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Article Citation - WoS: 27Citation - Scopus: 30The Distributions of Sum, Minima and Maxima of Generalized Geometric Random Variables(Springer, 2015) Tank, Fatih; Eryilmaz, SerkanGeometric distribution of order as one of the generalization of well known geometric distribution is the distribution of the number of trials until the first consecutive successes in Bernoulli trials with success probability . In this paper, it is shown that this generalized distribution can be represented as a discrete phase-type distribution. Using this representation along with closure properties of phase-type distributions, the distributions of sum, minima and maxima of two independent random variables having geometric distribution of order are obtained. Numerical results are presented to illustrate the computational details.Article Citation - WoS: 10Citation - Scopus: 14Optimal Age Replacement Policy for Discrete Time Parallel Systems(Springer, 2023) Eryilmaz, Serkan; Tank, FatihIn the case of discrete age replacement policy, a system whose lifetime is measured by the number cycles is replaced preventively after a specific number of cycles or correctively at failure, whichever occurs first. Under the discrete setup, the policy has been mostly considered for single unit systems. In this paper, a discrete time age replacement policy is studied for a parallel system that consists of components having discretely distributed lifetimes. In particular, the necessary conditions for the unique and finite replacement cycle that minimize the expected cost rate are obtained. The theoretical results are illustrated with numerical examples to observe the effect of the cost values and the mean lifetime of the components on the optimal replacement cycle.
