This paper examines a replacement model in which system replacement is carried out either at a random time T or upon failure, rather than at a fixed time. The lifetime model is formulated for parallel systems with a random number of components, where each component follows a power series distribution. The optimization problem is investigated under a preventive age-replacement policy. The planned replacement time is assumed to be a random variable with a distribution from the proportional hazard rate family, and the lifetimes of system components are modeled using a Weibull distribution. The number of components is assumed to follow geometric, logarithmic, or zero-truncated Poisson distributions. The total expected cost rate function is used as the optimization criterion. Both analytical and numerical results are obtained, and the applicability of the proposed approach is demonstrated using a real data set.

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