Serkan Eryilmaz

Atilim University · Department of Industrial Engineering

It is important to elicit information about the potential power output of a wind turbine and a wind farm consisting of specified number of wind turbines before installation of the turbines. Such information can be used to estimate the potential power output of the wind farm which will be built in a specific region. The output power of a wind turbin...

This article presents a signature-based representation for the expected cost rate of age-based preventive maintenance policy for a binary coherent system consisting of independent exponential components, and then specializes the method to consecutive k-out-of-n system and its generalizations. According to the age-based preventive maintenance policy...

In 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 weighted k-out-of- n system which is also known as a threshold system in reliability literature. The system under concern consists of [Formula: see text] identical wind turbines and [Formu...

Integrating multiple wind farms into power systems may reduce the fluctuation in total power output of wind farms and hence it decreases the system risk resulting from the wind speed variability. In this paper, a wind power system consisting of two wind farms is modeled and analyzed considering the dependence between wind speeds at two sites. In pa...

This paper is concerned with two optimization problems for a parallel system that consists of dependent components. First, the problem of finding the number of elements in the system that minimizes the mean cost rate of the system is considered. The second problem is concerned with the optimal replacement time of the system. Previous work assumes t...

According to the well-known age replacement policy, the system is replaced preventively at time t or correctively at system failure, whichever occurs first. For a coherent system consisting of components having common failure time distribution which has increasing failure rate, we present necessary conditions for the existence of the unique optimal...

For 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 d1 , and the system fails upon the occurrence of the first shock above a critical level d2 (> d1 ). In this paper, the distribution of the lifetime of such a system is studied when the times between...

Reliability assessment of system suffering from random shocks is attracting a great deal of attention in recent years. Excluding internal factors such as aging and wear-out, external shocks which lead to sudden changes in the system operation environment are also important causes of system failure. Therefore, efficiently modeling the reliability of...

The study of compound sums have always been very popular in the literature. Many models in insurance and engineering have been represented and solved by compound sums. In this paper, two different bivariate compound sums are proposed and studied. The phase-type distribution is applied to obtain the probability generating function of the bivariate s...

The reliability of a weighted-k-out-of-n system that consists of three-state components is studied. The system is assumed to comprise n three-state components, namely, perfect functioning, partial working, and complete failure and functions if the total weight of all the working components is at least k. Reliability expressions are presented when t...

An extreme shock model when there is a change in the distribution of the magnitudes of shocks is defined and studied. Such a model is useful in practice since a sudden change in environmental conditions may cause a larger shock. In particular, the reliability and mean time to failure of the system is obtained by assuming that the times between arri...

In this article, we develop a general statistical inference procedure for the probability of successful startup p in the case of startup demonstration tests when only the number of trials until termination of the experiment are observed. In particular, we define a class of startup demonstration tests and present expectation-maximization (EM) algori...

Consider a k-out-of-n system which is subject to shocks that occur at random times. Each shock causes failure of random number of components, and hence the system's lifetime corresponds to one of the arrival times of shocks. The reliability and mean time to failure of the system are studied when the times between shocks follow a phase type distribu...

A new mixed shock model is introduced and studied. According to the model, for two fixed critical values d1 and d2 such that d1<d2, the system under concern fails upon the occurrence of k consecutive shocks of size at least d1 or a single large shock of size at least d2. The new model combines run and extreme shock models. Reliability properties of...

The wind power generated by a wind plant has a stochastic nature due to randomness in the wind speed. Although the empirical distribution of the wind power has been extensively studied by using data sets in different regions, several works focused on theoretical distribution of the wind power produced by wind turbines. In this paper, the theoretica...

In 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....

This paper investigates a discrete‐time risk model that involves exchangeable dependent loss generating claim occurrences and compound binomially distributed aggregate loss amounts. First, a general framework is presented to derive the distribution of a surplus sequence using the model. This framework is then applied to obtain the distribution of a...

In this paper, a new shock model called Marshall–Olkin run shock model is defined and studied. According to the model, two components are subject to shocks that may arrive from three different sources, and component i fails when it is subject to k consecutive critical shocks from source i or k consecutive critical shocks from source 3, i=1,2. Relia...

This paper is concerned with a system consisting of multiple types of components and having (k1,k2,…,km)-out-of-n structure. The (k1,k2,…,km)-out-of-n system is a system consisting of ni components of type i, i=1,2,…,m, and functions if at least k1 components of type 1, k2 components of type 2, … km components of type m work, n=∑i=1ⁿni. The exact a...

