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Kai Yao - One of the best experts on this subject based on the ideXlab platform.

  • Uncertain Random Renewal Processes
    Uncertain Renewal Processes, 2019
    Co-Authors: Kai Yao
    Abstract:

    An uncertain random Process is a spectrum of uncertain random variables indexed by the time. The uncertain random Renewal Processes, as a family of sample-discontinuous uncertain random Processes, occur with the Renewals of an uncertain random system. This chapter introduces the uncertain random Renewal Process, the uncertain random Renewal Reward Process, and the uncertain random alternating Renewal Process.

  • Stochastic Renewal Processes
    Uncertain Renewal Processes, 2019
    Co-Authors: Kai Yao
    Abstract:

    A stochastic Process is essentially a spectrum of random variables indexed by the time. The stochastic Renewal Process is a stochastic Process, which counts the number of Renewals that a stochastic system incurs. This chapter introduces the stochastic Renewal Process, the stochastic Renewal Reward Process, and the stochastic alternating Renewal Process. The results in this chapter are well known, so the references are not provided. In this chapter, the interarrival times and the Rewards are assumed to have continuous probability distributions unless otherwise stated.

  • First hitting time of uncertain random Renewal Reward Process and its application in insurance risk Process
    Soft Computing, 2018
    Co-Authors: Kai Yao
    Abstract:

    The Renewal Reward Process is used to record the cumulative Rewards of a system, which is widely applied in the queuing problems and insurance pricing problems. This paper studies a type of Renewal Reward Processes with random inter-arrival times and uncertain Rewards from the point of view of first hitting time. The analytic expressions of the chance distribution and the expected value of the first hitting time are derived, and a numerical method for calculating the chance distribution is designed based on the Monte-Carlo simulation. Besides, the concept of first hitting time is applied to the insurance risk Process and is employed to model the ruin index of an insurance company.

  • Renewal Reward Process With Uncertain Interarrival Times and Random Rewards
    IEEE Transactions on Fuzzy Systems, 2018
    Co-Authors: Kai Yao, Jian Zhou
    Abstract:

    Renewal Reward Process is used to measure the cumulative occasional Rewards up to some given time. So far, two basic types of Renewal Reward Processes with only random parameters or with only uncertain parameters have been proposed. This paper aims at proposing a new type of Renewal Reward Process, which has uncertain interarrival times and random Rewards in the framework of the chance theory. The chance distribution of such an uncertain random Renewal Reward Process is obtained, and the Reward rate is derived. A Renewal Reward theorem is verified to show that the Reward rate converges in distribution to an uncertain variable, which is highly related to the interarrival times and the expected values of random Rewards.

  • Uncertain Random Renewal Reward Process With Application to Block Replacement Policy
    IEEE Transactions on Fuzzy Systems, 2016
    Co-Authors: Kai Yao, Jian Zhou
    Abstract:

    As a mixture of uncertain variable and random variable, uncertain random variable is an important tool to describe indeterminacy phenomena. In order to model the evolution of uncertain random phenomena, a concept of uncertain random Process has been proposed, and an uncertain random Renewal Process has been designed as an example. This paper aims to propose a new type of uncertain random Process, called uncertain random Renewal Reward Process, in which the interarrival times and the Rewards are assumed to be random variables and uncertain variables, respectively. The chance distribution of the Renewal Reward Process is obtained, and the Reward rate is derived. A Renewal Reward theorem is verified, which shows that the Reward rate converges in distribution to an uncertain variable derived from the random interarrival times and the uncertain Rewards. As an application, this paper also proposes an uncertain random block replacement problem and formulates an unconstrained optimization model by using the uncertain random Renewal Reward Process.

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 $.

  • Limit theorem for a semi - Markovian stochastic model of type (s,S)
    Hacettepe Journal of Mathematics and Statistics, 2018
    Co-Authors: Zulfiye Hanalioglu, Tahir Khaniyev
    Abstract:

    In this study, a semi-Markovian inventory model of type $(s,S)$ is considered and the model is expressed by means of Renewal-Reward Process $(X(t))$  with an asymmetric triangular distributed interference of chance and delay. The ergodicity of the Process $X(t)$  is proved and the exact expression for the ergodic distribution is obtained. Then, two-term asymptotic expansion for the ergodic distribution is found for standardized Process $W(t)\equiv (2X(t)) / (S-s)$. Finally, using this asymptotic expansion, the weak convergence theorem for the ergodic distribution of the Process $W(t)$ is proved and the explicit form of the limit distribution is found.

