Showing papers on "Renewal theory published in 2003"

The trend-renewal process for statistical analysis of repairable systems

Abstract: The most commonly used models for the failure process of a repairable system are nonhomogeneous Poisson processes, corresponding to minimal repairs, and renewal processes, corresponding to perfect repairs. This article introduces and studies a more general model for recurrent events, the trend-renewal process (TRP). The TRP is a time-transformed renewal process having both the ordinary renewal process and the nonhomogeneous Poisson process as special cases. Parametric inference in the TRP model is studied, with emphasis on the case in which several systems are observed in the presence of a possible unobserved heterogeneity between systems.

Block replacement policies for a two-component system with failure dependence

Abstract: Block replacement and modified block replacement policies for two-component systems with failure dependence and economic dependence are considered in this paper. Opportunistic maintenance policies are also considered. Where tractable, long-run costs per unit time are calculated using renewal theory based arguments; otherwise simulation studies are carried out. The management implications for the adoption of the various policies are discussed. The usefulness of the results in the paper is illustrated through application to a particular two-component system.

Renewal process with fuzzy interarrival times and rewards

Abstract: This paper considers a renewal process in which the interarrival times and rewards are characterized as fuzzy variables A fuzzy elementary renewal theorem shows that the expected number of renewals per unit time is just the expected reciprocal of the interarrival time Furthermore, the expected reward per unit time is provided by a fuzzy renewal reward theorem Finally, a numerical example is presented for illustrating the theorems introduced in the paper

Sequential detection of targets in multichannel systems

Abstract: It is supposed that there is a multichannel sensor system which performs sequential detection of a target. Sequential detection is done by implementing a generalized Wald's sequential probability ratio test, which is based on the maximum-likelihood ratio statistic and allows one to fix the false-alarm rate and the rate of missed detections at specified levels. We present the asymptotic performance of this sequential detection procedure and show that it is asymptotically optimal in the sense of minimizing the expected sample size when the probabilities of erroneous decisions are small. We do not assume that the observations are independent and identically distributed (i.i.d.). The first-order asymptotic optimality result holds for general statistical models that are not confined to the restrictive i.i.d. assumption. However, for i.i.d. and quasi-i.i.d. cases, where the log-likelihood ratios can be represented in the form of sums of random walks and slowly changing sequences, we obtain much stronger results. Specifically, using the nonlinear renewal theory we are able to obtain both tight expressions for the error probabilities and higher order approximations for the average sample size up to a vanishing term. The performance of the multichannel sequential detection algorithm is illustrated by an example of detection of a deterministic signal in correlated (colored) Gaussian noise. In this example, we provide both the results of theoretical analysis and the results of a Monte Carlo experiment. These results allow us to conclude that the use of the sequential detection algorithm substantially reduces the required resources of the system compared to the best nonsequential algorithm.

Matrix-Exponential Distributions: Calculus and Interpretations via Flows

Abstract: By considering randomly stopped deterministic flow models, we develop an intuitively appealing way to generate probability distributions with rational Laplace–Stieltjes transforms on [0,∞). That approach includes and generalizes the formalism of PH-distributions. That generalization results in the class of matrix-exponential probability distributions. To illustrate the novel way of thinking that is required to use these in stochastic models, we retrace the derivations of some results from matrix-exponential renewal theory and prove a new extension of a result from risk theory. Essentially the flow models allows for keeping track of the dynamics of a mechanism that generates matrix-exponential distributions in a similar way to the probabilistic arguments used for phase-type distributions involving transition rates. We also sketch a generalization of the Markovian arrival process (MAP) to the setting of matrix-exponential distribution. That process is known as the Rational arrival process (RAP).

Finitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theorem

Abstract: We consider the class of graph-directed constructions which are connected and have the property of finite ramification. By assuming the existence of a fixed point for a certain renormalization map, it is possible to construct a Laplace operator on fractals in this class via their Dirichlet forms. Our main aim is to consider the eigenvalues of the Laplace operator and provide a formula for the spectral dimension, the exponent determining the power-law scaling in the eigenvalue counting function, and establish generic constancy for the counting-function asymptotics. In order to do this we prove an extension of the multidimensional renewal theorem. As a result we show that it is possible for the eigenvalue counting function for fractals to require a logarithmic correction to the usual power-law growth.

A further extension of the generalized burn-in model

Abstract: In this paper, the general ized burn-in and replacement model considered by Cha (2001) is further extended to the case in which the probability of Type II failure is time dependent. Two burn-in procedures are considered and they are compared in cases when both the procedures are applicable. Under some mild conditions on the failure rate function r(t) and the Type II failure probability function p(t), the problems of determining optimal burn-in time and optimal replacement policy are considered.

Multiple-input heavy-traffic real-time queues

Abstract: A single queueing station that serves K input streams is considered. Each stream is an independent renewal process, with customers having random lead times. Customers are served by processor sharing across streams. Within each stream, two disciplines are considered--earliest deadline first and first-in, first-out. The set of current lead times of the K streams is modeled as a K-dimensional vector of random counting measures on $\mathbb{R}$, and the limit of this vector of measure-valued processes is obtained under heavy traffic conditions.

