Compound Poisson Approximation for Multiple Runs in a Markov Chain

We consider a sequence X ₁, ..., X _n of r.v.'s generated by a stationary Markov chain with state space A = {0, 1, ..., r}, r 1. We study the overlapping appearances of runs of k _i consecutive i's, for all i = 1, ..., r, in the sequence X ₁,..., X _n. We prove that the number of overlapping appearances of the above multiple runs can be approximated by a Compound Poisson r.v. with compounding distribution a mixture of geometric distributions. As an application of the previous result, we introduce a specific Multiple-failure mode reliability system with Markov dependent components, and provide lower and upper bounds for the reliability of the system.     

Fast formula of a reliability ofm-consecutive-k-out-of-n:F system with cyclek

Anm-consecutive-k-out-of-n:F system is a system ofn linearly arranged components which fails if and only if at leastm non-overlapping sequences ofk components fail, when there arek distinct components with failure probabilitiesq _i fori=1,...,k and where the failure probability of thej-th component (j=rk+i (1 ≤i ≤k) isq _j =q _i , we call this system by anm-consecutive-k-out-of-n:F system with cycle (or period)k. In this paper we give a formula of the failure probability ofm-consecutive-k-out-of-n:F system with cyclek via the failure probability of consecutive-k-out-of-n:F system.     

Reliability formula &; limit law of the failure time of “<Emphasis Type="Italic">m</Emphasis>-consecutive-<Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis>:<Emphasis Type="Italic">F</Emphasis> system”

An “m-consecutive-k-out-of-n:F system” consists of n components ordered on a line; the system fails if and only if there are at least m nonoverlapping runs of k consecutive failed components. In this paper, we give a recursive formula to compute the reliability of such a system. Thereafter, we state two asymptotic results concerning the failure time Z _n of the system. The first result concerns a limit theorem for Z _n when the failure times of components are not necessarily with identical failure distributions. In the second one, we prove that, for an arbitrary common failure distribution of components, the limit system failure distribution is always of the Poisson class.      

A limit theorem of certain repairable systems

Many large engineering systems can be viewed (or imbedded) as a series system in time. In this paper, we introduce the structure of a repairable system and the reliabilities of these large systems are studied systematically by studying the ergodicities of certain non-homogeneous Markov chains. It shows that if the failure probabilities of components satisfy certain conditions, then the reliability of the large system is approximately exp (-) for some >0. In particular, we demonstrate how the repairable system can be used for studying the reliability of a large linearly connected system. Several practical examples of large consecutive-k-out-of-n:F systems are given to illustrate our results. The Weibull distribution is derived under our natural set-up.This research work was partially supported by the National Science Council of the Republic of China.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant A-9216, and by the National Science Council of the Republic of China.     

Optimal Consecutive-k-out-of-(2k+1): G Cycle

We present a complete proof for the invariant optimal assignment for consecutive-k-out-of-(2k+1): G cycle, which was proposed by Zuo and Kao in 1990 with an incomplete proof, pointed out recently by Jalali, Hawkes, Cui and Hwang.     

The Hazard Rate and the Reversed Hazard Rate Orders, with Applications to Order Statistics

In this paper we first point out a simple observation that can be used successfully in order to translate results about the hazard rate order into results about the reversed hazard rate order. Using it, we derive some interesting new results which compare order statistics in the hazard and in the reversed hazard rate orders; as well as in the usual stochastic order. We also simplify proofs of some known results involving the reversed hazard rate order. Finally, a few further applications of the observation are given.     

Bivariate Markov chain embeddable variables of polynomial type

The primary aim of the present article is to provide a general framework for investigating the joint distribution of run length accumulating/enumerating variables by the aid of a Markov chain embedding technique. To achieve that we introduce first a class of bivariate discrete random variables whose joint distribution can be described by the aid of a Markov chain and develop formulae for their joint probability mass function, generating functions and moments. The results are then exploited for the derivation of the distribution of a bivariate run-related statistic. Finally, some interesting uses of our results in reliability theory and educational psychology are highlighted. Research supported by General Secretary of Research and Technology of Greece under grand PENED 2001.     

