10 - Semi-Markov Failure Rate Process

Abstract

In this chapter the reliability function is defined by the stochastic failure rate process with a nonnegative and right-continuous trajectories. Equations for the conditional reliability functions of an object are derived under the assumption that the failure rate is a semi-Markov process with an at most countable state space. A proper theorem is presented. The equations for Laplace transforms of conditional reliability functions for the finite space semi-Markov random walk failure rate process are presented in the chapter. The countable linear systems of equations for the appropriate Laplace transforms allow us to find the reliability functions for the Poisson and the Furry-Yule failure rate processes. Frequently, the random tasks and environmental conditions cause a random load of an object in its operation. The failure rate of an object, depending on the random load, may be the random process, too. This chapter presents the limit theorem concerning a failure rate under the assumption that it is a linear function of a random load process with an ergodic mean.