  • ASYMPTOTIC RESULTS FOR AN INVENTORY MODEL OF TYPE (s,S) WITH ASYMMETRIC TRIANGULAR DISTRIBUTED INTERFERENCE OF CHANCE AND DELAY
    gazi university journal of science, 2018
    Co-Authors: Zulfiye Hanalioglu, Tahir Khaniyev
    Abstract:

    In this study, a semi – Markovian inventory model of type (s,S)  is considered and the model is expressed by a modification of a Renewal – Reward Process (X(t))  with an asymmetric triangular distributed interference of chance and delay. The ergodicity of the Process X(t)  is proved under some weak conditions. Additionally, exact expressions and three – term asymptotic expansions are found for all the moments of the ergodic distribution. Finally, obtained asymptotic results are compared with exact results for a special case.

  • Investigation of Boundary Functionals for Renewal-Reward Process with a Generalized Reflecting Barrier
    2016
    Co-Authors: Tahir Khaniyev, Basak Gever, Zulfiye Hanalioglu
    Abstract:

    In this study, Renewal - Reward Process with a generalized reflecting barrier (X(t)) and its three boundary functionals are mathematically constructed. Next, the asymptotic expansions are obtained for the first four moments of these boundary functionals of the Process X(t).

Dug Hun Hong - One of the best experts on this subject based on the ideXlab platform.

  • Renewal AND Renewal Reward THEORIES FOR T-INDEPENDENT FUZZY RANDOM VARIABLES
    Journal of applied mathematics & informatics, 2015
    Co-Authors: Dug Hun Hong
    Abstract:

    Recently, Wang et al. [Computers and Mathematics with Ap-plications 57 (2009) 1232-1248.] and Wang and Watada [Information Sci-ences 179 (2009) 4057-4069.] studied the Renewal Process and Renewal Reward Process with fuzzy random inter-arrival times and Rewards under the T-independence associated with any continuous Archimedean t-norm. But, their main results do not cover the classical theory of the random elementary Renewal theorem and random Renewal Reward theorem when fuzzy random variables degenerate to random variables, and some given assumptions relate to the membership function of the fuzzy variable and the Archimedean t-norm of the results are restrictive. This paper improves the results of Wang and Watada and Wang et al. from a mathematical per-spective. We release some assumptions of the results of Wang and Watada and Wang et al. and completely generalize the classical stochastic Renewal theorem and Renewal Rewards theorem.

  • Renewal Reward Process for t t related fuzzy random variables on mathbb r p mathbb r q rp rq
    Fuzzy Optimization and Decision Making, 2014
    Co-Authors: Dug Hun Hong
    Abstract:

    In this paper, following our previous studies, we investigate the Renewal Rewards Process with respect to the necessity, credibility, chance measure and the expected value in which the random inter-arrival times and random Rewards are characterized as weighted fuzzy numbers under $$t$$ t -norm-based fuzzy operations on $$\mathbb {R}^{p}$$ R p and $$\mathbb {R}^{q}\,\,p,\,q \ge 1,$$ R q p , q ? 1 , respectively. Many versions of $$T$$ T -related fuzzy Renewal Rewards theorems are proved by using the law of large numbers for weighted fuzzy variables on $$\mathbb {R}^{p}$$ R p . An application example is provided to illustrate the utility of the results.

  • Renewal Reward Process for $$T$$ T -related fuzzy random variables on $$(\mathbb {R}^{p}, \mathbb {R}^{q})$$
    Fuzzy Optimization and Decision Making, 2014
    Co-Authors: Dug Hun Hong
    Abstract:

    In this paper, following our previous studies, we investigate the Renewal Rewards Process with respect to the necessity, credibility, chance measure and the expected value in which the random inter-arrival times and random Rewards are characterized as weighted fuzzy numbers under $$t$$ t -norm-based fuzzy operations on $$\mathbb {R}^{p}$$ R p and $$\mathbb {R}^{q}\,\,p,\,q \ge 1,$$ R q p , q ≥ 1 , respectively. Many versions of $$T$$ T -related fuzzy Renewal Rewards theorems are proved by using the law of large numbers for weighted fuzzy variables on $$\mathbb {R}^{p}$$ R p . An application example is provided to illustrate the utility of the results.

  • Renewal Reward Process for $$T$$T-related fuzzy random variables on $$(\mathbb {R}^{p}, \mathbb {R}^{q})$$(Rp,Rq)
    Fuzzy Optimization and Decision Making, 2014
    Co-Authors: Dug Hun Hong
    Abstract:

    In this paper, following our previous studies, we investigate the Renewal Rewards Process with respect to the necessity, credibility, chance measure and the expected value in which the random inter-arrival times and random Rewards are characterized as weighted fuzzy numbers under $$t$$ t -norm-based fuzzy operations on $$\mathbb {R}^{p}$$ R p and $$\mathbb {R}^{q}\,\,p,\,q \ge 1,$$ R q p , q ? 1 , respectively. Many versions of $$T$$ T -related fuzzy Renewal Rewards theorems are proved by using the law of large numbers for weighted fuzzy variables on $$\mathbb {R}^{p}$$ R p . An application example is provided to illustrate the utility of the results.

