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This paper is concerned with a coherent system consisting of active components and equipped with warm standby components. In particular, we study the random quantity which denotes the number of surviving warm standby components at the time of system failure. We represent the distribution of the corresponding random variable in terms of system signature and discuss its potential utilization with a certain optimization problem.

In recent years, the study of reliability properties of consecutive k-out-of-n systems has attracted a great deal of attention from both theoretical and practical perspectives. In this paper we consider linear and circular consecutive k-out-of-n systems. It is assumed that lifetimes of components of the systems are independent but their probability distributions are non-identical. We study the reliability properties of the residual lifetimes of such systems under the condition that at least (n−r+1), r ≤ n , components of the system are operating. We also investigate the probability that a specific number of components of the above-mentioned system operate at time t, t > 0 , under the condition that the system is alive at time t.

In this article, we study reliability properties of m-consecutive-k-out-of-n: F systems with exchangeable components. We deduce exact formulae and recurrence relations for the signature of the system. Closed form expressions for the survival function and the lifetime distribution as a mixture of the distribution of order statistics are established as well. These representations facilitate the computation of several reliability characteristics of the system for a given exchangeable joint distribution or survival function. Finally, we provide signature-based stochastic ordering results for the system's lifetime and investigate the IFR preservation property under the formulation of m-consecutive-k-out-of-n: F systems. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011

In recent years, consecutive systems were shown to have many applications in various branches of science such as engineering. This paper is a study on the stochastic and aging properties of residual lifetime of consecutive k-out-of-n systems under the condition that n−r+1, r≤n, components of the system are working at time t. We consider the linear and circular consecutive k-out-of-n systems and propose a mean residual lifetime (MRL) for such systems. Several properties of the proposed MRL is investigated. The mixture representation of the MRL of the systems with respect to the vector of signatures of the system is also studied.

In this paper we study circular consecutive k-out-of-n systems consisting of exchangeable components. We derive explicit expressions for both unconditional and conditional survival functions for 2 k + 1 ≥ n , while signature based mixture representations for general k are obtained. The applications and computational results concerned with mean residual life function and stochastic ordering are presented.

Let { 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.

A consecutive k-within-m-out-of-n:F system consists of n linearly ordered components and fails if and only if there are m consecutive components which include among them at least k failed components. This system model generalizes both consecutive k-out-of-n:F and k-out-of-n:F systems. In this article, we study the dynamic reliability properties of consecutive k-within-m-out-of-n:F system consisting of exchangeable dependent components. We also obtain some stochastic ordering results and use them to get simple approximation formulae for the survival function and mean time to failure of this system.

Let {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,m to be the maximum number of 1s within any m consecutive trials in {X i } i=1 n . The random variable S n,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,m for 2m ≥ n. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.

The signature of a system is a useful tool in a variety of applications including the evaluation of the reliability characteristics of systems, and the comparison of the performance of competing systems. We study the evaluation and application of signatures of systems involving two common failure criteria which are common in real life applications. The failure or survival of these systems generally depends on the number of consecutively failed or working components, or total number of failed or working components in the whole system. We provide a method for obtaining the signatures of such systems. Applications of the results are also presented.

This paper presents an overview of the research performed on reliability studies of consecutive k-out-of- n and related systems during the last decade. In particular, methods for reliability evaluation, importance and optimal arrangements of components, lifetime distribution, and stochastic orderings of such systems are presented, also research results on related systems are summarized.

The (n, f, k): F(G) system consists of n components and the system fails (works) if and only if there are at least f failed (working) components or at least k consecutive failed (working) components. These system models can be used in electronic equipment, automatic payment systems in banks, and furnace systems. In this paper we introduce and study the (n, f, k):F and (n, f, k): G systems consisting of weighted components. Recursive equations are presented for reliability evaluation of these new models. We also provide some conditions on the weights to represent weighted-(n, f, k) systems as usual (n, f, k) systems.

In this paper we study stress–strength reliability for a general coherent system. The exact expression as well as bounds and approximations for system reliability are presented. We also illustrate the estimation procedure for exponential stress–strength distributions.

