A study on a stochastic process with state classifications based on sojourn times

Abstract

The indexes for measuring the performances of systems, availability, reliability and others are important issues in the system design and analysis. Although there are many indices being used, they still cannot meet the various requirements in both theory and practice, so the study on these indices is a significant task. In this paper, we develop some new useful indexes classified into two categories, by introducing a stochastic process in terms of two threshold values on sojourn time. The two-category indices are named as point state and interval state probability measures, respectively. The corresponding computation formulae including some special cases are presented. These introduced indices can be used to describe the degrees of stable, semi-stable and unstable for the operation system. We also believe that the indices can be used in reliability engineering, survival analysis, actuarial study, financial analysis, economy and others, and this work definitely enriches the contents of theory of stochastic processes.

Keywords:

• Sojourn time
• alternative renewal process
• point state probability
• interval state probability
• renewal equation
• convolution

Acknowledgement

The authors would like to thank two anonymous referees and editor for their valuable suggestions on the paper.