A four-unit repairable system with three subsystems in series and one subsystem containing n components in parallel under environmental failure

Abstract

In this paper a four-unit repairable system with three subsystems in series and one subsystem containing n components in parallel has been studied with specific focus to understand the effect of environmental failure on this system. The causes of these environmental failures include failures due to human error, maintenance error, overloading etc. It has been assumed that the system is susceptible to these environmental failures from any given working state. The state transition diagram was prepared and mathematical modeling of the system was done by defining differential-difference equations for each state and their respective boundary conditions. These equations were solved by using supplementary variable technique and applying Laplace Transformation (LT) to obtain LT of upstate and downstate probabilities of the system. Abel’s lemma was applied to find steady state probabilities for up-state and down-state of the system. In order to complete numerical computations, a particular case, assuming repair rates to follow exponential distribution was taken to identify upstate probability for that case. Certain numerical values of failure and repair rates were taken to find out effect of increase in magnitude of above-mentioned environmental failures on availability of system. Further, certain values for cost and revenue parameters per unit time were assumed and used with upstate probability, at steady state, to ascertain variation in profit function over time due to increase in magnitude of these environmental failures.