A note on "Continuous review perishable inventory systems: models and heuristics"

Abstract: In a recent paper, Lian and Liu (2001) consider a continuous review perishable inventory model with renewal arrivals, batch demands and zero lead times. However, the main analytical result they provide holds only for some special cases such as Poisson arrivals with exponential interarrival times. In this note we generalize Theorem 1 of Lian and Liu (2001) for the case where the arrivals follow an arbitrary renewal process.

Renewal Theory for Functionals of a Markov Chain with Compact State Space

Abstract: Motivated by multivariate random recurrence equations we prove a new analogue of the Key Renewal Theorem for functionals of a Markov chain with compact state space in the spirit of Kesten. We simplify and modify Kesten's proof.

Optimal strategy policy in batch arrival queue with server breakdowns and multiple vacations

Abstract: This paper considers a like-queue production system in which server vacations and breakdowns are possible. The decision-maker can turn a single server on at any arrival epoch or off at any service completion. We model the system by an M[x]/M/1 queueing system with N policy. The server can be turned off and takes a vacation with exponential random length whenever the system is empty. If the number of units waiting in the system at any vacation completion is less than N, the server will take another vacation. If the server returns from a vacation and finds at least N units in the system, he immediately starts to serve the waiting units. It is assumed that the server breaks down according to a Poisson process and the repair time has an exponential distribution. We derive the distribution of the system size through the probability generating function. We further study the steady-state behavior of the system size distribution at random (stationary) point of time as well as the queue size distribution at departure point of time. Other system characteristics are obtained by means of the grand process and the renewal process. Finally, the expected cost per unit time is considered to determine the optimal operating policy at a minimum cost. The sensitivity analysis is also presented through numerical experiments.

A class of tests for renewal process versus monotonic and nonmonotonic trend in repairable systems data

Abstract: A class of statistical tests for trend in repairable systems data based on the general null hypothesis of a renewal process is proposed. This class does in particular include a test which is attractive for general use by having good power properties against both monotonic and nonmonotonic trends. Both the single system and the several systems cases are considered.

Diffusion in random environment and the renewal theorem

Abstract: According to a theorem of S. Schumacher and T. Brox, for a diffusion $X$ in a Brownian environment it holds that $(X_t-b_{\log t})/\log^2t\to 0 $ in probability, as $t\to\infty$, where $b_{\cdot}$ is a stochastic process having an explicit description and depending only on the environment. We compute the distribution of the number of sign changes for $b$ on an interval $[1,x]$ and study some of the consequences of the computation; in particular we get the probability of $b$ keeping the same sign on that interval. These results have been announced in 1999 in a non-rigorous paper by P. Le Doussal, C. Monthus, and D. Fisher and were treated with a Renormalization Group analysis. We prove that this analysis can be made rigorous using a path decomposition for the Brownian environment and renewal theory. Finally, we comment on the information these results give about the behavior of the diffusion.

A structural approximation method to generate the optimal auto-sleep schedule for computer systems

Abstract: This paper addresses a problem of energy reduction on computer systems, namely the problem of determining the optimal auto-sleep schedule on which the system is changed into a low-power state. For this problem, we provide a stochastic model for a computer system with autosleep function. In our modeling framework, user requests arrive at the system in accordance with a general renewal process and are served by generally distributed service times. To obtain the optimal auto-sleep schedule minimizing the expected power consumption per unit time in the steady state, the phase-type approximation is applied to the renewal arrival process. In numerical examples, we investigate accuracy of the phase-type approximation through a simulation study.

Internet Traffic Modeling using the Index of Variability.

Abstract: In this paper, we propose a novel and mathematically rigorous measure of variability, called the index of variability (Hv τ ), that fully and accurately captures the degree of variability of a typical network traffic process at each time scale and is analytically tractable for many popular traffic models. Using this proposed measure, we then analyzed two traffic models: the Two-State Markov Modulated Poisson Process (MMPP) and the renewal process with hyperexponential interarrival time distributions of order two (RPH2). Two-state MMPP models are popular in modeling the superposition of packet voice streams. The results show that the traffic variability can exhibit a non-monotonic behavior. In addition, the results suggest that renewal processes with interarrival times hyperexponentially distributed are suitable for modeling network traffic processes with high variability over a broad range of time scales.

