## Elements of the Theory of Markov Processes and Their ApplicationsGraduate-level text and reference in probability, with numerous scientific applications. Nonmeasure-theoretic introduction to theory of Markov processes and to mathematical models based on the theory. Appendixes. Bibliographies. 1960 edition. |

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Page 49

It can also be shown that the

moments of the nth generation and those of the first generation. If we let M = (m!')

denote the first-moment matrix, then differentiation of (1.122) with respect to s ...

It can also be shown that the

**relation**m?' = [m!')" (1.1.24) exists between the firstmoments of the nth generation and those of the first generation. If we let M = (m!')

denote the first-moment matrix, then differentiation of (1.122) with respect to s ...

Page 205

This

McKendrick Threshold Theorem): Let a:(0) = aco and y(0) = yo represent the

numbers of susceptible and infected individuals in the population at the start of

the ...

This

**relation**is expressed by the following theorem. Theorem 4.2 (Kermack-McKendrick Threshold Theorem): Let a:(0) = aco and y(0) = yo represent the

numbers of susceptible and infected individuals in the population at the start of

the ...

Page 302

From the recurrence

from (6.27) and (6.28) application of the Laplace transformation yields p(s) = ?(M(

t)} = iro (6.34) In order to obtain the asymptotic

...

From the recurrence

**relation**(6.25) we obtain p(s) = poss)[p(s)]** (6.33) Hence,from (6.27) and (6.28) application of the Laplace transformation yields p(s) = ?(M(

t)} = iro (6.34) In order to obtain the asymptotic

**relation**that M(t) satisfies, we must...

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### Contents

Introduction | 1 |

Processes Discrete in Space and Continuous in Time | 57 |

Processes Continuous in Space and Time | 129 |

Copyright | |

9 other sections not shown

### Other editions - View all

Elements of the Theory of Markov Processes and Their Applications A. T. Bharucha-Reid Limited preview - 2012 |

Elements of the Theory of Markov Processes and Their Applications Albert T. Bharucha-Reid Limited preview - 1997 |

### Common terms and phrases

absorber addition applications approach arrival associated assume assumptions becomes birth boundary branching processes called cascade coefficients collision concerned condition consider constant continuous counter death defined denote density derive described determined developed differential equation diffusion discussion distribution function electron energy epidemic equal exists expected expression finite fluctuation given gives growth Hence independent individuals initial condition integral interest interval introduce Kolmogorov equations Laplace transform length limit machine Markov Markov chain Markov processes Math mathematical mean method moments necessary nucleon obtain particle particular photon Poisson population positive primary problem Proof properties queueing radiation random variable reaction refer relation represent respectively satisfies shown simple ſº solution Statist Stochastic Processes Theorem theory tion transition probabilities zero