Showing papers on "Renewal theory published in 1996"

A quasi renewal process and its applications in imperfect maintenance

Abstract: This paper proposes a quasi renewal process, and its application in maintenance theory is discussed. The properties of this quasi renewal process are studied and its renewal function is derived. Three imperfect maintenance models are proposed and they model imperfect maintenance in a way that, after maintenance, the lifetime of a unit will decrease to a fraction of its immediate previous one. According to the theory of the quasi-renewal process developed in this paper, the expected maintenance cost rate and availability are obtained and optimum maintenance policies are discussed for these three models. Finally, a class of related optimization problems is discussed and a numerical example is presented

On the use of renewal theory in the analysis of ARQ protocols

Abstract: An automatic-repeat-request (ARQ) Go-Back-N (GBN) protocol with unreliable feedback and time-out mechanism is studied, using renewal theory. Transmissions on both the forward and the reverse channels are assumed to experience Markovian errors. The exact throughput of the protocol is evaluated, and simulation results, that confirm the analysis, are presented. A detailed comparison of the proposed method and the commonly used transfer function method reveals that the proposed approach is simple and potentially more powerful.

Point process approaches to the modeling and analysis of self-similar traffic .I. Model construction

Abstract: We propose four fractal point processes (FPPs) as novel approaches to modeling and analyzing various types of self-similar traffic: the fractal renewal process (FRP), the superposition of several fractal renewal processes (Sup-FRP), the fractal-shot-noise-driven Poisson process (FSNDP), and the fractal-binomial-noise-driven Poisson process (FBNDP). These models fall into two classes depending on their construction. A study of these models provides a thorough understanding of how self-similarity arises in computer network traffic. We find that (i) all these models are (second-order) self-similar in nature; (ii) the Hurst parameter alone does not fully capture the burstiness of a typical self-similar process; (iii) the heavy-tailed property is not a necessary condition to yield self-similarity; and (iv) these models permit parsimonious modeling (using only 2-5 parameters) and fast simulation. Simulation verifies that these models exhibit a fractal behavior over a wide range of time scales.

On nonhomogeneous models for volcanic eruptions

Abstract: Recently, a special nonhomogeneous Poisson process known as the Weibull process has been proposed by C-H. Ho for fitting historical volcanic eruptions. Revisiting this model, we learn that it possesses some undesirable features which make it an unsatisfactory tool in this context. We then consider the entire question of a nonstationary model in the light of availability and completeness of data. In our view, a nonstationary model is unnecessary and perhaps undesirable. We propose the Weibull renewal process as an alternative to the simple (homogeneous) Poisson process. For a renewal process the interevent times are independent and distributed identically with distribution function F where, in the Weibull renewal process, F has the Weibull distribution, which has the exponential as a special situation. Testing for a Weibull distribution can be achieved by testing for exponentiality of the data under a simple transformation. Another alternative considered is the lognormal distribution for F. Whereas the homogeneous Poisson process represents purely random (memoryless) occurrences, the lognormal distribution corresponds to periodic behavior and the Weibull distribution encompasses both periodicity and clustering, which aids us in characterizing the volcano. Data from the same volcanoes considered by Ho were analyzed again and we determined there is no reason to reject the hypothesis of Weibull interevent times although the lognormal interevent times were not supported. Prediction intervals for the next event are compared with Ho's nonhomogeneous model and the Weibull renewal process seems to produce more plausible results.

Analysis of a two-component series system with a geometric process model

Abstract: In this article, a geometric process model is introduced for the analysis of a two-component series system with one repairman. For each component, the successive operating times form a decreasing geometric process with exponential distribution, whereas the consecutive repair times constitute an increasing geometric process with exponential distribution, but the replacement times form a renewal process with exponential distribution. By introducing two supplementary variables, a set of partial differential equations is derived. These equations can be solved analytically or numerically. Further, the availability and the rate of occurrence of failure of the system are also determined. © 1996 John Wiley & Sons, Inc.

Stochastic renewal model of low-flow streamflow sequences

Abstract: It is shown that runs of low-flow annual streamflow in a coastal semiarid basin of Central California can be adequately modelled by renewal theory. For example, runs of below-median annual streamflows are shown to follow a geometric distribution. The elapsed time between runs of below-median streamflow are geometrically distributed also. The sum of these two independently distributed geometric time variables defines the renewal time elapsing between the initiation of a low-flow run and the next one. The probability distribution of the renewal time is then derived from first principles, ultimately leading to the distribution of the number of low-flow runs in a specified time period, the expected number of low-flow runs, the risk of drought, and other important probabilistic indicators of low-flow. The authors argue that if one identifies drought threat with the occurrence of multiyear low-flow runs, as it is done by water supply managers in the study area, then our renewal model provides a number of interesting results concerning drought threat in areas historically subject to inclement, dry, climate. A 430-year long annual streamflow time series reconstructed by tree-ring analysis serves as the basis for testing our renewal model of low-flow sequences.

