基于交替更新过程的模糊随机破产模型

假设索赔额、盈余额和更新过程均是在模糊随机环境中,并且将索赔过程定义为在交替更新过程.当索赔额和时间间隔是服从不同的指数分布时,本文建立了交替更新过程下的模糊随机破产模型,并给出了最终破产概率公式与最终破产机会均值公式.     

Limit Theorems for Stopped Functionals of Markov Renewal Processes

Some results for stopped random walks are extended to the Markov renewal setup where the random walk is driven by a Harris recurrent Markov chain. Some interesting applications are given; for example, a generalization of the alternating renewal process.     

Warranty cost analysis: Increasing warranty repair times

In this study, we model the warranty servicing costs under nonrenewing and renewing free repair warranties. We assume nonzero increasing repair times with the warranty cost depending on the length of the repair time. An increasing geometric process is used to model the consecutive repair times. We introduce the generalised alternating renewal process, which is an alternating process with cycles consisting of an item's operational time followed by the corresponding repair time. We derive analytical results for a generalised alternating renewal process with a finite time horizon and use them to evaluate the warranty costs over the warranty period and over the life cycle of the product under the nonrenewing free repair warranty and renewing free repair warranty. Properties of the model are demonstrated through a simulation study and through the application to warranty claims data from an automotive manufacturer.     

Simulation of First-Passage Times for Alternating Brownian Motions

The first-passage-time problem for a Brownian motion with alternating infinitesimal moments through a constant boundary is considered under the assumption that the time intervals between consecutive changes of these moments are described by an alternating renewal process. Bounds to the first-passage-time density and distribution function are obtained, and a simulation procedure to estimate first-passage-time densities is constructed. Examples of applications to problems in environmental sciences and mathematical finance are also provided.AMS 2000 Subject Classification: 60J65, 60G40, 93E30     

Upper and lower bounds for the solutions of Markov renewal equations

Upper and lower bounds are studied for the solutions of Markov renewal equations. Some of their special cases are derived under specific marginal conditons and in an alternating environment. The method to construct the bounds is also explained in detail. At the end, these bounds are applied to a shock model and an age-dependent branching process under Markovian environment.     

Occupation times for the finite buffer fluid queue with phase-type ON-times

In this short communication we study a fluid queue with a finite buffer. The performance measure we are interested in is the occupation time over a finite time period, i.e., the fraction of time the workload process is below some fixed target level. Using an alternating renewal sequence, we determine the double transform of the occupation time; the occupation time for the finite buffer M/G/1 queue with phase-type jumps follows as a limiting case.     

Supply chain inventory optimization with two customer classes in discrete time

In this paper we consider a single-item inventory system where two demand classes with different service requirements are satisfied from a common inventory. A critical level, reorder point, order quantity or (s, q, k) policy is in use. The time axis is divided into discrete time units, which is a common characteristic of many real-life supply-chain processes. The inventory process within the lead time of a replenishment order is modelled as a sequence of (1) an ordinary renewal process and (2) two alternating renewal processes. Approximations are developed for the demand class-specific fill rates and the probability distribution of the waiting time of low priority customer orders. This waiting time distribution is used for the inventory allocation in a two-stage supply chain.     

Exact transient analysis of a planar random motion with three directions

Consider a planar random motion with constant velocity and three directions forming the angles ~ /6, 5 ~ /6 and 3 ~ /2 with the x -axis, such that the random times between consecutive changes of direction perform an alternating renewal process. We obtain the probability law of the bidimensional stochastic process which describes location and direction of the motion. In the Markovian case when the random times between consecutive changes of direction are exponentially distributed, the transition densities of the motion are explicitly given. These are expressed in term of a suitable modified two-index Bessel function.     

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.     

Diffusion processes with delay at the endpoints of a segment

A continuous semi-Markov process with a segment as the range of values is considered. This process coincides with a diffusion process inside the segment, i.e., up to the first hitting time of the boundary of the segment and at any time when the process leaves the boundary. The class of such processes consists of Markov processes with reflection at the boundaries (instantaneously or with a delay) and semi-Markov processes with intervals of constancy on some boundary. We derive conditions of existence of such a process in terms of a semi-Markov transition generating function on the boundary. The method of imbedded alternating renewal processes is applied to find a stationary distribution of the process. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 284–297.     

The stochastic behavior of a buffer with non-identical input lines

A system consisting of a buffer, N input lines leading to it and one line leading out is considered. Successive active and idle periods on the input lines constitute an alternating renewal process of a special kind. While in previous work the case of identical input lines was considered, the present paper gives a solution to the general case of non-identical input lines. This provides a tool for the analysis of arbitrarily complicated networks of buffers. The paper contains results regarding the traffic pattern on the output lineas well as the content of the buffer and the maximum content of the buffer during intervals of non-emptiness.     

