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Systems reliability modelling and evaluation
made during the Polygon and Mode expeditions designed to explore the oceanic equivalent of weather systems in the atmosphere. Because the scale of these highly energetic eddies is typically i00km, they can only be resolved by the largest 'eddy resolving' models of the general circulation (EGCMs). In models with lower spatial resolution, their effects are parameterized, often in the most crude way. Holland's chapter on 'Oceanic general circulation models' (44pp) devotes particular attention to progress that is being made in coping with this long term problem. Forecasting the weather in the ocean remains such a distant possibility that its operational implementation is not seriously contemplated. Without this prognostic treadmill, oceanographers have been able to concentrate greater attention on diagnostic studies. These are reviewed in valuable chapters by Sarkisyan ('The diagnostic calculation of large-scale oceanic circulation', 96 pp) and Veronis ('Use of tracers in circulation studies', 20 pp). It is surprising that these and all other authors in this book fail to grasp the nettle of pseudoisentropic trajectories, preferring to work in classical pressure or depth coordinates. The pioneering work of Parr and Montgomery in the 1930s, and of their tiny band of disciples, has still not penetrated the world of computer modellers. The second part of the book is devoted to six chapters on geology or, more accurately, on sediment transport. The preoccupation here is with the flow in the bottom boundary layer of the sea, a subject in which the methods of modelling turbulent boundary layers, developed mainly by meteorologists art combined with empirical results concerning sediment particles. The chapters by Smith ('Modelling of sediment transport on continental shelves', 38 pp) and Owen ('Problems in the modelling of transport, erosion and deposition of cohesive sediments', 21 pp) illustrate the progress that is being made. The reader interested primarily in numerical methods will find the chapter by Taylor and Dyer on 'Theoretical models of flow near the bed and their implications for sediment transport' (21 pp), with its novel treatment of the undulating sea bed, particularly stimulating. The third part (comprising three chapters on chemistry) opens with Millero on 'Thermodynamic models for the state of metal ions in seawater' (40 pp), which is part of a larger paper presented at a Symposium on Chemistry of the Mediterranean Sea. This chapter concentrates on the chemical assumptions behind the numerous models reviewed, but does not give any indication of the numerical problems incurred in
constructing the models. It is therefore unlikely to be of interest to readers who are not involved in this branch of marine chemistry, but it will naturally be invaluable to a mathematician considering entering the field. The same can be said of Lerman's review of 'Migrational processes and chemical reactions in interstitial waters' (43 pp), which tackles the complexities of the top metre or so of the ocean sediments, proposing a model with two fluxes, one on deposited particles and one in the interstitial water. Moving to yet larger scales, we come to the third chemical paper 'Sedimentary cycling models of global processes' (46 pp) by MacKenzie and Wollast, who are concerned with box models which 'involve only simple mathematical formulation, but require the critical judgement of an experienced practicioner'. Their main concern is to present a summary of the sources, sinks and transport routes for a number of elements, with particular emphasis on mercury. The final part of the book, with six chapters devoted to biological modelling, follows a similar progression from small to large scale. The emphasis here is on incorporating results from controlled laboratory experiments into models based on empirical--usually speculative--equations. Dugdale starts with a short review of nutrient uptake (his chapter has the rather odd title 'Modeling', 18 pp). This leads naturally into the much longer (50 pp) chapter 'Modeling the productivity of phytoplankton' by Platt, Denman and Jassby, who in a stimulating preamble sound a useful warning note: 'We agree with Lucas (1964) that whatever the original motivation of the modeler, the model will eventually be used, by him or by somebody else, for making predictions (or even management decisions) about the system. This being the case, it is wise to proceed from the outset, whether we seek a model that is primarily descriptive or primarily explanatory, on the assumption that the final goal of our work is prediction. This is particularly important in fields, including phytoplankton productivity, that have an economic component.' This chapter, and the one that follows by Steele ('Zooplankton dynamics', 33 pp) are among the most thoughtful in the whole book, providing clear accounts for the non-expert reader as well as excellent reviews of a field that is progressing rapidly at present. Perhaps it is because biologists have had to rely more on words rather than mathematics in making their ideas clear; whatever the reason, this final part of the book contains more quotable remarks than the other three put together. Vinogradov and Menshatkin ('The modeling of open-sea ecosystems', 32 pp) make the important
remark that '... the application of mathematical modeling in oceanographic research has substantially modified our approach to the collection and treatment of samples and to the organization of fieldwork'. This, rattier than forecasting, is the principal legacy of the Richardson revolution for oceanographers. The book ends with a couple of case studies, on coastal upwelling (Walsh: 'A biological sketch-book for an eastern boundary current', 45 pp, replete with unreferenced quotations from T. S. Eliot) and on the San Francisco Bay estuary (Di Toro et al.: 'Estuarine phytoplankton biomass models-verification analyses and preliminary applications'), in which modelling and field observations were integrated. It is not easy to sum up the merits of a book with such a diversity of subject and style. The editors appear to have left the quality of their book in the hands of the authors. The decision to let one use the title 'Modeling' for his chapter seems rather careless, and the index is the worst I have ever tried to use. It has important omissions (try 'bacteria') and some inappropriate inclusions (FORTRAN). These failings on the part of the editors are unfair to the authors, who have provided the oceanographer with a mass of new results expertly reviewed and an up to date assessment of the impact of modelling in the major disciplines of marine research. The book is recommended, not only for those actively engaged in modelling, but for all oceanographers, because the requirements of computer models will increasingly influence all aspects of ocean exploration in the future.
