Showing papers on "Renewal theory published in 1971"

On a theory of search

Abstract: SUMMARY Stochastic models are developed to study the success of a search by what is termed a predator for randomly or contagiously located points, called prey. The model is specified by giving the distribution of the searching time to locate a prey, school or cluster of prey and the joint distribution of handling time and numbers caught from each cluster sighted. If a stationary distribution of prey is assumed and their removal is ignored, then the process is analogous to a cumulative renewal process. The mean numbers of prey or clusters of prey and their variances are readily obtained by use of limit theorems of renewal theory.

Further remarks on the ruin problem in case the epochs of claims form a renewal process

Abstract: In a paper [13] in this journal the author has further developed the ruin theory—first presented by E. Sparre Andersen [1] to the XVth International Congress of Actuaries, New York, 1957—in case the epochs of claims form a renewal process and the gross risk premium is positive. It turned out that much of the Cramer treatment [6] of the classical ruin theory (Poisson case) could be generalized to the more general situation. In the present paper the author points out that the identity shown by him in an earlier paper [12] for the Poisson case is also capable of being generalized. Some comments concerning the case with a negative gross risk premium are also included. I.a. the Cramer asymptotic formula for the ruin probability for an infinite period is generalized also in that case. Further information about the contents of the paper may be inferred from the following section headings. The paragraph 5.4 and the sections 6–7 are found in Part II to appear in SAT 1971:3–4.

Stochastic point processes and multicoincidences

Abstract: In this paper stochastic point processes (PP) are studied by means of coincidence probabilities (CP). After a definition and short review of these CP, it is shown that they provide a complete statistical description of the PP. All the statistics of the resulting shot noise are derived and time intervals between successive occurrences are studied and related to CP. Some new applications, including the "generalized" renewal process, illustrate the coincidence approach. The method applies very well to such cases as nonstationary or multidimensional PP.

Discrete Renewal Processes

Abstract: Discrete renewal processes until recently have not been applied to the mathematical modeling of physical processes. Analyses of such renewal processes have proceeded on the basis of generating functions but the results are often too complicated to be of use. This paper presents an alternative approach to discrete renewal theory and calculates many of the more complex statistics of such processes.

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)

Selective interaction of a poisson and renewal process: the dependency structure of the intervals between responses

Abstract: This paper studies the dependency structure of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b)). A new approach to the intervals between events in a stationary point process, based on the idea of an average event, is introduced. Average event initial conditions (as opposed to equilibrium initial conditions previously determined) for the renewal inhibited Poisson process are obtained and event stationarity of the resulting response process is established. The joint distribution and correlation between pairs of contiguous synchronous intervals is obtained; further, the joint distribution of non-contiguous pairs of synchronous intervals is derived. Finally, the joint distributions of pairs of contiguous synchronous and asynchronous intervals are related, and a similar but more general stationary point result is conjectured.

Superimposed non-stationary renewal processes

Abstract: For superposition of independent, stationary renewal processes, it is well known that the distribution of waiting time between events for the superimposed process is approximately exponential if the number of processes involved is sufficiently large, (see Khintchine (1960), Ososkov (1956)). We assume that all component processes have the same age t, and we generalize the classical result to show that even for t finite (non-stationary case), the limiting waiting time distribution (as the number of processes increases) is exponential with a scale parameter which depends on t through the average of the individual process renewal densities. For moderate values of t, this scale parameter can differ greatly from its limit due to the generally slow convergence of the renewal density function to its limit. While it has been generally assumed that this waiting time distribution is approximately exponential regardless of t, the explicit formulation of the necessary normalizations and conditions seems new. While a variety of papers on superimposed point processes have appeared recently, all assume stationarity (see 4inlar (1968) for references). Franken (1963) considered the time dependent behavior of superimposed renewal processes under the highly restrictive and generally unrealistic assumption that for each n, the superimposed process is itself a renewal process. Among other things, this requires the individual processes to have distributions which depend on n and which generally do not resemble any standard form. We believe that letting the individual processes have some standard form and imposing no particular form on the superimposed process is more realistic.

Bayesian Modification of the Failure Rate Function.

Abstract: : The failure rate function is widely used in such theoretical frameworks as dynamic programming and control theory, reliability and redundancy, queueing and renewal theory; which have been developed for such diverse applications as modeling research and development, priority schemes for time sharing computers, demand for replacement parts, and design of space vehicles. The parameters of the function are often uncertain; furthermore the fact of non-failure up to a point z yields information on the parameters. As Lucas has pointed out, specification of an initial prior distribution g(a) on the parameter vector a leads directly via Bayesian analysis from the original failure rate function h(z/a) to a modified failure rate function H*(z) which may then be used in the analysis as if it were deterministic. Of great interest are results about the general shape of H*(z) (increasing, decreasing, unimodal, convex, etc.) in terms of H(z/a) and g(a). (Author)

Selective interaction of a Poisson and renewal process: the spectrum of the intervals between responses

Abstract: This paper is concerned with the spectrum of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b), (1971)). A generating function for all the pairwise joint distributions of the synchronous intervals following an average response is obtained and leads directly to the associated serial correlations. It is shown that these correlations are equivalent to those predicted on different assumptions by the general stationary point theory. The results are then used to obtain the interval spectrum, and to exhibit a relationship between the sum of the serial correlations and the variance-time function. Explicit results for the spectrum of the renewal inhibited Poisson process are given for gamma inhibitory distributions, and the qualitative behavior is determined. Possible further developments are briefly discussed.

A note on transforms of renewal and other models applied to the collective risk theory

Abstract: 1. Renewal models applied to the risk theory In this note the interval between the (n—1)th, and the nth event in a random process—the interoccurrence time—will be denoted by τ n for n = 2, 3 ... , and the interval from the starting point of the process to the time point of the first event by τ1. τ has here been chosen for the notations, in order to differentiate from t, usually used for the expected number of claims; the mean of n in a renewal process is, for finite values of τ, different from τ, and for τ→∞ linear in τ; τ is not necessarily measured on the absolute parameter scale. The renewal process is based on the assumptions that τ n are mutually independent variables, equally distributed with the distribution function Prob [τ n ⩽ τ] = K(τ) , n = 1, 2 ... , where K(τ) shall be defined below. The complex Fourier transform of dK(τ) shall be designated by k(z).