Age-usage semi-Markov models

This paper presents a non-homogeneous age-usage semi-Markov model with a measurable state space. Several probability functions useful to assess the system’s reliability are investigated. They satisfy the same family of equations we call indexed Markov renewal equations. Sufficient conditions to assure the existence and uniqueness of their solutions are provided. The numerical analysis of these equations is executed through the construction of a process discrete in time and space, which is shown to converge to the continuous one in the Skorohod topology. An algorithm useful for solving the discretized system of equations is presented by using a matrix representation.     

On the Existence and Uniqueness of Solution of MRE and Applications

In this paper, we study the existence and uniqueness of the solution for Markov renewal equation (MRE) of a semi-Markov process with countable state space. This method and its proof are based on an iterative scheme. A numerical solution is also given as well as a case study on system reliability assessment.     

Reliability Measures of Semi-Markov Systems with General State Space

The aim of this paper is to present a systematic modeling of reliability and related measures: availability, maintainability, failure rate, rate of occurrence of failures, mean times, etc., known in the literature under the term dependability. This model includes the continuous and discrete time semi-Markov processes with general state space. This is one of the most general models in reliability theory since it includes as particular cases the Markov and renewal processes.     

Fuzzy random renewal process and renewal reward process

So far, there have been several concepts about fuzzy random variables and their expected values in literature. One of the concepts defined by Liu and Liu (2003a) is that the fuzzy random variable is a measurable function from a probability space to a collection of fuzzy variables and its expected value is described as a scalar number. Based on the concepts, this paper addresses two processes—fuzzy random renewal process and fuzzy random renewal reward process. In the fuzzy random renewal process, the interarrival times are characterized as fuzzy random variables and a fuzzy random elementary renewal theorem on the limit value of the expected renewal rate of the process is presented. In the fuzzy random renewal reward process, both the interarrival times and rewards are depicted as fuzzy random variables and a fuzzy random renewal reward theorem on the limit value of the long-run expected reward per unit time is provided. The results obtained in this paper coincide with those in stochastic case or in fuzzy case when the fuzzy random variables degenerate to random variables or to fuzzy variables.     

A novel optimal preventive maintenance policy for a cold standby system based on semi-Markov theory

A novel optimal preventive maintenance policy for a cold standby system consisting of two components and a repairman is described herein. The repairman is to be responsible for repairing either failed component and maintaining the working components under certain guidelines. To model the operational process of the system, some reasonable assumptions are made and all times involved in the assumptions are considered to be arbitrary and independent. Under these assumptions, all system states and transition probabilities between them are analyzed based on a semi-Markov theory and a regenerative point technique. Markov renewal equations are constructed with the convolution of the cumulative distribution function of system time in each state and corresponding transition probability. By using the Laplace transform to solve these equations, the mean time from the initial state to system failure is derived. The optimal preventive maintenance policy that will provide the optimal preventive maintenance cycle is identified by maximizing the mean time from the initial state to system failure, and is determined in the form of a theorem. Finally, a numerical example and simulation experiments are shown which validated the effectiveness of the policy.     

Reliability of several component sets with inspections at random times

We consider a random process which represents a system of components with constant failure rates and subjected to inspections at times defining a renewal process. We give an analytic method for calculating the reliability function, its Laplace transform and the mean time to failure (MTTF). These formulas are computable if the Laplace transform of the inter-arrival law of the renewal process is explicit. We also study the asymptotic behavior of the reliability and determine the asymptotic failure rate of the system.     

基于半马尔科夫过程的逻辑门可靠性分析

动态故障树分析方法是在静态故障树的基础上拓展而来的自上而下的图形化演绎技术,可以很好地对具有复杂失效行为和交互作用的系统进行建模,进而分析系统的可靠性。本文从动态故障树逻辑门的可靠性建模与分析入手,结合半马尔科夫过程原理,将动态逻辑门转化为半马尔科夫链。其次给出在半马尔科夫链中动态逻辑门输出事件的发生概率和系统可靠性的计算公式。提出各种逻辑门到半马尔科夫链的通用转化模型,通过更改通用模型中的相关参数,将逻辑门转化为半马尔科夫链。最后,基于半马尔科夫过程求解动态逻辑门输出事件的发生概率,以动态优先与门、顺序相关门和备件门为例,并给出系统可靠性的计算公式。     

Interval Reliability,Corrections and Developments of “Reliability Measures of Semi-Markov Systems with General State Space”

This paper present the interval reliability of a semi-Markov systems in general state space in continuous and discrete-time cases. We get also, as a particular case the interval reliability for the alternating renewal process. This is a continuation of the paper Limnios (Methodol Comput Appl Probab 14(4):895–917, 2012) where the interval reliability is referred as interval availability and also Propositions 2.1 and 3.1 below replace Propositions 3.4 and 5.3 respectively.     