Consecutive expansions of -out-of- systems

We introduce consecutive expansions of k–out–of–n systems, which have the property that components are totally ordered by the node criticality relation and with respect to well-known structural importance measures. We propose some formulae to easily compute these measures and study the hierarchies induced for them for large systems.     

Some random approximations and random fixed point theorems for 1-set-contractive random operators

In this paper, we will prove that the random version of Fan's Theorem (Math. Z. 112 (1969), 234-240) is true for 1-set-contractive random operator , where is a weakly compact separable closed ball in a Banach space and is a measurable space. This class of 1-set-contractive random operator includes condensing random operators, semicontractive random operators, LANE random operators, nonexpansive random operators and others. As applications of our theorems, some random fixed point theorems of non-self-maps are proved under various well-known boundary conditions.

     

Reliability analysis for the consecutive-k-out-of-n: F system with repairmen taking multiple vacations

Commonly studied models of the consecutive-k-out-of-n: F repairable systems in the existing literatures were considering the systems which had one repairman without vacation or infinite repairmen without vacations. In addition to those models, multiple repairmen without vacations are studied occasionally. However, technical personnel are very short in some fields. Some failed components cannot be repaired in time. This paper deals with the phenomenon of waiting for repair by supposing R repairmen with multiple vacations in the system. Using the pairs (i, |j|), the factor that the R repairmen taking multiple vacations was embedded into the classical C(k, n: F) system. Reliability indexes are presented. Finally, the Runge–Kutta method was used to a special case, and the experimental results demonstrate the necessity and validity of the new model.     

Reliability Approximation for Markov Chain Imbeddable Systems

In the present article, a simple method is developed for approximating the reliability of Markov chain imbeddable systems. The approximating formula reduces the problem to the reliability assessment of smaller systems with structure similar to the original systems. Two specific reliability structures which have attracted considerable research interest recently (r-within-consecutive-k-out-of-n system and two dimensional r-within-k₁ × k₂-out-of-n₁ × n₂ system) are studied by the new approach and numerical calculations are carried out, which reveal the high quality of our approximations. Several possible extensions and generalizations are also presented in brief.     

Algorithm for a general discrete k-out-of-n: G system subject to several types of failure with an indefinite number of repairpersons

A discrete k-out-of-n: G system with multi-state components is modelled by means of block-structured Markov chains. An indefinite number of repairpersons are assumed and PH distributions for the lifetime of the units and for the repair time are considered. The units can undergo two types of failures, repairable or non-repairable. The repairability of the failure can depend on the time elapsed up to failure. The system is modelled and the stationary distribution is built by using matrix analytic methods. Several performance measures of interest, such as the conditional probability of failure for the units and for the system, are built into the transient and stationary regimes. Rewards are included in the model. All results are shown in a matrix algorithmic form and are implemented computationally with Matlab. A numerical example of an optimization problem shows the versatility of the model.     

Lifetime distribution and estimation problems of consecutive-k-out-of-n:F systems

Explicit formula is given for the lifetime distribution of a consecutive-k-out-of-n:F system. It is given as a linear combination of distributions of order statistics of the lifetimes of n components. We assume that the lifetimes are independent and identically distributed. The results should make it possible to treat the parametric estimation problems based on the observations of the lifetimes of the system. In fact, we take up, as some examples, the cases where the lifetimes of the components follow the exponential, the Weibull, and the Pareto distributions, and obtain feasible estimators by moment method. In particular, it is shown that the moment estimator is quite good for the exponential case in the sense that the asymptotic efficiency is close to one.This research was partially supported by the ISM Cooperative Research Program (94-ISM-CRP-5).     

Consecutive k-out-of-n systems with maintenance

A consective k-out-of-n system consists of n linearly or cycliccally ordered components such that the system fails if and only if at least k consecutive components fail. In this paper we consider a maintained system where each component is repaired independently of the others according to an exponential distribution. Assuming general lifetime distributions for system's components we prove a limit theorem for the time to first failure of both linear and circular systems.     

