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.     

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.     

Active redundancy allocation for a k-out-of-n:F system of dependent components

We characterize active redundancy through compensator transform and use the reverse rule of order 2 (RR₂) property between compensator processes to investigate the problem of where to allocate a spare in a k-out-of-n:F system of dependent components through active redundancy.     

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.     

Reliability of a consecutive (r, s)-out-of-(m, n):F lattice system with conditions on the number of failed components in the system

A consecutive(r, s)-out-of-(m, n):F lattice system which is defined as a two-dimensional version of a consecutive k-out-of-n:F system is used as a reliability evaluation model for a sensor system, an X-ray diagnostic system, a pattern search system, etc. This system consists of m × n components arranged like an (m, n) matrix and fails iff the system has an (r, s) submatrix that contains all failed components. In this paper we deal a combined model of a k-out-of-mn:F and a consecutive (r, s)-out-of-(m, n):F lattice system. Namely, the system has one more condition of system down, that is the total number of failed components, in addition to that of a consecutive (r, s)-out-of-(m, n):F lattice system. We present a method to obtain reliability of the system. The proposed method obtains the reliability by using a combinatorial equation that does not depend on the system size. Some numerical examples are presented to show the relationship between component reliability and system reliability.     

Combined m-consecutive and k-out-of-n sliding window systems

This paper proposes a new model that generalizes the linear multi-state sliding window system. In this model the system consists of n linearly ordered multi-state elements. Each element can have different states: from complete failure up to perfect functioning. A performance rate is associated with each state. The system fails if at least one of the following two conditions is met: (1) there exist at least m consecutive overlapping groups of r adjacent elements having the cumulative performance lower than V; (2) there exist at least k arbitrarily located groups of r adjacent elements having the cumulative performance lower than W. An algorithm for system reliability evaluation is suggested which is based on an extended universal moment generating function. Examples of evaluating system reliability and elements’ reliability importance indices are presented. Optimal sequencing of system elements is demonstrated.     

Bounds for increasing multi-state consecutive k-out-of-r-from-n: F system with equal components probabilities

The consecutive k-out-of-r-from-n: F system was generalized to multi-state case. This system consists of n linearly ordered components which are at state below j if and only if at least k_j components out of any r consecutive are in state below j. In this paper we suggest bounds of increasing multi-state consecutive-k-out-of-r-from-n: F system (k₁ ? k₂ ? ? ? k_M) by applying second order Boole–Bonferroni bounds and applying Hunter–Worsley upper bound. Also numerical results are given. The programs in V.B.6 of the algorithms are available upon request from the authors.     

Parallel and k-out-of-n: G systems with nonidentical components and their mean residual life functions

A system with n independent components which has a k-out-of-n: G structure operates if at least k components operate. Parallel systems are 1-out-of-n: G systems, that is, the system goes out of service when all of its components fail. This paper investigates the mean residual life function of systems with independent and nonidentically distributed components. Some examples related to some lifetime distribution functions are given. We present a numerical example for evaluating the relationship between the mean residual life of the k-out-of-n: G system and that of its components.     

On the GI/M/1/N queue with multiple working vacations—analytic analysis and computation

We consider finite buffer single server GI/M/1 queue with exhaustive service discipline and multiple working vacations. Service times during a service period, service times during a vacation period and vacation times are exponentially distributed random variables. System size distributions at pre-arrival and arbitrary epoch with some important performance measures such as, probability of blocking, mean waiting time in the system etc. have been obtained. The model has potential application in the area of communication network, computer systems etc. where a single channel is allotted for more than one source.     

Reliability analysis of a k-out-of-n:G repairable system with single vacation

This paper analyzes a k-out-of-n:G   repairable system with one repairman who takes a single vacation, the duration of which follows a general distribution. The working time of each component is an exponentially distributed random variable and the repair time of each failed component is governed by an arbitrary distribution. Moreover, we assume that every component is “as good as new” after being repaired. Under these assumptions, several important reliability measures such as the availability, the rate of occurrence of failures, and the mean time to first failure of the system are derived by employing the supplementary variable technique and the Laplace transform. Meanwhile, their recursive expressions are obtained. Furthermore, through numerical examples, we study the influence of various parameters on the system reliability measures. Finally, the Monte Carlo simulation and two special cases of the system which are (n-1)$(n-1)$-out-of-n:G repairable system and 1-out-of-n:G repairable system are presented to illustrate the correctness of the analytical results.     