As a generalization of k-out-of-n:F and consecutive k-out-of-n:F systems, the consecutive k-within-m-out-of-n:F sys- tem consists of n linearly ordered components such that the system fails iff there are m consecutive components which include among them at least k failed components. In this article, the reliability properties of consecutive k-within-m-out-of-n:F systems with exchangeable components are studied. The bounds and approximations for the survival function are provided. A Monte Carlo estimator of system signature is obtained and used to approximate survival function. The results are illustrated and numerics are provided for an exchangeable multivariate Pareto distribution. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 56: 503-510

Let \({\{ X_{i}\} _{i\geq 1}}\) be an infinite sequence of recurrent partially exchangeable binary random variables. We study the exact distributions of two run statistics (total number of success runs and the longest success run) in \({\{ X_{i}\} _{i\geq1}}\) . Since a flexible class of models for binary sequences can be obtained using the concept of partial exchangeability, as a special case of our results one can obtain the distribution of runs in ordinary Markov chains, exchangeable and independent sequences. The results also enable us to study the distribution of runs in particular urn models.

Waiting time distributions associated with a sequence of binary trials have been widely studied and applied in various areas of probability and statistics. In this paper we study simple and compound waiting time random variables associated with random threshold models which make the elements of sequence dependent. The exact distributions are derived and the results are illustrated for random thresholds based on order statistics.

In this paper we consider mean of success run lengths appearing in a sequence of binary trials. We derive the exact and limiting distributions of mean success run length for i.i.d. Bernoulli trials. The exact distribution of the corresponding random variable is also derived for a sequence of Markov-dependent Bernoulli trials. In addition, a combinatorial formula for the distribution of any success run statistic defined on Markov-dependent trials is presented.

Reliability evaluation has a vital importance at all stages of processing and controlling modern engineering systems. In most of these modern systems the components have different contributions to the system, which are defined as weights. Among the other weighted systems, a linear consecutive-weighted-k-out-of-n:F system consists of n components, where each component has its own weight and reliability. The system fails if and only if the total weight of the failed consecutive components is at least k. The aim of this paper is to study the reliability of linear consecutive-weighted-k-out-of-n:F system consisting of independent & nonidentical and nonhomogeneous Markov dependent components. Exact formulae are provided for computing the reliability for above mentioned cases. Approximation formulae for the reliability are also presented.

Let {Z i }i≥1 be an arbitrary sequence of trials with two possible outcomes either success (1) or failure (0). General expressions for the exact distributions of runs, both success and failure, in Z 1, . . . , Z n are presented. Our method is based on the use of joint distribution of success and failure run lengths and unifies the results on distribution of runs. As a special case of our results we obtain the distributions of runs for various binary sequences. As illustrated in the paper the results enable us to derive the distribution of runs for binary trials arising in urn models.

Recently, Makri, Philippou and Psillakis (2007b) studied the exact distribution of success run statistics defined on an urn model. They derived the exact distributions of various success run statistics for a sequence of binary trials generated by the Pólya-Eggenberger sampling scheme. In our study we derive the joint distributions of run statistics defined on the multicolor urn model using a simple unified combinatorial approach and extend some of the results of Makri, Philippou and Psillakis (2007b). As a consequence of our results, we obtain the joint distributions of success and failure runs defined on the two-color urn model. The results enable us to compute the characteristics of particular consecutive-type systems and start-up demonstration tests.

Consecutive successes total failures (CSTF) is a well known start-up demonstration test procedure in which a unit under test is accepted when successive start-up attempts produce a specified number of consecutive successes before a specified number of failures; otherwise the unit is rejected, and in both cases testing (the experiment) is terminated. The CSTF procedure is studied here assuming that the probability of a successful start-up is a random variable. Under this assumption, the outcomes of the attempted start-ups are dependent random variables following an exchangeable model. The unconditional, and the conditional probability distributions of the waiting time (length of the test) are derived and studied in this situation. The first two moments of unconditional waiting time distribution are calculated. Numerical and graphical illustrations are provided, and comparisons are made with the corresponding results for the i.i.d. model.

The stress–strength models have been widely used for reliability design of systems. In these models the reliability is defined as the probability that the strength is larger than the stress. The analysis is then based on the binary reliability theory since there are two possible states for the system. In this paper, we study the stress–strength reliability in a different framework assigning more than two states to the system depending on the difference between strength and stress values. In other words, the stress–strength reliability is studied under multi-state system modeling. System state probabilities are evaluated and estimated under various assumptions on the system. The multicomponent form is also studied and some results are provided for large systems.

Summary This paper concerns the joint behaviour of precedence and exceedance statistics in random threshold models. Joint distributions of precedence and exceedance statistics, both exact and asymptotic, are obtained, and the results are illustrated for random thresholds based on order statistics and record values.

