Discrete-time stochastic modeling and optimization for reliability systems with retrial and cold standbys

As an effective means to improve system reliability, cold standby redundancy design has been applied in many fields. Studies on the reliability of systems with retrial mechanisms mainly focus on the assumption of continuous time, but for some engineering systems whose states cannot be continuously monitored, it is of great theoretical and practical value to establish and analyze the reliability model of the discrete-time cold standby repairable retrial system. In this paper, the lifetime, repair time, and retrial time of each component were described by geometric distribution, and the reliability and optimal design models of a discrete-time cold standby retrial system were developed. Two different models were proposed based on two types of priority rules. According to the discrete-time Markov process theory, the transition probability matrix of the system states was given. The availability, reliability function, mean time to first failure (MTTFF) of the system, and other performance measures were obtained using the iterative algorithm of the difference equation, and the generative function method, algorithms for calculating stationary probability, and transient probability of the system were designed. The particle swarm optimization (PSO) algorithm was used to determine the optimal values of the repair rate and retrial rate corresponding to the minimum value of the cost-benefit ratio. Moreover, numerical analysis was performed to show the influence of each parameter on the system reliability and the cost-benefit ratio. The reliability measures of the systems with and without retrial mechanism were analytically compared.