Two renewal theorems for general random walks tending to infinity

Abstract: Necessary and sufficient conditions for the existence of moments of the first passage time of a random walk S n into [x, ∞) for fixed x≧ 0, and the last exit time of the walk from (−∞, x], are given under the condition that S n →∞ a.s. The methods, which are quite different from those applied in the previously studied case of a positive mean for the increments of S n , are further developed to obtain the “order of magnitude” as x→∞ of the moments of the first passage and last exit times, when these are finite. A number of other conditions of interest in renewal theory are also discussed, and some results for the first time for which the random walk remains above the level x on K consecutive occasions, which has applications in option pricing, are given.

A lost sales (S - 1,S) perishable inventory system with renewal demand

Abstract: This article analyzes a one-to-one ordering perishable inventory model with renewal demands and exponential lifetimes. The leadtimes are independently and exponentially distributed and the demands that occur during stock out periods are lost. Although the items are assumed to decay at a constant rate, the output process is not renewal and the Markov renewal techniques are successfully employed to obtain the operating characteristics. The problem of minimizing the long run expected cost rate is discussed and numerical values of optimal stock level are also provided.

First crossing of basic counting processes with lower non-linear boundaries: A unified approach through pseudopolynomials (I)

Abstract: The paper is concerned with the distribution of the level N of the first crossing of a counting process trajectory with a lower boundary. Compound and simple Poisson or binomial processes, gamma renewal processes, and finally birth processes are considered. In the simple Poisson case, expressing the exact distribution of N requires the use of a classical family of Abel-Gontcharoff polynomials. For other cases convenient extensions of these polynomials into pseudopolynomials with a similar structure are necessary. Such extensions being applicable to other fields of applied probability, the central part of the present paper has been devoted to the building of these pseudopolynomials in a rather general framework.

Non‐parametric estimation for semi‐Markov kernels with application to reliability analysis

Abstract: The authors consider an irreducible Markov renewal process (MRP) with a finite number of states. Their aim is to derive estimators of a censored MRP with a finite number of states either in a fixed time T or in the Nth jump. The estimators given here are seen to be of the Kaplan-Meier type. The asymptotic properties these estimators are given. The reliability of a semi-Markov system model is examined numerically by the estimators, and a comparison is made with estimators obtained by Lagakos et al.

Preventive maintenance modelling

Abstract: Develops practical models for preventive maintenance policies using Bayesian methods of statistical inference. Considers the analysis of a delayed renewal process and a delayed alternating renewal process with exponential times to failure. This approach has the advantage of generating predictive distributions for numbers of failures and downtimes rather than relying on estimated renewal functions. Demonstrates the superiority of this approach in analysing situations with non‐linear cost functions, which arise in reality, by means of an example.

Maintenance Policies for Multicomponent Systems: An Overview

Abstract: We present an overview of some recent developments in the area of mathematical modelling of maintenance decisions for multicomponent systems. We do not claim to be complete, but rather we expose some ideas both in modelling and in solution procedures which turned out to be useful in understanding and supporting complex maintenance management decision problems. The mathematical tools that are used mainly stem from applied probability theory, renewal theory, and Markov decision theory.

Closed Form Performance Distributions of a Discrete Time GIg/D/1/N Queue with Correlated Traffic

Abstract: A novel approach is applied to the study of a queue with general correlated traffic, in that the only features of the traffic which are taken into account are the usual measures of its correlation: the traffic is modelled as a batch renewal process. The batch renewal process is a precise tool for investigation into effects of correlation because it is the least biased choice of process which is completely determined by the infinite sets of measures of the traffic’s correlation (e.g. indices of dispersion, covariances or correlation functions).

Life Estimation from Pooled Discrete Renewal Counts

Abstract: We study a problem arising in the analysis of field reliability data generated by a repair process that replaces at a depot individual components on line-replaceable units that then are returned to service. Data collected in such a repair scenario is usually limited to counts of the number of components replaced each month without regard to age. We construct a pooled discrete renewal process model for this scenario and study a maximum likelihood-like estimation of the parameters in this model

Policies without Memory for the Infinite-Armed Bernoulli Bandit under the Average-Reward Criterion

Abstract: We consider a bandit problem with infinitely many Bernoulli arms whose unknown parameters are i.i.d. We present two policies that maximize the almost sure average reward over an infinite horizon. Neither policy ever returns to a previously observed arm after switching to a new one or retains information from discarded arms, and runs of failures indicate the selection of a new arm. The first policy is nonstationary and requires no information about the distribution of the Bernoulli parameter. The second is stationary and requires only partial information; its optimality is established via renewal theory. We also develop e-optimal stationary policies that require no information about the distribution of the unknown parameter and discuss universally optimal stationary policies.

