On δ-shock model in a multi-state system

Abstract

Shock models and multi-state systems have both been studied in the reliability literature. Suppose a system experiences shocks that occur randomly and their occurrence causes a change in the system performance. Suppose these shocks occur randomly over discrete time periods, and ${\delta }_1$ and ${\delta }_2$ are two critical thresholds. The system’s efficiency decreases when the time interval between two consecutive shocks falls between ${[\delta }_1,{\delta }_2)$ and the system fails when the time interval is less than ${\delta }_1$. In this article, the failure time of such a system is studied under two cases. In the first case, the system's failure only depends on the time interval falling below ${\delta }_1$ whereas in the second case, the failure of the system can also occur due to decreasing of system's efficiency. We derive explicit expressions for the distribution and the mean time to failure of the system under the two above mentioned cases.

Keywords:

• Multi-state system
• $\mathrm{\delta }$-shock model
• geometric distribution
• geometric distribution of order k

MATHEMATICS SUBJECT CLASSIFICATION:

• 60C05
• 60E05
• 62N05