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Convolution of geometrics and a reliability problem. (English) Zbl 0947.62071
Summary: In single-shot expensive tests of a destructive nature, an inverse sampling scheme is often pursued in order to use the available resources efficiently. This is particularly relevant for evaluating reliabilities for systems that are subjected to test-analyze-and-fix programs at successive stages, which cause a change in the failure probabilities across different stages. This note presents an elementary derivation of the distribution of the number of failures under this construct. A numerical illustration is presented by means of a discrete reliability growth model used in the literature. A correspondence with the well-studied pure birth process is pointed out.

MSC:
62N05 Reliability and life testing
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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[1] Duane, J.T., Learning curve approach to reliability monitoring, IEEE trans. aerospace, 2, 563-566, (1964)
[2] Dubman, M.; Sherman, B., Estimation of parameters in a transient Markov chain arising in a reliability growth model, Ann. math. statist., 40, 1542-1556, (1969) · Zbl 0198.23303
[3] Feller, W., 1968. An Introduction to Probability Theory and its Applications, vol. 1, 3rd ed., Wiley, New York. · Zbl 0155.23101
[4] Johnson, N.L., Kotz, S., Balakrishnan, N., 1994. Continuous Univariate Distributions, vol. 1, 2nd ed., Wiley, New York. · Zbl 0811.62001
[5] Margolin, B.H., Winokur, H.S. Jr., 1967. Exact moments of the order statistics of the geometric distribution and their relation to inverse sampling and reliability of redundant systems. J. Amer. Statist. Assoc. 62, 915-925.
[6] Mathai, A., 1982. Storage capacity of a dam with gamma type inputs. Ann. Inst. Statist. Math. 34 (Part A), 591-597. · Zbl 0505.60099
[7] Sen, A., Bhattacharyya, G.K., 1993. A piecewise exponential model for reliability growth and associated inferences. In: Basu, A.P. (Ed.), Advances in Reliability. North-Holland, Amsterdam, pp. 331-355.
[8] Sen, A.; Fries, A., Estimation in a discrete reliability growth model under an inverse sampling scheme, Ann. inst. statist. math., 49, 211-229, (1997) · Zbl 0896.62105
[9] Taylor, H.M., Karlin, S., 1984. An Introduction to Stochastic Modeling. Academic Press, New York. · Zbl 0946.60002
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