A unified treatment of single component replacement models

Abstract: In this paper we discuss a general framework for single component replacement models. This framework is based on the regenerative structure of these models and by using results from renewal theory a unified presentation of the discounted and average finite and infinite horizon cost models is given. Moreover, we present sufficient conditions for the numerator of the cost function to have a vanishing singular part. Finally, some well-known replacement models are discussed, and making use of the previous results an easy derivation of their cost functions is presented.

Analysis of Open Discrete Time Queueing Networks: A Refined Decomposition Approach

Abstract: This paper deals with approximate analysis methods for open queueing networks. External and internal flows from and to the nodes are characterized by renewal processes with discrete time distributions of their interarrival times. Stationary distributions of the waiting time, the queue size and the interdeparture times are obtained using efficient discrete time algorithms for single server (GI/G/1) and multi-server (GI/D/c) nodes with deterministic service. The network analysis is extended to semi-Markovian representations of each flow among the nodes, which include parameters of the autocorrelation function.

The Value of Information in an (R,s,Q) Inventory Model

Abstract: In this paper we compare three methods for the determination of the reorder point s in an (R; s; Q) inventory model subject to a service level constraint. The three methods di er in the modelling assumptions of the demand process which in turn leads to three di erent approximations for the distribution function of the demand during the lead time.The rst model is most common in the literature, and assumes that the time axis is divided in time units (e.g. days).It is assumed that the demands per time unit are independent and identically distributed random variables.The second model monitors the customers individually.In this model it is assumed that the demand process is a compound renewal process, and that the distribution function of the interarrival times as well as that of the demand per customer are approximated by the rst two moments of the associated random variable.The third method directly collects information about the demand during the lead time plus undershoot, avoiding convolutions of stochastic random variables and residual lifetime distributions.Consequently, the three methods require di erent types of information for the calculation of the reorder point in an operational setting.The purpose of this paper is to derive insights into the value of information; therefore it compares the target service level with the actual service level associated with the calculated reorder point.It will be shown that the performance of the rst model (discrete time model) depends on the coe cient ofvariation of the interarrival times. Furthermore, because we use asymptotic relations in the compound renewal model, we derive some bounds for the input parameters within which this model applies. Finally we show that the aggregated information model is superior to the other two models.

A parallel Poisson generator using parallel prefix

Abstract: In this paper, we use the renewal theory to develop a Poisson random number algorithm without restart. A parallel Poisson random number generator is designed based on this algorithm and prefix computation. This generator iteratively produces m Poisson random numbers with mean μ in average time complexity O([mμn]f(n,p)) on EREW PRAM, where f(n,p) is the time for computing an n-element parallel prefix on p processors in each iteration, assuming that parallel uniform random numbers can be generated at the rate of one number per unit time per processor. If n is selected near mμ, it achieves linear speedup when p is small and the average time complexity is O(log(mμ)) when p is O(mμ).

Sample Quantiles of Additive Renewal Reward Processes

Abstract: The distribution of the sample quantiles of random processes is important for the pricing of some of the so-called financial ‘look-back' options. In this paper a representation of the distribution of the α-quantile of an additive renewal reward process is obtained as the sum of the supremum and the infimum of two rescaled independent copies of the process. This representation has already been proved for processes with stationary and independent increments. As an example, the distribution of the α-quantile of a randomly observed Brownian motion is obtained.

On the Rate of Convergence of the Ross Approximation to the Renewal Function

Abstract: Consider a renewal process [ N ( t ), t>0]. For fixed t > 0 and each n ≥ 1, let y n ,1 , …, Y n,n be independent exponentials each having mean t / n , independent of the renewal process. Ross [2] developed a recursion for the sequence of approximations m n = EN ( Y n ,1 + … + Y n,n ) that converges to m ( t )if the renewal function m (·) = EN (·) is continuous at t > 0. In this note, we derive an upper bound on the rate of convergence of this sequence under mild conditions on m near t . Tightness of this bound is discussed in terms of regularity conditions on m .

Security systems and renewal processes

Abstract: An interesting feature of successive inter-event times in a Poisson renewal process, when observed through a superimposed (possibly random) grid, can be interpreted as an extended form of the 'inspection paradox'. Probabilistic measures are determined and an estimation and testing procedure is outlined which could be used to examine possible departure from randomness (Poisson form). The problem arose from study of security systems and the results have important applications in that field.