Busy-period analysis of a one-server two-unit system subject to non-negligible inspection time

Abstract

This paper deals with the busy-period analysis of a single-server two-unit cold standby redundant repairable system subject to inspection. Inspection policy models deal with stochastically failing systems in which failure is detected by inspection only. Previous authors assume that the time required for inspection is zero, while the current study deals with preparedness models when duration of inspection is non-negligible. A preparedness model is considered in which the failure of a unit is detected by actual inspection but a system break-down can be detected instantaneously without inspection. The failure time of a unit is assumed to be exponentially distributed with parameter γ. Initially, a unit is switched on and the other one is kept as a cold standby. The system breaks down if a unit fails while the other is still under repair. This paper characterizes the system by the probabilities of its being in the up or the down state, sets up integral equations for these probabilities by identifying suitable regenerative stochastic processes and employs the Laplace-transform technique to solve these equations. Measures of system effectiveness, namely: 1. 1. Point-wise availability of the system in (0, t], 2. 2. s-expected up-time of the system in (0, t], 3. 3. s-expected busy-period of the service facility in (0, t], 4. 4. Steady-state availability of the system, have been derived to obtain the s-expected net gain per unit time. Finally, numerical results pertaining to four particular cases are presented.