Dependability for Systems with a Partitioned State Space

Probabilistic 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.

1 Stochastic processes for dependability assessment.- 1.1 Markov and semi-Markov processes for dependability assessment.- 1.2 Example systems.- 1.2.1 Markov models.- 1.2.2 Semi-Markov models.- 2 Sojourn times for discrete-parameter 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 three-unit 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 discrete-parameter 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 continuous-parameter 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 three-unit 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 continuous-parameter 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 two-unit 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 continuous-time 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 continuous-time 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 three-unit power transmission system revisited.- 8 Sojourn times for finite semi-Markov 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 semi-Markov 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 semi-Markov 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 two-unit power transmission system revisited.- 10.3.2 The two-unit semi-Markov 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 semi-Markov 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 time-horizon sojourn times for finite semi-Markov processes.- 12.1 The double Laplace transform of finite-horizon 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.