J. Woods
Systems reliability modelling and evaluation C. Singh a n d R. Billinton H u t c h i n s o n Publishing G r o u p , London, 1977, £14.00 Designers of complex engineering and commercial systems are frequently concerned with the problem of system reliability and its estimation from analytical studies or by statistical procedures. This book is concerned with the qualitative evaluation of reliability by building mathematical models of the actual system, which can then be analysed or numerically manipulated to generate suitable reliability models. The central theme of this well-written book is reliability modelling and evaluation. Validation of models, data collection or evaluation of alternative reliability models are not the subject of this text. The main concept used by the authors is the frequency balancing, or more simply, frequency
Appl. Math. Modelling, 1978, Vol 2, June 139
and duration approach to reliability; where the duration of time spent in a particular state and the frequency of encountering that state are computed from fundamental probability theory. This method produces the same linear, differential or intero-differential equations as would be obtained via conventional transitional probability techniques. The advantage of this approach is demonstrated by the simplicity with which frequency, cycle time and mean duration (and hence mean time to failure etc.) can be computed. The book is almost a self-contained unity, beginning with an introduction to the appropriate areas of stochastic processes which include expectation, transform methods, special distributions, Markov chains (first passage times) etc. All of these topics can easily be found in a standard text on stochastic processes. This introductory section is like the remainder of the book--clearly written, brief in detail (almost to the point of superficiality) and of an introductory nature. The frequency and duration approach to reliability is developed very effectively in chapter three, to derive expressions for the mean cycle time and mean duration of a state for Markov chains. The emphasis is on frequency as expectation of the encounter rate of a state. The mean cycle rate is then obtained as the reciprocal of the mean frequency. An alternative approach is given, whereby expressions for the mean times between failure (MTBF) etc. are computed by renewal theory, and the frequency relationships are then deduced from MTBF. In chapter four maintained system reliability is considered, in which restorative action (repairs) is initiated immediately after component failure. Maintained systems are considered since reliability models and their validation is more complex than for non-maintained systems. Various system descriptions are used, including the state space approach, where the state space system describes the state of the components and the environment in which they operate. This approach involves enumerating all possible state and interstate probability transitions. The frequency technique is then used to compute frequency cycle times and mean duration. Both independent and dependent state conditions (through Bayes' rule) are considered. Complex systems can be decomposed into simpler subsystems and by successive application of the conditional probability rule, reliability measures of the overall system can be computed. An alternative to the state space approach is the block transfer or
140
network approach which can be used to give a shorter solution but gives neither any internal system insight nor is it applicable to dependent failures or when repairs are involved. The various techniques of network reduction used in systems theory for analysing series or parallel systems are derived. When the system is too complex for these decompositions the cut set or tie set methods are used. Considerable space in the book is devoted to these techniques and an algorithm, suitable for computer implementation, that determines minimum cut sets is derived. This part of the book reflects the authors' own research interests. Chapter five is devoted to the state space approach for large scale (dimensions) systems with dependent or repair modes and those which involve graded modes of operation. Transfer between states for transient behaviour as well as for components with fluctuating environment are considered. Some state reduction techniques-generally called mergability--are developed. State truncation techniques or neglecting those states whose contribution to reliability is insignificant are also investigated. Development of sequential truncation and its merits are discussed at length. The majority of reliability models assume that the forward and reverse times of components are exponentially distributed, i.e. constant interstate transition rate. Whilst the forward are likely to be exponentially distributed, the reverse are not--leading to nonMarkovian models. Some of the difficulties and techniques of reliability estimation for the non-Markovian process is discussed in chapter six, together with some applications. This chapter is one of the most useful in the book and should prove useful in reliability modelling of weapon systems, vertical transport systems and commercial projects such as banking. A concluding chapter on simulation techniques applicable to reliability measures is included. Since not all reliability problems can be solved analytically, simulation provides a flexible solution which can take into account factors which conventional modelling cannot. Unfortunately this chapter is extremely brief and of little use to the practising exponent of reliability or to the student; reference to a thorough and complete text on simulation techniques would be more appropriate.
