Stationary Renewal Process Research Articles

Infinite divisibility and the waiting-time paradox

It is shown that the interarrival time Z covering the point zero in a stationary renewal process generated by X has the form with Y nonnegative and independent of X, if and only if X is infinitely divisible. In the special case that X has a compound-exponential distribution there is a similar decomposition of the stationary waiting time. These results shed some new light on the waiting-time paradox

Stationary renewal processes

Let (t n = z 0 + · + z n) N be renewal instants sequence. Suppose that the random variables ( z n ) N are independent and the corresponding renewal process is stationary (in weak sense) with a finite positive intensity. Then there exists an integer d ≥ 1 such that the sequence ( t nd ) n≥0 forms a stationary recurrent renewal process and z m = 0 almost surely if m ≠ nd, ∀n ϵ N .

$L_2$ Rates of Convergence for Attractive Reversible Nearest Particle Systems: The Critical Case

Reversible nearest particle systems are certain one-dimensional interacting particle systems whose transition rates are determined by a probability density $\beta(n)$ with finite mean on the positive integers. The reversible measure for such a system is the distribution $\nu$ of the stationary renewal process corresponding to this density. In an earlier paper, we proved under certain regularity conditions that the system converges exponentially rapidly in $L_2(\nu)$ if and only if the system is supercritical. This in turn is equivalent to $\beta(n)$ having exponential tails. In this paper, we consider the critical case, and give moment conditions on $\beta(n)$ which are separately necessary and sufficient for the convergence of the process in $L_2(\nu)$ at a specified algebraic rate. In order to do so, we develop conditions on the generator of a general Markov process which correspond to algebraic $L_2$ convergence of the process. The use of these conditions is also illustrated in the context of birth and death chains on the positive integers.

Optimal random replacement policies for repairable systems

For a stationary renewal process with preventive repair optimal policies of preventive repair are established within the class of age-dependent replacement policies with random repair time, provided the expected number of renewals for a fixed time period is given. Under certain conditions the distribution of the random preventive repair time is the degenerate one. Nevertheless, there exist conditions that yield optimal replacement policies with nondegenerate repair time distributions, i.e. random replacement times

Exponential $L_2$ Convergence of Attractive Reversible Nearest Particle Systems

Nearest particle systems are continuous-time Markov processes on $\{0, 1\}^Z$ in which particles die at rate 1 and are born at rates which depend on their distances to the nearest particles to the right and left. There is a natural parametrization of these systems with respect to which they exhibit a phase transition. When the process is attractive and reversible, the critical value $\lambda_c$ above which a nontrivial invariant measure exists can be computed exactly. This invariant measure is the distribution $v$ of a stationary discrete time renewal process. Under a mild regularity assumption, we prove that the following three statements are equivalent: (a) The nearest particle system converges to equilibrium exponentially rapidly in $L_2(v)$. (b) The density of the interarrival times in the renewal process has exponentially decaying tails. (c) The nearest particle system is supercritical in the sense that $\lambda > \lambda_c$. Under an additional second-moment assumption, we prove that the critical exponent associated with the exponential convergence is 2. The proof of exponential convergence is based on an unusual comparison of the nearest particle system with an infinite system of independent birth and death chains. To carry out this comparison, a new representation is developed for a stationary renewal process with a log-convex renewal sequence in terms of a sequence of i.i.d. random variables.

Non‐parametric estimation for renewal processes from event count data

AbstractA procedure is developed for estimating the interevent time distribution of a stationary renewal process that uses only data in the form of counts (the number of events occurring in each of several adjacent disjoint intervals) that are predominantly zeros and ones. Some properties of the procedure are derived, and the procedure's performance is assessed via Monte Carlo simulation.

A new estimation approach based on periodic inspection

The authors present a novel approach for estimating the failure-time distribution using count data, the number of failures per interval, from periodic inspection of a standby redundant system. The procedure is based on a result for stationary renewal processes that related the forward recurrence time to the interevent time distribution. The procedure performed well in a simulation study for five Weibull distributions and is general enough to work for any failure-time distribution.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The square‐wave spectral density of a stationary renewal process

The power spectral density of a random square wave is a promising tool in the qualitative study of stationary point processes. This is illustrated for renewal processes and their superpositions.

