Renewal theory

About: Renewal theory is a research topic. Over the lifetime, 2381 publications have been published within this topic receiving 54908 citations.

Correlated fractional counting processes on a finite-time interval

Abstract: We present some correlated fractional counting processes on a finite-time interval. This will be done by considering a slight generalization of the processes in Borges et al. (2012). The main case concerns a class of space-time fractional Poisson processes and, when the correlation parameter is equal to 0, the univariate distributions coincide with those of the space-time fractional Poisson process in Orsingher and Polito (2012). On the one hand, when we consider the time fractional Poisson process, the multivariate finite-dimensional distributions are different from those presented for the renewal process in Politi et al. (2011). We also consider a case concerning a class of fractional negative binomial processes.

Functional approximation theorems for controlled renewal processes

Abstract: We prove a functional law of large numbers and a functional central limit theorem for a controlled renewal process, that is, a point process which differs from an ordinary renewal process in that the ith interarrival time is scaled by a function of the number of previous i arrivals. The functional law of large numbers expresses the convergence of a sequence of suitably scaled controlled renewal processes to the solution of an ordinary differential equation. Likewise, the functional central limit theorem establishes that the error in the law of large numbers converges weakly to the solution of a stochastic differential equation. Our proofs are based on martingale and time-change arguments.

A general framework for some asymptotic reliability formulas

Abstract: The authors prove that certain reliability formulas which link asymptotic availability, mean normal operation time, mean time between failures, mean number of failures over a period of time and asymptotic Vesely rate, and which are well known in the case of modelling using a Markov jump process or an alternating renewal process, are also true in the context of more general modelling.

Renewal Processes and Their Computational Aspects

Abstract: In this chapter, we review classical renewal theory and focus on the computational aspects for the renewal function. As well known, the renewal processes have an important role in understanding the discrete event systems arising in queueing theory, production and inventory control, design of communication systems, performance evaluation in computer science and product warranty estimation and also in reliability and maintenance modeling. On the other hand, from the practical perspective, since the computation of the renewal function is not so easy, the system analyst tends to treat the renewal function via the simplest method for him or her. In fact, a large number of authors have discussed the computation problems of the renewal function. Nevertheless, no articles reporting those results Fin a systematic way have appeared in the literature. In this chapter, we survey the computational aspects of the renewal function based on the most recent results obtained up to the present stage. A comprehensive bibliography in this research area is also provided.

Optimum Test Procedure Under Stress

Abstract: A means of scheduling a test procedure is presented when one component, for example, out of n has failed. Each component has a given probability of failure. The time to examine each component is a random variable with known probability distribution. However, there is only a limited amount of time available for search. What test procedure should be utilized to maximize the probability of locating the failed component within the given time? The problem is solved using some renewal theory results and the functional equation approach of dynamic programming.