Intensity functions for nonhomogeneous Poisson processes

When analysing reliability data from a repairable system e.g. a car, a fridge, etc, reliability specialists are interested in the full life history of a system so that they may identify problems that have affected its life and apply solutions, if possible, to other new systems so that these new systems do not suffer from the same problems. The basis of reliability analysis is to understand how systems are failing by modeling their life history so that prediction of the likely future reliability of systems is possible. This is the basis of reliability growth (development) testing where systems under development are tested to find latent design and manufacturing defects so that they can be eliminated from the full-scale production system. The simplest way of identifying structure in the system's life history is to plot the data. Three plots are commonly used: plotting failures on a time line; plotting the failure number against cumulative time, and plotting an estimate of the intensity, the rate of occurrence of failures (ROCOF) against cumulative time. These plots appear in Ascher and Feingold [31] and Crowder et al. [61] among others. These plots will help to identify structure such as an increase or decrease in the interfailure times of a system over time (more commonly known as reliability growth and reliability decay); evidence of cycles in the data—these may be short or long cycles; periods of high or low reliability (again related to usage); gaps in the time stream (due to change in the reporting mechanism, management, etc) and outliers (possible abnormal operating conditions). This review lists the intensity functions and their associated expected number of failures, which is the mean value parameter of the nonhomogeneous Poisson process (NHPP). The simplest and best known intensities, those based on mortality rates, those based on hazards of well-known probability distributions, the software related intensities, and those based on differential equations and extensions are discussed under their appropriate classification headings. Keywords: nonhomogeneous Poisson process; intensity functions; ROCOFs ; mortality rates; probability distributions; SRGMs ; classification

[1]  G. Härtler.  The nonhomogeneous Poisson process — A model for the reliability of complex repairable systems , 1989 .

[2]  Amrit L. Goel,et al.  Time-Dependent Error-Detection Rate Model for Software Reliability and Other Performance Measures , 1979, IEEE Transactions on Reliability.

[3]  Kenneth J. Ryan.  Some flexible families of intensities for non‐homogeneous Poisson process models and their Bayes inference , 2003 .

[4]  I. W. Burr.  Cumulative Frequency Functions , 1942 .

[5]  T. L. Landers,et al.  Robustness of a semi-parametric proportional intensity model , 1991 .

[6]  G. Pulcini,et al.  Bayes inference for a non-homogeneous Poisson process with power intensity law (reliability) , 1989 .

[7]  J. T. Duane.  Learning Curve Approach to Reliability Monitoring , 1964, IEEE Transactions on Aerospace.

[8]  F. T. Wright,et al.  Tests for positive jumps in the intensity of a Poisson process: a power study , 1990 .

[9]  Larry Lee.  Comparing Rates of Several Independent Weibull Processes , 1980 .

[10]  Bev Littlewood,et al.  Stochastic Reliability-Growth: A Model for Fault-Removal in Computer-Programs and Hardware-Designs , 1981, IEEE Transactions on Reliability.

[11]  J. Cozzolino.  Probabilistic models of decreasing failure rate processes , 1968 .

[12]  Andrew G. Glen,et al.  The Arctangent Survival Distribution , 1997 .

[13]  John D. Musa,et al.  A theory of software reliability and its application , 1975, IEEE Transactions on Software Engineering.

[14]  C. Maclean,et al.  Estimation and testing of an exponential polynomial rate function within the nonstationary Poisson process , 1974 .

[15]  D. P. Gaver,et al.  Analytical hazard representations for use in reliability, mortality, and simulation studies , 1979 .

[16]  Steven E. Rigdon.  Properties of the Duane plot for repairable systems , 2002 .

[17]  Ming Zhao,et al.  ON SOME RELIABILITY GROWTH MODELS WITH SIMPLE GRAPHICAL INTERPRETATIONS , 1993 .

[18]  Benjamin Gompertz,et al.  XXIV. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. In a letter to Francis Baily, Esq. F. R. S. &c , 1825, Philosophical Transactions of the Royal Society of London.

[19]  Lawrence M. Leemis,et al.  TECHNICAL NOTE: Nonparametric estimation and variate generation for a nonhomogeneous Poisson process from event count data , 2004 .

[20]  Shigeru Yamada,et al.  Reliability growth models for hardware and software systems based on nonhomogeneous Poisson processes: A survey , 1983 .

[21]  Bev Littlewood.  Rationale for a Modified Duane Model , 1984, IEEE Transactions on Reliability.

[22]  Mitsuru Ohba,et al.  Software Reliability Analysis Models , 1984, IBM J. Res. Dev..

[23]  Max Engelhardt,et al.  Sequential Probability Ratio Tests for the Shape Parameter of a Nonhomogeneous Poisson Process , 1982, IEEE Transactions on Reliability.

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

[25]  Wilhelm Kremer,et al.  Birth-Death and Bug Counting , 1983, IEEE Transactions on Reliability.

[26]  Lawrence M. Leemis,et al.  Nonparametric Estimation of the Cumulative Intensity Function for a Nonhomogeneous Poisson Process from Overlapping Realizations , 2000 .

[27]  L. Leemis,et al.  Nonparametric Estimation of the Cumulative Intensity Function for a Nonhomogeneous Poisson Process from Overlapping Realizations , 2000 .

[28]  J. Lawless.  Regression Methods for Poisson Process Data , 1987 .

[29]  Hunter B. Fraser,et al.  Explaining mortality rate plateaus , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[30]  D. S. Bai,et al.  Efficient Sequential Estimation in a Nonhomogeneous Poisson Process , 1987, IEEE Transactions on Reliability.

[31]  Bo Henry Lindqvist,et al.  Statistical Modeling and Analysis of Repairable Systems , 2006, 0708.0362.

[32]  D. Zelterman.  A statistical distribution with an unbounded hazard function and its application to a theory from demography. , 1992, Biometrics.

[33]  O. Aalen.  A linear regression model for the analysis of life times. , 1989, Statistics in medicine.

[34]  D. Willis,et al.  The statistics of a particular non-homogeneous Poisson process , 1964 .

[35]  Huei‐Yaw Ke,et al.  Integrated Bayesian reliability assessment during equipment development , 1999 .