C. J. Harris
Annals of systems research, Vol 5 Edited b y B. van Rootselaar A p u b l i c a t i o n of t h e N e t h e r l a n d s S o c i e t y for S y s t e m s R e s e a r c h ,
Appl. Math. Modelling, 1978, Vol 2, June
Martinus Nijhoff Social Sciences Division, Leiden, The Netherlands, 1976, 142 pp. Dfl 38 An integral part of the explosion of interest generated in systems theory during the last two decades, has been the founding of many journals devoted to elaborating particular aspects of the subject area. Most of these journals have become highly specialist in that the articles contained therein are written for the professional systems theorist, and this makes their content less accessible to other scientists. In short, systems theory is no longer completely dependent for its momentum on other disciplines, and it is developing a style and language quite unique from these areas such as engineering, biology and mathematics which fostered it. Historians of science would not find any of this very surprising for the hallmark of our established scientific disciplines is based on such specialization. Indeed, Thomas Kuhn j argues that a science is only reaching maturity when the specialist article and journal become the order of the day. The science then enters a phase in which the majority of its proponents work within the theoretical structure of 'paradigm' already established and this activity is called by Kuhn, 'normal science'. Systems theory is no exception to this historical cycle and the Annals of Systems Research is only one of the latest in a line of specialist journals which show how the subject area is developing and diffusing on a global scale. The Annals is the publication of The Netherland's Society for Systems Research and as such, its contributions show the emerging interest in this field in Holland in particular, and in western Europe in general. At first sight, it might appear difficult to attempt a coherent review of one volume of a particular journal, for the degree of diversity of its contributions might mitigate against any unified treatment. But if Kuhn is right and a paradigm is beginning to emerge in the field, then a consensus about the key ideas in systems theory must exist, and this will be evident in the articles. In any case, the rationale for systems theory is the existence of common modes of representation for such seemingly diverse systems as machines, animals, societies, methods of thinking and so on, and in the past development qof the subject area, such communalities have become central. In particular, the idea that every system has a structure which can be interpreted statically and a behaviour
1T. Kuhn 'The structure of scientific revolutions', Chicago, 1970
constructing the models. It is therefore unlikely to be of interest to readers who are not involved in this branch of marine chemistry, but it will naturally be invaluable to a mathematician considering entering the field. The same can be said of Lerman's review of 'Migrational processes and chemical reactions in interstitial waters' (43 pp), which tackles the complexities of the top metre or so of the ocean sediments, proposing a model with two fluxes, one on deposited particles and one in the interstitial water. Moving to yet larger scales, we come to the third chemical paper 'Sedimentary cycling models of global processes' (46 pp) by MacKenzie and Wollast, who are concerned with box models which 'involve only simple mathematical formulation, but require the critical judgement of an experienced practicioner'. Their main concern is to present a summary of the sources, sinks and transport routes for a number of elements, with particular emphasis on mercury. The final part of the book, with six chapters devoted to biological modelling, follows a similar progression from small to large scale. The emphasis here is on incorporating results from controlled laboratory experiments into models based on empirical--usually speculative--equations. Dugdale starts with a short review of nutrient uptake (his chapter has the rather odd title 'Modeling', 18 pp). This leads naturally into the much longer (50 pp) chapter 'Modeling the productivity of phytoplankton' by Platt, Denman and Jassby, who in a stimulating preamble sound a useful warning note: 'We agree with Lucas (1964) that whatever the original motivation of the modeler, the model will eventually be used, by him or by somebody else, for making predictions (or even management decisions) about the system. This being the case, it is wise to proceed from the outset, whether we seek a model that is primarily descriptive or primarily explanatory, on the assumption that the final goal of our work is prediction. This is particularly important in fields, including phytoplankton productivity, that have an economic component.' This chapter, and the one that follows by Steele ('Zooplankton dynamics', 33 pp) are among the most thoughtful in the whole book, providing clear accounts for the non-expert reader as well as excellent reviews of a field that is progressing rapidly at present. Perhaps it is because biologists have had to rely more on words rather than mathematics in making their ideas clear; whatever the reason, this final part of the book contains more quotable remarks than the other three put together. Vinogradov and Menshatkin ('The modeling of open-sea ecosystems', 32 pp) make the important
