Dependability for Systems With a Partitioned State Space Markov and SemiMarkov Theory and Computational Implementation
 List Price: $109.00
 Binding: Paperback
 Publisher: Springer Verlag
 Publish date: 08/01/1994
Description:
1 Stochastic processes for dependability assessment. 1.1 Markov and semiMarkov processes for dependability assessment. 1.2 Example systems. 1.2.1 Markov models. 1.2.2 SemiMarkov models. 2 Sojourn times for discreteparameter Markov chains. 2.1 Distribution theory for sojourn times and related variables. 2.1.1 Key results: sojourn times in a subset of the state space. 2.1.2 Distribution theory for variables related to the sojourn time vector. 2.1.3 The joint distribution of sojourn times in A1 and A2 by the generalised renewal argument. 2.1.4 Tabular summary of results about sojourn times and related variables. 2.2 An application: the sequence of repair events for a threeunit power transmission model. 2.2.1 The number of major repair events Rn in a repair sequence of lenght n. 2.2.2 Numerical results. 2.2.3 Implementation with MATLAB. 2.2.4 MATLAB code. 3 The number of visits until absorption to subsets of the state space by a discreteparameter Markov chain: the multivariate case. 3.1 The probability generating function of M and the probability mass function of L. 3.2 Further results for n ? {2, 3}. 3.3 Tabular summary of results in Sections 3.1 and 3.2. 3.4 A power transmission reliabilty application. 3.4.1 Numerical results. 3.4.2 MATLAB code. 4 Sojourn times for continuousparameter Markov chains. 4.1 Distribution theory for sojourn times. 4.2 Some further distribution results related to sojourn times. 4.3 Tabular summary of results in Sections 4.1 and 4.2. 4.4 An application: further dependability characteristics of the threeunit power transmission model. 4.4.1 Numerical results. 4.4.2 MATLAB code. 5 The number of visits to a subset of the state space by a continuousparameter irreducible Markov chain during a finite time interval. 5.1 The variable $${M_{{A_1}}}(t)$$. 5.1.1 The main result. 5.1.2 The proof of Theorem 5.1. 5.2 An application: the number of repairs of a twounit power transmission system during a finite time interval. 5.2.1 Numerical results and implementation issues. 5.2.2 MATLAB code. 6 A compound measure of dependability for continuoustime Markov models of repairable systems. 6.1 The dependability measure and its evaluation by randomization. 6.2 The evaluation of ?(k, i, n). 6.2.1 The auxiliary absorbing Markov chain X(k). 6.2.2 The closed form expression for ?(k, i, n). 6.3 Application and computational experience. 6.3.1 Computational implementation. 6.3.2 Application: two parallel units with a single repairman. 6.3.3 Implementation in MATLAB. 6.3.4 MATLAB code. 7 A compound measure of dependability for continuoustime absorbing Markov systems. 7.1 The dependability measure. 7.2 Proof of Theorem 7.1. 7.2.1 Proof outline. 7.2.2 An auxiliary result. 7.2.3 Proof details. 7.3 Application: the Markov model of the threeunit power transmission system revisited. 8 Sojourn times for finite semiMarkov processes. 8.1 A recurrence relation for the Laplace transform of the vector of sojourn times. 8.2 Laplace transforms of vectors of sojourn times. 8.2.1 S is partitioned into three subsets (n = 2). 8.2.2 S is partitioned into four subsets (n = 3). 8.3 Proof of Theorem 8.1. 9 The number of visits to a subset of the state space by an irreducible semiMarkov process during a finite time interval: moment results. 9.1 Preliminaries on the moments of $${M_{{A_1}}}(t)$$. 9.2 Main result: the Laplace transform of the measures U'. 9.3 Proof of Theorem 9.2. 9.4 Reliability applications. 9.4.1 The alternating renewal process. 9.4.2 Two units in parallel with an arbitrary change out time distribution. 10 The number of visits to a subset of the state space by an irreducibe semiMarkov process during a finite time interval: the probability mass function. 10.1 The Laplace transform of the probability mass function of $${M_{{A_1}}}(t)$$. 10.1.1 A recurrence relation in the Laplace transform domain. 10.1.2 The direct computation of Laplace transforms. 10.2 Numerical inversion of Laplace transforms using Laguerre polynomials and fast Fourier transform. 10.2.1 A summary of the numerical Laplace transform inversion scheme. 10.2.2 The inversion scheme in the NAG implementation. 10.3 Reliability applications. 10.3.1 The Markov model of the twounit power transmission system revisited. 10.3.2 The twounit semiMarkov model revisited. 10.4 Implementation issues. 10.4.1 The NAG library. 10.4.2 Input data file and FORTRAN 77 code for the Markov model. 10.4.3 MATLAB implementation of the Laplace transform inversion algorithm. 10.4.4 MATLAB code. 11 The number of specific service levels of a repairable semiMarkov system during a finite time interval: joint distributions. 11.1 A recurrence relation for h(t; m1, m2) in the Laplace transform domain. 11.2 A computation scheme for the Laplace transforms. 12 Finite timehorizon sojourn times for finite semiMarkov processes. 12.1 The double Laplace transform of finitehorizon sojourn times. 12.2 An application: the alternating renewal process. 12.2.1 Laplace transforms. 12.2.2 Symbolic inversion with MAPLE and computational experience. 12.2.3 MAPLE code. Postscript. References.
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