  • a note on Renewal Reward Process with fuzzy Rewards
    Journal of the Korean Data and Information Science Society, 2005
    Co-Authors: Dug Hun Hong, Jeongjin Kim
    Abstract:

    In recently, Popova and Wu(1999) proved a theorem which presents the long-run average fuzzy Reward per unit time. In this note, we improve this result. Indeed we will show uniform convergence of a Renewal Reward Processes with respect to the level modeled as a fuzzy random variables.

Huaili Yun - One of the best experts on this subject based on the ideXlab platform.

  • fuzzy Renewal Process fuzzy Renewal Reward Process and their applications
    IEEE International Conference on Fuzzy Systems, 2004
    Co-Authors: Ruiqing Zhao, Wansheng Tang, Huaili Yun
    Abstract:

    This work extends the work in Zhao and Liu on fuzzy Renewal Process and Renewal Reward Process from continuous case to more general case. Fuzzy elementary Renewal theorem and fuzzy Renewal Reward theorem are developed and shown how they can be applied to maintenance policies. Three kinds of maintenance policies- fuzzy age replacement policy, fuzzy block replacement policy and fuzzy inspection policy are discussed. The optimality functions studied are the long run expected costs per unit of time. Finally, a hybrid intelligent algorithm is employed to solve the models proposed in maintenance policies and to arrive at optimum policies. A numerical example is enumerated and the result indicates that the algorithm is effective.

  • FUZZ-IEEE - Fuzzy Renewal Process, fuzzy Renewal Reward Process and their applications
    2004 IEEE International Conference on Fuzzy Systems (IEEE Cat. No.04CH37542), 1
    Co-Authors: Ruiqing Zhao, Wansheng Tang, Huaili Yun
    Abstract:

    This work extends the work in Zhao and Liu on fuzzy Renewal Process and Renewal Reward Process from continuous case to more general case. Fuzzy elementary Renewal theorem and fuzzy Renewal Reward theorem are developed and shown how they can be applied to maintenance policies. Three kinds of maintenance policies- fuzzy age replacement policy, fuzzy block replacement policy and fuzzy inspection policy are discussed. The optimality functions studied are the long run expected costs per unit of time. Finally, a hybrid intelligent algorithm is employed to solve the models proposed in maintenance policies and to arrive at optimum policies. A numerical example is enumerated and the result indicates that the algorithm is effective.

  • FUZZ-IEEE - SPSA Algorithm for Fuzzy Block Replacement Policy
    The 14th IEEE International Conference on Fuzzy Systems 2005. FUZZ '05., 1
    Co-Authors: Guoyong Wang, Ruiqing Zhao, Huaili Yun
    Abstract:

    In this paper, some latest results on uncertain Renewal Processes including fuzzy Renewal Process, fuzzy Renewal Reward Process are introduced. And then the application of these results in fuzzy block replacement policy is provided. In order to solve the model, a simultaneous perturbation stochastic approximation (SPSA) algorithm based on fuzzy simulation technique is designed to search the optimal solution. Finally, a numerical example is presented

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

  • Limit theorem for a semi - Markovian stochastic model of type (s,S)
    Hacettepe University, 2019
    Co-Authors: Hanalioglu Zulfiye, Khaniyev Tahir
    Abstract:

    In this study, a semi-Markovian inventory model of type (s,S) is considered and the model is expressed by means of Renewal-Reward Process (X(t)) with an asymmetric triangular distributed interference of chance and delay. The ergodicity of the Process X(t) is proved and the exact expression for the ergodic distribution is obtained. Then, two-term asymptotic expansion for the ergodic distribution is found for standardized Process W(t) equivalent to (2X(t))/(S - s). Finally, using this asymptotic expansion, the weak convergence theorem for the ergodic distribution of the Process W(t) is proved and the explicit form of the limit distribution is found

  • Asymptotic Expansions For The Moments Of The Renewal-Reward Process With A Normal Distributed Interference Of Chance
    Ministry Communications & High Technologies Republic Azerbaijan, 2018
    Co-Authors: Hanalioglu Z., Unver N. Fescioglu, Khaniyev Tahir
    Abstract:

