## Proceedings of the International School of Physics "Enrico Fermi". |

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Page 32

The process J(t) is said to be semi-Markov. When all the transition rate functions r

\lk(x) are constants, i.e. independent of x, the process J(t) is Markov. The process

J(t) is then said to be a finite Markov

The process J(t) is said to be semi-Markov. When all the transition rate functions r

\lk(x) are constants, i.e. independent of x, the process J(t) is Markov. The process

J(t) is then said to be a finite Markov

**chain**in continuous time, and will be ...Page 33

For most

from any other state in a finite number of steps, i.e. for every pair to, n there is

some r such that = P[J(k + r) = n\J(k) = to] > 0. The

irreducible.

For most

**chains**of interest, a = (amn) is such that every state can be reachedfrom any other state in a finite number of steps, i.e. for every pair to, n there is

some r such that = P[J(k + r) = n\J(k) = to] > 0. The

**chain**is then said to beirreducible.

Page 34

For this reason the unique limiting distribution eT is called the steady-state

distribution of the

aperiodic transition probability matrix is said to be ergodic. Certain discrete-time

Markov ...

For this reason the unique limiting distribution eT is called the steady-state

distribution of the

**chain**, or the ergodic distribution of the**chain**. An irreducibleaperiodic transition probability matrix is said to be ergodic. Certain discrete-time

Markov ...

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### Contents

System Eeliabujty | 3 |

Statistical Theory of Eeliablitt | 8 |

Definitions and characterizations | 12 |

Copyright | |

39 other sections not shown

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algorithm approach associated assume assumption Bayesian boundary points chain coherent system complex conjugate prior consider correctness defined denote detected discussed edited equations equivalence class ergodic errors example exponential distribution failure rate Fault Tree Analysis function gamma given human reliability IEEE Trans IFEA implementation increasing independent input domain integration interval likelihood Markov Markov chain matrix mean method modules monotone month2 N. D. Singpurwalla number of failures number of system NUMITEMS observed obtained operational output parameters phase Poisson Poisson process possible predictive prior distribution probability problem procedure Proschan R. E. Barlow random variables reliability growth models reliability theory renewal theory repair requirements sample sect sequence Software Eng software reliability software reliability models specification Stat statistical stochastic stochastic process subsection system failure system reliability techniques theorem tion tt tt values vector zero