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R Jiang - One of the best experts on this subject based on the ideXlab platform.

  • A practical approximation of Weibull Renewal Function for solving relevant optimization problems
    Quality Engineering, 2020
    Co-Authors: R Jiang, Zhigao Chen
    Abstract:

    Some optimization problems in reliability and maintenance need to iteratively evaluate the Renewal Function (RF) of a lifetime distribution and/or its integral. For such problems, an accurate RF ap...

  • A novel two-fold sectional approximation of Renewal Function and its applications
    Reliability Engineering & System Safety, 2020
    Co-Authors: R Jiang
    Abstract:

    Abstract The Renewal Function (RF) has many applications such as reliability analysis, maintenance policy optimization and inventory planning. The RFs of most distribution Functions do not have closed-form expressions while such expressions are desired for most of applications. Several models that aim to approximate RF over the entire time range have been developed in the literature, but their accuracy is not high enough. To fill this gap, this paper proposes a two-fold sectional approximation, which is obtained through smoothly connecting two limiting relations. The proposed approximation is simple, applicable for ordinary lifetime distributions (e.g., Weibull and lognormal distributions), and accurate for the distributional parameters in the usual range. The variance of Renewals derived from the approximation is fairly accuracy and the integral of the approximation has a closed-form expression for the Weibull distribution. The approximation is useful for solving the optimization problems that involve RF or/and its integral, such as optimization of block replacement policy.

  • a simple approximation for the Renewal Function with an increasing failure rate
    Reliability Engineering & System Safety, 2010
    Co-Authors: R Jiang
    Abstract:

    This paper proposes a simple approximation for the Renewal Function of a failure distribution with an (equivalently) increasing failure rate. The approximation is a linear combination of the cumulative distribution and hazard Functions, and the coefficients are Functions of the shape parameter of the distribution. The approximation is applied to the Weibull, gamma and lognormal distributions, and it is shown that the approximation is accurate for t up to a certain value of larger than the characteristic life. The approximation is useful for maintenance policy analysis and optimization where the Renewal Function needs to be evaluated.

  • A simple approximation of the Weibull Renewal Function
    2009 IEEE International Conference on Industrial Engineering and Engineering Management, 2009
    Co-Authors: R Jiang
    Abstract:

    This paper proposes a simple approximation for the Renewal Function of the Weibull distribution with an increasing failure rate (i.e., the shape parameter being larger than one). It is developed for using in optimization of preventive maintenance policies. The approximation is a weighted geometer average of the cumulative distribution and hazard Functions with the weight being a Function of the shape parameter. The approximation is accurate for t up to a certain value of larger than the scale parameter.

  • A Gamma-normal series truncation approximation for computing the Weibull Renewal Function
    Reliability Engineering & System Safety, 2008
    Co-Authors: R Jiang
    Abstract:

    Abstract This paper presents a series truncation approximation for computing the Weibull Renewal Function. In the proposed model, the n-fold convolution of the Weibull Cdf is approximated by a mixture of the n-fold convolutions of Gamma and normal Cdfs. The mixture weight can be optimally determined and fitted into a very accurate linear Function of Weibull shape parameter β . Major advantages of the proposed model include: (a) The proposed model and its parameters can be directly written out. Using the proposed model, the Renewal density and variance Functions can be easily evaluated. (b) The proposed model includes Gamma and normal series truncation models as its special cases. It is easy to be implemented in Excel. The series converges fairly fast. (c) Over the range of β ∈ ( 0.87 , 8.0 ) , the maximum absolute error is smaller than 0.01; and over β ∈ ( 3.0 , 8.0 ) , the maximum absolute error is smaller than 0.0037. (d) The model can be easily extended to non-Weibull case with some additional work.

Tahir Khaniyev - One of the best experts on this subject based on the ideXlab platform.

  • a semi markovian Renewal reward process with gamma g distributed demand
    Turkish Journal of Mathematics, 2020
    Co-Authors: Aslı Bektaş Kamışlık, Busra Alakoc, Tülay Kesemen, Tahir Khaniyev
    Abstract:

    We consider a classical semi-Markovian stochastic model of type $ s,S $ with Logistic distributed demand random variables. Logistic distribution is a member of special distribution class known as $\Gamma g $ that encounters in many real-life applications involving extreme value theory. The objective of this study is to observe some major characteristics of a stochastic process $X t $ which represents semi-Markovian Renewal reward process of type $ s,S $. We used new approximation results for Renewal Function that allow us to obtain three-term asymptotic expansion for ergodic distribution Function and for $n^{th}$ order moments of ergodic distribution of the process $X t $.

  • A semi-Markovian Renewal reward process with $\Gamma g $ distributed demand
    TURKISH JOURNAL OF MATHEMATICS, 2020
    Co-Authors: Aslı Bektaş Kamışlık, Busra Alakoc, Tülay Kesemen, Tahir Khaniyev
    Abstract:

    We consider a classical semi-Markovian stochastic model of type $ s,S $ with Logistic distributed demand random variables. Logistic distribution is a member of special distribution class known as $\Gamma g $ that encounters in many real-life applications involving extreme value theory. The objective of this study is to observe some major characteristics of a stochastic process $X t $ which represents semi-Markovian Renewal reward process of type $ s,S $. We used new approximation results for Renewal Function that allow us to obtain three-term asymptotic expansion for ergodic distribution Function and for $n^{th}$ order moments of ergodic distribution of the process $X t $.

