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Stochastic modelling and analysis: A computational approach,
Insurance: Mathematics and Economics 6 (1987) 233-234 North-Holland
233
Book review Henk C. Thijms, Stochastic modelling and analysis: A computational approach, Wiley Series in Probability and Mathematical Statistics (Wiley, New York, 1985) pp. xii + 418, fi19.95. ‘Many real-world phenomena require the analysis of a system in a probabilistic rather than a deterministic setting. Stochastic models are becoming increasingly important for understanding or making a performance evaluation of complex systems in a broad spectrum of fields such as operations research, computer science, telecommunications and engineering.’ The author has carefully selected a set of topics that are interesting and applicable. The first chapter treats renewal processes with applications to inventoc,/production and reliability. Special attention is given to the Poisson process, and to renewal processes generated by Erlangian distributions and to renewal processes with a reward structure. I found it just remarkable how much the author squeezes out of the observation that a process under consideration. The application to a production/inventory control with variable production rate and service level constraints is far reaching and combines intuition with sound probabilistic analysis. The other building block of the theory of stochastic processes and its applications is formed by Markov chain theory, which is treated in the second chapter. A fine variety of simple illustrations is provided with emphasis on queueing theory. An intuitive transition from discrete to continuous time Markov chains then allows the treatment of birth-death processes and continuous time Markov chains with a reward structure. A set of examples from queueing, reliability and capacity theory illustrates the use of continuous time Markov chains. The second chapter ends with a discussion of Jackson-type results on stochastic networks and insensitivity. In a third chapter the author treats Markov decision processes and their applicatious. A scru_I__lsllllpit:
---_____, lc;llt;wal
__---__ puL;t;3s
0167-6687/87/$3.50
‘_ IS
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f._ m
.I.:nt:
.
.-
tinuous and revealing comparison is made of three different but related algorithms: the policy-iteration algorithm, the linear programming algorithm and the value-iteration algorithm; all three are illustrated with examples from queueing and from insurance. An extension to semi-Markov decision processes is provided and exemplified with queueing theory. The chapter closes with some tailormade algorithms for practical applications having a specific structure. The final chapter covers algorithms and approximations for queueing models. After treating some fundamental relations the s,ingle-server queue with full access of arrivals is investigated with special emphasis on a two-moment approximation for the waiting times, in particular the GI/G/l with phase-type or with deterministic arrivals. The author then turns to similar queGeing systems with limited access crf XI-&& again with a wealth of numerical results and approximations. Finally multi-server queueing systems are considered with Poisson arrivals and with finite or infinitely many seirvers, again using i.a. two-moment approximations based on M/M/c and M/D/c. Each chapter ends with a set of bibliographical notes followed by a selected set of references to the enormously extended literature, and by an impressive list of solid problems. There are four appendixes: The computation of probabilities and expectations by conditioning, useful probability distribution functions, some results from Laplace transform theory, and successive overrelaxation methods for solving linear equations. An author index followed by a subject index finishes the textbook. Among the many interesting features of this welcome book are the following: - a careful discussion of useful practical computational methods for solving stochastic models; - a wide variety of realistic examples illustrating the basic models and the associated solution methods; - the inclusion of interesting and thought-provoking problems referring to a wide variety of
0 1987, Elsevier Scaence Publishers B.V. (North-Hoiland)
Book review
234
