Counting the number of renewals during a random interval in a discrete-time delayed renewal process

In a discrete-time delayed renewal process, we study the distribution of the number of renewals during a random interval. We obtain closed-form expressions for the probability mass function and binomial moments of this number for various distributions of the random interval and interrenewal times.     

The correlation of the backward and forward recurrence times in a renewal process

The joint complementary distribution function is used to obtain the covariance function of the backward and forward recurrence times in an ordinary renewal process for both the time dependent and the steady state cases. Hence, a closed form expression for the steady state correlation of the backward and forward recurrence times is obtained and special cases are investigated     

Random fuzzy renewal process

This paper attempts to discuss a random fuzzy renewal process based on random fuzzy theory. The interarrival times are characterized as nonnegative random fuzzy variables which is a more reasonable consideration in the real world. Under this setting, the rate of the random fuzzy renewal process is discussed and a random fuzzy elementary renewal theorem is presented. Furthermore, the expected value of renewals in an arbitrary interval is investigated and Blackwell’s theorem in random fuzzy sense is also established.     

The moments of forward recurrence time

The moments of the forward recurrence time of an ordinary renewal process are derived in terms of the renewal function and the moments of the common lifetime distribution. The covariance between the forward recurrence time and the number of renewals is also obtained. Asymptotic formulae as the process is allowed to run on for a fixed long time are given.     

A generalization of the inspection paradox in an ordinary renewal process

The inspection paradox in an ordinary renewal process is investigated in the very general case of the time dependent problem and arbitrary distribution of inter–renewal times with finite mean and a probability density function     

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.     

Fisher information in window censored renewal process data and its applications

Suppose we have a renewal process observed over a fixed length of time starting from a random time point and only the times of renewals that occur within the observation window are recorded. Assuming a parametric model for the renewal time distribution with parameter θ, we obtain the likelihood of the observed data and describe the exact and asymptotic behavior of the Fisher information (FI) on θ contained in this window censored renewal process. We illustrate our results with exponential, gamma, and Weibull models for the renewal distribution. We use the FI matrix to determine optimal window length for designing experiments with recurring events when the total time of observation is fixed. Our results are useful in estimating the standard errors of the maximum likelihood estimators and in determining the sample size and duration of clinical trials that involve recurring events associated with diseases such as lupus.     

Capital renewal as a real option

We consider the timing of replacement of obsolete subsystems within an extensive, complex infrastructure. Such replacement action, known as capital renewal, must balance uncertainty about future profitability against uncertainty about future renewal costs. Treating renewal investments as real options, we derive an optimal solution to the infinite horizon version of this problem and determine the total present value of an institution’s capital renewal options. We investigate the sensitivity of the infinite horizon solution to variations in key problem parameters and highlight the system scenarios in which timely renewal activity is most profitable. For finite horizon renewal planning, we show that our solution performs better than a policy of constant periodic renewals if more than two renewal cycles are completed.     

离散更新过程的几何分布特征

本文证明了离散更新过程在随机区间内的更新次数的两上几何分布特征,这一结果与不可靠服务员的离散排队系统密切相关。     

The expected discounted penalty function under a renewal risk model with stochastic income

In this paper, we consider a renewal risk model with stochastic premiums income. We assume that the premium number process and the claim number process are a Poisson process and a generalized Erlang (n) processes, respectively. When the individual stochastic premium sizes are exponentially distributed, the Laplace transform and a defective renewal equation for the Gerber-Shiu discounted penalty function are obtained. Furthermore, the discounted joint distribution of the surplus just before ruin and the deficit at ruin is given. When the claim size distributions belong to the rational family, the explicit expression of the Gerber-Shiu discounted penalty function is derived. Finally, a specific example is provided.     

