Renewal theory

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

Bounds and approximations for the renewal function.

Abstract: : The thesis gives a number of approximations and bounds for the renewal function in an ordinary renewal process. Each approximation and bound is calculated for the uniform, gamma and hyperexponential distributions and compared with the renewal function for these cases. They are also calculated for the log-normal distribution and compared with results of the simulation of the renewal function. Results are tabulated and shown graphically. (Author)

Regenerative processes in the infinite mean cycle case

Abstract: A class of non-negative alternating regenerative processes is considered, where the process stays at zero random time (waiting period), then it jumps to a random positive level and hits zero after some random period (life period), depending on the evolution of the process. It is assumed that the waiting time and the lifetime belong to the domain of attraction of stable laws with parameters in the interval (1/2, 1]. An integral representation for the distribution functions of the regenerative process is obtained, using the spent time distributions of the corresponding alternating renewal process. Given the asymptotic behaviour of the process in the regenerative cycle, different types of limiting distributions are proved, applying some new results for the corresponding renewal process and two limit theorems for the convergence in distribution.

Evaluation of the uncertainty affecting reliability models

Abstract: Purpose – Reliability models are generally estimated from small samples. This paper seeks to calculate the uncertainty affecting reliability parameters in function of the sample size.Design/methodology/approach – The confidence intervals are calculated on the basis of Monte Carlo simulations and using the variance‐covariance matrix; the two methods are compared.Findings – Numerical results for the estimation of uncertainty have been obtained for standard reliability models, non‐homogeneous Poisson process and generalized renewal process.Originality/value – For the generalized renewal process, the article points out the influence of the age correction factor on the number of repairs authorized and on uncertainty. The surface plot of the likelihood function with respect to parameters is a convenient tool to interpret the uncertainty.

Two tandem queues with general renewal input I: diffusion approximation and integral representations

Abstract: We consider two tandem queues with exponential servers. Arrivals to the first queue are governed by a general renewal process. If the arrivals were also exponentially distributed, this would be a simple example of a Jackson network. However, the structure of the model is much more complicated for general arrivals. We analyze the joint steady-state queue length distribution for this network, in the heavy traffic limit, where the arrival rate is only slightly less than the service rates. We formulate and solve the boundary value problem for the diffusion approximation to this model. We obtain simple integral representations for the (asymptotic) steady-state queue length distribution.

A unified treatment of single component replacement models

Abstract: In this paper we discuss a general framework for single component replacement models. This framework is based on the regenerative structure of these models and by using results from renewal theory a unified presentation of the discounted and average finite and infinite horizon cost models is given. Moreover, we present sufficient conditions for the numerator of the cost function to have a vanishing singular part. Finally, some well-known replacement models are discussed, and making use of the previous results an easy derivation of their cost functions is presented.