In this paper, a generalized class of run shock models associated with a bivariate sequence {(X i , Y i )} i≥1 of correlated random variables is defined and studied. For a system that is subject to shocks of random magnitudes X 1 , X 2 , ... over time, let the random variables Y 1 , Y 2 , ... denote times between arrivals of successive shocks. The...

δ-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 δ. 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...

Mean residual life is a useful dynamic characteristic to study reliability of a system. It has been widely considered in the literature not only for single unit systems but also for coherent systems. This article is concerned with the study of mean residual life for a coherent system that consists of multiple types of dependent components. In parti...

The number of failed components in a failed or operating system is a very useful quantity in terms of replacement and maintenance strategies. These quantities have been studied in several papers for a system consisting of identical components. In this paper, the number of failed components at the time when the system fails and the number of failed...

Series and parallel systems consisting of two dependent components are studied under bivariate shock models. The random variables N1 and N2 that represent respectively the number of shocks until failure of component 1 and component 2 are assumed to be dependent and phase-type. The times between successive shocks are assumed to follow a continuous p...

A coherent system that consists of n independent components and equipped with r cold standby components is considered. A generalized mixture representation for the survival function of such a system is obtained, and it is used to examine reliability properties of the system. In particular, the effect of adding r standby components to a given set of...

In most real life situations, the system's components contribute differently in different performance levels. Such a situation can be modeled by systems with multi-state components having more than one working status, e.g. perfect functioning, and partial working. In this paper, a multi-state system that consists of two types of three-state compone...

Marginal and joint reliability importance measures have been found to be useful in optimal system design. Various importance measures have been defined and studied for a variety of system models. The results in the literature are mostly based on the assumption that the components within the system are independent or identical. The present paper is...

Let Xii=1ⁿ be a sequence of n dependent binary trials such that the first n1 in Xii=1ⁿ are of type 1 and follow an exchangeable joint distribution denoted by L1, and the last n2 elements in Xii=1ⁿ are of type 2 and follow an exchangeable joint distribution denoted by L2, where n1+n2=n. That is, the trials within the same group are exchangeable depe...

A new scheme-distribution-based representation is presented for the cumulative distribution function of the number of success runs of length k in a sequence of exchangeable binary trials. By utilizing this new representation, some stochastic ordering results are obtained to compare success runs. The results are illustrated for beta-binomial distrib...

This paper is concerned with a parallel system that have a random number of units. The distribution of the number of units is assumed to follow a power series class of distributions which contains well-known distributions such as modified or truncated Poisson, geometric, and logarithmic distributions. Optimal number of units and replacement time fo...

This paper is concerned with a two dimensional discrete time risk model based on exchangeable dependent claim occurrences. In particular, we obtain a recursive expression for the finite time non-ruin probability under such a dependence among claim occurrences. For an illustration, we define a bivariate compound beta-binomial risk model and present...

Shock models are of great interest in engineering reliability. Among the others, the δ-shock model has been widely studied in the literature. In this model, the system breaks down due to the arrivals of two successive shocks which are too close to each other. That is, the system fails when the time between two consecutive shocks falls below a fixed...

The effect of adding cold standby redundancy to a system at system and component levels provides a useful information in reliability design. For a series (parallel) system adding cold standby redundancy at the component (system) level yields longer system lifetime. In this paper, the effect of adding cold standby redundancy to a general coherent st...

The stress-strength model has attracted a great deal of attention in reliability analysis, and it has been studied under various modeling assumptions. In this paper, the stress-strength reliability is studied for both single unit and multicomponent systems when stress and strength distributions are of phase type. Phase type distributions, besides t...

This paper introduces a new class of bivariate lifetime distributions. Let {Xi}i ≥ 1 and {Yi}i ≥ 1 be two independent sequences of independent and identically distributed positive valued random variables. Define T1 = min (X1, …, XM) and T2 = min (Y1, …, YN), where (M, N) has a discrete bivariate phase-type distribution, independent of {Xi}i ≥ 1 and...

In this article, the influence of a cold standby component to the reliability of weighted k-out-of-n: G systems consisting of two different types of components is studied. Weighted k-out-of-n: G systems are generalization of k-out-of-n systems that has attracted substantial interest in reliability theory because of their various applications in eng...