  • On the Stationary Distribution for a Fuzzy Inventory Model of Type (s,S) with Inverse Gaussian Distributed Demands
    Iranian Journal of Science and Technology Transactions A: Science, 2018
    Co-Authors: Tahir Khaniyev, Fikri Gökpınar, Tagi Hanalioglu, I. Burhan Turksen
    Abstract:

    In this study, we consider a fuzzy inventory model of type ( s , S ) with random demands having an inverse Gaussian distribution. We first show the monotonicity of the Renewal Function with respect to mean parameter. Thus we obtain the membership Function of the fuzzy Renewal Function when the amount of demands is a random variable having an inverse Gaussian distribution with a fuzzy mean parameter by using the monotonicity property of Renewal Function. Making use of the membership Function of the Renewal Function, we obtain the membership Function of the fuzzy ergodic distribution of this process. We also present some numerical results obtained by using this membership Function.

  • Investigation of fuzzy inventory model of type (s, S) with Nakagami distributed demands
    Journal of Intelligent & Fuzzy Systems, 2015
    Co-Authors: I. Burhan Turksen, Tahir Khaniyev, Fikri Gökpınar
    Abstract:

    In this study, a fuzzy inventory model of type (s, S) is considered under Nakagami distribution of demands. We first obtain the membership Function of the fuzzy Renewal Function when the amount of demand has Nakagami distribution with a fuzzy spread parameter. By using fuzzy Renewal Function, we obtain the fuzzy ergodic distribution of this process. Also some numerical results are obtained with the use of this membership Function.

  • Asymptotic properties of the straight line estimator for a Renewal Function
    2015
    Co-Authors: Esra Gökpınar, Tahir Khaniyev, Hamza Gamgam
    Abstract:

    In estimation problems in Renewal Function, when the distribution is not known, nonparametric estimators of Renewal Function are used. Frees (1986a, Warranty analysis and Renewal Function estimation, Naval Res. Logist. Quart, 33, 361-372) proposed the nonparametric estimator of Renewal Function for large values of t. Frees’s estimator is easy to apply in practice. It is a preferred estimator for large values of t. However, its statistical properties still have not been investigated in detailed. For this reason, in this study, we investigate asymptotic properties of this estimator such as consistency, asymptotic unbiasedness and asymptotic normality. Also Monte Carlo simulation study is given to assess the performance of this estimator according to value of Renewal Function. Simulation results indicate that in the large values of t, Frees estimator is sufficiently close to the Renewal Function for the Gamma distribution with various parameters.

Laurence A. Baxter - One of the best experts on this subject based on the ideXlab platform.

  • Nonparametric confidence intervals for the Renewal Function with censored data
    Journal of Nonparametric Statistics, 1995
    Co-Authors: Laurence A. Baxter, Linxiong Li
    Abstract:

    An asymptotic nonparametric method for constructing confidence intervals for the Renewal Function using censored data is presented. The method is based on the fact that the productlimit estimator of the Renewal Function converges weakly to a Gaussian process as the sample size increases

  • Non-parametric Confidence Intervals for the Renewal Function and the Point Availability
    Scandinavian Journal of Statistics, 1994
    Co-Authors: Laurence A. Baxter
    Abstract:

    A large sample non-parametric method for constructing confidence intervals for the Renewal Function and the point availability is investigated. The method is based on a linearization and on the fact that the empirical distribution Function converges weakly to a Gaussian process as the sample size increases. The technique is illustrated by the analysis of some hitherto unpublished data. Two of the most important Functions arising in Renewal theory are the Renewal Function, the expected number of Renewals in a given interval, and the point availability, the probability that a system modelled by an alternating Renewal process (ARP) is in a particular state at a specified time. See, for example, Karlin & Taylor (1975, Ch. 5), Ross (1970, Ch. 3) and Cox (1962) for a discussion of applications of these Functions. If the Functional forms of the distribution Functions of the random variables generating the processes are known, and observations of the random variables are available, point estimates of these Functions are readily constructed. Further, approximate (large sample) confidence intervals may, in principle, be calculated by an application of the delta method, assuming that the parameter estimates are asymptotically normally distributed. If, however, as is sometimes the case, the Functional forms of the underlying distribution Functions are unknown, a non-parametric approach is required. Frees (1986a, b, 1988) discussed some non-parametric estimators of the Renewal Function and constructed a non-parametric confidence interval for this quantity. See Schneider et al. (1990) for a study of these estimators. In this paper, we propose an alternative non-parametric confidence interval for the Renewal Function which is easier to compute than that of Frees (1986a) and which is appreciably narrower. In addition, we derive an analogous non-parametric confidence interval for the point availability. To the best, of our knowledge, this is the first non-parametric interval estimator of the point availability to have ,been proposed. Our methodology is based on the analysis of Harel et al. (1994), who prove that the empirical Renewal Function converges weakly to a Gaussian process as the sample size increases. A numerical study shows that our proposed confidence intervals are easy to compute, requiring only a few seconds of CPU time on a Sun Sparc Station, and are fairly narrow for moderate sample sizes.

