Renewal Function Research Articles

Superposition of renewal processes

This paper extends the asymptotic results for ordinary renewal processes to the superposition of independent renewal processes. In particular, the ordinary renewal functions, renewal equations, and the key renewal theorem are extended to the superposition of independent renewal processes. We fix the number of renewal processes, p, and study the asymptotic behavior of the superposition process when time, t, is large. The key superposition renewal theorem is applied to the study of queueing systems.

A General Optimization Model for Multi-Tool Manufacturing Systems

A machining center is a single numerically-controlled machine that incorporates several time-saving features and performs a multiplicity of operations. The life of cutting tools used in machining centers is modeled using probabilistic methods because of the stochastic nature and variability of tool-wear data. An approximate expression for the renewal function is used to establish an optimum scheduled tool replacement interval and an optimum set of cutting conditions to obtain a minimum production cost. A general optimization model for the multi-tool machining center is formulated considering practical machine constraints. Premature failure cost is not considered constant in this study. It is estimated using the tool cost, the expected cost of unplanned downtime, and the expended cost of product damage by premature tool failure. A user friendly computer program was developed to solicit information from a production engineer regarding cutting operations for the various tools used at the machining center. A sensitivity analysis on the results of the optimization process is presented.

The renewal function for an alternating renewal process, which has a Weibull failure distribution and a constant repair time

Abstract A series expansion is developed for the n th order distribution functions of the Weibull distribution and the coefficients are found by solving recurrence equations. The availability and the renewal function of an alternating renewal process, which consists pf a Weibull failure distribution and a constant repair time, are written as related double series. The distribution functions and the unavailability are calculated for several values of the parameters and the rate of convergence of the series is examined.

A simple approximation to the renewal function : Eric Smeitink and Rommert Dekker. IEEE Trans. Reliab.39(1), 71 (1990)

A simple approximation to the renewal function : Eric Smeitink and Rommert Dekker. IEEE Trans. Reliab.39(1), 71 (1990)

On the optimality of the (S — 1,S) policy

The one-for-one (S - 1,S) inventory policy, which calls for a replenishment order after each demand equal in magnitude to the size of the demand, is often advocated for controlling the stock levels of expensive, slow-moving items. In particular, this policy has frequently been promoted for use in recoverable-item inventory systems. An important managerial question is: When is the (S - 1,S) policy optimal? Results in this article provide guidance for the selection of this policy. Conditions under which it is not economical to batch demands are developed, provided the renewal function of the renewal process of demand sizes is concave. This includes the important case of unit-sized demands. The inventory system considered is one with continuous review, constant lead times, general interarrival and discrete demand distributions, complete backlogging, and linear holding and penalty costs per unit per unit time. Examples are given for the special case when the demand process is Poisson and when the demand process is a particular compound Poisson distribution known as the stuttering Poisson distribution.

A Replacement Model for Multi-Tool Transfer Lines

Transfer lines have long been used for machining a single product at high production rates. This study deals with the transfer line, in particular, estimating the production time per part and the effect of using probability models in describing tool and machine lives. The production time is estimated using three major causes of line stoppage. An aspiration level criterion is used to establish a scheduled tool replacement interval. The aspiration decision parameter is system reliability. It is shown how a scheduled tool replacement interval could be established to obtain this reliability level. A methodology for selecting the replacement interval for a group of tools is developed and the impact of changing this interval on tool changing costs and tool failure costs is discussed. In this study, only a single premature tool failure is assumed to occur; however, the results are compared with multiple tool failures using the renewal function. It is shown that this single premature failure analysis is equivalent to the renewal function approach for machining systems having a reliability greater than 50 percent and tool failure variabilities having a coefficient of variation less than 1.0. Using the model developed in this paper, an example problem is presented. For this problem, electromechanical equipment failure and tool wear failures are modeled using the exponential and the Weibull probability distributions, respectively. Scheduled tool replacement intervals are developed for a range of target reliabilities. The part cycle time and operating cost for the transfer line based on these reliabilities are computed. These results are compared to the minimum cost reliability level. The minimum cost operating condition is dependent on the ratio of the scheduled replacement costs over the failure replacement costs. For the example presented, the reliability level at minimum cost is much lower than the desired target value of .9. This result illustrates the importance of conducting a careful analysis and using the results to help guide operating practices toward more efficient tool change practices and maintenance policies.

