Background A standby system is defined as a system consisting of several components, where only one component operates at a time while the others remain in standby mode. The system fails when the applied stress exceeds the strength of the active component, at which point one of the standby components is activated. It is assumed that the inactive components do not undergo any change in their properties while in standby mode, and the system is considered to have failed when all its components have failed. Methods In this study, a mathematical formula of the standby system was derived for more than one component. Y is subject to independent random variable Ψ with Distribution of stress, which is considered a combination of two exponentials and the stress. It follows the following distributions: one-parameter exponential, two-parameter exponential, one parameter Lindley, and two-parameter generalized exponential. Hence, the dependent functions of the standby system are found R (1), R (2), R (3), R (4), R 4 depending on the mathematical formulas derived for each of the four distributions mentioned above. Results In this section, we will obtain the stress-strength of standby Reliability system R 4, based on four different distributions the above mentioned, the standby Reliability functions R (2) for distributions are given by adding Marginal Reliability R (1) and R (2), R (3) given by adding R (1), R (2) and R (3). Also R 4 given by adding R (1), R (2), R (3) and R (4), and parameters estimator by maximum likelihood. Conclusions a simulation study will be conducted to see the behavior of R (1), R (2), R (3), R (4), R 4 of a standby system for four different distributions, in this simulation study will be conducted to see the estimator for all distributions and compare the results by using one important statistical criteria mean square error (MSE). Then results will be discussed to see which one of the estimators is the best for each one of the distributions separately.
Ahmed et al. (Mon,) studied this question.