In this paper, we study the compound random variable S=∑t=1NYt when there is a dependence between a random variable N and a sequence of random variables {Yt}t≥1. Such a compound random variable has been found to be useful in several fields including actuarial science, risk management, and reliability. In particular, we develop some results on distr...

The notion of signature has been widely applied for the reliability evaluation of technical systems that consist of binary components. Multi-state system modeling is also widely used for representing real life engineering systems whose components can have different performance levels. In this article, the concept of survival signature is generalize...

In this paper, a system that consists of n independent components each having two dependent subcomponents (Ai, Bi), i = 1, …, n is considered. The system is assumed to compose of components which have two correlated subcomponents (Ai, Bi), and functions iff both systems of subcomponents A1, A2, …, An and B1, B2, …, Bn work under certain structural...

In 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 a...

A finite sequence of binary random variables is called a weak exchangeable sequence of order m if the sequence consists of m random vectors such that the elements within each random vector are exchangeable in the usual sense and the different random vectors are dependent. The exact and asymptotic joint distributions of the m-dimensional random vect...

In this paper, we study a discrete time risk model based on exchangeable dependent claim occurrences. In particular, we obtain expressions for the finite time non-ruin probability, and the joint distribution of the time to ruin, the surplus immediately before ruin, and the deficit at ruin. An illustration of the results is given and some implicatio...

Reliability analysis of consecutive k-out-of-n systems and their generalizations has attracted a great deal of attention in the literature. Such systems have been used to model telecommunication networks, oil pipeline systems, vacuum systems in accelerators, spacecraft relay stations, etc. In this paper, nonrecursive closed form equations are prese...

Let {xi(n), n >= 1} be a sequence of independent trials with three possible outcomes 0, 1, 2 labeled as failure, success of type I and success of type II, respectively. Suppose that at each time a success of type I (type II) occurs in {xi(n), n >= 1} a random reward of type I (type II) is received. We obtain distributions of the number of trials un...

Let {Yi}i≥1 be a sequence of {0,1} variables which forms a Markov chain with a given initial probability distribution and one-step transition probability matrix. Define Nn to be the number of trials until the nth success ("1") in {Yi}i≥1. In this paper, we study the distribution of the random variable T= ∑ i=1NnXi, where {Xi}i≥1 is a sequence of in...

For degraded multi-state systems, it has been assumed in the literature that, for any given system, the instantaneous degradation rates are fixed. This paper attempts to study a three-state degraded system that have random degradation rates among its states. In particular, a reliability model for such a three-state system is presented assuming that...

In this paper, a general formula for computing the joint reliability importance of two components is obtained for a binary coherent system that consists of exchangeable dependent components. Using the new formula, the joint reliability importance can be easily calculated if the path sets of the system are known. As a special case, an expression for...

Let be a sequence of independent trials with three possible outcomes 0, 1, 2 labeled as failure, success of type I and success of type II, respectively. Suppose that at each time a success of type I (type II) occurs in a random reward of type I (type II) is received. We obtain distributions of the number of trials until either the sum of consecutiv...

The 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,… a sequence of independent and identically distributed discrete rando...

Most practical systems consist of multiple types of components although the components perform the same task within the system. The analysis of such systems is more challenging than the systems with single type of components. In this paper, we present expressions for the survival function of the failure time, and mean time to failure of the system...

In this paper, we study a three-state k-out-of- n system with n independent components ( k = (k1,k2) ). Each component can be in a perfect functioning state (state “2”), partially working (state “1”), or failed (state “0”). We assume that, at time t = 0 , n1 components are in a partially working state while the rest n2 components are fully function...

In this paper, a new class of lifetime distributions which is obtained by compounding arbitrary continuous lifetime distribution and discrete phase-type distribution is introduced. In particular, the class of exponential-phase type distributions is studied with some details.

Let (Xi, Yi), i = 1, ..., n be a pair where the first coordinate Xi represents the lifetime of a component, and the second coordinate Yi denotes the utility of the component during its lifetime. Then the random variable Y[r: n] which is known to be the concomitant of the rth order statistic defines the utility of the component which has the rth sma...

In 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...

A three-state k-out-of-n system with n independent components is considered, where the vector k of integers is determined by given fixed scalars k1 and k2 such that k1,k2≤n. The mean number of components of each type either in a perfect functioning state or in a partially working state at the time of the system failure and at a time while the syste...

