Renewal theory

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

The GR X n /G n /∞ system: system size

Abstract: An important property of most infinite server systems is that customers are independent of each other once they enter the system. Though this non-interacting property (NIP) has been instrumental in facilitating excellent results for infinite server systems in the past, the utility of this property has not been fully exploited or even fully recognized. This paper exploits theNIP by investigating a general infinite server system with batch arrivals following a Markov renewal input process. The batch sizes and service times depend on the customer types which are regulated by the Markov renewal process. By conditional approaches, analytical results are obtained for the generating functions and binomial moments of both the continuous time system size and pre-arrival system size. These results extend the previous results on infinite server queues significantly.

Comparison of Expected Failure Times for Several Replacement Policies

Abstract: Planned replacement policies are used to reduce the incidence of system failures, or to return a failed system to work. New better than used aging classes are commonly used in reliability theory to model situations in which the lifetime of a new unit is "better" than the lifetime of a used one. The purpose of this paper is to establish comparisons of expected failure times of an age (block) replacement policy, and a renewal process with no planned replacements when the lifetime of the unit is NBUE. As we will see, age and block replacement policies improve the stochastic behavior compared with the renewal process with no planned replacements when the underlying distribution is NBUE. Some interpretations, applications, and a discussion about some related results are included

Moments-Based Approximation to the Renewal Function

Abstract: Numerous approximations to the renewal function that have been proposed so far are based on the knowledge of the inter-arrival time distribution function F(t) of the renewal process. This article proposes a simple and easy to evaluate approximation to the renewal function based only on the first three moments of F(t) but not on the functional form of F(t). In this sense, the approximation is nonparametric and yields exact renewal function for several important distributions like K 2 (also called Coxian-2 distribution) and mixture of two exponentials. An iterative procedure to improve the approximation is also proposed and is shown to converge to the renewal function. For the application of the iterative procedure, a new method of fitting a K 2 distribution to match the given moments is developed. Comparisons of the present method with the benchmark approximations available are made. As an application, an optimal replacement problem is used to test the approximation.

Renewal theory and source coding

Abstract: Renewal theory provides a wavy to derive fundamental results about source coding and is useful in the analysis and design of many lossless data compression algorithms We consider two very different applications of renewal theory to source coding The first one results in a variable-length counterpart to the asymptotic equipartition property for unifilar Markov sources The second application leads to the first analysis of variable-to-fixed length codes with plurally parsable dictionaries

On the Ruin Probability Under a Class of Risk Processes

Abstract: In this paper a class of risk processes in which claims occur as a renewal process is studied. A clear expression for Laplace transform of the finite time ruin probability is well given when the claim amount distribution is a mixed exponential. As its consequence, a well-known result about ultimate ruin probability in the classical risk model is obtained.