Bayesian interval estimation and predictive analysis in a nonhomogeneous poison process with delayed s-shaped intensity function

Abstract

In the 21st century, software reliability is a significant issue as computers are the most preferred system in almost every global sector. A software is reliable if it can perform its functions for a specified period under specified conditions without causing system failure. A software neither wears out nor burns out and does not fail unless flaws within cause a failure in its dependent system. As such, software reliability testing is performed in the development phase to correct the flaws within the software. Among the non-homogeneous Poisson processes (NHPP) software reliability growth models (SRGMs) proposed and used in software reliability assessment is the Delayed S-shaped model with two unknown parameters 𝛼 and 𝛽, that must be estimated. Most research works have fitted the model to software failure data and obtained point and interval estimates of the unknown parameters using the Maximum Likelihood Estimation (MLE) and Bayesian approaches. However, the construction of Bayesian credible sets for the parameters of this model and the comparison of their accuracy with the traditional Wald confidence intervals based on simulation has not been explored. Predictive analysis on the model has been explored using the Bayesian method with gamma-distributed informative prior. More optimal methods can be developed based on the priors assigned to the unknown parameters to enhance accuracy in modifying, debugging, and determining when to terminate software testing processes. This study introduced a non-informative prior given by 1/αβ and also used 1/α prior existing in the literature and gamma-distributed informative prior to construct Bayesian credible intervals, compare them with Wald confidence intervals using interval lengths and coverage probabilities, and perform predictive analysis. Markov Chain Monte Carlo (MCMC) via Metropolis-Hastings (MH) within Gibbs was used to sample the parameters from their respective conditional posterior distributions. Bayesian approach was also used to address four prediction issues closely associated with software reliability testing. The issues have been outlined as Propositions I, II, III, and IV for the case of non-informative priors, and I.1, II.1, III.1, and IV.1 for the case of the informative prior. The study found that the Bayesian method with gamma-distributed informative and 1/αβ priors yielded more precise interval estimates than the Wald confidence intervals. Moreover, the study developed methods for addressing the outlined single-sample prediction problems and illustrated them using secondary software failure data. The methods developed in this study can be used in software quality assessment