Queues with superposition arrival processes in heavy traffic

To help provide a theoretical basis for approximating queues with superposition arrival processes, we prove limit theorems for the queue-length process in a Σ GI i / G/ s model, in which the arrival process is the superposition of n independent and identically distributed stationary renewal processes each with rate n −1. The traffic intensity ρ is allowed to approach the critical value one as n increases. If n(1− ρ) 2 → c, 0 < c < ∞, then a limit is obtained that depends on c. The two iterated limits involving ρ and n, which do not agree, are obtained as c → 0 and c → ∞.

A Short-Cut Method for Estimation in Renewal Processes

This article proposes a simple, noniterative estimator of the lifetime distribution from stationary renewal processes data, which is a variation of the Kaplan-Meier estimator. The method is compared with the nonparametric maximum likelihood estimator using real and simulated data, and the results show that as long as the censoring level is not too high the two estimates are remarkably close. A heuristic asymptotic argument is given to support the simulation results and to justify the rationale behind the estimator. The estimator's connection with the Kaplan–Meier estimator is exploited to suggest a variance estimate and associated confidence interval, which, through a simulation study, are found to be conservative.

Eruption patterns of the chilean volcanoes Villarrica, Llaima, and Tupungatito

The historical eruption records of three Chilean volcanoes have been subjected to many statistical tests, and none have been found to differ significantly from random, or Poissonian, behaviour. The statistical analysis shows rough conformity with the descriptions determined from the eruption rate functions. It is possible that a constant eruption rate describes the activity of Villarrica; Llaima and Tupungatito present complex eruption rate patterns that appear, however, to have no statistical significance. Questions related to loading and extinction processes and to the existence of shallow secondary magma chambers to which magma is supplied from a deeper system are also addressed. The analysis and the computation of the serial correlation coefficients indicate that the three series may be regarded as stationary renewal processes. None of the test statistics indicates rejection of the Poisson hypothesis at a level less than 5%, but the coefficient of variation for the eruption series at Llaima is significantly different from the value expected for a Poisson process. Also, the estimates of the normalized spectrum of the counting process for the three series suggest a departure from the random model, but the deviations are not found to be significant at the 5% level. Kolmogorov-Smirnov and chi-squared test statistics, applied directly to ascertaining to which probabilityP the random Poisson model fits the data, indicate that there is significant agreement in the case of Villarrica (P=0.59) and Tupungatito (P=0.3). Even though theP-value for Llaima is a marginally significant 0.1 (which is equivalent to rejecting the Poisson model at the 90% confidence level), the series suggests that nonrandom features are possibly present in the eruptive activity of this volcano.

Transport properties of the one-dimensional stochastic Lorentz model: I. Velocity autocorrelation function

Point scatterers are placed on the real line such that the distances between scatterers are independent identically distributed random variables (stationary renewal process). For a fixed configuration of scatterers a particle performs the following random walk: The particle starts at the pointx with velocityυ, ¦υ¦=1. In between scatterers the particle moves freely. At a scatterer the particle is either transmitted or reflected, both with probability 1/2. For given initial conditions of the particle the velocity autocorrelation function is a random variable on the scatterer configurations. If this variable is averaged over the distribution of scatterers, it decays not faster thant −3/2.

Attractive Nearest Particle Systems

We consider certain Markov processes with state space $\{0, 1\}^z$ which were introduced and first studied by Spitzer. In these systems, deaths occur at rate one independently of the configuration, and births occur at rate $\beta(\ell, r)$ where $\ell$ and $r$ are the distances to the nearest particles to the left and right, respectively. In his paper, Spitzer gave a necessary and sufficient condition for this process to have a reversible invariant measure, and showed that such a measure must be a stationary renewal process. It was that fact which motivated the study of these systems. Assuming that the process is attractive in the sense of Holley, we give conditions under which (a) the pointmass on the configuration "all zeros" is invariant, and (b) the reversible renewal process is the only nontrivial invariant measure which is translation invariant. As an application, these results allow us to determine exactly the values of $\lambda > 0$ and $p > 0$ for which the process with $\beta(\ell, r) = \lambda(1/\ell + 1/r)^p$ is ergodic.