Optimal adaptive replacement in a renewal process

In this paper we present an exact method for computing the Weibull renewal function and its derivative for application in maintenance optimization. The computational method provides a solid extension to previous work by which an approximation to the renewal function was used in a Bayesian approach to determine optimal replacement times. In the maintenance scenario, under the assumption an item is replaced by a new one upon failure, the underlying process between planned replacement times is a renewal process. The Bayesian approach takes into account failure and survival information at each planned replacement stage to update the optimal time until the next planned replacement. To provide a simple approach to carry out in practice, we limit the decision process to a one‐step optimization problem in the sequential decision problem. We make the Weibull assumption for the lifetime distribution of an item and calculate accurately the renewal function and its derivative. A method for finding zeros of a function is adapted to the maintenance optimization problem, making use of the availability of the derivative of the renewal function. Furthermore, we develop the maximum likelihood estimate version of the Bayesian approach and illustrate it with simulated examples. The maintenance algorithm retains the adaptive concept of the Bayesian methodology but reduces the computational need. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

A class of delayed renewal risk processes with a threshold dividend strategy

This paper considers a class of delayed renewal risk processes with a threshold dividend strategy. The main result is an expression of the Gerber-Shiu expected discounted penalty function in the delayed renewal risk model in terms of the corresponding Cerber-Shiu function in the ordinary renewal model. Subsequently, this relationship is considered in more detail in both the stationary renewal risk model and the ruin probability.     

On a transient renewal function of dominated variation

We show that under a certain condition the renewal function of a transient renewal process varies dominatedly if and only if the interarrival times distribution varies dominatedly. This way the result of [1] is generalized.     

On the discounted penalty function in the discrete time stationary renewal risk model

In this paper we consider the discrete time stationary renewal risk model. We express the Gerber-Shiu discounted penalty function in the stationary renewal risk model in terms of the corresponding Gerber-Shiu function in the ordinary model. In particular, we obtain a defective renewal equation for the probability generating function of ruin time. The solution of the renewal equation is then given. The explicit formulas for the discounted survival distribution of the deficit at ruin are also derived.     

带利率的更新风险模型的破产概率的上界估计

本研究了在常利率条件下普通更新风险模型的破产概率问题．采用一种递推的方法给出了这种情况下破产概率的一个上界估计．     

The moments of the forward recurrence times of an alternating renewal process

Expressions for the moments of the forward recurrence times of an alternating renewal process are presented. The alternating phase-type renewal process is considered in detail.     

Lagging and leading coupled continuous time random walks, renewal times and their joint limits

Subordinating a random walk to a renewal process yields a continuous time random walk (CTRW), which models diffusion and anomalous diffusion. Transition densities of scaling limits of power law CTRWs have been shown to solve fractional Fokker-Planck equations. We consider limits of CTRWs which arise when both waiting times and jumps are taken from an infinitesimal triangular array. Two different limit processes are identified when waiting times precede jumps or follow jumps, respectively, together with two limit processes corresponding to the renewal times. We calculate the joint law of all four limit processes evaluated at a fixed time t.     

On stationary renewal reward processes where most rewards are zero

We consider a stationary version of a renewal reward process, i.e., a renewal process where a random variable called a reward is associated with each renewal. The rewards are nonnegative and I.I.D., but each reward may depend on the distance to the next renewal. We give an explicit bound for the total variation distance between the distribution of the accumulated reward over the interval (0,L] and a compound Poisson distribution. The bound depends in its simplest form only on the first two joint moments of T and Y (or I{Y > 0}), where T is the distance between successive renewals and Y is the reward. If T and Y are independent, and LE(Y) (or LP(Y > 0)) is bounded or Y binary valued, then the bound is O(E(Y)) as E(Y) → 0 (or O(P(Y > 0)) as P(Y > 0) → 0). To prove our result we generalize a Poisson approximation theorem for point processes by Barbour and Brown, derived using Stein's method and Palm theory, to the case of compound Poisson approximation, and combine this theorem with suitable couplings. Received: 1 March 1999 / Revised version: 2 August 1999 /?Published online: 31 May 2000     

First passage times for Markov renewal processes and applications

This paper proposes a uniformly convergent algorithm for the joint transform of the first passage time and the first passage number of steps for general Markov renewal processes with any initial state probability vector. The uniformly convergent algorithm with arbitrarily prescribed error can be efficiently applied to compute busy periods, busy cycles, waiting times, sojourn times, and relevant indices of various generic queueing systems and queueing networks. This paper also conducts a numerical experiment to implement the proposed algorithm.