Toward the evaluation of P(X_(t) > Y_(t)) when both X_(t) and Y_(t) are inactivity times of two systems

ABSTRACT

The inactivity time, also known as reversed residual life, has been a topic of increasing interest in the literature. In this investigation, based on the comparison of inactivity times of two devices, we introduce and study a new estimate of the probability of the inactivity time of one device exceeding that of another device. The problem studied in this paper is important for engineers and system designers. It would enable them to compare the inactivity times of the products and, hence to design better products. Several properties of this probability are established. Connections between the target probability and the reversed hazard rates of the two devices are established. In addition, some of the reliability properties of the new concept are investigated extending the well-known probability ordering. Finally, to illustrate the introduced concepts, many examples and applications in the context of reliability theory are included.

KEYWORDS:

• Bias sampling
• mean inactivity time
• reliability
• reliability ordering
• reversed hazard rate
• U-statistics

Acknowledgments

The authors thank the editor and the anonymous referee for their valuable and constructive comments, which have greatly improved the contents of the paper.