Optimum Markov replacement policies for using the most reliable components. (English) Zbl 0821.62066
Summary: Consider a system consisting of \(n\) components, where in case of any failure two different types of components of unknown reliability are available for an immediate replacement, to keep all \(n\) components working. Under the assumption of independent exponentially distributed failure times, replacement decision rules are considered for the situation where past experience cannot be utilized. First it is shown that once all of the components are of the same type, the other type should not be used any further. For the resulting Markov chains with two absorbing states, optimum replacement policies are derived, including ‘look-ahead-one-failure’ Bayes rules.
MSC:
| 62N05 | Reliability and life testing |
| 62F15 | Bayesian inference |
| 60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |
| 90B25 | Reliability, availability, maintenance, inspection in operations research |
Keywords:
look-ahead-one-failure Bayes rules; type selections for replacements; birth and death process; absorbing Markov chain; independent exponentially distributed failure times; optimum replacement policiesReferences:
| [1] | Alam, K., Selection from Poisson processes, Ann. Inst. Statist. Math., 23, 411-418 (1971) · Zbl 0255.62031 |
| [2] | Alam, K.; Thompson, J. R., A problem of ranking and estimation with Poisson processes, Technometrics, 15, 801-808 (1973) · Zbl 0269.62080 |
| [3] | Barlow, R. E.; Proschan, F., Statistical Theory of Reliability and Life Testing (1981), To Begin With, Silver Spring |
| [4] | Berger, J. O., Statistical Decision Theory and Bayesian Analysis (1985), Springer: Springer New York · Zbl 0572.62008 |
| [5] | Dixon, D. O.; Bland, R. P., A Bayes solution to the problem of ranking Poisson parameters, Ann. Inst. Statist. Math., 23, 119-124 (1971) · Zbl 0257.62020 |
| [6] | Goel, P. S., A note on the non-existence of subset selection procedures for Poisson populations, (Mimeo. Series, 303 (1972), Dept. of Statistics, Purdue Univ) |
| [7] | Gupta, S. S.; Huang, D.-Y., On subset selection procedures for Poisson populations and some applications to the multinomial selection problems, (Gupta, R. P., Applied Statistics (1975), North-Holland: North-Holland Amsterdam), 97-109 · Zbl 0302.62010 |
| [8] | Gupta, S. S.; Leong, Y.-K.; Wong, W.-Y., On subset selection procedures for Poisson populations, Bull. Malaysian Math. Soc., 2, 89-110 (1979) · Zbl 0444.62034 |
| [9] | Gupta, S. S.; Panchapakesan, S., Multiple Decision Procedures (1979), Wiley: Wiley New York · Zbl 0516.62030 |
| [10] | Gupta, S. S.; Wong, W.-Y., On subset selection procedures for Poisson processes and some applications to the binomial and multinomial problems, (Henn, R.; etal., Operations Research Verfahren, Vol. 23 (1977), Meisenheim am Glan: Meisenheim am Glan Anton Hain), 49-70 · Zbl 0404.62067 |
| [11] | Hochberg, Y.; Tamhane, A. C., Multiple Comparison Procedures (1987), Wiley: Wiley New York · Zbl 0731.62125 |
| [12] | Iosifescu, M., Finite Markov Processes and Their Applications (1980), Wiley: Wiley New York · Zbl 0436.60001 |
| [13] | Karlin, S.; Taylor, H. M., A First Course in Stochastic Processes (1975), Academic Press: Academic Press New York · Zbl 0315.60016 |
| [14] | Karlin, S.; Taylor, H. M., A Second Course in Stochastic Processes (1981), Academic Press: Academic Press New York · Zbl 0469.60001 |
| [15] | Miescke, K. J., Optimum replacement policies for using the most reliable components, J. Statist. Plann. Inference, 26, 267-276 (1990) · Zbl 0760.62092 |
| [16] | Taylor, H.; Karlin, S., An Introduction to Stochastic Modelling (1984), Academic Press: Academic Press New York |
| [17] | Wilson, J., Robust Bayesian selection of the best Poisson population, (Bofinger, E.; Dudewicz, E. J.; Lewis, G. J.; Mengersen, K., The Frontiers of Modern Statistical Inference Procedures, II (1992), American Sciences Press: American Sciences Press Columbus, OH), 289-302 · Zbl 0825.62388 |
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