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A differential equation for a class of discrete lifetime distributions with an application in reliability. A demonstration of the utility of computer algebra. (English) Zbl 1337.60224

Summary: It is shown that the probability generating function of a lifetime random variable \(T\) on a finite lattice with polynomial failure rate satisfies a certain differential equation. The interrelationship with Markov chain theory is highlighted. The differential equation gives rise to a system of differential equations which, when inverted, can be used in the limit to express the polynomial coefficients in terms of the factorial moments of \(T\). This can then be used to estimate the polynomial coefficients. Some special cases are worked through symbolically using computer algebra. A simulation study is used to validate the approach and to explore its potential in the reliability context.

MSC:

60K10 Applications of renewal theory (reliability, demand theory, etc.)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J22 Computational methods in Markov chains
90B25 Reliability, availability, maintenance, inspection in operations research
62M05 Markov processes: estimation; hidden Markov models
65C40 Numerical analysis or methods applied to Markov chains
65C50 Other computational problems in probability (MSC2010)

Software:

MACSYMA; Maxima
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Full Text: DOI

References:

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