## Dependability for Systems with a Partitioned State Space: Markov and Semi-Markov Theory and Computational ImplementationProbabilistic models of technical systems are studied here whose finite state space is partitioned into two or more subsets. The systems considered are such that each of those subsets of the state space will correspond to a certain performance level of the system. The crudest approach differentiates between 'working' and 'failed' system states only. Another, more sophisticated, approach will differentiate between the various levels of redundancy provided by the system. The dependability characteristics examined here are random variables associated with the state space's partitioned structure; some typical ones are as follows • The sequence of the lengths of the system's working periods; • The sequences of the times spent by the system at the various performance levels; • The cumulative time spent by the system in the set of working states during the first m working periods; • The total cumulative 'up' time of the system until final breakdown; • The number of repair events during a fmite time interval; • The number of repair events until final system breakdown; • Any combination of the above. These dependability characteristics will be discussed within the Markov and semi-Markov frameworks. |

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### Contents

Stochastic processes for dependability assessment | 1 |

Sojourn times for discreteparameter Markov chains | 14 |

The number of visits until absorption to subsets of the state space by | 53 |

Copyright | |

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AA,A AA,Al AA2A AA2Ai aA2T AAlA2 AAlAl absorbing Markov chain AGB ABB AGB ABB"1 abg Apple Macintosh assumed Chapter closed form expression column vector component computation continuous-time Corollary 2.6 cumulative distribution function defined dti dt2 equations Figure finite fix(clock follows given IFAIL initial probability vector input('Enter irreducible joint distribution Laguerre polynomials lamt Laplace transform domain Laplace transform inversion Lemma Macintosh SE/30 MAi(t MAl(t Markov chain Markov model Markov process MATLAB MATLAB implementation matrix exponential MB(to notation number of visits O.OOOODO O.OOOODO obtained parameter partitioned probability generating function probability mass function PROOF OF COROLLARY proof of Theorem Qa,a QA2a QA2Ai random recurrence relation reliability renewal argument repair events right hand side Section semi-Markov processes sojourn space subroutine subsets Table TEMP2 TG(to Theorem 2.1 theory transition probability matrix transition rate matrix values variable WRITE(6 WRITEdO