remark that '... the application of mathematical modeling in oceanographic research has substantially modified our approach to the collection and treatment of samples and to the organization of fieldwork'. This, rattier than forecasting, is the principal legacy of the Richardson revolution for oceanographers. The book ends with a couple of case studies, on coastal upwelling (Walsh: 'A biological sketch-book for an eastern boundary current', 45 pp, replete with unreferenced quotations from T. S. Eliot) and on the San Francisco Bay estuary (Di Toro et al.: 'Estuarine phytoplankton biomass models-verification analyses and preliminary applications'), in which modelling and field observations were integrated. It is not easy to sum up the merits of a book with such a diversity of subject and style. The editors appear to have left the quality of their book in the hands of the authors. The decision to let one use the title 'Modeling' for his chapter seems rather careless, and the index is the worst I have ever tried to use. It has important omissions (try 'bacteria') and some inappropriate inclusions (FORTRAN). These failings on the part of the editors are unfair to the authors, who have provided the oceanographer with a mass of new results expertly reviewed and an up to date assessment of the impact of modelling in the major disciplines of marine research. The book is recommended, not only for those actively engaged in modelling, but for all oceanographers, because the requirements of computer models will increasingly influence all aspects of ocean exploration in the future.
J. Woods
Systems reliability modelling and evaluation C. Singh a n d R. Billinton H u t c h i n s o n Publishing G r o u p , London, 1977, £14.00 Designers of complex engineering and commercial systems are frequently concerned with the problem of system reliability and its estimation from analytical studies or by statistical procedures. This book is concerned with the qualitative evaluation of reliability by building mathematical models of the actual system, which can then be analysed or numerically manipulated to generate suitable reliability models. The central theme of this well-written book is reliability modelling and evaluation. Validation of models, data collection or evaluation of alternative reliability models are not the subject of this text. The main concept used by the authors is the frequency balancing, or more simply, frequency
Appl. Math. Modelling, 1978, Vol 2, June 139
and duration approach to reliability; where the duration of time spent in a particular state and the frequency of encountering that state are computed from fundamental probability theory. This method produces the same linear, differential or intero-differential equations as would be obtained via conventional transitional probability techniques. The advantage of this approach is demonstrated by the simplicity with which frequency, cycle time and mean duration (and hence mean time to failure etc.) can be computed. The book is almost a self-contained unity, beginning with an introduction to the appropriate areas of stochastic processes which include expectation, transform methods, special distributions, Markov chains (first passage times) etc. All of these topics can easily be found in a standard text on stochastic processes. This introductory section is like the remainder of the book--clearly written, brief in detail (almost to the point of superficiality) and of an introductory nature. The frequency and duration approach to reliability is developed very effectively in chapter three, to derive expressions for the mean cycle time and mean duration of a state for Markov chains. The emphasis is on frequency as expectation of the encounter rate of a state. The mean cycle rate is then obtained as the reciprocal of the mean frequency. An alternative approach is given, whereby expressions for the mean times between failure (MTBF) etc. are computed by renewal theory, and the frequency relationships are then deduced from MTBF. In chapter four maintained system reliability is considered, in which restorative action (repairs) is initiated immediately after component failure. Maintained systems are considered since reliability models and their validation is more complex than for non-maintained systems. Various system descriptions are used, including the state space approach, where the state space system describes the state of the components and the environment in which they operate. This approach involves enumerating all possible state and interstate probability transitions. The frequency technique is then used to compute frequency cycle times and mean duration. Both independent and dependent state conditions (through Bayes' rule) are considered. Complex systems can be decomposed into simpler subsystems and by successive application of the conditional probability rule, reliability measures of the overall system can be computed. An alternative to the state space approach is the block transfer or