    In this study, a Renewal-Reward Process with a normal distributed interference of chance is mathematically constructed. The ergodicity of this Process is discussed. The exact formulas for the nth order moments of the ergodic distribution of the Process are obtained, when the interference of chance has a truncated normal distribution with parameters (a, sigma(2)). Using these results, we derive the asymptotic expansions with three terms for the nth order moments of the ergodic distribution, when a -> infinity. Finally, the accuracy of the approximation formulas for the nth order moments of the ergodic distribution are tested by the Monte Carlo simulation method.[Hanalioglu, Z.] Karabuk Univ, Dept Actuarial Sci & Risk Management, TR-78050 Karabuk, Turkey; [Unver, N. Fescioglu; Khaniyev, T.] TOBB Univ Econ & Technol, Dept Ind Engn, TR-06560 Ankara, Turkey; [Khaniyev, T.] Azerbaijan Natl Acad Sci, Inst Control Syst, Baku, Azerbaija

  • The Class of L boolean AND D and Its Application to Renewal Reward Process
    Amer Inst Physics, 2018
    Co-Authors: Kamislik, Ash Bektas, Kesemen Tulay, Khaniyev Tahir
    Abstract:

    6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA), AUG 15-17, 2017, Budapest, HUNGARYThe class of L boolean AND D is generated by intersection of two important subclasses of heavy tailed distributions: The long tailed distributions and dominated varying distributions. This class itself is also an important member of heavy tailed distributions and has some principal application areas especially in Renewal, Renewal Reward and random walk Processes. The aim of this study is to observe some well and less known results on Renewal functions generated by the class of L boolean AND D and apply them into a special Renewal Reward Process which is known in the literature a semi Markovian inventory model of type (s, S). Especially we focused on Pareto distribution which belongs to the L boolean AND D subclass of heavy tailed distributions. As a first step we obtained asymptotic results for Renewal function generated by Pareto distribution from the class of L boolean AND D using some well-known results by Embrechts and Omey [1]. Then we applied the results we obtained for Pareto distribution to Renewal Reward Processes. As an application we investigate inventory model of type (s, S) when demands have Pareto distribution from the class of L boolean AND D. We obtained asymptotic expansion for ergodic distribution function and finally we reached asymptotic expansion for nth order moments of distribution of this Process.[Kamislik, Ash Bektas] Recep Tayyip Erdogan Univ, Dept Math, TR-53020 Rize, Turkey; [Kesemen, Tulay] Karadeniz Tech Univ, Dept Math, TR-61080 Trabzon, Turkey; [Khaniyev, Tahir] TOBB Univ Econ & Technol, Dept Ind Engn, TR-06560 Ankara, Turke

  • Asymptotic Results for an Inventory Model of Type (s, S) with Asymmetric Triangular Distributed Interference of Chance and Delay
    Gazi Univ, 2018
    Co-Authors: Hanalioglu Zulfiye, Khaniyev Tahir
    Abstract:

    In this study, a semi - Markovian inventory model of type (s, S) is considered and the model is expressed by a modification of a Renewal - Reward Process (X(t)) with an asymmetric triangular distributed interference of chance and delay. The ergodicity of the Process X(t) is proved under some weak conditions. Additionally, exact expressions and three - term asymptotic expansions are found for all the moments of the ergodic distribution. Finally, obtained asymptotic results are compared with exact results for a special case.[Hanalioglu, Zulfiye] Karabuk Univ, Dept Actuarial Sci & Risk Management, Karabuk, Turkey; [Khaniyev, Tahir] TOBB Univ Econ & Technol, Dept Ind Engn, TR-06560 Ankara, Turke

  • Asymptotic approach for a Renewal-Reward Process with a general interference of chance
    Taylor & Francis Inc, 2016
    Co-Authors: Aliyev Rovshan, Ardic Ozlem, Khaniyev Tahir
    Abstract:

    In this study, a Renewal-Reward Process with a discrete interference of chance is constructed and considered. Under weak conditions, the ergodicity of the Process X(t) is proved and exact formulas for the ergodic distribution and its moments are found. Within some assumptions for the discrete interference of chance in general form, two-term asymptotic expansions for all moments of the ergodic distribution are obtained. Additionally, kurtosis coefficient, skewness coefficient, and coefficient of variation of the ergodic distribution are computed. As a special case, a semi-Markovian inventory model of type (s, S) is investigated.[Aliyev, Rovshan] Baku State Univ, Dept Probabil Theory & Math Stat, Baku, AZ, Azerbaijan; [Ardic, Ozlem; Khaniyev, Tahir] TOBB Univ Econ & Technol, Dept Ind Engn, TR-06560 Ankara, Turkey; [Aliyev, Rovshan; Khaniyev, Tahir] Azerbaijan Natl Acad Sci, Inst Cybernet, Baku, AZ, Azerbaija