The material is chosen with taste and insight, and will be of interest to a wide audience. Original sources are given for many of the ideas, so that the bibliography is comprehensive and useful. ... Definitions are vague, regularity conditions only hinted at, and proofs are complicated but incomplete. Geometrical probability has become unfashionable because of its seeming lack of rigour. Kendall and Moran do nothing to correct this impression.

Most statisticians will have heard of Buffon's needle as a method of estimating π, and many will be familiar will be familiar with probability questions regarding the length of random arcs on a circle and the distribution of points occurring at random along a fixed line. The present work is the first volume in English to try and give a systematic account of the application of probability to such geometrical situations. Suitable applications have greatly increased in number since the war and the corresponding theory has been scattered in a wide number of journals. Probably the most famous application is to the field of atomic physics but the authors also include examples from those of astronomy, biology, crystallography and others. ... Throughout, the treatment is mathematical and rigorous. The authors are to be commended on collecting together so many problems from a wide range of references; as a treatise on the application of probability to these problems, the work will appeal to those statisticians with a strong mathematical background.

Geometrical probabilities - the chances of various configurations arising out of the random distribution of points, linear forms and geometrical figures - are, like metrical geometry, essentially a matter of physics. The failure of mathematicians to grasp this fully and their attempt to supplant induction by "pure reason" led to the famous but nowadays slightly naive "paradoxes" of Bertrand, Wilson and others. [E.g. four lines are drawn at random in an infinite plane. E is the event that a given one of them runs through the interior of the triangle enclosed by the other three. In the figure formed by any four lines, two lines have the property and two do not, so that plainly P{E} = 1/2, from symmetry - yet, equally plainly, any triangle is finite and the plane is infinite, so P{E} = 0 is true!] This has meant that the few preceding texts by Czuber (1884), Morgan Crofton (1884 and 1885) and Deltheil (1926) have started off on the wrong foot and have been unnecessarily obscure in their initial stages as a consequence. Further, the physical basis of the problems of geometrical probability gives a physical insight into their solution which is easily lost in an abstract formulation. The present book adopts a more realistic approach though it does not wholly succeed in exorcising the influence of its predecessors. For example, in some cases the underlying probabilities are deduced, at one remove from reality, from invariance to rotation and translation rather than directly from the physical specification.

Geometrical probability has exercised great fascination on research workers in diverse fields. In several instances the solutions obtained in quite unrelated contexts have been shown to be equivalent. There has been a long felt need for a unified text-book presentation of results that could come under the heading of geometrical probability. Professors M G Kendall and P A P Moran must be thanked for writing just such a book that fulfils this need. The monograph is divided into five chapters : Distribution of geometric elements, distribution of points in Euclidean space, random lines in a plane and in space, random planes and random rotations and problems of coverage. The bibliography at the end collects more than 180 articles on geometrical probability scattered in the literature.

Statistical geometry is a relatively old branch of pure mathematics (pursued by Barbier, Cartan, Cauchy, Crofton, Czuber, Glaisher, Minkowski, Sylvester, and others mainly in the second half of the nineteenth century), to which applied mathematicians have given a new lease of life within the last twenty years or so. Current applications extend over a very varied field: this book contains examples from and references to acoustics, astronomy, atomic physics, biology, bombing, botany, crystallography, ecology, epidemiology, forestry, geology, gravitational theory, haematology, harmonic analysis, molecular theory, Monte Carlo methods, numerical analysis, phytosociology, projectometry, sampling theory, sedimentation, theory of liquids and traffic studies; and this by no means exhausts the list of applications. As the authors remark in their introduction, these applications "require for their solution all that was discovered in the past about geometrical probabilities and a great deal more besides. The subject, in fact, has been reborn." Apart from texts on the related subject of integral geometry, there has been no book on statistical geometry since Deltheil's Probabilités Géométriques (1926); and Kendall and Moran's cloth-covered monograph is the first treatment to be published in English. It is therefore very welcome.

