Volume 9, Issue 1

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On the NHPP with Underlying Distributions of the Location–Scale Family

Guest Submission

Vasiliy V. Krivtsov
Ford Motor Company

In most reliability-related publications on the Non-Homogeneous Poisson Process (NHPP), authors discuss various NHPP models by specifying a parametric form of the rate of occurrence of failures (ROCOF). An alternative (often forgotten) way of defining an NHPP is by specifying its underlying distribution. In this article, we consider this latter method with the underlying distribution belonging to the location-scale family. Using the fact that NHPP's ROCOF numerically coincides with the hazard function of the underlying distribution, parameter estimation of such an NHPP model reduces to the estimation of the cumulative hazard function of the underlying distribution. We use ReliaSoft's data analysis toolset to illustrate the point.


The Non-Homogeneous Poisson Process (NHPP) is widely used to model the failure process of repairable systems. An overwhelming majority of publications on the reliability applications of the NHPP — including Ascher and Feingold (1984), Crowder et al. (1991) and Rausand and Hoyland (1994) — discuss various NHPP models by specifying a parametric form of ROCOF. With that, they mainly consider two monotonic forms of the NHPP ROCOF. The first one is the log-linear model due to Cox and Lewis (1966) and the other one is the power law model due to Crow (1974).

However, there is an alternative way of defining an NHPP, that is, by specifying its underlying distribution, which is also referred to as time-to-first-failure (TTFF) distribution. This is to say that the TTFF distribution fully defines all subsequent failures under the NHPP model.

Indeed, if F(t) is the cumulative distribution function (CDF) of the TTFF distribution and R(t) is the respective reliability function, then the distribution of the time to the second failure (and all successive failures) is introduced in terms of conditional distributions. For example, the CDF of the time to the second failure distribution is given by:

In this respect, Thompson (1981) remarks, "it is a nonintuitive fact that is casting doubt on the NHPP as a realistic model for repairable systems. Use of an NHPP model implies that if we are able to estimate the failure rate of the time to the first failure, such as for a specific types of automobiles, we at the same time have an estimate of the ROCOF of the entire life of the automobile."

Treating NHPP as a particular case of the Generalized Renewal Process, Krivtsov (2007) proves Thompson's point as to formal correspondence between ROCOF and the hazard function of the time-to-first-failure distribution and shows that a variety of traditional lifetime distributions besides Weibull (i.e.,, Lognormal, Normal, Log-Logistic, etc.) can be used to specify the respective NHPP models. Clearly, a Weibull-distributed TTFF corresponds to the popular powerlaw NHPP.

In this article, we consider an NHPP estimation procedure for the case when its TTFF distribution belongs to the family of location-scale distributions. The proposed procedure can be easily executed using ReliaSoft's data analysis toolset.

Statistical Estimation of NHPP with Underlying Distributions of the Location-Scale Family

Let the number of repairs, N (t), to occur in (0, t] be given by N(t) = max{k|Tk <= t} for k = 1, 2, .... The Cumulative Intensity Function (CIF) in (0, t ] is then given by:

Crowder et al. (1991) give a natural estimator of Λ(t):

where N is a number of systems, and the total observed number of repairs in (0, t] is:

It can be shown (Krivtsov, 2007) that the above nonparametric estimator could be used both for the CIF of the point process and the cumulative hazard function (CHF) of the underlying distribution:


Once the nonparametric estimate of the CHF has been obtained, one can use standard statistical procedures, e.g.,, hazard papers or probability papers (Nelson, 1982), for the estimation of the underlying distribution (and the respective NHPP) parameters. It must be stressed that the failure data used in Eqn. 1 includes all failures, not only first ones.

In Figure 1, consider the CIF of a point process related to arrival times (in seconds) of the demands upon a computer-aided engineering server. Evidently, the traditional power-law NHPP does not seem to fit the data well.

Figure 1: NHPP with the underlying Weibull distribution (power-law ROCOF)

Keeping in mind Eqn. 1, we use the following transformation.

to plot the data in the lognormal probability paper, which seems to show a reasonable fit — see Figure 2.

Figure 2: Lognormal probability plot

Having obtained the estimates of the lognormal distribution parameters, we now use the inverse transformation to return to the CIF domain and compare the NHPP with the lognormal underlying distribution versus that with the Weibull underlying distribution — see Figure 3. Evidently, the former exhibits a better fit.

Figure 3: NHPP with the underlying Weibull distribution (power-law ROCOF) versus NHPP with the underlying lognormal distribution

It is easy to see that the described procedure can be extended to any distribution of the location-scale family, i.e.,, whose CDF can be estimated using the probability paper method.


Ascher, H. and Feingold, H. (1984). Repairable Systems Reliability. New York: Marcel Dekker.

Cox, D.R. and Lewis, P.A. (1966). The Statistical Analysis of Series of Events. London: Methuen.

Crow, L.H. (1974). Reliability Analysis for Complex Repairable Systems. In Proschan, F. and Serfling, R.J. (Ed.), Reliability and Biometry, (pp. 379 - 410). SIAM.

Crowder, M.J., Kimber, A.C., Smith, R.L. and Sweeting, T.J. (1991). Statistical Analysis of Reliability Data. London: Chapman and Hall.

Krivtsov, V.V. (2007). Practical Extensions to NHPP Application in Repairable System Reliability Analysis. Reliability Engineering & System Safety, Vol. 92(5), 560-562.

Nelson, W. (1982). Applied Life Data Analysis. New York: John Wiley & Sons.

Rausand, M. and Hoyland, A. (1994). System Reliability Theory: Models, Statistical Methods, and Applications. New York: Wiley.

Thompson, W.A. Jr. (1981). On the Foundation of Reliability. Technometrics, 23, 1-13.

About the Author

Vasiliy Krivtsov is a practitioner and a consultant in Reliability Engineering, Risk Analysis and Applied Statistics presently employed by Ford Motor Company as a Staff Technical Specialist. He holds M.S. and Ph.D. degrees in Electrical Engineering from Kharkov Polytechnic Institute, Ukraine and a Ph.D. in Reliability Engineering from the University of Maryland, USA. Dr. Krivtsov is the author and coauthor of over 50 professional publications, including a book on Reliability Engineering and Risk Analysis, 9 patented inventions and 2 Ford corporate secret inventions. He is an editor of the Elseveir's Reliability Engineering and System Safety journal and is a member of the IEEE Reliability Society. Prior to Ford, Dr. Krivtsov held the position of Associate Professor of Electrical Engineering in Ukraine, and that of a Research Affiliate at the University of Maryland Center for Reliability Engineering. Further information on Dr. Krivtsov's professional activity is available at www.krivtsov.net.

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