Abstract A system is subject to random shocks over time. Let c1 and c2 be two critical levels such that c1 < c2. A shock with a magnitude between c1 and c2 has a partial damage on the system, and the system transits into a lower partially working state upon the occurrence of each shock in (c1, c2). A shock with a magnitude above c2 has a catastroph...

We consider two different sets of exchangeable samples which are assumed to be dependent. A single set of observations is obtained from these two dependent samples. The distribution of single order statistic, and the joint distribution of the minimum and an arbitrary order statistic are derived. The results are illustrated in the context of reliabi...

This 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 presen...

Multi-state systems have attracted great attention due to their wide applications in engineering. They have been effectively used in modeling various systems such as power supply systems and transportation systems. In this paper, phase type modeling is proposed for dynamic assessment of nonrepairable multi-state systems when the system degrades acc...

Let \(\left\{ X_{t},t\ge 1\right\} \) be a sequence of random variables with two possible values as either “1” (success) or “0” (failure). Define an independent sequence of random variables \(\left\{ D_{i},i\ge 1\right\} \). The random variable \(D_{i}\) is associated with the success when it occupies the ith place in a run of successes. We define...

A coherent system of order n that consists two different types of dependent components is considered. The lifetimes of the components in each group are assumed to follow an exchangeable joint distribution, and the two random vectors, which represent the lifetimes of the components in each group are also assumed to be dependent. Under this particula...

This paper investigates the stress–strength reliability in the presence of fuzziness. The fuzzy membership function is defined as a function of the difference between stress and strength values, and the fuzzy reliability of single unit and multicomponent systems are calculated. The inclusion of fuzziness in the stress–strength interference enables...

This paper is concerned with the computation of the Barlow-Proschan importance measure for systems involving two common failure criteria, and consisting of statistically independent and identical components. The failure or survival of these systems generally depends on the number of consecutively failed or working components, or the total number of...

This paper is concerned with dynamic reliability modeling of three-state systems consisting of three-state $s$-independent components. The components and the systems are assumed to be in three states: perfect functioning, partial performance, and complete failure. Survival functions of such systems are studied in different state subsets. It is show...

A generalized $k$ -out-of-$n$ system which is denoted by $((n_{1},ldots,n_{N}),f,k)$ consists of $N$ modules ordered in a line or a circle, and the $i$th module is composed of $n_{i}$ components in parallel $(n_{i}geq 1,i=1,ldots,N)$. The system fails iff there exist at least $f$ failed components or at least $k$ consecutive failed modules. In this...

This 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...

A binary weighted-k-out-of-n:G system is a system that consists of n binary components, and functions if and only if the total weight of working components is at least k. The performance of such a system is characterized by its total weight/capacity. Therefore, the evaluation of the capacity of the system is of special importance for understanding...

In this article we present several results pertaining to the stochastic comparison of the lifetimes of two reliability systems with exchangeable components. More specifically, we provide signature-based sufficient and necessary conditions for establishing hazard rate and reverse hazard rate orderings. Finally, focusing on the class of consecutive-t...

The purpose of this article is to develop a Monte-Carlo simulation algorithm for computing mean time to failure (MTTF) of weighted-k-out-of-n:G and linear consecutive-weighted-k-out-of-n:G systems. Our algorithm is based on the use of appropriately defined stochastic process which represents the total weight of the system at time t. These stochasti...

In 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...

This paper is concerned with dynamic reliability analysis of three-state kk-out-of-nn:G systems. It is assumed that the components and the systems can be in three states: perfect functioning, partial performance and complete failure. Using the concept of permanent, we study marginal and joint survival functions for the lifetime of two different thr...

In this paper, we study joint reliability importance (JRI) in a k -out-of- n: G structure consisting of exchangeable dependent components. We obtain a closed-form formula for the JRI of multiple components of a k -out-of- n: G system with dependent components. We illustrate the results for the k -out-of- n: G model under stress-strength setup. The...

In this paper, a multivariate copula based modeling methodology for dynamic reliability modeling of weighted-k-out-of-n systems is applied. The system under consideration is assumed to have n dependent components each having its own weight. It has a performance level of at least k when the total weight of operating components is k or above. Copula...

In this paper, we study a multi-state weighted k-out-of-n:G system model in a dynamic setup. In particular, we study the random time spent by the system with a minimum performance level of k. Our method is based on ordering the lifetimes of the system's components in different state subsets. Using this ordering along with the Monte-Carlo simulation...