practical application areas such as inventory/ production control, maintenance, reliability and queueing; - an impressive set of exercises requiring the student to write computer codes for the cakulation of a sohrtion to a raised problem; - an ample supply of thoroughIy tested algorithms for specific examples. As a theoreticaIIy inclined reader I was looking forward how the author would succeed in the objecGve stated in the preface: ’ . . .I have deliberately chosen to otit proofs that are not necessary for an understanding of the theory* but I have tried to give as many p +n +~*%&Iusefd in_ explanations as possiblw iv rauva~~ sights into the working of the theorv’. On a number of places the au’ihor conceals some deep and far-reaching theoretical results; for example, most of renewal theory is stated under the extra assumption that the occurrence time distribution is absolutely continuous. The more generaI statements with an arbitrary distribution hoId most of *he time, however* ?nd a specialization to the Iattice case wouId have provided a number of rest&s in regenerative theory that now look extraneous. (For a typicaI case, see p. 113.) Nobody is helped by the phrase on page 124 ‘invoking a famous insensitivity result from teletraffic theory for loss systems, we have that . . . ’ &r&&QGt @&
J&St_ f&P ra&
actll?ay
is_ sme
arguments become mathematically dangerous if made too intuitive, like the ‘loose definition’ of a continuous Markov chain on page 101; or the computation of the steady state distribution at time ao by conditioning upon the state at time 00 - 1, as on page 91. Sometimes the author avoids the introduction of new concepts needed on page x while he sees no harm to give them on page y > x. The useful classification of the states of a discrete time
Markov chain into transient and recurrent states is not given on pages 88 etc. but crops up on page 178. In the mean time the unicity of the steady state probabilities for the ergodic, aperiodic case rests upon assumptions containing the undefined concept of a regeneration state. It is hard to see what is actually gained this way, especially as the uniqueness property is vitally used on pages 94 and 112. The notion of irreducibility is postponed to page 134 but is implicit in the foregoing pages. The above examples are only meant to warn the mathematically inclined reader against oversimplifications. The attentive reader will benefit tremendously from the i&$&e and computational approach offered by the author. The book fills a gap in the exi&.ng l.ite&ure; it can be viewed as a companion of the algorithmically inclined book by M.F. Neuts: Matrix-Geometric Solutions in Stochastic Models-An Algorithmic Approach (The Johns Hopkins University Press, Baltimore, MD, 1981) or of the algebraically oriented book of J.J. Hunter: Mathematical Techniques of Lv’;h.- V-OL 2, Ciscrete The Models: Applied Probauiu~~, Techniques and Applications (Academic Press, New York, 1983). For readers interested in insurance mathematics there are a few good examples from their area of interest: renewal theory is used to derive asymptotic expressions for ruin and waiting time probabiities; discrete tumc l+$arkov &in theory is applied to a vehicle insurance problem; Markov decision theory is illustrated in deriving optimal no-claim limits for which the long-run average cost per year is minimal. The wealth of examples can easily be extended by suitable adaptation of the terminology in the examples from queueing theory or inventory control. Jozef L. Teugels Katholieke Universiteit Leuven
233
Book review Henk C. Thijms, Stochastic modelling and analysis: A computational approach, Wiley Series in Probability and Mathematical Statistics (Wiley, New York, 1985) pp. xii + 418, fi19.95. ‘Many real-world phenomena require the analysis of a system in a probabilistic rather than a deterministic setting. Stochastic models are becoming increasingly important for understanding or making a performance evaluation of complex systems in a broad spectrum of fields such as operations research, computer science, telecommunications and engineering.’ The author has carefully selected a set of topics that are interesting and applicable. The first chapter treats renewal processes with applications to inventoc,/production and reliability. Special attention is given to the Poisson process, and to renewal processes generated by Erlangian distributions and to renewal processes with a reward structure. I found it just remarkable how much the author squeezes out of the observation that a process under consideration. The application to a production/inventory control with variable production rate and service level constraints is far reaching and combines intuition with sound probabilistic analysis. The other building block of the theory of stochastic processes and its applications is formed by Markov chain theory, which is treated in the second chapter. A fine variety of simple illustrations is provided with emphasis on queueing theory. An intuitive transition from discrete to continuous time Markov chains then allows the treatment of birth-death processes and continuous time Markov chains with a reward structure. A set of examples from queueing, reliability and capacity theory illustrates the use of continuous time Markov chains. The second chapter ends with a discussion of Jackson-type results on stochastic networks and insensitivity. In a third chapter the author treats Markov decision processes and their applicatious. A scru_I__lsllllpit:
---_____, lc;llt;wal
__---__ puL;t;3s
0167-6687/87/$3.50
‘_ IS
___l-_ll_l txllcJalu~:u
f._ m
.I.:nt:
.
