Showing papers on "Renewal theory published in 1974"
TL;DR: In this paper, a unified theory for studying renewal processes in two dimensions is developed, and explicit expressions for the renewal density, the two-dimensional renewal function, the correlation between the marginal univariate renewal counting processes, and other related quantities are derived.
Abstract: In this paper a unified theory for studying renewal processes in two dimensions is developed. Bivariate generating functions and bivariate Laplace transforms are the basic tools used in generalizing the standard theory of univariate renewal processes. An example involving a bivariate exponential distribution is presented. This is used to illustrate the general theory and explicit expressions for the two-dimensional renewal density, the two-dimensional renewal function, the correlation between the marginal univariate renewal counting processes, and other related quantities are derived.
66 citations
TL;DR: In this paper, the authors developed asymptotic results concerning the joint distribution of the bivariate renewal counting process (Nx(1), Ny(2)), the distribution of two-dimensional renewal counting processes Nxy and Nx y, and the moments of the minimum of two correlated normal random variables.
Abstract: In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (Nx(1), Ny(2)), the distribution of the two-dimensional renewal counting process Nxy and the two-dimensional renewal function Nx y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended. TWO-DIMENSIONAL RENEWAL THEORY; RENEWAL FUNCITIONS; ASYMPTOTIC DISTRIBUTIONS; MINIMUM OF TWO CORRELATED NORMAL RANDOM VARIABLES
56 citations
TL;DR: In this paper, Thorin has shown that the Wiener-hopf technique, originally developed by Cramer (1955) in the case of a Poisson process, can be used in this more general case, and Takacs (1970) has derived results similar to those of Thorin by an entirely new technique.
Abstract: During the latest few years much attention has been given to the study of the ruin problem of a risk business when the epochs of claims form a renewal process. The study of this problem was initiated by E. S. Andersen (1957). Thorin has then in a series of papers (Thorin, 1970, 1971a, 1971b) shown that the Wiener-Hopf technique, originally developed by Cramer (1955) in the case of a Poisson process, can be used in this more general case, and Takacs (1970) has derived results similar to those of Thorin by an entirely new technique.
43 citations
TL;DR: The paper shows that computation of three sample quantities suffices to generate point and interval estimates for twelve system descriptors of simulated queuing systems.
Abstract: This paper presents methods for computing point and interval estimates of descriptors of simulated queuing systems. The methods, which rely on the renewal process representation of certain processes, extend the results of Crane and Iglehart, and the author. In particular, the paper shows that computation of three sample quantities suffices to generate point and interval estimates for twelve system descriptors.
39 citations
TL;DR: This first part of a two-part paper investigates the renewal process and, subsequently, the alternating renewal process, which constitute the mathematical model of a renewable item.
Abstract: After a brief introduction to the general concept of regenerative processes and to some applications of these processes to reliability theory, this first part of a two-part paper investigates the renewal process and, subsequently, the alternating renewal process. Both these processes are basic regenerative processes and constitute the mathematical model of a renewable item. The investigation deals carefully with those quantities and theorems which are of particular interest for reliability theory and which will be used in the second part of the paper. A brief review of the literature dealing with repairable systems containing redundancy, as well as a description of an alternative investigation method is given in an appendix.
33 citations
TL;DR: In this paper, it was shown that a necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes.
Abstract: Theorem: A necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes. A complete proof of this theorem is given; also it is shown how the theorem follows from the corresponding one for the superposition of two stationary renewal processes.
31 citations
TL;DR: In this paper, two Markov renewal limit theorems are given with particular attention to the roles of periodicity and transient states, and applied to semi-Markov and semi-regenerative processes.
Abstract: : In an irreducible Markov renewal process either all states are periodic or none are. In the former case they all have the same period. Periodicity and the period can be determined by direct inspection from the semi-Markov kernel defining the process. The periodicity considerably increases the complexity of the limits in Markov renewal theory especially for transient initial states. Two Markov renewal limit theorems are given with particular attention to the roles of periodicity and transient states. The results are applied to semi-Markov and semi-regenerative processes. (Author)
28 citations
25 citations
TL;DR: In this article, the state space is partitioned into good and bad states and the distribution of sojourn times on the good set and that of exit times from this set have a simple renewal theoretic relationship.
Abstract: To treat the transient behavior of a system modeled by a stationary Markov process in continuous time, the state space is partitioned into good and bad states. The distribution of sojourn times on the good set and that of exit times from this set have a simple renewal theoretic relationship. The latter permits useful bounds on the exit time survival function obtainable from the ergodic distribution of the process. Applications to reliability theory and communication nets are given.
25 citations
01 Jan 1974
TL;DR: In a pure loss GI/G/ 1 queueing system, necessary and sufficient conditions are given for the output to be a renewal process.
Abstract: In a pure loss GI/G/ 1 queueing system, necessary and sufficient conditions are given for the output to be a renewal process. These conditions involve dependence between the service distribution and the renewal function of the arrival process: for example, if pr {service time 0, then it is sufficient for the renewal function to be that of a quasi-Poisson process with index ξ.
