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Article Citation - WoS: 22Citation - Scopus: 23On Residual Lifetime of Coherent Systems After the rth Failure(Springer, 2013) Eryilmaz, SerkanIn this article we study the residual lifetime of a coherent system after the rth failure, i.e. the time elapsed from the rth failure until the system failure given that the system operates at the time of the rth failure. We provide a mixture representation for the corresponding residual lifetime distribution in terms of signature. We also obtain some stochastic ordering results for the residual lifetimes.Article Citation - WoS: 2Citation - Scopus: 1Computing 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: 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: 2Citation - Scopus: 1Age replacement policies for discrete and continuous heterogeneous k-out-of-n systems(Springer, 2024) Eryilmaz, Serkan; Bulanik, IremThis paper studies age replacement policy for the k-out-of-n system that consists of independent but nonidentical components. Both continuously and discretely distributed components' lifetimes are considered. The failed components are replaced by new components and non-failed components are rejuvenated. Because the components are non-identical, the acquisition and rejuvenation costs of the components are chosen differently. The policy and the associated optimization problem are presented for general k and n, and 2-out-of-3 systems are studied in detail. The findings of the present paper extend the results in the literature from parallel systems to k-out-of-n systems.Article Citation - WoS: 11Citation - Scopus: 12A New Mixed Δ-Shock Model With a Change in Shock Distribution(Springer, 2023) Chadjiconstantinidis, Stathis; Tuncel, Altan; Eryilmaz, SerkanIn this paper, reliability properties of a system that is subject to a sequence of shocks are investigated under a particular new change point model. According to the model, a change in the distribution of the shock magnitudes occurs upon the occurrence of a shock that is above a certain critical level. The system fails when the time between successive shocks is less than a given threshold, or the magnitude of a single shock is above a critical threshold. The survival function of the system is studied under both cases when the times between shocks follow discrete distribution and when the times between shocks follow continuous distribution. Matrix-based expressions are obtained for matrix-geometric discrete intershock times and for matrix-exponential continuous intershock times, as well.Article Citation - WoS: 13Citation - Scopus: 13Reliability Assessment of a Discrete Time Cold Standby Repairable System(Springer, 2021) Kan, Cihangir; Eryilmaz, SerkanThis paper is concerned with the study of a discrete time repairable system consisting of one active and one standby component. The lifetime and repair time are assumed to have discrete phase-type distributions. The system's lifetime is represented as a compound random variable. A matrix-based expression for the probability generating function of the system's lifetime is obtained based on the phase characteristics of lifetime and repair time distributions. The probability generating function is then used to obtain the distribution of the system's lifetime. Reliability and hazard rate functions are computed and evaluated for some particular choices of lifetime and repair time distributions. The limiting behavior of the hazard rates is also investigated.Article Citation - WoS: 19Citation - Scopus: 14Assessment of Shock Models for a Particular Class of Intershock Time Distributions(Springer, 2022) Kus, Coskun; Tuncel, Altan; Eryilmaz, SerkanIn this paper, delta and extreme shock models and a mixed shock model which combines delta-shock and extreme shock models are studied. In particular, the interarrival times between successive shocks are assumed to belong to a class of matrix-exponential distributions which is larger than the class of phase-type distributions. The Laplace -Stieltjes transforms of the systems' lifetimes are obtained in a matrix form. Survival functions of the systems are approximated based on the Laplace-Stieltjes transforms. The results are applied for the reliability evaluation of a certain repairable system consisting of two components.Article Citation - WoS: 45Citation - Scopus: 49Life Behavior of -Shock Models for Uniformly Distributed Interarrival Times(Springer, 2014) Eryilmaz, Serkan; Bayramoglu, KonulIn this paper we study the life behavior of -shock models when the shocks occur according to a renewal process whose interarrival distribution is uniform. In particular, we obtain the first two moments of the corresponding lifetime random variables for general interarrival distribution, and survival functions when the interarrival distribution is uniform.Article Citation - WoS: 4Citation - Scopus: 4Relative Behavior of a Coherent System With Respect To Another Coherent System(Springer, 2015) Eryilmaz, Serkan; Tutuncu, G. YazgiIn this paper, two independent coherent systems with different structures, and different types of components are considered. The remaining lifetime and the remaining number of working components of system I after the failure of the system II when we know that the system II fails before the system I are studied. In particular, signature-based expressions are obtained for the distribution of these conditional random variables. Illustrative examples are provided.Article Citation - WoS: 7Citation - Scopus: 8A new look at dynamic behavior of binary coherent system from a state-level perspective(Springer, 2014) Eryilmaz, SerkanIn this paper we study lifetime properties of binary coherent systems from a state-level perspective. We define and study a system whose performance levels are determined by its total number of working components and structure. That is, the more working components the better performance level for the system. This enables us to make a more detailed analysis of a binary system. We obtain the distributions of the time that is spent by the system in a specific state subset and a specific state. Our analysis is based on the use of system signature. We also define an optimization problem concerned with the determination of the number of warm standby components.
