Renewal Processes and Repairable Systems

Abstract

In this thesis we discuss the following topics: 1. Renewal reward processes The marginal distributions of renewal reward processes and its version, which we call in this thesis instantaneous reward processes, are derived. Our approach is based on the theory of point processes, especially Poisson point processes. The idea is to represent the renewal reward processes and its version as functionals of Poisson point processes. Important tools we use are the Palm formula and the Laplace functional of Poisson point processes. The results are presented in the form of Laplace transforms. An application of the instantaneous reward processes to the study of traffic is given. Some asymptotic properties of the renewal reward processes are reconsidered. A proof of the expected-value version of the renewal reward theorem using the Tauberian theorem is given. A second order term in the expected-value version of the renewal reward theorem is obtained. Similar results for the instantaneous reward processes are investigated. Asymptotic normality of the instantaneous reward processes is proved. The covariance structure of renewal processes, which can be considered as a special case of renewal reward processes, is derived. As an addition, we study system reliability in a stress-strength model, where the amplitudes of stresses can be considered as rewards. We consider renewal and Cox processes as models for the occurrences of the stresses. Using our result on renewal reward processes we investigate the effect of dependence between stress and strengths on system reliability. 2. Integrated renewal processes The marginal probability density function of an integrated homogeneous Poisson Process is known in the literature. It is natural to generalize the integrated homogeneous Poisson process into integrated non homogeneous Poisson, Cox, and renewal processes. In this thesis we derive expressions for the marginal distributions of integrated Poisson and Cox processes using conditioning arguments, and derive the marginal distributions of integrated renewal processes using the theory of point processes. The results are presented in the form of Laplace transforms. Asymptotic properties of the integrated renewal processes are also investigated. An application to the study of traffic is given. 3. Total downtime of repairable systems An expression for the cumulative distribution function of the total downtime of a repairable system, which is regarded as a single component, under an assumption that the failure and the repair times of the system are independent has been derived by several authors using different methods. We use a different method (using point processes) to compute the distribution function of the total downtime. We also consider a more general situation where we allow dependence of the failure and the repair times of the system. The covariance structure and asymptotic properties of the total downtime for the independent case are also known in the literature. We derive the similar results for the dependent case. Examples are given to see the effect of dependence between the failure and the repair times on the total downtime. We also discuss the total downtime of repairable systems consisting of n 2 stochastically independent components. We derive an expression for the marginal distribution of the total uptime of the system for the case the failure and the repair times of each component are exponentially distributed. For arbitrary failure or repair times of the components we derive an expression for the mean of the total uptime.