The internal states of the 3-component standby system

Abstract

The study analyzes the internal states of a 3-component system with one active element and two spares in cold standby (pure replacement policy without repair). Elements of the system are assumed to have exponentially distributed lifetimes, however, special attention is paid to systems composed of components with different failure rates. The analysis is developed as a continuous-time Markovian process with stationary transition probabilities. Probabilities that exactly i components have failed by time t are calculated based on three levels of information: for systems in unknown condition, for systems known to be in UP-condition, and for systems whose condition was not observed for some amount of time. A key part is the investigation of conditional probabilities of i components having failed by time t for a system known to be UP, the conditional limiting distribution as t-oo, and relation to the system failure rate. State probabilities for systems not monitored continuously for being UP are shown to be bounded between those corresponding to systems that are either observed constantly or not at all.