Let { X i } i ≥ 1 be a an infinite sequence of exchangeable multi-state trials with m ( m ≥ 2 ) possible values. The exact distributions of runs, particularly the total number of runs, longest and shortest runs, in { X i } i ≥ 1 are derived. Applications associated with multi-color Polya urn sampling scheme and statistical process control are also provided.

This work presents some additional results to (2007. Extension of runs to the continuous-valued sequences, Statistics & Probability Letters, 77, 383–388) which extended the definition of runs into the continuous-valued sequences.

The random variables ξ 1 , ξ 2 , … , ξ n are said to be exchangeable (or symmetric) if for each n, P { ξ 1 ⩽ x 1 , … , ξ n ⩽ x n } = P { ξ π ( 1 ) ⩽ x 1 , … , ξ π ( n ) ⩽ x n } for any permutation π = ( π ( 1 ) , … , π ( n ) ) of { 1 , 2 , … , n } and any x i ∈ R , i = 1 , … , n , i.e. the joint distribution of ξ 1 , ξ 2 , … , ξ n is invariant under permutation of its arguments. In this study, run statistics are considered in the situation for which the elements of an exchangeable sequence ξ 1 , ξ 2 , … , ξ n are binary with possible values “1” (success) or “0” (failure). The exact distributions of various run statistics are derived using the fact that the conditional distribution of any run statistic given the number of successes is identical to the corresponding distribution in the independent and identically distributed case.

Let \(X_{1},\ldots,X_{n}\) be independent and identically distributed random variables with continuous distribution function. Denote by \(X_{1:n} \leq \cdots \leq X_{n:n}\) the corresponding order statistics. In the present paper, the concept of \(\varepsilon\)-neighbourhood runs, which is an extension of the usual run concept to the continuous case, is developed for the sequence of ordered random variables \(X_{1:n}\leq\cdots\leq X_{n:n}.\)

Let (X 1, Y 1), (X 2, Y 2),… be a sequence of independent and identically distributed (i.i.d.) pairs of random variables with two possible outcomes at each coordinate. Runs in the second coordinate are considered until the appearance of the first success in the first coordinate.

In this paper we study reliability properties of consecutive-k-out-of-n systems with exchangeable components. For 2k ≦ n, we show that the reliability functions of these systems can be written as negative mixtures (i.e. mixtures with some negative weights) of two series (or parallel) systems. Some monotonicity and asymptotic properties for the mean residual lifetime function are obtained and some ordering properties between these systems are established. We prove that, under some assumptions, the mean residual lifetime function of the consecutive-k-out-of-n: G system (i.e. a system that functions if and only if at least k consecutive components function) is asymptotically equivalent to that of a series system with k components. When the components are independent and identically distributed, we show that consecutive-k-out-of-n systems are ordered in the likelihood ratio order and, hence, in the mean residual lifetime order, for 2k ≦ n. However, we show that this is not necessarily true when the components are dependent.

In this note, new run definitions which we call ɛ - neighbourhood and ɛ - increasing runs are introduced. These definitions are the rational extension of runs to the continuous-valued sequences. Distributional properties of the introduced run statistics are studied.

Shewhart-type distribution-free control charts are considered for the known in-control median of a continuous process distribution based on the Wilcoxon signed-rank statistic and some runs type rules. The new charts are more attractive to the practitioner than a basic Shewhart-type signed-rank chart proposed by Bakir (2004), as they offer more desirable (smaller) false alarm rates and (larger) in-control average run-lengths, and can be easily implemented. In addition to being nonparametric, that is with a known and stable in-control performance for all continuous distributions, a simulation study indicates that the proposed charts can have better out-of-control performance than the Shewhart X-bar chart and the basic signed-rank chart for the normal distribution and for some heavy-tailed distributions such as the double exponential and the Cauchy. A numerical example is provided.

Distributional properties of spacings and exceedances based on progressive type II censored order statistics are investigated. Recurrence relations between moments of concomitants for the model of progressive type II censoring are given. We unify and extend some results on spacings, exceedances and concomitants for the model of progressive censoring.

Concomitants of order statistics are considered for the situation in which the random vectors (X 1, Y 1), (X 2, Y 2),…, (X n , Y n ) are independent but otherwise arbitrarily distributed. The joint and marginal distributions of the concomitants of order statistics and stochastic comparisons among the concomitants of order statistics are studied in this situation.

The number of success runs for nonhomogeneous markov dependent trials are represented as the sum of Bernoulli trials and the expected value of runs are obtained by using this representation. The distribution and bounds for the distribution of the longest run are derived for markov dependent trials.