Renewal Processes and Repairable Systems

Abstract: In this thesis we discuss the following topics: 1. Renewal reward processes The marginal distributions of renewal reward processes and its version, which we call in this thesis instantaneous reward processes, are derived. Our approach is based on the theory of point processes, especially Poisson point processes. The idea is to represent the renewal reward processes and its version as functionals of Poisson point processes. Important tools we use are the Palm formula and the Laplace functional of Poisson point processes. The results are presented in the form of Laplace transforms. An application of the instantaneous reward processes to the study of traffic is given. Some asymptotic properties of the renewal reward processes are reconsidered. A proof of the expected-value version of the renewal reward theorem using the Tauberian theorem is given. A second order term in the expected-value version of the renewal reward theorem is obtained. Similar results for the instantaneous reward processes are investigated. Asymptotic normality of the instantaneous reward processes is proved. The covariance structure of renewal processes, which can be considered as a special case of renewal reward processes, is derived. As an addition, we study system reliability in a stress-strength model, where the amplitudes of stresses can be considered as rewards. We consider renewal and Cox processes as models for the occurrences of the stresses. Using our result on renewal reward processes we investigate the effect of dependence between stress and strengths on system reliability. 2. Integrated renewal processes The marginal probability density function of an integrated homogeneous Poisson Process is known in the literature. It is natural to generalize the integrated homogeneous Poisson process into integrated non homogeneous Poisson, Cox, and renewal processes. In this thesis we derive expressions for the marginal distributions of integrated Poisson and Cox processes using conditioning arguments, and derive the marginal distributions of integrated renewal processes using the theory of point processes. The results are presented in the form of Laplace transforms. Asymptotic properties of the integrated renewal processes are also investigated. An application to the study of traffic is given. 3. Total downtime of repairable systems An expression for the cumulative distribution function of the total downtime of a repairable system, which is regarded as a single component, under an assumption that the failure and the repair times of the system are independent has been derived by several authors using different methods. We use a different method (using point processes) to compute the distribution function of the total downtime. We also consider a more general situation where we allow dependence of the failure and the repair times of the system. The covariance structure and asymptotic properties of the total downtime for the independent case are also known in the literature. We derive the similar results for the dependent case. Examples are given to see the effect of dependence between the failure and the repair times on the total downtime. We also discuss the total downtime of repairable systems consisting of n 2 stochastically independent components. We derive an expression for the marginal distribution of the total uptime of the system for the case the failure and the repair times of each component are exponentially distributed. For arbitrary failure or repair times of the components we derive an expression for the mean of the total uptime.

Optimal maintenance of systems subject to deterioration of the renewal type

Abstract: We review some of the recent work on optimal maintenance of devices subject to wear of the cumulative renewal process type. The optimality considered using the total discounted cost over the infinite horizon, and the long-run average cost criterion.

Nonparametric estimation of some quantities of interest from renewal theory

Abstract: In this paper, confidence intervals are given for two quantities of importance related to renewal processes For each quantity, two confidence intervals are discussed One confidence interval is given for general, all-purpose use Another confidence interval is given which is easier to compute, but not of general use The case where data are subject to right censorship is also considered Some numerical comparisons are made © 2003 Wiley Periodicals, Inc Naval Research Logistics 50: 638–649, 2003

Non-linear Renewal Theory with Stationary Perturbations

Abstract: A non-linear renewal theorem is obtained from random walks that are perturbed by an approximately stationary sequence. As corollaries, the limiting joint distribution of the excess over the boundary and last perturbation are obtained along with an approximation to expected first passage times. The results are illustrated by an analysis of a sequential probability ratio test when data are subject to both censoring and staggered entry.

Asymptotic Optimal Control of Multi-Class G/G/1 Queues with Feedback

Abstract: In this chapter, we extend the well-known Klimov problem, i.e. the multiclass M/G/1 queueing control problem, to a general G/G/1 setting. Specifically, we consider a single server multi-class queueing system with Bernoulli feedback, where the arrival stream is a renewal process and the service times per class are i.i.d. random variables of general distribution, independent of the arrival process and other classes. A linear holding cost is incurred by each job in the system each unit time. The problem is to determine the optimal control so as to minimize the long-run average holding cost. We show that as the traffic intensity is close to 1, the control policy specified in Klimov [10] is strongly asymptotically optimal in the sense that its absolute difference from the optimal value is bounded from above while the optimal value is approaching infinity. To establish such strong asymptotic optimality, we present a method of measuring the tightness of control policies to optimality by developing close bounds for the corresponding system performance. We further give an example to show that typical Brownian approximations may not be sufficient to lead to solutions that are strongly asymptotically optimal, due to the loss of non-heavy traffic information in the Brownian limit.

On the rate of convergence to equilibrium for countable ergodic markov chains

Abstract: Using elementary methods, we prove that for a countable Markov chain P of ergodic degree d > 0 the rate of convergence towards the stationary distribution is subgeometric of order n −d , provided the initial distribution sat- isfies certain conditions of asymptotic decay. An example, modelling a renewal process and providing a markovian approximation scheme in dynamical system theory, is worked out in detail, illustrating the relationships between conver- gence behaviour, analytic properties of the generating functions associated to transition probabilities and spectral properties of the Markov operator P on the Banach space l1. Explicit conditions allowing to obtain the actual asymptotics for the rate of convergence are also discussed.