On the concavity of the infinitesimal renewal function

The expected time of the first passage above a level of a process with stationary independent non-negative increments is studied. It is called infinitesimal renewal function. Results analogous to Brown's (1980) concavity result on the conventional renewal function, for the infinitesimal renewal function, are proved. Some connections with Brown's (1981) conjecture and shock models are pointed out.

Comparison of Nonparametric Estimators for the Renewal Function

This paper addresses the problem of nonparametric estimation of the renewal function. Two estimators are discussed. The first estimator, introduced by Frees, is based on the sum of the convolutions without replacement of the empirical distribution function. We suggest a polynomial time algorithm to compute this estimator. The second estimator is based on the renewal function of the empirical distribution. We show how this estimator may be computed efficiently by solving a discretized renewal equation. In a simulation study the estimations are compared with respect to bias, mean-squared error and computing time

A simple approximation to the renewal function (reliability theory)

The authors present a simple, easy-to-understand approximation to the renewal function that is easy to implement on a personal computer. The key idea is that, for small values of time, the renewal function is almost equal to the cumulative distribution function of the interrenewal time, whereas for larger values of time an asymptotic expansion depending only on the first and second moment of the interrenewal time can be used. The relative error is typically smaller than a few percent for Weibull interrenewal times. The simple approximation methods works very well with one term if not too much accuracy is required (e.g. in the block replacement problem) or if the interrenewal (failure) distribution is not exactly known (e.g. only the first two moments are known). Although the accuracy of the simple approximation can be improved by increasing the number of terms, this strategy is not advocated since speed and simplicity are lost. If high accuracy is required, it is better to use another approximating method (e.g. power series expansion or cubic splines method).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Comparing Renewal Processes, with Application to Reliability Modeling of Highly Reliable Systems

Elementary methods are used to obtain upper and lower bounds for the difference between two renewal functions corresponding to two different life distributions and to obtain various order relationships between them that follow from order relationships on the underlying life distributions. The bounds are used to approximate the value of an unknown renewal function by the value of a known renewal function and to bound the error of approximation by expressions involving the difference between the two underlying life distributions. They are useful mainly in the region where time is small compared with the means of the two distributions, and they therefore supplement the information that can be obtained for large time using the elementary renewal theorem. These results find major application in reliability modeling of highly reliable systems.

Comparing Renewal Processes, With Application to Reliability Modeling of Highly Reliable Systems

Elementary methods are used to obtain upper and lower bounds for the difference between two renewal functions corresponding to two different life distributions and to obtain various order relationships between them that follow from order relationships on the underlying life distributions. The bounds are used to approximate the value of an unknown renewal function by the value of a known renewal function and to bound the error of approximation by expressions involving the difference between the two underlying life distributions. They are useful mainly in the region where time is small compared with the means of the two distributions, and they therefore supplement the information that can be obtained for large time using the elementary renewal theorem. These results find major application in reliability modeling of highly reliable systems.

A simple approximation for IFR weibull renewal functions

The renewal function is useful in reliability theory and is often needed to compute optimal maintenance policies. Although the Weibull is frequently the distribution of choice for reliability estimation, it is seldom used in maintenance models. One reason for this is that the renewal function for the Weibull is intractable. Existing methods used to numerically compute or to approximate this function typically involve algorithms that are O( n 2) in complexity and have large storage requirements. Although the renewal function converges to a linear expression as t → ∞, these methods become increasingly expensive as t becomes large. In this paper, we present a simple and accurate approximation that has computational complexity that is O( n) with O(1) storage requirements. The approximation is based on the limiting expression of the renewal function and makes use of several well known relations for IFR renewal functions. We demonstrate the usefulness and the accuracy of the approximation with several examples.