Geometric distribution of order \(k\) as one of the generalization of well known geometric distribution is the distribution of the number of trials until the first \(k\) consecutive successes in Bernoulli trials with success probability \(p\). In this paper, it is shown that this generalized distribution can be represented as a discrete phase-type...

In this paper, we introduce and study geometric distribution of order kk 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 kk is called geometric distribution of order kk wi...

Modeling statistical dependence between two systems or components is an important problem in reliability theory. Such a problem has been well studied for binary systems and components. In the present paper, we provide a way for modeling s-dependence between two multi-state components. Our method is based on the use of copulas which are very popular...

In this paper we study the life behavior of $\delta $ δ -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 d...

A weighted-k-out-of-n:G system is a system that consists of n binary components, each with its own positive weight, and operates only when the total weight of working components is at least k. Such a structure is useful when the components have different contributions to the performance of the entire system. This paper is concerned with both margin...

In this paper, we obtain exact expression for the distribution of the time to failure of discrete time cold standby repairable system under the classical assumptions that both working time and repair time of components are geometric. Our method is based on alternative representation of lifetime as a waiting time random variable on a binary sequence...

This article is concerned with the reliability analysis of a weighted-k-out-of-n:G system consisting of two types of components. The system is assumed to have n components which are classified into two groups with respect to their weight and reliability, and it is assumed to operate if the total weight of all working components exceeds a prespecifi...

In 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 expres...

In this paper, we investigate the effect of a single cold standby component on the performance of a coherent system. In particular, we focus on coherent systems which may fail at the time of the first component failure in the system. We obtain signature based expressions for the survival function and mean time to failure of the coherent systems sat...

The purpose of this paper is to show the usefulness of system signature for computing some important reliability indices of repairable systems. In particular, we obtain signature-based expressions for stationary availability, rate of occurrence of failure, and mean time to the first failure of repairable systems. Using these expressions we compute...

This paper is concerned with the distribution of runs associated with claim indicators in a compound binomial risk model. We study the total number of claims, the longest run without claim, the shortest run without claim and the total number of runs up to a fixed period before the occurrence of a ruin. These quantities are potentially useful for an...

In this paper, we study parallel and consecutive-k-out-of-n:F systems consisting of components which are subject to random deterioration with time. The random deterioration in resistance of a component is defined through a stochastic process. We obtain lifetime distribution of a parallel system via classical probabilistic techniques. The lifetime d...

In 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...

The signature of a system is a useful concept not only in the analysis of binary coherent systems but also in network reliability. Computation of system signature is a well-defined combinatorial problem. This article is concerned with the computation of signature vectors of series and parallel systems consisting of modules. We derive simple formula...

In this paper, we obtain an expression between the sums of the marginal distributions of the order statistics and the common marginal distribution of an exchangeable random sequence. We also derive an expression between the sums of the joint distribution of two order statistics and the two dimensional joint distribution of an exchangeable random se...

We study the joint reliability importance (JRI) of two components in Lin/m/Con/k/n:F systems. A Lin/m/Con/k/n:F system consists of n linearly ordered binary components, and the system fails iff there are at least m nonoverlapping runs of k consecutive failed components (n ≥ mk). In particular, we obtain expressions for the JRI in Lin/m/Con/k/n:F sy...

There are various systems consisting of components which may have different contribution to the performance of the system. Such systems can be modeled systems with weighted components. In this paper, we study the mean instantaneous performance of this type of systems after successive component failures. The mean instantaneous performance is a usefu...

The study of stress-strength reliability in a time-dependent context needs to model at least one of the stress or strength quantities as dynamic. We study the stress-strength reliability for the case in which the strength of the system is decreasing in time and the stress remains fixed over time; that is, the strength of the system is modeled as a...

A k-out-of-n:G system consists of n components, and operates if at least k of its components operate. Its reliability properties have been widely studied in the literature from different perspectives. This paper is concerned with the reliability analysis of a k-out-of-n:G system equipped with a single warm standby unit. We obtain an explicit expres...

Measuring the relative importance of components in a mechanical system is useful for various purposes. In this article, we study Birnbaum and Barlow-Proschan importance measures for two frequently studied system designs: linear consecutive k -out-of- n and m -consecutive- k -out-of- n systems. We obtain explicit expressions for the component import...

A multistate k-out-of-n system model is an extension of binary k-out-of-n system model by allowing multiple performance levels for the system and its components. Various definitions of multistate k-out-of-n system model have been proposed in the literature. Previous studies on these systems mostly focus on reliability evaluation algorithms. The pre...