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 13
... given by = = q ( " } ( j ) = Σ q ( i ) p ! " ) i = 0 ( 1.12 ) Hence , given the q ( i ) and the n - step transition probabilities p ) , the q ( ) ( j ) can be calculated . A problem of special interest is the behavior of q ( n ) ( j ) ...
... given by = = q ( " } ( j ) = Σ q ( i ) p ! " ) i = 0 ( 1.12 ) Hence , given the q ( i ) and the n - step transition probabilities p ) , the q ( ) ( j ) can be calculated . A problem of special interest is the behavior of q ( n ) ( j ) ...
Page 80
... given by Ο 0 0 0 0 0 ... 0 -λ 2 0 00 ... A = 0 0 -21 21 0 0 0 0 0 -31 31 0 -- ... Hence the Kolmogorov equations for the birth process are given by dPis ( t ) dt = —λ¡P¿¿ ( t ) + λ ; -1P¿‚¿ - 1 ( t ) dP ( t ) dt = -λ¿P¿s ( t ) + λ¿Pi + 1 ...
... given by Ο 0 0 0 0 0 ... 0 -λ 2 0 00 ... A = 0 0 -21 21 0 0 0 0 0 -31 31 0 -- ... Hence the Kolmogorov equations for the birth process are given by dPis ( t ) dt = —λ¡P¿¿ ( t ) + λ ; -1P¿‚¿ - 1 ( t ) dP ( t ) dt = -λ¿P¿s ( t ) + λ¿Pi + 1 ...
Page 182
... given that there are x individuals in S1 at time t , is μ At + o ( At ) . με 3. The probability of a unit increase in the population size of S2 in the interval ( t , t + At ) , given that there are y individuals in S2 at time t , is 2 ...
... given that there are x individuals in S1 at time t , is μ At + o ( At ) . με 3. The probability of a unit increase in the population size of S2 in the interval ( t , t + At ) , given that there are y individuals in S2 at time t , is 2 ...
Contents
Introduction | 1 |
THEORY | 7 |
Processes Discrete in Space and Continuous in Time | 53 |
Copyright | |
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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 |
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absorber applications arrival assume assumptions asymptotic birth process birth-and-death process boundary branching processes cascade process cascade theory coefficient collision consider counter defined denote the number denote the probability derive determined deterministic differential equation diffusion equations diffusion processes distribution function E₁ electron electron-photon cascades energy epidemic equilibrium exists expression Feller finite functional equation given Hence initial condition integral equation interval 0,t ionization joins the queue Kendall Kolmogorov equations Laplace transform Laplace-Stieltjes transform Let the random limiting distribution Markov chain Markov processes Math matrix Mellin transform Monte Carlo methods nonnegative nucleon o(At obtain P₁ parameter particle photon Phys Poisson process population probability distribution Proc product density queueing process queueing system Ramakrishnan random variable random variable X(t recurrent repairman satisfies Statist stochastic model Stochastic Processes t₁ Takács Theorem tion transition probabilities x₁ zero