Inequalities and bounds for variances of point processes and fibre processes

For the variances of numbers of points and total fibre lengths in compact convex sets, corresponding to point processes and fibre processes, bounds are proved. They base on monotonicity and convexity properties of a function of geometrical nature occuring in the formula for the variance when using the reduced second moment measure. As particular cases, stationary renewal processes. Cox processes and Neyman-Scott processes are considered. For certain fibre process models Poisson point processes and segment processes are used as approximative models.

Stationary semi-Markov models in risk and queueing theories

Abstract For the semi-Markov model in risk and queueing theories, we introduce the natural extension of the concept of stationarity in classical models (i.e. the passage of a renewal process to a general stationary renewal process for the arrivals). It is shown that some interesting relations can be obtained between positive capital risk models using a new concept of duality called *-duality.

Renewal processes decomposable into i.i.d. components

Let N be a stationary renewal process with a probability density function f(t). Suppose that N can be expressed as the superposition of a finite number of i.i.d. stationary components N(1), …, N(p) (p≧2). Then, under a supplementary condition on f(t), N and N(1), …, N(p) are all Poisson. This is proved by using recurrence relations given in Ito (1978) for the probability distribution of i.i.d. components of a superposition process.

Renewal processes decomposable into i.i.d. components

Let N be a stationary renewal process with a probability density function f(t). Suppose that N can be expressed as the superposition of a finite number of i.i.d. stationary components N (1), …, N (p) (p≧2). Then, under a supplementary condition on f(t), N and N (1), …, N (p) are all Poisson. This is proved by using recurrence relations given in Ito (1978) for the probability distribution of i.i.d. components of a superposition process.

Firing rate of a retinal neuron are not predictable from interspike interval statistics

The intervals between successive action potentials (impulses, or "spikes") produced the maintained firing of a neuron (ISIs) are often treated as if they were independent on each other; that is, an impulse train is considered as a stationary renewal process. If this is so, the variability of the mean rate of firing impulses in a sequence of temporal windows should be predictable from the distribution of ISIs. This was found not to be the case for the maintained firing of retinal ganglion cells in goldfish. Although some evident nonstationarity sometimes resulted in greater variability of the observed rate distributions than those predicted (for relatively long temporal windows), as a general rule the observed rate distributions were considerable less dispersed than would be predicted by sampling of the ISI distributions. This was taken as evidence of long-term serial dependency between successive ISIs; however, two standard test for dependency (autocorrelations and serial correlograms failed to to reveal structure of sufficiently long duration to account for the effect noted.

A statistical description of the occurrence of premature ventricular contractions based on the poisson process

A stochastic model characterizing the occurrence of premature ventricular contractions (PVCs) is described and tested for consistency with clinical data. Single-lead EKG signals from 21 patients hospitalized in the CCU were recorded and the intervals between ectopic beats were stored in the computer for a subsequent time series analysis. The PVC occurrence could be adequately described by locally stationary renewal processes for 86% of the patients, and for 52% consistency with the more specific Poisson process was established. Adopting the Poisson process as a pertinent model, a statistically efficient procedure for the on-line detection of PVC rate changes was developed. The proposed detection scheme performed without significant degradation for the much wider class of data consistent with nonspecific renewal processes. The detection procedure provides for the partition of the record of PVC occurrence into a sequence of segments with constant rates, thereby greatly reducing the amount of data for monitoring.

A characterization of stationary renewal processes and of memoryless distributions

We prove first that a renewal process is stationary if and only if the distributions of the age and the residual waiting time coincide for every t>0, and for 0≦x<t. From there it follows a characterization of memoryless distributions (i.e., either exponential or geometric,) in the case of an ordinary renewal process.