Renewal theory

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

Linear Bounds on the Renewal Function

Abstract: Lower and upper linear bounds are obtained on the renewal function of an ordinary renewal process. We show that any linear function, with slope equal to the reciprocal of the mean of the time between renewals, when iterated in the renewal equation will converge on the renewal function. However, convergence may not be monotonic for all t. We find the “best” linear bounds, which we define to be the sharpest bounds which, when iterated, converge monotonically to $M( t )$ for all t. We also show that the failure rate function of an equilibrium distribution has less variation than the failure rate of the underlying distribution. This fact is used to show that the bounds in this paper are an improvement over previously published linear bounds.

Some results on block replacement policies and renewal theory

Abstract: Age and block replacement policies are commonly used in order to reduce the number of in-service failures. The focus in this paper is on the block replacement policies, about which relatively less is known than age replacement policies. Several new results which connect the properties of block replacement policies with the properties of the corresponding renewal function and the excess lifetimes are obtained. Some applications and the relationships between these new results and some known results are included.

Sampling Bias in Population Studies—How to Use the Lexis Diagram

Abstract: Modified versions of the lifetime distribution are often used in survival analysis. The modifications depend on how we choose individuals for the study and on the assumptions on the behaviour of the population. A rigorous point process description of the Lexis diagram is used to make the sampling mechanisms and the preconditions transparent. The point process description gives a framework to handle all possible sampling patterns. The set-up is generalized so it can handle more complicated life descriptions than just lifetimes, and the diability model is used as an example. Two set-ups can be used. Conditional on the birthtimes, the lifetime distribution is left truncated and subject to either right censoring or right truncation. Assuming that the birthtimes can be described by a Poisson process the modifications are length bias and the recurrence time distribution known from renewal theory.

Approximations for the renewal function

Abstract: This paper describes an accurate, computable approximation for evaluating the renewal function (RF). The method uses Pade approximants to compute the RF near the origin and switches to the asymptotic values farther from the origin. There is a polynomial switch-over function in terms of the coefficient of variation of the distribution, enabling one to determine a priori if the asymptotic value can be used instead of computing the Pade approximant. The results are tested with the truncated Gaussian distribution. The method yields a set of approximants to the RF that are re-usable, and can be used to compute the derivative and the integral of the RF. Results for the RF are within 1% of the optimal solution for most coefficients of variation.