## Matrix-geometric Solutions in Stochastic Models: An Algorithmic ApproachTopics include matrix-geometric invariant vectors, buffer models, queues in a random environment and more. |

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### Contents

Preface | 1 |

Probability Distributions of Phase Type | 41 |

QuasiBirthandDeaih Processes | 81 |

The GIPHl Queue and Related Models | 142 |

Buffer Models | 215 |

Queues in a Random Environment | 254 |

310 | |

329 | |

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### Common terms and phrases

absorption algorithmic analysis applications arrival process arrival rate behavior blocks buffer busy period classical clearly column components computation conditional probability consider corresponding defined deftned denote differential equations dimension eigenvalue elementary embedded Markov chain Erlang distributions evaluate example exponential servers exponentially distributed finite follows formula GI/PH/l queue given input queue interarrival invariant probability vector invariant vector irreducible iterative Laplace-Stieltjes transform left eigenvector left transitions Lemma M. F. Neuts Markov process Markov renewal process mathematical matrix Q matrix-geometric maximal eigenvalue methods minimal nonnegative solution nonnegative matrix nonsingular number of customers numerical solution obtained parameters partitioned PH-distribution phase type Poisson process positive recurrent probability distributions process Q Proof QBD process queueing models queueing theory random readily representation satisfies service rate service time distribution solving spectral radius stationary probability vector stochastic matrix structure theorem theory transition probability matrix Unit values waiting time distributions zero

### Popular passages

Page 321 - Queue with a General Class of Service-Time Distribution by the Method of Generating Functions,

Page 323 - Neuts, MF , Moment formulas for the Markov renewal branching process.

Page 321 - Gl/M/c queue with a different service rate for customers who need not wait — an algorithmic solution Cahiers Centre Etud.

Page 310 - Avi-Itzhak, B. and Heyman, DP, "Approximate Queueing Models for Multiprogramming Computer Systems," Technical Memorandum, MM-7K-1713-15, Bell Telephone Laboratories, 1971.

Page 323 - The second moments of the absorption times in the Markov renewal branching process./.