Inference in nonhomogeneous Poisson process models, with applications to software reliability

Abstract

Nonhomogeneous Poisson Process (NHPP) models are commonly used to model recurrent events (failures or repairs) in repairable systems which fail or break down many times during their lifetime. NHPP models having an intensity of the form ..\(t; 7J) = v .A0 (t; {3), for scalar v > 0 and each component of the vector {3 positive-valued, are widely used in modelling the times of occurrences of failures in the debugging phase of software development. In software system reliability applications, these models are used to predict future behaviour of the occurrence of failures and to provide information for making decisions on when to stop testing. In this thesis, we have addressed statistical issues pertaining to parameter estimation, model verification, and interval prediction for NHPP models having an intensity of the above form. In Chapter 2 , we assess the maximum likelihood estimation procedure used to obtain estimates for v and B for specific models belonging to the above general family of NHPP models. In particular, we study conditions under which a finite, positive-valued maximum likelihood estimate for v is obtained, consider choices of parameterization to facilitate estimation, and consider the effects of total test time on these matters. In Chapter 3, we propose a new approach for testing the goodness of fit for NHPP models of the above general form. We also suggest two alternative models that include specific software reliability models of interest. These models are of use for testing the goodness of fit of their submodels. In Chapter 4, we propose a frequentist approach for providing approximate interval predictors of N2 = N(T1,T2], the number of events in the future time interval (Ti,T2], based on the observed data up to time T1. We also use this method to assess the effect of data accumulation on prediction of N3 = N(T1,oo], the number of remaining events to be eventually observed given data has been observed up to time T1. We also discuss how to obtain Bayesian prediction intervals and compare them with the frequentist-based prediction intervals for N3 = N(T1,oo] in some examples. In Chapter 5, we discuss research areas to be investigated further. The problems presented here are not unique to the software reliability context. In fact, the results of this thesis may be extended to various reliability applications in which NHPP models of the above form are of use.