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Book Part Shocks, Scans, and Reliability Systems(Springer New York, 2024) Eryilmaz,S.This chapter summarizes the close connection between one of the widely studied shock models known as δ-shock model and runs/scans. Under discrete time setting, i.e., when the shocks occur according to a binomial process, the linkage between the lifetime of the system under the shock model and the waiting time for the first scan is presented. Such a useful connection may create a new perspective to study the reliability properties of the system under the δ-shock model. © Springer Science+Business Media, LLC, part of Springer Nature 2024.Article Citation - WoS: 2Citation - Scopus: 2Computing Waiting Time Probabilities Related To (k1< k2< ..., kl< Pattern(Springer, 2023) Chadjiconstantinidis, Stathis; Eryilmaz, SerkanFor a sequence of multi-state trials with l possible outcomes denoted by {1, 2, ..., l}, let E be the event that at least k(1) consecutive is followed by at least k(2) consecutive 2s,..., followed by at least k(l) consecutive ls. Denote by T-r the number of trials for the rth occurrence of the event E in a sequence of multi-state trials. This paper studies the distribution of the waiting time random variable T-r when the sequence consists of independent and identically distributed multi-state trials. In particular, distributional properties of T-r are examined via matrix-geometric distributions.Article Citation - WoS: 65Citation - Scopus: 76Generalized δ-shock model via runs(Elsevier Science Bv, 2012) Eryilmaz, SerkanAccording to the delta-shock model, the system fails when the time between two consecutive shocks falls below a fixed threshold delta. This model has a potential application in various fields such as inventory, insurance and system reliability. In this paper, we study run-related generalization of this model such that the system fails when k consecutive interarrival times are less than a threshold delta. The survival function and the mean value of the failure time of the system are explicitly derived for exponentially distributed interarrival times. We also propose a new combined shock model which considers both the magnitudes of successive shocks and the interarrival times. (C) 2011 Elsevier B.V. All rights reserved.Article Citation - WoS: 8Citation - Scopus: 11Joint Distribution of Run Statistics in Partially Exchangeable Processes(Elsevier Science Bv, 2011) Eryilmaz, SerkanLet {X-i}(i >= 1) be an infinite sequence of recurrent partially exchangeable random variables with two possible outcomes as either "1" (success) or "0" (failure). In this paper we obtain the joint distribution of success and failure run statistics in {X-i}(i >= 1). The results can be used to obtain the joint distribution of runs in ordinary Markov chains, exchangeable and independent sequences. (C) 2010 Elsevier B.V. All rights reserved.Article Citation - WoS: 11Citation - Scopus: 12Discrete Time Shock Models Involving Runs(Elsevier Science Bv, 2015) Eryilmaz, SerkanIn this paper, three different discrete time shock models are studied. In the first model, the failure occurs when the additively accumulated damage exceeds a certain level while in the second model the system fails upon the local damage caused by the consecutively occurring shocks. The third model is a mixed model and combines the first and second models. The survival functions of the systems under these models are obtained when the occurrences of the shocks are independent, and when they are Markov dependent over the periods. (C) 2015 Elsevier B.V. All rights reserved.Article Citation - WoS: 14Citation - Scopus: 17Compound Geometric Distribution of Order k(Springer, 2017) Koutras, Markos V.; Eryilmaz, SerkanThe distribution of the number of trials until the first k consecutive successes in a sequence of Bernoulli trials with success probability p is known as geometric distribution of order k. Let T (k) be a random variable that follows a geometric distribution of order k, and Y (1),Y (2),aEuro broken vertical bar a sequence of independent and identically distributed discrete random variables which are independent of T (k) . In the present article we develop some results on the distribution of the compound random variable Y-t.Article Citation - WoS: 13Citation - Scopus: 12q-geometric and q-binomial Distributions of Order k(Elsevier Science Bv, 2014) Yalcin, Femin; Eryilmaz, SerkanIn this paper, we generalize geometric and binomial distributions of order k to q-geometric and q-binomial distributions of order k using Bernoulli trials with a geometrically varying success probability. In particular, we derive expressions for the probability mass functions of these distributions. For q = 1, these distributions reduce to geometric and binomial distributions of order k which have been extensively studied in the literature. (C) 2014 Elsevier B.V. All rights reserved.
