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Article Citation - WoS: 12Citation - Scopus: 14The Markov Discrete Time Δ-Shock Reliability Model and a Waiting Time Problem(Wiley, 2022) Chadjiconstantinidis, Stathis; Eryilmaz, Serkandelta-shock model is one of the widely studied shock models in reliability theory and applied probability. In this model, the system fails due to the arrivals of two consecutive shocks which are too close to each other. That is, the system breaks down when the time between two successive shocks falls below a fixed threshold delta. In the literature, the delta-shock model has been mostly studied by assuming that the time between shocks have continuous distribution. In the present paper, the discrete time version of the model is considered. In particular, a proper waiting time random variable is defined based on a sequence of two-state Markov dependent binary trials and the problem of finding the distribution of the system's lifetime is linked with the distribution of the waiting time random variable, and we study the joint as well as the marginal distributions of the lifetime, the number of shocks and the number of failures associated with these binary trials.Article Citation - WoS: 11Citation - Scopus: 10Modeling of Claim Exceedances Over Random Thresholds for Related Insurance Portfolios(Elsevier, 2011) Eryilmaz, Serkan; Gebizlioglu, Omer L.; Tank, FatihLarge claims in an actuarial risk process are of special importance for the actuarial decision making about several issues like pricing of risks, determination of retention treaties and capital requirements for solvency. This paper presents a model about claim occurrences in an insurance portfolio that exceed the largest claim of another portfolio providing the same sort of insurance coverages. Two cases are taken into consideration: independent and identically distributed claims and exchangeable dependent claims in each of the portfolios. Copulas are used to model the dependence situations. Several theorems and examples are presented for the distributional properties and expected values of the critical quantities under concern. (C) 2011 Elsevier B.V. All rights reserved.Article Citation - WoS: 8Citation - Scopus: 9Reliability-Based Evaluation of Hybrid Wind-Solar Energy System(Sage Publications Ltd, 2021) Devrim, Yilser; Eryilmaz, SerkanIn this article, a hybrid system that consists of a specified number of wind turbines and solar modules is considered. In particular, the system is modeled using weightedk-out-of-nsystem which is also known as a threshold system in reliability literature. The system under concern consists ofn1identical wind turbines andn2identical solar modules, and each turbine and module can be in one of two states as working or failed. The probability that the entire hybrid system withn=n1+n2components produces power at minimum levelkis computed and evaluated. The importance of single-wind turbine and solar module is also calculated to measure which renewable energy component is more critical and important. Extensive numerical results that are based on real data set are presented to illustrate the model.Article Citation - WoS: 26Citation - Scopus: 28Reliability Assessment for Discrete Time Shock Models Via Phase-Type Distributions(Wiley, 2021) Eryilmaz, Serkan; Kan, CihangirIn this paper, particular shock models are studied for the case when the times between successive shocks and the magnitudes of shocks have discrete phase-type distributions. The well-known shock models such as delta shock model, extreme shock model, and the mixed shock model which is obtained by combining delta and extreme shock models are considered. The probability generating function and recursive equation for the distribution of the system's lifetime are obtained for the cases when the interarrival times between shocks and the magnitudes of shocks are independent and when they are dependent. System reliability is computed for particular interarrival distributions such as geometric, negative Binomial and generalized geometric distributions.Article Citation - WoS: 11Citation - Scopus: 13Geometric Distribution of Order k With a Reward(Elsevier Science Bv, 2014) Eryilmaz, SerkanIn this paper, we introduce and study geometric distribution of order k with a reward. In a sequence of binary trials, suppose that each time a success occurs a random reward is received. The distribution of the number of trials until the sum of consecutive rewards is equal to or exceeds the level k is called geometric distribution of order k with a reward. We obtain expressions for the probability mass function of this distribution. (C) 2014 Elsevier B.V. All rights reserved.Article Citation - WoS: 61Citation - Scopus: 64Computing Optimal Replacement Time and Mean Residual Life in Reliability Shock Models(Pergamon-elsevier Science Ltd, 2017) Eryilmaz, SerkanIn this paper, matrix-based methods are presented to compute the optimal replacement time and mean residual lifetime of a system under particular class of reliability shock models. The times between successive shocks are assumed to have a common continuous phase-type distribution. The system's lifetime is represented as a compound random variable and some properties of phase-type distributions are utilized. Extreme shock model, run shock model, and generalized extreme shock model are shown to be the members of this class. Graphical illustrations and numerical examples are presented for the run shock model when the interarrival times between shocks follow Erlang distribution. (C) 2016 Elsevier Ltd. All rights reserved.Article Citation - WoS: 30Citation - Scopus: 34Discrete Time Shock Models in a Markovian Environment(Ieee-inst Electrical Electronics Engineers inc, 2016) Eryilmaz, SerkanThis paper deals with two different shock models in a Markovian environment. We study a system from a reliability point of view under these two shock models. According to the first model, the system fails if the cumulative shock magnitude exceeds a critical level, while in the second model the failure occurs when the cumulative effect of the shocks in consecutive periods is above a critical level. The shock occurrences over discrete time periods are assumed to be Markovian. We obtain expressions for the failure time distributions of the system under the two model. Illustrative computational results are presented for the survival probabilities and mean time to failure values of the system.Article Citation - WoS: 6Citation - Scopus: 6Reliability Analysis of Systems With Components Having Two Dependent Subcomponents(Taylor & Francis inc, 2017) Eryilmaz, SerkanIn this article, a system that consists of n independent components each having two dependent subcomponents (A(i), B-i), i = 1, ... ,n is considered. The system is assumed to compose of components that have two correlated subcomponents (A(i), B-i), and functions iff both systems of subcomponents A(1),A(2), ... ,A(n) and B-1, B-2, ... , B-n work under certain structural rules. The expressions for reiiabiiity and mean time to failure of such systems are obtained. A sufficient condition to compare two systems of bivariate components in terms of stochastic ordering is also ordering presented.Article Citation - WoS: 29Citation - Scopus: 34System Reliability Under Δ-Shock Model(Taylor & Francis inc, 2018) Tuncel, Altan; Eryilmaz, Serkandelta-shock model is one of the widely studied shock models in reliability. Under this model, the system fails when the time between two consecutive shocks falls below a fixed threshold . In this paper, the survival function and the mean time to failure of the system are obtained when the times between successive shocks follow proportional hazard rate model.Article Citation - WoS: 27Citation - Scopus: 31Generalized Extreme Shock Models and Their Applications(Taylor & Francis inc, 2020) Bozbulut, Ali Riza; Eryilmaz, SerkanIn the classical extreme shock model, the system fails due to a single catastrophic shock. In this paper, by assuming different arrival patterns of the shocks, two new types of extreme shock models are introduced. In these models, m possible sources may exert shocks on the system. Both models reduce to the classical extreme shock model when m = 1. Assuming phase-type distribution for times between successive shocks, we obtain survival functions and mean time to failure values of the system under new models. Two different optimization problems are also considered to determine the optimal number of sources.
