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In this paper, state space of a time-homogeneous Markov process is split into several ordered subspaces. Then, there are three kinds of transitions between states—transitions from a higher-order subspace to a lower-order subspace, transitions within the same subspace, and transitions from a lower-order subspace to a higher-order subspace. Considering time interval omission problem and first passage time considered, we define some related counting processes for the Markov process and discuss their associated probabilities, expectations and generating functions by using Laplace transform. The main results are presented in matrix forms. The relationships among counting processes and their special cases are also discussed briefly. Some simple numerical examples are presented to illustrate the established results. These results will be useful in different problems arising in reliability, economics and social science fields. [ABSTRACT FROM AUTHOR]

Linear Hawkes processes are widely used in many fields and means are the basic and critical information of them. However, there is little research on linear Hawkes processes' means. In this paper, we present a numerical method based on the Laplace transform and inverse Laplace transform for means of linear Hawkes processes. The advantage of this method is that whatever the kernel function is, we can always obtain the numerical solutions of means for a linear Hawkes process. In addition, this numerical method provides the basic information of linear Hawkes processes by means. As an application, the numerical method is applied in a WeChat network model. [ABSTRACT FROM AUTHOR]

Many practical systems are subject to neglected down time, i.e., if the down time is less than a given threshold, then the system can be considered as working during the down time. Otherwise, the system is considered as in the failure state. This paper investigates the availability and maintenance policy for inspected systems with neglected down time. The instantaneous and steady-state availabilities of the system are derived analytically. Furthermore, a maintenance model is developed to find the optimal inspection interval, , which minimizes the long-run average cost rate. Based on the developed maintenance model, the analytical properties of the optimal inspection policy are explored. Numerical examples on ventilator system calculating the availability and optimal inspection policy are given to illustrate the applicability of the proposed model. [ABSTRACT FROM AUTHOR]

Availability analysis is an important issue in many practical fields. This paper investigates the availability for general repairable systems with repair time threshold. Based on practical applications, a repair time threshold is introduced. If the period of a repair is less than a predefined time threshold, then the system may be considered as working during this period, i.e., the effect of the repair could be neglected. Otherwise, if the period of a repair is longer than the given threshold, then the system is considered as working from the beginning of the system failure until the repair time exceeding the threshold, i.e., the time point of the system down could be delayed. We consider both constant and random repair time threshold. This paper valuates the user-perceived availability, when the user does not experience any service interruption because the duration of repair is too short. The results can be applied in reliability engineering, queueing theory and many other fields. A numerical example for ventilator system is presented to demonstrate the application of the developed approach. [ABSTRACT FROM AUTHOR]

To accurately describe the performance of repairable systems operating under alternative environments, for example, mild/harsh, working/idling, maximum/minimum level demand etc., a Semi-Markov process with a finite state space and two different Semi-Markov kernels is introduced. The state set of the system which is regarded as acceptable might depend on the environments. Two important reliability indices, the availability and time to the first system failure, are obtained via Markov renewal theory, transform and matrix methods. The results and numerical examples are also provided for two special cases: (1) when sojourn times under alternative environments are constants and (2) when sojourn times under environments have exponential distributions. [ABSTRACT FROM AUTHOR]

Phased-mission systems (PMS) can be widely found in a lot of practical application areas. Reliability evaluations and analysis for this kind of systems become important issues. The reliability of PMS is typically defined as the probability that the system successfully accomplishes the missions of all phases. However, thek-out-of-nsystem success criterion for PMS has not been investigated. In this paper, according to this criterion, we develop two new models, which are static and dynamic, respectively. The assumptions for these two models are described in detail as well. The system reliabilities for both models are presented for the first time by employing finite Markov chain imbedding approach (FMCIA). In terms of FMCIA, we define different state spaces for the two models, and transition probability matrices are obtained. Then some numerical examples are given to illustrate the application of FMCIA. Finally, some discussions are made and conclusions are summarized. [ABSTRACT FROM AUTHOR]

This paper studies a cold standby repairable system with two identical components and one repairman having multiple vacations applying matrix-analytic methods. The lifetime of the component follows a phase-type distribution. The repair times and the vacation times of the repairman are governed by different phase-type distributions, respectively. For this system, the Markov process governing the system is constructed. The system is studied in a transient and stationary regime, the availability, the reliability, the rates of occurrence of the different types of failures, and the working probability of the repairman are calculated, respectively. A numerical application is performed to illustrate the calculations. [ABSTRACT FROM AUTHOR]

