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Order statistics, Poisson processes and repairable systems

Published online by Cambridge University Press:  14 July 2016

Douglas R. Miller
Douglas R. Miller*
Affiliation:
University of Missouri - Columbia
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Abstract

Necessary and sufficient conditions are presented under which the point processes equivalent to order statistics of n i.i.d. random variables or superpositions of n i.i.d. renewal processes converge to a non-degenerate limiting process as n approaches infinity. The limiting process must be one of three types of non-homogeneous Poisson process, one of which is the Weibull process. These point processes occur as failure-time models in the reliability theory of repairable systems.

Keywords

ORDER STATISTICSSUPERPOSITION OF RENEWAL PROCESSESPOINT PROCESSNON-HOMOGENEOUS POISSON PROCESSWEIBULL PROCESSCONVERGENCE OF PROCESSEXTREME-VALUE DISTRIBUTIONRELIABILITY THEORY
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 3 , September 1976 , pp. 519 - 529
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Ascher, H. E. (1968) Evaluation of repairable system reliability using the ‘bad-as-old’ concept. IEEE Trans. Reliability R-17, 103–110.Google Scholar
[2] Barlow, R. E. and Hunter, L. C. (1960) Optimum preventive maintenance policies. Operat. Res. 9, 90–100.Google Scholar
[3] Barlow, R. E. and Proschan, F. (1975a) Statistical Theory of Reliability and Life Testing, Vol. I: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
[4] Barlow, R. E. and Proschan, F. (1975b) Theory of maintained systems: distribution of time to first system failure. FSU Statistics Report No. M.330.Google Scholar
[5] Bassin, W. M. (1973) A Bayesian optimal overhaul interval model for Weibull restoration process. J. Amer. Statist. Assoc. 68, 575–578.Google Scholar
[6] Brown, M. (1975) The first passage time distribution for a parallel exponential system with repair. In Reliability and Fault Tree Analysis, ed. Barlow, R. E., Fussel, J. B. and Singpurwalla, N. D. SIAM, Philadelphia.Google Scholar
[7] Çinlar, E. (1971) Superposition of point processes. In Stochastic Point Processes: Statistical Analysis, Theory and Applications, ed. Lewis, P. A. W. Wiley, New York, 549–606.Google Scholar
[8] Crow, L. H. (1974) Reliability analysis for complex repairable systems. In Reliability and Biometry, ed. Proschan, F. and Serfling, R. J. SIAM, Philadelphia.Google Scholar
[9] Daley, D. J. and Vere-Jones, D. (1971) A summary of the theory of point processes. In Stochastic Point Processes: Statistical Analysis, Theory and Applications, ed. Lewis, P. A. W. Wiley, New York, 299–383.Google Scholar
[10] Dehaan, L. (1970) On regular variation and its applications to the weak convergence of sample extremes. Mathematical Centre Tracts, Amsterdam.Google Scholar
[11] Dwass, M. (1964) Extremal processes. Ann. Math. Statist. 35, 1718–1725.CrossRefGoogle Scholar
[12] Gnedenko, B. V. (1943) Sur la distribution limitée du terme maximum d'une serie aléatoire. Ann. Math. 44, 423–453.CrossRefGoogle Scholar
[13] Grigelionis, B. (1963) On the convergence of sums of random step processes to Poisson processes. Theor. Prob. Appl. 8, 177–182.CrossRefGoogle Scholar
[14] Karlin, S. (1966) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
[15] Lamperti, J. (1964) On extreme order statistics. Ann. Math. Statist. 35, 1726–1737.Google Scholar
[16] Ross, S. M. (1974a) Multicomponent reliability systems. ORC 74–4, Operations Research Center, Berkeley.Google Scholar
[17] Ross, S. M. (1974b) On the time to first failure in multicomponent exponential reliability systems. ORC 74–8, Operations Research Center, Berkeley.Google Scholar
[18] Smirnoff, N. V. (1949) Limit distributions for the terms of a variational series (English Translation; Amer. Math. Soc. Transl. (1), 67, (1952)).Google Scholar
[19] Thompson, W. A. (1969) Applied Probability. Holt, Rinehart and Winston, New York.Google Scholar
[20] Whitt, W. (1973) Representation and convergence of point processes on the line. Yale University Technical Report.Google Scholar