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Article Citation - WoS: 5Component Importance in Coherent Systems With Exchangeable Components(Cambridge Univ Press, 2015) Eryilmaz, SerkanThis paper is concerned with the Birnbaum importance measure of a component in a binary coherent system. A representation for the Birnbaum importance of a component is obtained when the system consists of exchangeable dependent components. The results are closely related to the concept of the signature of a coherent system. Some examples are presented to illustrate the results.Article Citation - WoS: 1Citation - Scopus: 2Computing Minimal Signature of Coherent Systems Through Matrix-Geometric Distributions(Cambridge Univ Press, 2021) Eryilmaz, Serkan; Eryılmaz, Serkan; Tank, Fatih; Eryılmaz, Serkan; Industrial Engineering; Industrial EngineeringSignatures are useful in analyzing and evaluating coherent systems. However, their computation is a challenging problem, especially for complex coherent structures. In most cases the reliability of a binary coherent system can be linked to a tail probability associated with a properly defined waiting time random variable in a sequence of binary trials. In this paper we present a method for computing the minimal signature of a binary coherent system. Our method is based on matrix-geometric distributions. First, a proper matrix-geometric random variable corresponding to the system structure is found. Second, its probability generating function is obtained. Finally, the companion representation for the distribution of matrix-geometric distribution is used to obtain a matrix-based expression for the minimal signature of the coherent system. The results are also extended to a system with two types of components.Article Citation - WoS: 6Citation - Scopus: 7DISTRIBUTIONS OF RANDOM VARIABLES INVOLVED IN DISCRETE CENSORED δ-SHOCK MODELS(Cambridge Univ Press, 2023) Chadjiconstantinidis, Stathis; Eryilmaz, SerkanSuppose that a system is affected by a sequence of random shocks that occur over certain time periods. In this paper we study the discrete censored delta-shock model, delta <= 1 , for which the system fails whenever no shock occurs within a -length time period from the last shock, by supposing that the interarrival times between consecutive shocks are described by a first-order Markov chain (as well as under the binomial shock process, i.e., when the interarrival times between successive shocks have a geometric distribution). Using the Markov chain embedding technique introduced by Chadjiconstantinidis et al. (Adv. Appl. Prob. 32, 2000), we study the joint and marginal distributions of the system's lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system. The joint and marginal probability generating functions of these random variables are obtained, and several recursions and exact formulae are given for the evaluation of their probability mass functions and moments. It is shown that the system's lifetime follows a Markov geometric distribution of order (a geometric distribution of order under the binomial setup) and also that it follows a matrix-geometric distribution. Some reliability properties are also given under the binomial shock process, by showing that a shift of the system's lifetime random variable follows a compound geometric distribution. Finally, we introduce a new mixed discrete censored delta -shock model, for which the system fails when no shock occurs within a -length time period from the last shock, or the magnitude of the shock is larger than a given critical threshold . gamma > 0. Similarly, for this mixed model, we study the joint and marginal distributions of the system's lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system, under the binomial shock process.Article Citation - WoS: 5Citation - Scopus: 8Discrete Scan Statistics Generated by Exchangeable Binary Trials(Cambridge Univ Press, 2010) Eryilmaz, SerkanLet {X-i}(i=1)(n) be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable S-n,S-m to be the maximum number of Is within any m consecutive trials in {X-i}(i=1)(n). The random variable S-n,S-m is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {X-i}(i=1)(n) consists of exchangeable binary trials. We provide simple closed-form expressions for both conditional and unconditional distributions of S-n,S-m for 2m >= n. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.Article Citation - WoS: 12Citation - Scopus: 13A NEW SHOCK MODEL WITH A CHANGE IN SHOCK SIZE DISTRIBUTION(Cambridge Univ Press, 2021) Eryilmaz, Serkan; Kan, CihangirFor a system that is subject to shocks, it is assumed that the distribution of the magnitudes of shocks changes after the first shock of size at least d(1), and the system fails upon the occurrence of the first shock above a critical level d(2) (> d(1)). In this paper, the distribution of the lifetime of such a system is studied when the times between successive shocks follow matrix-exponential distribution. In particular, it is shown that the system's lifetime has matrix-exponential distribution when the intershock times follow Erlang distribution. The model is extended to the case when the system fails upon the occurrence of l consecutive critical shocks.