.-
tinuous and revealing comparison is made of three different but related algorithms: the policy-iteration algorithm, the linear programming algorithm and the value-iteration algorithm; all three are illustrated with examples from queueing and from insurance. An extension to semi-Markov decision processes is provided and exemplified with queueing theory. The chapter closes with some tailormade algorithms for practical applications having a specific structure. The final chapter covers algorithms and approximations for queueing models. After treating some fundamental relations the s,ingle-server queue with full access of arrivals is investigated with special emphasis on a two-moment approximation for the waiting times, in particular the GI/G/l with phase-type or with deterministic arrivals. The author then turns to similar queGeing systems with limited access crf XI-&& again with a wealth of numerical results and approximations. Finally multi-server queueing systems are considered with Poisson arrivals and with finite or infinitely many seirvers, again using i.a. two-moment approximations based on M/M/c and M/D/c. Each chapter ends with a set of bibliographical notes followed by a selected set of references to the enormously extended literature, and by an impressive list of solid problems. There are four appendixes: The computation of probabilities and expectations by conditioning, useful probability distribution functions, some results from Laplace transform theory, and successive overrelaxation methods for solving linear equations. An author index followed by a subject index finishes the textbook. Among the many interesting features of this welcome book are the following: - a careful discussion of useful practical computational methods for solving stochastic models; - a wide variety of realistic examples illustrating the basic models and the associated solution methods; - the inclusion of interesting and thought-provoking problems referring to a wide variety of
0 1987, Elsevier Scaence Publishers B.V. (North-Hoiland)
Book review
234
practical application areas such as inventory/ production control, maintenance, reliability and queueing; - an impressive set of exercises requiring the student to write computer codes for the cakulation of a sohrtion to a raised problem; - an ample supply of thoroughIy tested algorithms for specific examples. As a theoreticaIIy inclined reader I was looking forward how the author would succeed in the objecGve stated in the preface: ’ . . .I have deliberately chosen to otit proofs that are not necessary for an understanding of the theory* but I have tried to give as many p +n +~*%&Iusefd in_ explanations as possiblw iv rauva~~ sights into the working of the theorv’. On a number of places the au’ihor conceals some deep and far-reaching theoretical results; for example, most of renewal theory is stated under the extra assumption that the occurrence time distribution is absolutely continuous. The more generaI statements with an arbitrary distribution hoId most of *he time, however* ?nd a specialization to the Iattice case wouId have provided a number of rest&s in regenerative theory that now look extraneous. (For a typicaI case, see p. 113.) Nobody is helped by the phrase on page 124 ‘invoking a famous insensitivity result from teletraffic theory for loss systems, we have that . . . ’ &r&&QGt @&
J&St_ f&P ra&
actll?ay
is_ sme
arguments become mathematically dangerous if made too intuitive, like the ‘loose definition’ of a continuous Markov chain on page 101; or the computation of the steady state distribution at time ao by conditioning upon the state at time 00 - 1, as on page 91. Sometimes the author avoids the introduction of new concepts needed on page x while he sees no harm to give them on page y > x. The useful classification of the states of a discrete time
Markov chain into transient and recurrent states is not given on pages 88 etc. but crops up on page 178. In the mean time the unicity of the steady state probabilities for the ergodic, aperiodic case rests upon assumptions containing the undefined concept of a regeneration state. It is hard to see what is actually gained this way, especially as the uniqueness property is vitally used on pages 94 and 112. The notion of irreducibility is postponed to page 134 but is implicit in the foregoing pages. The above examples are only meant to warn the mathematically inclined reader against oversimplifications. The attentive reader will benefit tremendously from the i&$&e and computational approach offered by the author. The book fills a gap in the exi&.ng l.ite&ure; it can be viewed as a companion of the algorithmically inclined book by M.F. Neuts: Matrix-Geometric Solutions in Stochastic Models-An Algorithmic Approach (The Johns Hopkins University Press, Baltimore, MD, 1981) or of the algebraically oriented book of J.J. Hunter: Mathematical Techniques of Lv’;h.- V-OL 2, Ciscrete The Models: Applied Probauiu~~, Techniques and Applications (Academic Press, New York, 1983). For readers interested in insurance mathematics there are a few good examples from their area of interest: renewal theory is used to derive asymptotic expressions for ruin and waiting time probabiities; discrete tumc l+$arkov &in theory is applied to a vehicle insurance problem; Markov decision theory is illustrated in deriving optimal no-claim limits for which the long-run average cost per year is minimal. The wealth of examples can easily be extended by suitable adaptation of the terminology in the examples from queueing theory or inventory control. Jozef L. Teugels Katholieke Universiteit Leuven