Characteristics of Birnbaum and Saunders model

Fatigue failure occurs when some dominant crack or cracks in a component extend to a critical level under the application of cyclic loading. By using the theory of stochastic processes Birnbaum and Saunders proposed a probability model to characterize the time (i.e. the number of cycles) required to propagate a fatigue crack past a critical value. The model is phenomenologically quite sound and provides a probabilistic interpretation of Miner's rule. In statistical literature a thorough treatment of the model is missing. For example, no work has been reported about the renewal and related functions of this model. This paper presents: (i) a summary of some known characteristics of the model; (ii) parameter estimation methods, and K– S test statistics for the model validation; (iii) the nature of hazard function in terms of the coefficient of life variation; (iv) the renewal function, renewal rate function and variance of number of renewals in graphical form; and (v) a comparison of a typical set of various functions.

Some results on renewal equations

In this paper solutions of renewal-type integral equations are studied. It is proved that a recursively spades;efined approximation to the solution has some nice convergence properties. Some simple bounds and other results on the renewal function and the renewal spades;,ensity are obtained.

Practical Problems in the Statistical Analysis of Reliability Data

Three examples of reliability data sets are discussed. Stochastic process models are used to describe repairable systems. The problems encountered in the analysis of failure time data are considered in particular when censoring is an important feature. Finally the assessment of system reliability is investigated by prediction of the renewal function.

A strong renewal theorem for generalized renewal functions in the infinite mean case

LetF(x) be a nonarithmetic c.d.f. on (0, ∞) such that 1 —F(x)=x−αL(x), whereL(x) is slowly varying and 0≤α≤1. Leta(x) be regularly varying with exponent β≥1. A strong renewal theorem (of Blackwell type) for generalized renewal functions of the form\(G(t) \equiv \sum\limits_{n = 0}^\infty {a(n) F^n (t)} \) is proved here, thus extending the recent work of Embrechts, Maejima and Omey [1] and that of Erickson [4].

A cost relationship between age and block replacement policies

The general cost structure of a unit on line is assumed to be governed by a stochastic process , where R(t) denotes the operating cost on [0, t)and ζ denotes the time of an unscheduled (or unplanned) replacement by a new unit at a cost c1. For an age replacement maintenance policy, scheduled (or planned) replacements occur whenever an operating unit reaches age T, whereas in the block replacement case, scheduled replacements occur every T units of time. Such scheduled replacements cost c2. The expected long-run cost per unit time can then be expressed in the form A(T)/Ε [min(ζ, T)] and B(T)/T respectively. Our main result shows that where U is the associated renewal function generated by ζ.

A cost relationship between age and block replacement policies

The general cost structure of a unit on line is assumed to be governed by a stochastic process , where R(t) denotes the operating cost on [0, t)and ζ denotes the time of an unscheduled (or unplanned) replacement by a new unit at a cost c 1. For an age replacement maintenance policy, scheduled (or planned) replacements occur whenever an operating unit reaches age T, whereas in the block replacement case, scheduled replacements occur every T units of time. Such scheduled replacements cost c2. The expected long-run cost per unit time can then be expressed in the form A(T)/Ε [min(ζ, T)] and B(T)/T respectively. Our main result shows that where U is the associated renewal function generated by ζ.

Computing the Phase-Type Renewal and Related Functions

This article presents a procedure for computing the renewal function, the renewal density, the integral of the renewal function, and the variance function of phase-type renewal processes. The procedure hinges on the computation of the state probability vector of a continuous-time Markov chain. This is accomplished by using a randomization approach that is simple, efficient, and numerically stable and does not require numerical integration. I discuss approximating arbitrary interrenewal distributions by phase-type distributions so that the procedure can be used to approximate renewal and related functions.

Estimating the Cost of a Warranty

The cost of certain types of warranties is closely related to functions that arise in renewal theory. The problem of estimating the warranty cost for a random sample of size n can be reduced to estimating these functions. In an earlier paper, I gave several methods of estimating the expected number of renewals, called the renewal function. This answered an important accounting question of how to arrive at a good approximation of the expected warranty cost. In this article, estimation of the renewal function is reviewed and several extensions are given. In particular, a resampling estimator of the renewal function is introduced. Further, I argue that managers may wish to examine other summary measures of the warranty cost, in particular the variability. To estimate this variability, I introduce estimators, both parametric and nonparametric, of the variance associated with the number of renewals. Several numerical examples are provided.