风险模型中重尾随机变量和的若干大偏差结果

进一步研究随机变量部分和与随机和的大偏差,其中S(n)=∑ni=1Xi,S(t)=∑N(t)i=1Xi(t>0).{Xn,n≥1}是一个独立同分布的随机变量(未必是非负的)序列具有共同的分布F(定义于R上)和有限期望μ=EX1.{N(t),t≥0}是一个非负的整数值的随机变量的更新计数过程且与{Xn,n≥1}相互独立.本文在假定F∈C条件下,进一步推广并改进了由Klüppelberg等和Kaiw等人给出的一些大偏差结果.这些结果可应用到某些金融保险方面的一些特定的问题中去.     

Analysis of a single server queue with semi-Markovian service interruption

In this paper we analyze a discrete-time single server queue where the service time equals one slot. The numbers of arrivals in each slot are assumed to be independent and identically distributed random variables. The service process is interrupted by a semi-Markov process, namely in certain states the server is available for service while the server is not available in other states. We analyze both the transient and steady-state models. We study the generating function of the joint probability of queue length, the state and the residual sojourn time of the semi-Markov process. We derive a system of Hilbert boundary value problems for the generating functions. The system of Hilbert boundary value problems is converted to a system of Fredholm integral equations. We show that the system of Fredholm integral equations has a unique solution. This revised version was published online in June 2006 with corrections to the Cover Date.     

Poisson冲击下的$k/n(G)$系统的可靠性分析

本文研究了一类Poisson冲击下的$k/n(G)$系统(即$k$-out-of-$n$: $G$系统). 假定冲击的到达数形成一个参数为$\lambda$的Poisson过程, 且冲击的量服从某一分布. 当每次冲击到达时, 对系统中工作的部件独立地产生影响. 进而假定每一部件以一定的概率故障, 概率值是冲击量的函数. 且各次冲击独立地对系统造成损失, 直到工作部件数少于$k$系统故障为止. 在这些假定下, 我们获得了系统的可靠度函数和系统的平均工作时间. 进一步, 假定系统是可修的, 系统中有一个维修工, 并根据``先坏先修’’的维修规则对故障部件进行维修. 在维修时间服从指数分布的假设下, 系统状态转移服从Markov过程. 对该系统我们建立了状态转移方程, 并求得了系统可用度、稳态下的平均工作时间、平均停工时间和系统失效频率等可靠性指标. 最后, 我们还给出了一个简单例子来演示讨论的模型.     

Nonparametric Estimation for Semi-Markov Processes Based on its Hazard Rate Functions

The problem of estimating the Markov renewal matrix and the semi-Markov transition matrix based on a history of a finite semi-Markov process censored at time T (fixed) is addressed for the first time. Their asymptotic properties are studied. We begin by the definition of the transition rate of this process and propose a maximum likelihood estimator for the hazard rate functions and then we show that this estimator is uniformly strongly consistent and converges weakly to a normal random variable. We construct a new estimator for an absolute continous semi-Markov kernel and give detailed derivation of uniform strong consistency and weak convergence of this estimator as the censored time tends to infinity. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Positive and Monotone Numerical Scheme for Volterra-Renewal Equations with Space Fluxes

We study a numerical method for solving a system of Volterra-renewal integral equations with space fluxes, that represents the Chapman-Kolmogorov equation for a class of piecewise deterministic stochastic processes. The solution of this equation is related to the time dependent distribution function of the stochastic process and it is a non-negative and non-decreasing function of the space. Based on the Bernstein polynomials, we build up and prove a non-negative and non-decreasing numerical method to solve that equation, with quadratic convergence order in space.     

