Showing papers on "Renewal theory published in 1968"

Estimation of the transition distributions of a markov renewal process

Abstract: The present paper is concerned with the estimation of the transition distributions of a Markov renewal process with finitely many states. A natural estimator of the transition distributions is defined and shown to be consistent. Limiting distributions of this estimator are derived. A density for a Markov renewal process is developed to permit the definition of maximum likelihood estimators for a renewal process and for a Markov renewal process.

A stochastic process whose successive intervals between events form a first order Markov chain — I

Abstract: In this paper we discuss a counter system whose output is a stochastic point process such that the time intervals between pairs of successive events form a first order Markov chain. Such processes may be regarded as next, in order of complexity, in a hierarchy of stochastic point processes, to “renewal” processes, which latter have been studied extensively. The main virtue of the particular system which is studied here is that virtually all its important statistical properties can be obtained in closed form and that it is physically realizable as an electronic device. As such it forms the basis for a laboratory generator whose output may be used for experimental work involving processes of this kind. Such statistical properties as the one and two-dimensional probability densities for the time intervals are considered in both the stationary and nonstationary state and also discussed are corresponding properties of the successive numbers arising in the stores of the counter system. In particular it is shown that the degree of coupling between successive time intervals may be adjusted in practice without altering the one dimensional probability density for the interval lengths. It is pointed out that operation of the counter system may also be regarded as a problem in queueing theory involving one server alternately serving two queues. A generalization of the counter system, whose inputs are normally a pair of statistically independent Poisson processes, to the case where one of the inputs is a renewal process is considered and leads to some interesting functional equations.

A Renewal Theoretic Approach to the Estimation of Future Demand for Replacement Parts

Abstract: This expository paper presents a renewal theoretic approach to the determination of the probability distribution of the demand for replacement parts in a specified future time interval. Ordinary, modified, and equilibrium renewal processes are discussed as models for physical processes in which parts are replaced as they fail. Some results from renewal theory are stated and then used to estimate the demand for replacement parts under a number of different conditions. Practical difficulties, and ways to circumvent them, are then discussed. The paper concludes with an illustrative example.

Letter to the Editor—Bruce's Spider and Renewal Theory

Abstract: The theory of an alternating renewal process with absorption at the renewal points is examined. The problem is discussed in the context of the behavior of Bruce's spider that attempted to climb a smooth wall but repeatedly slipped back and had to start again; but it provides a model for many other processes, of which two typical examples are given. Results are obtained under fairly general distributional assumptions for the distribution of the time to absorption and for the probabilities, conditional upon no absorption having taken place, of the process being in one phase or the other of the alternating process.

Clock Error Statistics as a Renewal Process

Abstract: A. Barnes Standards A model ensemble specifying the distribution of clock readings about their average is given which leads to an integral equation of renewal type. Renewal processes are often described by the random times of replacement of mechanical or electrical components. In our case, clocks in the ensemble may repeatedly read correctly and "renew" themselves. Solutions of the equation are discussed and are related to other error statist ics.