Bayesian Reliability Analysis of The Power Law Process and Statistical Modeling of Computer and Network Vulnerabilities with Cybersecurity Applicationn

Abstract

As most of mankind now lives in an era of high dependence on multiple technologies and complex systems to store and manage sensitive information, researchers are constantly urged to obtain and improve measurements and methodologies that have the ability to evaluate systems reliability and security. The objectives of the present dissertation are to improve the Bayesian reliability estimation of a software package where the Power Law Process, also known as Non- Homogeneous Poisson Process, is the underlying failure model and to develop a set of statistical models evaluating computer operating systems vulnerabilities. Furthermore, we develop a reliability function of a computer network system using the Common Vulnerability Scoring System framework. In the context of software reliability, we propose a Bayesian Reliability analysis approach of the Power Law Process under the Higgins-Tsokos loss function for modeling software failure times. We demonstrate, using real data, that the shape parameter of the Power Law Process behaves as a random variable. Based on Monte Carlo simulations and using real data, we show that the Bayesian estimate of the shape parameter and the proposed estimate of the scale parameter perform better compared to approximate maximum likelihood estimates, while they are sensitive to a prior selection. Using this result, we obtained a Bayesian reliability estimate of the Power Law Process. The results of this study have the potential to contribute not only to the reliability analysis field but also to other fields that employ the Power Law Process. We further illustrate the robustness of the Higgins-Tsokos loss function verses the commonly used squared-error loss function in Bayesian Reliability analysis of the Power Law Process. Based on extensive Monte Carlo simulations and using real data, the Bayesian estimate of the shape parameter and the proposed estimate of the scale parameter were not only as robust as the Bayesian estimates under the squared-error loss function, but also performed better. The reliability function of the Power Law Process is a function of the intensity function; therefore the relative efficiency is used to compare the intensity function estimates. The intensity function using the Bayesian estimate of the shape parameter under the Higgins-Tsokos loss function and its influence on the scale parameter estimate is more efficient than using the Bayesian estimate under the squared-error loss function. Moreover, using Monte Carlo simulations for different sample sizes, we show the efficiency and best performance of Bayesian reliability analysis under the Higgins-Tsokos loss function, recognizing that it is sensitive to selections of its parameters values and the shape parameter’s prior density function. An interactive user interface application was developed to simply, without any prior coding knowledge required of the user, compute and visualize the Bayesian and maximum likelihood estimates of the intensity and reliability functions of the Power Law Process for a given dataset. In addition, we propose a new approach using Copula theory to obtain Bayesian estimates of the Power Law Process intensity function parameters. We first demonstrate, using real data, the random behaviors of the shape and scale parameters of the Power Law Process. We then show the applicability of Copula theory in capturing the dependency structure of the subject parameters and develop a bivariate probability distribution that best characterizes their bivariate probabilistic behaviors. Copula-based Bayesian analysis, under the squared-error loss function and the developed bivariate probability distribution, was studied, where Copula-based Bayesian estimates of the shape and scale parameters of the Power Law Process are obtained simultaneously, considering both parameters unknown and ran