140
network approach which can be used to give a shorter solution but gives neither any internal system insight nor is it applicable to dependent failures or when repairs are involved. The various techniques of network reduction used in systems theory for analysing series or parallel systems are derived. When the system is too complex for these decompositions the cut set or tie set methods are used. Considerable space in the book is devoted to these techniques and an algorithm, suitable for computer implementation, that determines minimum cut sets is derived. This part of the book reflects the authors' own research interests. Chapter five is devoted to the state space approach for large scale (dimensions) systems with dependent or repair modes and those which involve graded modes of operation. Transfer between states for transient behaviour as well as for components with fluctuating environment are considered. Some state reduction techniques-generally called mergability--are developed. State truncation techniques or neglecting those states whose contribution to reliability is insignificant are also investigated. Development of sequential truncation and its merits are discussed at length. The majority of reliability models assume that the forward and reverse times of components are exponentially distributed, i.e. constant interstate transition rate. Whilst the forward are likely to be exponentially distributed, the reverse are not--leading to nonMarkovian models. Some of the difficulties and techniques of reliability estimation for the non-Markovian process is discussed in chapter six, together with some applications. This chapter is one of the most useful in the book and should prove useful in reliability modelling of weapon systems, vertical transport systems and commercial projects such as banking. A concluding chapter on simulation techniques applicable to reliability measures is included. Since not all reliability problems can be solved analytically, simulation provides a flexible solution which can take into account factors which conventional modelling cannot. Unfortunately this chapter is extremely brief and of little use to the practising exponent of reliability or to the student; reference to a thorough and complete text on simulation techniques would be more appropriate.
C. J. Harris
Annals of systems research, Vol 5 Edited b y B. van Rootselaar A p u b l i c a t i o n of t h e N e t h e r l a n d s S o c i e t y for S y s t e m s R e s e a r c h ,
Appl. Math. Modelling, 1978, Vol 2, June
Martinus Nijhoff Social Sciences Division, Leiden, The Netherlands, 1976, 142 pp. Dfl 38 An integral part of the explosion of interest generated in systems theory during the last two decades, has been the founding of many journals devoted to elaborating particular aspects of the subject area. Most of these journals have become highly specialist in that the articles contained therein are written for the professional systems theorist, and this makes their content less accessible to other scientists. In short, systems theory is no longer completely dependent for its momentum on other disciplines, and it is developing a style and language quite unique from these areas such as engineering, biology and mathematics which fostered it. Historians of science would not find any of this very surprising for the hallmark of our established scientific disciplines is based on such specialization. Indeed, Thomas Kuhn j argues that a science is only reaching maturity when the specialist article and journal become the order of the day. The science then enters a phase in which the majority of its proponents work within the theoretical structure of 'paradigm' already established and this activity is called by Kuhn, 'normal science'. Systems theory is no exception to this historical cycle and the Annals of Systems Research is only one of the latest in a line of specialist journals which show how the subject area is developing and diffusing on a global scale. The Annals is the publication of The Netherland's Society for Systems Research and as such, its contributions show the emerging interest in this field in Holland in particular, and in western Europe in general. At first sight, it might appear difficult to attempt a coherent review of one volume of a particular journal, for the degree of diversity of its contributions might mitigate against any unified treatment. But if Kuhn is right and a paradigm is beginning to emerge in the field, then a consensus about the key ideas in systems theory must exist, and this will be evident in the articles. In any case, the rationale for systems theory is the existence of common modes of representation for such seemingly diverse systems as machines, animals, societies, methods of thinking and so on, and in the past development qof the subject area, such communalities have become central. In particular, the idea that every system has a structure which can be interpreted statically and a behaviour
1T. Kuhn 'The structure of scientific revolutions', Chicago, 1970