Cigdem Cengiz - One of the best experts on this subject based on the ideXlab platform.

  • Nonparametric estimation of a Renewal Function in the case of censored sample
    Bitlis Eren University Journal of Science and Technology, 2019
    Co-Authors: Cigdem Cengiz
    Abstract:

    A Renewal process is a counting process which counts the number of Renewals that occurs as a Function of time, wherein the durations between successive Renewals are random variables independent of one another, with identical F distributions. The mean value Function data is frequently needed in applications of Renewal processes. For the Renewal Function, open expressions depending on distribution Function F can be calculated from each other. However, even though the distribution Function F is known, the Renewal Function cannot be obtained analytically except for a few distributions. In this study, in the case that F is totally unknown, life table management and Kaplan-Meier estimator were used depending on random right-censored sampling for the estimation of F value. Then, for the estimation of the Renewal Function value in the random right-censored data, nonparametric estimators were proposed and the problem of how to calculate these estimators were discussed.

  • GAMMA Renewal Function in Censored Data
    Bitlis Eren University Journal of Science and Technology, 2015
    Co-Authors: Cigdem Cengiz, Halil Aydogdu
    Abstract:

    In this study, the Renewal process whose times between the intervals are gamma-diffused has been examined. In the situation where the sampling is Randomly Right Censored, a Parametric Estimator is recommended, which depends on the Maximum Likelihood Estimators of the unknown parameters of the gamma diffusion; and the statistical characteristics of these estimators have been investigated.

  • estimation of weibull Renewal Function for censored data
    International Journal of Scientific and Technological Research, 2015
    Co-Authors: Cigdem Cengiz, Ayse Metin Karakas
    Abstract:

    In this study, the weibull Renewal process in which interRenewal times are weibull distributed is considered. In case of random right censored sample, a parametric estimator for the value of weibull Renewal Function is proposed based on the maximum likelihood estimators of the unknown parameters of weibull distribution. Key Words: Weibull distribution, Renewal process, Renewal Function, censoring

U.r. Krieger - One of the best experts on this subject based on the ideXlab platform.

  • Nonparametric Estimation of the Renewal Function by Empirical Data
    Stochastic Models, 2006
    Co-Authors: Natalia M. Markovich, U.r. Krieger
    Abstract:

    We consider an estimate of the Renewal Function (rf) H(t) using a limited number of independent observations of the interarrival times for an unknown interarrival-time distribution (itd). The nonparametric estimate is derived from the rf-representation as a series of distribution Functions (dfs) of consecutive arrival times using a finite summation and approximations of the latter by empirical dfs. Due to the limited number of observed interarrival times, the estimate is accurate just for closed time intervals [0, t]. An important aspect is given by the selection of an optimal number of terms k of the finite sum. Here two methods are proposed: (1) an a priori choice of k as Function of the sample size l which provides almost surely (a.s.) the uniform convergence of the estimate to the rf for light- and heavy-tailed itds if the time interval is not too large, and (2) a data-dependent selection of k by a bootstrap method. To evaluate both the efficiency of the estimate and the selection methods of k, a Mont...

  • Estimating Basic Characteristics of Arrival Processes in Telecommunication Networks by Empirical Data
    Telecommunication Systems, 2002
    Co-Authors: Natalia M. Markovitch, U.r. Krieger
    Abstract:

    Considering the realistic teletraffic analysis in advanced telecommunication networks, the estimation of basic characteristics of arrival processes by empirical data is an important subject of current research. Using independent observations of the interarrival times between events and the mean numbers of events in intervals of fixed length, we propose methods to estimate the intensity of a nonhomogeneous arrival stream, particularly a Poisson process, and the Renewal Function of a Renewal process. We formulate the estimation task as stochastically ill-posed problem and apply procedures for the stabilization of the estimates.

  • Estimation of the Renewal Function by empirical data-a Bayesian approach
    1st European Conference on Universal Multiservice Networks. ECUMN'2000 (Cat. No.00EX423), 2000
    Co-Authors: N.m. Markovitch, U.r. Krieger
    Abstract:

    Considering traffic measurements in advanced telecommunication networks. We study some basic estimation issues using nonparametric techniques. Using only a very limited number of independent observations of the interarrival times between the recorded events of interest, we propose a simple histogram-type estimate to restore the Renewal Function of an underlying Renewal process with its unknown interarrival time distribution. Due to the limited number of the observed interarrival times the estimate is accurate just for closed, but sufficiently wide observation intervals. To provide the minimum of the mean-squared error for different closed time intervals of the estimation, the optimal number of terms of the histogram-type estimate is selected by a Bayesian principle. Furthermore, the effectiveness of the estimate is evaluated by a simulation study for several light- and heavy-tailed interarrival-time distributions including the exponential, Gamma and Weibull distribution with a shape parameter less than one.