On the Foundations of Reliability

Under the framework of a stochastic point process of failures, this paper discusses basic ways to characterize reliability. The distinction between the failure rate of a process, useful for repairable systems, and the failure rate of a distribution, useful for nonrepairable systems, is drawn. The paper then concentrates on modeling the wearout characteristics of repairable system reliability. Neither the homogeneous Poisson nor the renewal processes will suffice for this purpose. The nonhomogeneous Poisson process is appealing as a general wearout model, but it too has nonintuitive features; for example, the distribution of first failure determines the entire process. This leads us to search for other alternatives and to consider the reliability characteristics of general point processes of failures. Results pertaining to the appealing alternative of superimposed processes are reviewed.

[1]  H. B. Mann.  Nonparametric Tests Against Trend , 1945 .

[2]  Saul Blumenthal,et al.  The Transient Reliability Behavior of Series Systems or Superimposed Renewal Processes , 1973 .

[3]  S. Blumenthal,et al.  Superimposed non-stationary renewal processes , 1971, Journal of Applied Probability.

[4]  L. H. Crow.  Confidence Interval Procedures for Reliability Growth Analysis , 1977 .

[5]  W. A. Thompson.  Applied probability , 1969 .

[6]  M. R. Leadbetter.  On basic results of point process theory , 1972 .

[7]  Werner Fieger.  Eine für beliebige Call-Prozesse geltende Verallgemeinerung der Palmschen Formeln. , 1965 .

[8]  Boris Vladimirovič Gnedenko,et al.  Mathematical methods in the reliability theory , 1969 .

[9]  G. Enderlein,et al.  Chiang, Ch. L.: Introduction to Stochastic Processes in Biostatistics. Wiley, & Sons, Inc., New York, London, Sydney 1968. 301 S., 1 Abb., 25 Tab., Preis 131 s , 1970 .

[10]  Marek Fisz,et al.  Characterization of Sample Functions of Stochastic Processes by Some Absolute Probabilities , 1961 .

[11]  Erhan Çinlar,et al.  Introduction to stochastic processes , 1974 .

[12]  Michel Loève,et al.  Probability Theory I , 1977 .

[13]  G. A. Barnard,et al.  Time Intervals between accidents—a note on Maguire, Pearson and Wynn's paper , 1953 .

[14]  Emanuel Parzen,et al.  Stochastic Processes , 1962 .

[15]  A. Mintz,et al.  Time intervals between accidents. , 1954 .

[16]  Z. A. Lomnicki.  Some Aspects of the Statistical Approach to Reliability , 1973 .

[17]  J. Kalbfleisch.  Statistical Inference Under Order Restrictions , 1975 .

[18]  A. Ya. Khinchin.  On Poisson Sequences of Chance Events , 1956 .

[19]  S. Blumenthal,et al.  A Comparison of the Bad as Old and Superimposed Renewal Models , 1976 .

[20]  D. Cox,et al.  The statistical analysis of series of events , 1966 .

[21]  M. H. Quenouille,et al.  Mathematical Methods in the Theory of Queueing , 1962, Mathematical Gazette.

[22]  H. Grimm.  Parzen,E.: Stochastic Prozesses. Holden-Day, Inc., San Francisco,London, Amster dam 1962 (2. Nachdruck, Mai 1964). XI + 324 S., Preis $ 11,95 , 1967 .

[23]  H. Ascher.  Evaluation of Repairable System Reliability Using the ``Bad-As-Old'' Concept , 1968 .

[24]  Richard E Barlow,et al.  Analysis of Time Between Failures for Repairable Components. , 1977 .

[25]  J. Doob.  Stochastic processes , 1953 .

[26]  R. Pyke,et al.  Tests for monotone failure rate , 1967 .

[27]  J. C. Burkill.  Functions of Intervals , 1924 .

[28]  C. L. Chiang,et al.  Introduction to stochastic processes in biostatistics. , 1968 .

[29]  R. Barlow,et al.  Optimum Preventive Maintenance Policies , 1960 .

[30]  H. Ascher.  Comments on "Models for Reliability of Repaired Equipment , 1979 .