## Computational ProbabilityWinfried K. Grassmann Springer Science & Business Media, 2013 M03 14 - 490 pages Great advances have been made in recent years in the field of computational probability. In particular, the state of the art - as it relates to queuing systems, stochastic Petri-nets and systems dealing with reliability - has benefited significantly from these advances. The objective of this book is to make these topics accessible to researchers, graduate students, and practitioners. Great care was taken to make the exposition as clear as possible. Every line in the book has been evaluated, and changes have been made whenever it was felt that the initial exposition was not clear enough for the intended readership. The work of major research scholars in this field comprises the individual chapters of Computational Probability. The first chapter describes, in nonmathematical terms, the challenges in computational probability. Chapter 2 describes the methodologies available for obtaining the transition matrices for Markov chains, with particular emphasis on stochastic Petri-nets. Chapter 3 discusses how to find transient probabilities and transient rewards for these Markov chains. The next two chapters indicate how to find steady-state probabilities for Markov chains with a finite number of states. Both direct and iterative methods are described in Chapter 4. Details of these methods are given in Chapter 5. Chapters 6 and 7 deal with infinite-state Markov chains, which occur frequently in queueing, because there are times one does not want to set a bound for all queues. Chapter 8 deals with transforms, in particular Laplace transforms. The work of Ward Whitt and his collaborators, who have recently developed a number of numerical methods for Laplace transform inversions, is emphasized in this chapter. Finally, if one wants to optimize a system, one way to do the optimization is through Markov decision making, described in Chapter 9. Markov modeling has found applications in many areas, three of which are described in detail: Chapter 10 analyzes discrete-time queues, Chapter 11 describes networks of queues, and Chapter 12 deals with reliability theory. |

### Contents

11 | |

42 | |

Numerical Methods for Computing Stationary Distributions of Finite Irreducible | 81 |

Stochastic Automata Networks 113 | 112 |

Matrix Analytic Methods | 153 |

Use of Characteristic Roots for Solving Infinite State Markov Chains 205 | 204 |

An Introduction to Numerical Transform Inversion and Its Application to Prob | 257 |

On Numerical Computations of Some DiscreteTime Queues | 365 |

The Product Form Tool for Queueing Networks | 409 |

Techniques for System Dependability Evaluation 445 | 444 |

Index | 481 |

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### Common terms and phrases

A₁ Abate and Whitt algorithm analysis analytic apply approach approximation arrival automaton balance equations block bounds calculate Chaudhry Choudhury Ciardo closed unit disk consider continuous-time convergence corresponding CTMC defined denote dependability diagonal discrete discrete-time eigenvalues elements Euler summation evaluated example exponential factor finite G/M/1 type Markov Gail given Grassmann Hence IEEE integral interval irreducible Laplace transform linear Markov chain Markovian multiple N₁ Neuts normalization constant numerical inversion Numerical Solution obtained open unit disk Operations Research optimal parameters Performance Petri nets positive recurrent probability vector problem product form queueing networks Queueing Systems Queueing Theory reward roots Rouché's Theorem scalar Section server solving Souza e Silva space station balance stationary policy steady-state Stewart stochastic Petri stochastic process synchronizing events techniques tensor product Theorem transient transition matrix transition probabilities Trivedi type Markov chain unit circle zeros