Capacity expansion with interrupted demand growth

In most long-range planning problems accurate forecasting of demand growth is often difficult. Sometimes unexpected shocks in the economy occur and disrupt the predicted growth rate of demand. For example, growth can unexpectedly cease for and indeterminate length of time due to crucial occurrnces such as wars, economic recessions or oil price shocks.In this paper, we examine the effects of unforeseen demand plateaus on the capacity expansion problem of sizing and timing production facilities. We assume that demand for capacity follows a stochastic process. That is, we allow for plateaus to happen at random future times and to continue for an uncertain duration. When the demand process is generated by an alternating renewal process, we show that demand plateaus result in a modified discounting rate and a modified import cost.     

Heavy-traffic limits for an infinite-server fork–join queueing system with dependent and disruptive services

We study an infinite-server fork–join queueing system with dependent services, which experiences alternating renewal service disruptions. Jobs are forked into a fixed number of parallel tasks upon arrival and processed at the corresponding parallel service stations with multiple servers. Synchronization of a job occurs when its parallel tasks are completed, i.e., non-exchangeable. Service times of the parallel tasks of each job can be correlated, having a general continuous joint distribution function, and moreover, the service vectors of consecutive jobs form a stationary dependent sequence satisfying the strong mixing (\(\alpha \)-mixing) condition. The system experiences renewal alternating service disruptions with up and down periods. In each up period, the system operates normally, but in each down period, jobs continue to enter the system, while all the servers will stop working, and services received will be conserved and resume at the beginning of the next up period. We study the impact of both the dependence among service times and these down times upon the service dynamics, the unsynchronized queueing dynamics, and the synchronized process, assuming that the down times are asymptotically negligible. We prove FWLLN and FCLT for these processes, where the limit processes in the FCLT possess a stochastic decomposition property and the convergence requires the Skorohod \(M_1\) topology.     

Delayed renewal process with uncertain interarrival times

Delayed renewal process is a special type of renewal process in which the first interarrival time is quite different from the others. This paper first proposes an uncertain delayed renewal process whose interarrival times are regarded as uncertain variables. Then it gives an uncertainty distribution of delayed renewal process and an elementary delayed renewal theorem.     

Distribution of order statistics of waiting times in an ordinary renewal process and the covariance of the renewal increments

The joint distribution of inter–renewal times and the number of renewals is used to derive joint and marginal distributions of order statistics of waiting times of an ordinary renewal process. Also, expressions are obtained for the covariance function of the number of renewals and of the renewal increments in an ordinary renewal counting process in terms of the renewal function     

更新理论推理过程及其应用

更新过程和马尔可夫更新过程中均有相对应的更新方程,实际问题中有许多变量都满足更新方程.但是在运用更新方程时,对于一些感兴趣的变量很难直接套用更新方程,这就使更新方程在实际问题的应用有许多困难.针对这个问题,总结归纳了应用更新方程的更新理论推理过程,给出了具体的推理方法和步骤,并举例进行了说明.     

Budget-constrained maintenance in the building industry

Given records of building component renewals which occurredduring the year and also theage profile of the building stock,the theory of alternating renewal processes in discrete timeis used to deduce the underlying probability mass functionsof the natural lifespan of building components and of delaydue to limited resource. In an investigation into the economicsof renewal delay, the expected number of component renewalsin each year of the life of a building is deduced from the convolutionsof the probability mass function of natural lifespan: theseexpected numbers depend on the chosen delay. The nett presentvalue of the expected cost associated with each specified delayis calculated, so that a judgement can be made as to the optimumdelay (if any). These calculations are conveniently performedusing a spreadsheet macro.     

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.     

M/G/∞ with alternating renewal breakdowns

We consider an M/G/ queue where the service station is subject to occasional interruptions which form an alternating renewal process ofup anddown periods. We show that under some natural conditions the random measure process associated with the residual service times of the customers is regenerative in the strict sense, and study its steady state characteristics. In particular we show that the steady state distribution of this random measure is a convolution of two distributions of (independent) random measures, one of which is associated with a standard M/G/ queue.     

On Hilbert functions of graded modules

This “splitting techniques” for MARKOV chains developed by NUMMELIN (1978a) and ATHREYA and NEY (1978b) are used to derive an imbedded renewal process in WOLD's point process with MARKOV-correlated intervals. This leads to a simple proof of renewal theorems for such processes. In particular, a key renewal theorem is proved, from which analogues to both BLACKWELL's and BREIMAN's forms of the renewal theorem can be deduced.