15 citations
TL;DR: Two novel developments in the paper are the formulation of the waited time between live births in terms of an absorbing semi-Markov process and the waiting time to conception in populations utilizing several methods of contraception as an absorbing Markov chain.
Abstract: A terminating nonhomogeneous renewal process is formulated and studied in relation to the development of quantitative methods in family planning evaluation. The motivation underlying the results presented here was to satisfy a need, which arose during computer studies, to complete and extend a stochastic model of human reproduction reported previously. The paper is divided into seven sections detailing the nature of changes made in previous work. Two novel developments in the paper are the formulation of the waiting time between live births in terms of an absorbing semi-Markov process and the waiting time to conception in populations utilizing several methods of contraception as an absorbing Markov chain. These formulations not only permit the accommodation of many family planning variables but also lead to greater conceptual simplification than formulations used in previous work.
TL;DR: In this paper, limit theorems for large deviations of renewal processes are presented, analogous to the classical Cramer and Feller large deviation theorem for sums of independent random variables.
TL;DR: For a broad class of one-sample rank-order statistics, weak convergence to Brownian motion processes is studied in this article, and a simple proof of the asymptotic normality of these statistics for random sample sizes is also presented.
Abstract: Analogous to the Donsker theorem on partial cumulative sums of independent random variables, for a broad class of one-sample rank order statistics, weak convergence to Brownian motion processes is studied here. A simple proof of the asymptotic normality of these statistics for random sample sizes is also presented. Some asymptotic results on renewal theory for one-sample rank order statistics are derived.
TL;DR: In this article, a unified theory for studying renewal processes in two dimensions is developed, and explicit expressions for the renewal density, the two-dimensional renewal function, the correlation between the marginal univariate renewal counting processes, and other related quantities are derived.
Abstract: In this paper a unified theory for studying renewal processes in two dimensions is developed. Bivariate generating functions and bivariate Laplace transforms are the basic tools used in generalizing the standard theory of univariate renewal processes. An example involving a bivariate exponential distribution is presented. This is used to illustrate the general theory and explicit expressions for the two-dimensional renewal density, the two-dimensional renewal function, the correlation between the marginal univariate renewal counting processes, and other related quantities are derived.
01 Feb 1974
TL;DR: A multicomponent reliability system in which each component is either up or down in accordance with an alternating renewal process is considered, and the following quantities are derived: the average rate of system failure; the average uptime of the system; and, the average downtime.
Abstract: : A multicomponent reliability system in which each component is either up (i.e., working) or down (i.e., failed) in accordance with an alternating renewal process is considered. For arbitrary structures the following quantities are derived: The average rate of system failure; the average uptime of the system; and, the average downtime of the system. Further results are also obtained in the special case where the system structure is either series or parallel. (Author)
TL;DR: In this paper, non-markovian kinetic equations are formulated in which the occurrence of collisions is described by a stochastic renewal process, and a simple physical meaning to be given to the non markovian character of the kinetic equations is given.
TL;DR: In this paper, the first time that the total amount of off time exceeds a specified time was derived for a modified alternating renewal process, which can be described as an alternating renewal.
TL;DR: The output probability distribution function, together with other quantities of interest, are derived with the help of renewal theory and give a generalization of the results of Ricciardi et al. (1966), which may be easily obtained explicitly by specialization.
Abstract: This paper considers the output properties of a neuron-like non-linear switching element with a stochastic dead time receiving stimuli in the form of a stationary renewal process. The output probability distribution function, together with other quantities of interest, are derived with the help of renewal theory. This gives a generalization of the results of Ricciardi et al. (1966), which may be easily obtained explicitly by specialization.
01 May 1974
TL;DR: In this paper, the models of renewal theory are reviewed, with particular emphasis on their practical applications, and bounds and approximations to various quantities are given, along with computational methods and results.
Abstract: : The models of renewal theory are reviewed, with particular emphasis on their practical applications. Bounds and approximations to various quantities are given, along with computational methods and results. Examples of how renewal theory is best used are discussed. (Author)
TL;DR: When a homogeneous renewal process is modulated by a step random signal, the mathematical expressions are derived of the probability density functions of the sums of r-successive inter point intervals, the intensity functions and the first order correlation coefficient of intervals.
Abstract: Integral density modulation of point processes is defined, and the properties of the modulated point processes are described. When a homogeneous renewal process is modulated by a step random signal, the mathematical expressions are derived of the probability density functions of the sums of r-successive inter point intervals, the intensity functions and the first order correlation coefficient of intervals. These quantities are calculated and illustrated for several parameter values. Modulated point sequences are generated by computer simulation method. The interval histograms and the serial correlation coefficients of counts and of intervals of the sequences are obtained. The results are compared with the theoretical results on the point processes modulated by the step random signal.