The two-element system with one restoring device optimum maintenance

The paper considers the problem of the optimum preventive maintenance of the system with two elements and one re¬storing device. The system 's behaviour is described by a semi-Markov process with complex descrete and continuous set of states. Whether the preventive maintenance is performed depends on the state of the system's elements and the level of past life of the element which performs the duties of the main element. The paper contains the solution to the system of integral equations relative to the stationary distribution of a Markov chain included in given semi-Markov process. This results makes it possible to find various-stationary functionals on the process trajectories and lead the task of optimum control to searching the extreme value of given function of two real variables     

Establishment of non-negative constraint method as a robust and efficient first-order reliability method

In structural reliability analysis, computation of reliability index or probability of failure is the main purpose. The Hasofer–Lind and Rackwitz–Fiessler (HL-RF) method is a widely used method in the category of first-order reliability methods (FORM). However, this method cannot be trusted for highly nonlinear limit state functions. Two proposed methods of this paper replace the original real valued constraint of FORM with a non-negative constraint, in all steps and during the whole procedure. First, the non-negative constraint is directly used to construct a non-negative Lagrange function and a search direction vector. Then, the first- and second-order Taylor approximation of the non-negative constraint are employed to compute step sizes of the first and second proposed methods, respectively. Contribution of the non-negative constraint and the effective approach of determining step sizes have led to the efficient computation of reliability index in nonlinear problems. The robustness and efficiency of two proposed methods are shown in various mathematical and structural examples of the literature.     

Mean time to failure and availability of semi-Markov missions with maximal repair

We analyze mean time to failure and availability of semi-Markov missions that consist of phases with random sequence and durations. It is assumed that the system is a complex one with nonidentical components whose failure properties depend on the mission process. The stochastic structure of the mission is described by a Markov renewal process. We characterize mean time to failure and system availability under the maximal repair policy where the whole system is replaced by a brand new after successfully completing a phase before the next phase starts. Special cases involving Markovian missions are also considered to obtain explicit formulas.     

A Bayesian model and numerical algorithm for CBM availability maximization

In this paper, we consider an availability maximization problem for a partially observable system subject to random failure. System deterioration is described by a hidden, continuous-time homogeneous Markov process. While the system is operational, multivariate observations that are stochastically related to the system state are sampled through condition monitoring at discrete time points. The objective is to design an optimal multivariate Bayesian control chart that maximizes the long-run expected average availability per unit time. We have developed an efficient computational algorithm in the semi-Markov decision process (SMDP) framework and showed that the availability maximization problem is equivalent to solving a parameterized system of linear equations. A numerical example is presented to illustrate the effectiveness of our approach, and a comparison with the traditional age-based replacement policy is also provided.     

Aggregated semi-Markov repairable systems with history-dependent up and down states

Some states in the aggregated semi-Markov repairable system with history-dependent up and down states are changeable in the sense that whether those physical states are up and down depends on the immediately preceding state of the system evolution process. Two reliability indices of the system, the frequency of failures and the time to the first system failure are given. The Laplace–Stieltjes transforms of several time distributions in a cycle, such as the up and down time, the total time the system is in the up, down and changeable states, the length of a single sojourn in the up, down and changeable states are derived. The means of them are also presented. Markov renewal theory, transform and matrix methods are employed for getting these performance measures. A numerical example is given to illustrate the results in the paper.     

Reliability analysis for devices subject to competing failure processes based on chance theory

This paper studies the reliability for devices subject to independent competing failure processes of degradation and shocks in an uncertain random environment. The continuous degradation is governed by an uncertain process, and external shocks arrive according to an uncertain random renewal reward process, in which the inter-arrival times of shocks and the shock sizes are assumed to be random variables and uncertain variables, respectively. The device reliability is defined as the chance measure that the uncertain degradation signals do not exceed a soft failure threshold L, and the uncertain random shocks do not cause the device failure. The device reliability is obtained by employing chance theory under four different shock patterns. Finally, a case study on a gas insulated transmission line is carried out to show the implementation of the proposed model.     

Numerical Bounds for Semi-Markovian Quantities and Application to Reliability

We propose new easily computable bounds for different quantities which are solutions of Markov renewal equations linked to some continuous-time semi-Markov process (SMP). The idea is to construct two new discrete-time SMP which bound the initial SMP in some sense. The solution of a Markov renewal equation linked to the initial SMP is then shown to be bounded by solutions of Markov renewal equations linked to the two discrete time SMP. Also, the bounds are proved to converge. To illustrate the results, numerical bounds are provided for two quantities from the reliability field: mean sojourn times and probability transitions.