Computer-Oriented Formulation of Transition-Rate Matrices via Kronecker Algebra

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Abstract
This paper formulates the differential equations typical of a Markov problem in system-reliability theory in a systematic way in order to generate computer-oriented procedures. The coefficient matrix of these equations (the transition rate matrix) can be obtained for the whole system through algebraic operations on component transition-rate matrices. Such algebraic operations are performed according to the rules of Kronecker Algebra. We consider system reliability and availability with stress dependence and maintenance policies. Theorems are given for constructing the system matrix in four cases: * Reliability and availability with on-line multiple or single maintenance. * Reliability and availability with system-state dependent failure rates. * Reliabilityand availability with standby components. * Off-line maintainability. The results are expres § ed in algebraic terms and as a consequence their implementation by a computer program is straightforward. We also obtain information about the structure of the matrices involved. Such information can considerably improve computational efficiency of the computer codes because it allows introducing special ideas and techniques developed for large-system analysis such as sparsity, decomposition, and tearing.

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