This is an excellent book, dealing with a highly fascinating topic, which has attracted in the last 200 years many excellent mathematicians, and which, after being for a time somewhat neglected, became again very popular in the last two decades. It is not east to define the subject strictly. One may try to define the topic as that part of probability theory which admits a geometric interpretation. this definition is however not quite adequate, because every statement on a finite number of random variables having a continuous probability distribution may be interpreted geometrically; however among such statements only those belong to geometric probability which are interesting to the point of view of geometry too. One may also define geometric probability as the application of probabilistic ideas to the study of geometrical objects and configurations. This definition is however not quite successful either, because not ready results of probability theory are applied to geometry, but special methods are developed to deal with special problems. ... The book consists of five chapters (1. Distributions of geometrical elements, 2. Distribution of points in euclidean space, 3. Random lines in a plane and in space, 4. Random planes and random rotations, 5. Problems of coverage). The style of the book is very clear and vivid; it gives the reader much pleasure. The authors were very successful in writing the book in such a manner that it can be understood without much preliminary knowledge, other than a basic knowledge of elementary geometry and the elements of probability theory. The emphasis is on giving a clear understanding of problems and results and of their practical significance and to show the main ideas of the available methods. Simple calculations are usually left to the reader and for difficult proofs the reader is often referred to the literature. However in most cases when it was possible to give a full proof in a few lines, this is done; in other cases the proof is only sketched. The topic is rich in many exciting and challenging unsolved problems, and many of these are pointed out by the authors. By all these merits it is sure that this book will be most useful to mathematicians and statisticians who are interested in doing research work in this field, as well as to scientists of all kinds, who are interested in the application of geometrical probability to their own field of work.

4.1. Review by: J F C Kingman.
Journal of the Royal Statistical Society. Series A (General) 132 (1) (1969), 106.

The theory of probability may be presented in many different ways, from the austere elegance of Loève to the lush vegetation of queueing theory. This book is probability 'à l'anglaise', and will not appeal to every taste. But it has great virtues, not the least of which is the rather precisely determined level of mathematical sophistication, low enough to be accessible to the applied mathematician but high enough to allow most results to be rigorously proved. The most notable feature is the wealth of detail condensed into about five hundred pages of readable text (together with an excellent bibliography). Thus the reader will find, in addition to the usual material on distributions and their manipulation, such useful tools of statistics and applied probability as Cochran's theorem, Campbell's theorem ("shot effect"), Spitzer's identity, and the Borel-Tanner distribution. Among stochastic processes considered, apart from simple Markov chains and processes, are less common models like Daniels's stiff chains and Hammersley's self-avoiding random walks. The author is at his best when describing particular processes of this sort; his foot is sometimes less sure on the heights of abstract theory.

In the preface the author says: "The book is designed to be read by honour students in about their third year." In the United States it will be an appropriate book for certain first-year graduate courses - say, courses serving students in applied areas such as statistics - although portions of the book are appropriate for undergraduate students. There are 10 chapters: 1. The Probabilities of Events. 2. Discrete Distributions. 3. Markov Processes. 4. Probability and Measure Theory. 5. Random Variables and Continuous Distributions. 6. The Characteristic Function. 7. Special Continuous Distributions. 8. Sequences and Sums. 9. The Arithmetic of Distributions and the Brownian Movement. 10. The Random Walk. The brevity of the table of contents hides one of the strengths of the book. Chapter 1, for instance, contains sections on the distribution of the number of inversions in a random permutation, on martingales, and on entropy. Hidden in Sections 4.15 and 4.16 is an introduction to geometric probability. Throughout, the broad scope of probability is highlighted nicely by directions such as these. The numerous references to appropriate items in the literature further add to the value of the book. The brevity of the table of contents contributes to what some may consider to be a defect in the book. I was not able to see an organizational scheme on which the book is constructed. Chances to help the student obtain a coherent view are missed. ... The book makes probability exciting. That certainly compensates for some shortcomings. I also liked the extensive bibliography, author index, and subject index.

This book is intended as an intermediate-level text from a somewhat applied viewpoint, although measure-theoretic probability is introduced in Chapter 4. Its treatment of standard material is (usually) reasonably detailed ... It is also, by nature, an encyclopaedia of probability, containing brief discussions of little known topics (e.g., intermittent convergence, self-avoiding random walks), as well as fascinating insights into well known ones, and concluding with a huge (if slightly dated) bibliography. The breadth of coverage and frequently novel arguments make the book a useful addition to the probabilistic literature. On the negative side, perhaps as a result of the bulk of material, the treatment of many important topics is unsatisfyingly brief (e.g., branching processes in under four pages), the reader being shunted off to the references rather soon. The reviewer also found the lack of section titles in the list of contents (only chapter headings are given) extremely frustrating, in spite of the subject index.