The marginal and joint distributions of the new sample rank of an order statistic and its concomitant are derived.

The behavior of a sequence of independent and identically distributed random vectors with respect to a random threshold is investigated.

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 for the system which minimizes the mean cost rate are computed. The results extend the results in the literature from Poisson distribution to a wider class of distributions.

Since the hydrologic risk is defined as a risk of exceedance of a critical level during a period, the number of the longest successive exceedances which is referred to as the longest run statistic shows the length of the longest hydrologic risk period. This paper is concerned with the longest run statistic associated with exceedances. In the derivation of a binary sequence, an independent sequence of random variables and a random threshold are considered. The elements of a binary sequence are assigned with respect to a random threshold. In such a consideration, the elements of the corresponding binary sequence are not independent, but are symmetrically dependent, i.e. exchangeable. The distribution of the longest run statistic is derived for the sequence of exchangeable binary variables and its hydrological use is discussed.

Exact and asymptotic distributions of some statistics based on spacing having minimal length are investigated. The behaviour of a sequence of independent identically distributed random variables with respect to a record threshold is studied.

Consecutive- k -out-of- n system and its generalisations have attracted substantial interest due to their applications to model telecommunications systems, oil pipeline systems, heating systems, etc. One of the most important generalisation of this system is an m -consecutive- k -out-of- n :F system. An m -consecutive- k -out-of- n :F system with overlapping runs is a system consisting of n linearly ordered components and fails if and only if there are at least m overlapping runs of k consecutive failures. For m = 1, the system is same with the usual consecutive- k -out-of- n :F system. In this paper, we study the reliability properties of such systems via system signature. In particular, we obtain a closed form expression for the signature of this system and use it to compute and evaluate several reliability characteristics of the system.

Consider a system with m elements which is used to fulfill tasks. Each task is sent to one element which fulfills a task and the outcome is either fulfillment of the task (''1'') or the failure of the element (''0''). Initially, m tasks are sent to the system. At the second step, a complex of length m"1 is formed and sent to the system, where m"1 is the number of tasks fulfilled at the first step, and so on. The process continues until all elements fail and the corresponding waiting time defines the lifetime of the binary sequence which consists of ''1'' or ''0''. We obtain a recursive equation for the expected value of this waiting time random variable.

Recently, Grabner et al. [Combinatorics of geometrically distributed random variables: run statistics, Theoret. Comput. Sci. 297 (2003) 261–270] and Louchard and Prodinger [Ascending runs of sequences of geometrically distributed random variables: a probabilistic analysis, Theoret. Comput. Sci. 304 (2003) 59–86] considered the run statistics of geometrically distributed independent random variables. They investigated the asymptotic properties of the number of runs and the longest run using the corresponding probability generating functions and a Markov chain approach. In this note, we reconsider the asymptotic properties of such statistics using another approach. Our approach of finding the asymptotic distributions is based on the construction of runs in a sequence of m-dependent random variables. This approach enables us to find the asymptotic distributions of many run statistics via the theorems established for m-dependent sequence of random variables. We also provide the asymptotic distribution of the total number of non-decreasing runs and the longest non-decreasing run.

Consider a system which has n independent components (or subsystems) each consisting of m dependent elements. Let X"i=(X"i^1,X"i^2,...,X"i^m), i=1,2,...,n denote the random strength vector of the ith component, where X"i^j denotes the random strength of the jth element of the ith component. The elements of the components are subjected to a common random stress Y=(Y"1,Y"2,...,Y"m) over time. In this paper, we setup a multivariate stress-strength model based on the conditional ordering between X"is and Y and evaluate the reliability of coherent structures in this setup.

In this article, we study a shock model in which the shocks occur according to a binomial process, i.e. the interarrival times between successive shocks follow a geometric distribution with mean 1/p. According to the model, the system fails when the time between two consecutive shocks is less than a prespecified level. This is the discrete time version of the so-called δ-shock model which has been previously studied for the continuous case. We obtain the probability mass function and probability generating function of the system’s lifetime. We also present an extension of the results to the case where the shock occurrences are dependent in a Markovian fashion.

In this paper we study multi-state systems consisting of two components when the number of system states and the number of states of each component are the same, i.e. the systems under consideration are homogeneous multi-state systems. In particular we evaluate multi-state series and cold standby systems assuming that the degradation in their components follow a Markov process. The behaviour of systems with respect to degradation rates is also investigated in terms of stochastic ordering.