Ebook: Dependability for Systems with a Partitioned State Space: Markov and Semi-Markov Theory and Computational Implementation

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.

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This work is concerned with the dependability characteristics of reliability models which have a partitioned state space. In applications these partitions will correspond to various degrees of system performance. Both discrete and continuous time systems are considered as well as Markov and semi-Markov systems. With these models one may develop many dependability characteristics including: the number of working periods during an interval, the number of repair periods until system breakdown, and the total time spent in the set of working states.
The author shows how the theory may be applied using numerous examples and with three computing packages: MATLAB, MAPLE, and the NAG Fortran 77 subroutine library. As a result, researchers and practioners concerned with analyzing and modelling the performance of systems, and engineers working on systems dependability will find much of interest here.

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This work is concerned with the dependability characteristics of reliability models which have a partitioned state space. In applications these partitions will correspond to various degrees of system performance. Both discrete and continuous time systems are considered as well as Markov and semi-Markov systems. With these models one may develop many dependability characteristics including: the number of working periods during an interval, the number of repair periods until system breakdown, and the total time spent in the set of working states.
The author shows how the theory may be applied using numerous examples and with three computing packages: MATLAB, MAPLE, and the NAG Fortran 77 subroutine library. As a result, researchers and practioners concerned with analyzing and modelling the performance of systems, and engineers working on systems dependability will find much of interest here.
Content:
Front Matter....Pages i-ix
Stochastic Processes for Dependability Assessment....Pages 1-13
Sojourn times for Discrete-Parameter Markov Chains....Pages 14-52
The Number of Visits Until Absorption to Subsets of the State Space by a Discrete-Parameter Markov Chain: the Multivariate Case....Pages 53-68
Sojourn Times for Continuous-Parameter Markov Chains....Pages 69-105
The Number of Visits to a Subset of the State Space by a Continuous-Parameter Irreducible Markov Chain During a Finite Time Interval....Pages 106-121
A Compound Measure of Dependability for Continuous-Time Markov Models of Repairable Systems....Pages 122-140
A Compound Measure of Dependability For Continuous- Time Absorbing Markov Systems....Pages 141-150
Sojourn Times for Finite Semi-Markov Processes....Pages 151-166
The Number of Visits to a Subset of the State Space by an Irreducible Semi-Markov Process During a Finite Time Interval: Moment Results....Pages 167-178
The Number of Visits to a Subset of the State Space by an Irreducible Semi-Markov Process during a Finite Time Interval: The Probability Mass Function....Pages 179-204
The Number of Specific Service Levels of a Repairable Semi-Markov System during a Finite Time Interval: Joint Distributions....Pages 205-211
Finite Time-Horizon Sojourn Times for Finite Semi-Markov Processes....Pages 212-230
Back Matter....Pages 235-239

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This work is concerned with the dependability characteristics of reliability models which have a partitioned state space. In applications these partitions will correspond to various degrees of system performance. Both discrete and continuous time systems are considered as well as Markov and semi-Markov systems. With these models one may develop many dependability characteristics including: the number of working periods during an interval, the number of repair periods until system breakdown, and the total time spent in the set of working states.
The author shows how the theory may be applied using numerous examples and with three computing packages: MATLAB, MAPLE, and the NAG Fortran 77 subroutine library. As a result, researchers and practioners concerned with analyzing and modelling the performance of systems, and engineers working on systems dependability will find much of interest here.
Content:
Front Matter....Pages i-ix
Stochastic Processes for Dependability Assessment....Pages 1-13
Sojourn times for Discrete-Parameter Markov Chains....Pages 14-52
The Number of Visits Until Absorption to Subsets of the State Space by a Discrete-Parameter Markov Chain: the Multivariate Case....Pages 53-68
Sojourn Times for Continuous-Parameter Markov Chains....Pages 69-105
The Number of Visits to a Subset of the State Space by a Continuous-Parameter Irreducible Markov Chain During a Finite Time Interval....Pages 106-121
A Compound Measure of Dependability for Continuous-Time Markov Models of Repairable Systems....Pages 122-140
A Compound Measure of Dependability For Continuous- Time Absorbing Markov Systems....Pages 141-150
Sojourn Times for Finite Semi-Markov Processes....Pages 151-166
The Number of Visits to a Subset of the State Space by an Irreducible Semi-Markov Process During a Finite Time Interval: Moment Results....Pages 167-178
The Number of Visits to a Subset of the State Space by an Irreducible Semi-Markov Process during a Finite Time Interval: The Probability Mass Function....Pages 179-204
The Number of Specific Service Levels of a Repairable Semi-Markov System during a Finite Time Interval: Joint Distributions....Pages 205-211
Finite Time-Horizon Sojourn Times for Finite Semi-Markov Processes....Pages 212-230
Back Matter....Pages 235-239
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