Review of Some Functionals of Compound Poisson Processes and Related Stopping Times

The paper reviews recent results of D. Perry, W. Stadje and S. Zacks, on functionals of stopping times and the associated compound Poisson process with lower and upper linear boundaries. In particular, formulae of these functionals are explicitly developed for the total expected discounted cost of discarded service in an M/G/1 queue with restricted accessibility; for the expected total discounted waiting cost in an M/G/1 restricted queue; for the shortage, holding and clearing costs in an inventory system with continuous input; for the risk in sequential estimation and for the transform of the busy period when the upper boundary is random.      

Fluid approximations for a processor-sharing queue

In this paper a fluid approximation, also known as a functional strong law of large numbers (FSLLN) for a GI/G/1 queue under a processor-sharing service discipline is established and its properties are analysed. The fluid limit depends on the arrival rate, the service time distribution of the initial customers, and the service time distribution of the arriving customers. This is in contrast to the known result for the GI/G/1 queue under a FIFO service discipline, where the fluid limit is piecewise linear and depends on the service time distribution only through its mean. The piecewise linear form of the limit can be recovered by an equilibrium type choice of the initial service distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

Nonparametric inference for sequential <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> systems

The k-out-of-n model is commonly used in reliability theory. In this model the failure of any component of the system does not influence the components still at work. Sequential k-out-of-n systems have been introduced as an extension of k-out-of-n systems where the failure of some component of the system may influence the remaining ones. We consider nonparametric estimation of the cumulative hazard function, the reliability function and the quantile function of sequential k-out-of-n systems. Furthermore, nonparametric hypothesis testing for sequential k-out-of-n-systems is examined. We make use of counting processes to show strong consistency and weak convergence of the estimators and to derive the asymptotic distribution of the test statistics.     

Approximation to the Sobolev classesW q r of functions of several variables by bilinear forms inL p for 2≤q≤p≤∞

We study the approximation of functions of several variables by bilinear forms that are the pairwise products of functions of fewer variables. The order of approximation of Sobolev classesW _q ^r by bilinear forms inL _p for 2≤q≤p≤∞ is found. Translated by N. K. Kulman Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 18–34, July, 1997.     

Reliability and covariance estimation of weighted k-out-of-n multi-state systems

In the literature of reliability engineering, reliability of the weighted k-out-of-n system can be calculated using component reliability based on the structure function. The calculation usually assumes that the true component reliability is completely known. However, this is not the case in practical applications. Instead, component reliability has to be estimated using empirical sample data. Uncertainty arises during this estimation process and propagates to the system level. This paper studies the propagation mechanism of estimation uncertainty through the universal generating function method. Equations of the complete solution including the unbiased system reliability estimator and the corresponding unbiased covariance estimator are derived. This is a unified approach. It can be applied to weighted k-out-of-n systems with multi-state components, to weighted k-out-of-n systems with binary components, and to simple series and parallel systems. It may also serve as building blocks to derive estimators of system reliability and uncertainty measures for more complicated systems.     

Reliability evaluation of multi-state consecutive k-out-of-r-from-n: G system

This paper proposes a model that generalizes the linear consecutive k-out-of-r-from-n: G system to multi-state case. In this model the system consists of n linearly ordered multi-state components. Both the system and its components can have different states: from complete failure up to perfect functioning. The system is in state j or above if and only if at least k_j components out of r consecutive are in state j or above. An algorithm is provided for evaluating reliability of a special case of multi-state consecutive k-out-of-r-from-n: G system. The algorithm is based on the application of the total probability theorem and on the application of a special case taken from the [Jinsheng Huang, Ming J. Zuo, Member IEEE and Yanhong Wu, Generalized multi-state k-out-of-n: G system, IEEE Trans. Reliab. 49(1) (2000) 105–111.]. Also numerical results of the formerly published test examples and new examples are given.