In this article, two new approaches are introduced to design attributes single plans, and the corresponding models are constructed separately. For Approach I, an algorithm is proposed to design sampling plans by setting a goal function to fulfill the two-point conditions on the operating characteristic curve. For Approach II, the plan parameters are solved by a nonlinear optimization model which minimizes the integration of the probability of acceptance in the interval from the producer's risk quality to the consumer's risk quality. Then numerical examples and discussions based on numerical computation results are given to illustrate the approaches, and tables of the designed plans under various conditions are provided. Moreover, a fact is given to be proved that there is a relation between the conventional design and the new approaches. [ABSTRACT FROM AUTHOR]

Supply chain management has received considerable attention in the literature and it is meaningful and important to be able to measure the reliability of supply chains. In the article, the suppliers in the supply chain systems are not independent of each other and the dependency relation may be either linear or nonlinear correlation. From the view of the distribution service process, a copula-based method is proposed for analyzing the reliability of supply chains. In this article, by introducing the model of k-out-of-n: G system into the studies of supply chains, an evaluation method is suggested and the reliability indexes are obtained. Finally, a numerical example is presented to illustrate the results obtained in this article. [ABSTRACT FROM AUTHOR]

This article studies reliability for a Markov repairable two-item cold standby system with neglected failures. In the system, if a failed time of the system is too short (less than a given critical value) to cause the system to fail, then the failed time may be omitted from the downtime record, i.e., the failure effect could be neglected. In ion-clwinnel modeling, this situation is called the time interval omission problem. The availability indices and the mean downtime are presented as two measures of reliability for this repairable system. Some numerical examples are shown to illustrate the results obtained in this article. [ABSTRACT FROM AUTHOR]

This article considers nonparametric estimation of reliable life based on ranked set sampling and its properties. It is proven analytically that the large sample efficiency of the reliable life estimator under the balanced ranked set sampling is higher than that under the simple random sampling of the same size, but the relative efficiency damps away as the reliable life moves away from the median on both directions. To improve the efficiency for the estimation of extreme reliable life, we then propose a reliable life estimator under a modified ranked set sampling protocol, its strong consistency and asymptotic normality are established. The proposed sampling is shown to be superior to the balanced ranked set sampling, and the relative advantage improves as the reliable life moves away from median. Finally, results of simulation studies for small sample as well as an application to a real data set are presented to illustrate some of the theoretical findings. [ABSTRACT FROM AUTHOR]

The computation of reliability characteristics of a system that consists of dependent components is sometimes difficult especially when the type of dependence is not known. This article introduces the copula method to calculate the reliability of dependent consecutive k-out-of-n: G systems. The components in these systems are dependent on each other and the dependency may be either linear or nonlinear. The copula is a popular tool for modeling the dependence structure of data. It contains the information about the dependency structure of a vector of random variables and can capture nonlinear dependence. Based on the copula theory, the article analyzes the consecutive k-out-of-n: G systems and gets the reliability indexes. Finally, some numerical examples are presented to illustrate the results obtained in this article. [ABSTRACT FROM AUTHOR]

Renewal-type equations are frequently encountered in the study of reliability, warranty analysis, replacement and maintenance policies, and inventory control. Renewal equations usually do not have analytical solutions, and hence, bounds or approximations are very useful. In this article, analytical bounds are studied based on a simple iterative procedure which provides some analytical results and nice convergence properties when the number of iteration increases. Bounds and approximations are also investigated for a recursive algorithm for numerical computation. In addition, some interesting monotonicity properties are introduced and discussed. The approximation error, which is important for determining the stopping rule of the iterative procedure and the numerical algorithm, is also studied. [ABSTRACT FROM AUTHOR]

The (n,f,k(i,j)):F(⟨ n,f,k(i,j)>:F) system consists of n components ordered in a line or circle, while the system fails if, and only if, there exist at least f failed components OR (AND) at least k consecutive failed components among components i,i + 1,..,j - 1,j. In this article, we present the system reliability formulae for these systems with product of matrices by means of a two-stage finite Markov chain imbedding approach, a technique first used by Cui et al. (2002). In addition, their dual systems, denoted by (n,f,k(i,j)):G and ⟨ n,f,k(i,j)>:G, are also introduced. Two numerical examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

A preface for the 35(2006) issue of "Communications in Statistics-Theory and Methods," is presented.