Renewal theory

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

On some distributional properties of a first-order nonnegative bilinear time series model

Abstract: We study a simple first-order nonnegative bilinear time-series model and give conditions under which the model is stationary. The probability density function of the stationary distribution (when it exists) is found. We also discuss the tail behaviour of the stationary distribution and calculate the probability density function by a numerical method. Simulation is used to check the calculation.

Reliability and Availability of Two General Multi-Unit Systems

Abstract: This paper considers k-unit systems with repair. Reliability and availability integral equations are set up using renewal theory. This paper treats a k-unit system with s-dependent failure rate and general repair time distribution, and a k-unit cold-standby system with Erlang failure time distribution and general repair time distribution.

Variance Estimation Based on Invariance Principles

Abstract: Consistent variance estimators for certain stochastic processes are suggested using the fact that (weak or strong) invariance principles may be available. Convergence rates are also derived, the latter being essentially determined by the approximation rates in the corresponding invariance principles. As an application, a change point test in a simple AMOC renewal model is briefly discussed, where variance estimators possessing good enough convergence rates are required.

Finite-size corrections to Poisson approximations of rare events in renewal processes

Abstract: Consider a renewal process. The renewal events partition the process into i.i.d. renewal cycles. Assume that on each cycle, a rare event called 'success’ can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence may be relatively slow, because each success corresponds to a time interval, not a point. In 1996, Altschul and Gish proposed a finite-size correction to a particular approximation by a Poisson point process. Their correction is now used routinely (about once a second) when computers compare biological sequences, although it lacks a mathematical foundation. This paper generalizes their correction. For a single renewal process or several renewal processes operating in parallel, this paper gives an asymptotic expansion that contains in successive terms a Poisson point approximation, a generalization of the Altschul-Gish correction, and a correction term beyond that.

The accuracy of call-congestion measurements for loss systems with renewal input

Abstract: The concept of a generalized renewal process is used to derive an asymptotic approximation for the variance of the observed proportion of unsuccessful attempts on a trunk group during a given time-interval. Calls are assumed to arrive according to a general renewal process, and those which are blocked leave the system and do not return (loss system). As an application of our result we examine the special case of an overflow input–an important example from telephone networks with alternate routing. Comparison of our results with values obtained from simulation indicates that the approximation is quite accurate for telephone traffic-engineering purposes.