Fluid model driven by an M/G/1 queue with multiple exponential vacations

This paper studies a fluid model driven by an M/G/1 queue with multiple exponential vacations. By introducing various vacation strategies to the fluid model, we can provide greater flexibility for the design and control of input rate and output rate. The Laplace transform of the steady-state distribution of the buffer content is expressed through the minimal positive solution to a crucial equation. Then the performance measure-mean buffer content, which is independent of the vacation parameter, is obtained. Finally, with some numerical examples, the parameter effect on the mean buffer content is presented.     

Analysis of the MAP/G/1/N queue with multiple vacations

This paper deals with a batch service queue and multiple vacations. The system consists of a single server and a waiting room of finite capacity. Arrival of customers follows a Markovian arrival process (MAP). The server is unavailable for occasional intervals of time called vacations, and when it is available, customers are served in batches of maximum size ‘b’ with a minimum threshold value ‘a’. We obtain the queue length distributions at various epochs along with some key performance measures. Finally, some numerical results have been presented.     

The influence of delivery times on repairable k-out-of-N systems with spares

The k-out-of-N structure is a popular type of redundancy in fault-tolerant systems with wide applications in computer and communication systems, and power transmission and distribution systems, among others, during the past several decades. In this paper, our interest is in such a reliability system with identical, repairable components having exponential life times, in which at least k out of N components are needed for the system to perform its functions. There is a single repairman who attends to failed components on a first-come-first-served basis. The repair times are assumed to be of phase type. The system has K spares which can be tapped to extend the lifetime of the system using a probabilistic rule. We assume that the delivery time of a spare is exponentially distributed and there could be multiple requests for spares at any given time. Our main goal is to study the influence of delivery times on the performance measures of the k-out-of-N reliability system. To that end, the system is analyzed using a finite quasi-birth-and-death process and some interesting results are obtained.     

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.      

Computing tolerance limits for the lifetime of a k-out-of-n:F system based on prior information and censored data

Exact guaranteed-coverage and expected-coverage Bayesian tolerance limits for the lifetime distribution of a k-out-of-n:F system are computed by solving nonlinear equations. The bounds are based on exponential component test data and available prior information concerning the expected component lifetime which is described by an inverted gamma distribution. The Bayesian tolerance limits are valid for single (right or left), double and progressive (standard or general) censoring, and even have frequentist validity in the noninformative case. The derived results allow the reliability engineer to judge the quality of a system prior to assembly, which offers obvious practical and economic benefits. Minimum and expected percentages of conforming systems are assessed by constructing suitable tolerance limits. Even though the viewpoints are different, the Bayesian tolerance limits that adopt the natural diffuse prior coincide numerically with recently published conditional tolerance limits in the double censoring case. The proposed Bayesian approach may be deemed as an extension of the existing frequentist methodology under double censoring that also takes into account the presence of prior information and general progressive censoring. The perspective developed simplifies and unifies the computation of tolerance limits with both frequentist and Bayesian interpretations, and also provides a probabilistic way of updating the tolerance limits in the light of new, relevant data, which is especially important in the dynamic analysis of a sequence of data. Moreover, the Bayesian approach is shown to outperform the frequentist viewpoint in terms of accuracy. In most situations, the use of substantial prior information significantly increases the accuracy level and considerably reduces the required number of failures to attain a specified degree of accuracy. Two illustrative numerical examples are studied, including the analysis of a system of water pumps for cooling a reactor. The results developed are extended to the Weibull case with unknown scale parameter and other probability models.     

Composition operators from F(p, q, s) spaces to the nth weighted-type spaces on the unit disc

In this note we characterize the boundedness and compactness of the composition operator from the general function space F(p, q, s) to the nth weighted-type space on the unit disk, where the nth weighted-type space has been recently introduced by Stevo Stevi?.     

Busy period analysis for the GI/M/1 queue with exponential vacations

We consider the busy period in the GI/M/1 queue with multiple exponential vacations. We derive the transform of the joint distribution of the length of a busy period, the number of customers served during the busy period, and the residual interarrival time at the instant the busy period ends.     

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.