Uniform convergence of fuzzy random renewal process

Recently, Zhao et al. (in Fuzzy Optimization and Decision Making 2007 6, 279–295) presented a fuzzy random elementary renewal theorem and fuzzy random renewal reward theorem in the fuzzy random process. In this paper, we study the convergence of fuzzy random renewal variable and of the total rewards earned by time t with respect to the extended Hausdorff metrics d _∞ and d ₁. Using this convergence information and applying the uniform convergence theorem, we provide some new versions of the fuzzy random elementary renewal theorem and the fuzzy random renewal reward theorem.     

Fuzzy random renewal process and renewal reward process

So far, there have been several concepts about fuzzy random variables and their expected values in literature. One of the concepts defined by Liu and Liu (2003a) is that the fuzzy random variable is a measurable function from a probability space to a collection of fuzzy variables and its expected value is described as a scalar number. Based on the concepts, this paper addresses two processes—fuzzy random renewal process and fuzzy random renewal reward process. In the fuzzy random renewal process, the interarrival times are characterized as fuzzy random variables and a fuzzy random elementary renewal theorem on the limit value of the expected renewal rate of the process is presented. In the fuzzy random renewal reward process, both the interarrival times and rewards are depicted as fuzzy random variables and a fuzzy random renewal reward theorem on the limit value of the long-run expected reward per unit time is provided. The results obtained in this paper coincide with those in stochastic case or in fuzzy case when the fuzzy random variables degenerate to random variables or to fuzzy variables.     

Uncertain calculus with renewal process

Uncertain calculus is a branch of mathematics that deals with differentiation and integration of function of uncertain processes. As a fundamental concept, uncertain integral has been defined with respect to canonical process. However, emergencies such as economic crisis and war occur occasionally, which may cause the uncertain process a sudden change. So far, uncertain renewal process has been employed to model these jumps. This paper will present a new uncertain integral with respect to renewal process. Besides, this paper will propose a type of uncertain differential equation driven by both canonical process and renewal process.     

A curious renewal process average

The lifetimes of a renewal process observed during a fixed interval (0, t] are smaller, on the average, than the process mean lifetime; it is shown that the mean observed lifetime has a particularly simple form.     

Optimal adaptive replacement in a renewal process

In this paper we present an exact method for computing the Weibull renewal function and its derivative for application in maintenance optimization. The computational method provides a solid extension to previous work by which an approximation to the renewal function was used in a Bayesian approach to determine optimal replacement times. In the maintenance scenario, under the assumption an item is replaced by a new one upon failure, the underlying process between planned replacement times is a renewal process. The Bayesian approach takes into account failure and survival information at each planned replacement stage to update the optimal time until the next planned replacement. To provide a simple approach to carry out in practice, we limit the decision process to a one‐step optimization problem in the sequential decision problem. We make the Weibull assumption for the lifetime distribution of an item and calculate accurately the renewal function and its derivative. A method for finding zeros of a function is adapted to the maintenance optimization problem, making use of the availability of the derivative of the renewal function. Furthermore, we develop the maximum likelihood estimate version of the Bayesian approach and illustrate it with simulated examples. The maintenance algorithm retains the adaptive concept of the Bayesian methodology but reduces the computational need. Copyright © 2014 John Wiley & Sons, Ltd.     

Random fuzzy shock models and bivariate random fuzzy exponential distribution

In this paper, a random fuzzy shock model and a random fuzzy fatal shock model are proposed. Then bivariate random fuzzy exponential distribution is derived from the random fuzzy fatal shock model. Furthermore, some properties of the bivariate random fuzzy exponential distribution are proposed. Finally, an example is given to show the application of the bivariate random fuzzy exponential distribution.     

Bernoulli thinning of a Markov renewal process

A Bernoulli thinning of a Markov renewal process is investigated. The properties of the thinned process are considered and are related to the properties of the original process. The parameters, moments and equilibrium of the thinned process are determined in terms of the parameters defining the underlying Markov renewal process. Results are illustrated by examples. © 1998 John Wiley & Sons, Ltd.     

Delayed renewal process with uncertain interarrival times

Delayed renewal process is a special type of renewal process in which the first interarrival time is quite different from the others. This paper first proposes an uncertain delayed renewal process whose interarrival times are regarded as uncertain variables. Then it gives an uncertainty distribution of delayed renewal process and an elementary delayed renewal theorem.     

A risk model with renewal shot-noise Cox process

In this paper we generalise the risk models beyond the ordinary framework of affine processes or Markov processes and study a risk process where the claim arrivals are driven by a Cox process with renewal shot-noise intensity. The upper bounds of the finite-horizon and infinite-horizon ruin probabilities are investigated and an efficient and exact Monte Carlo simulation algorithm for this new process is developed. A more efficient estimation method for the infinite-horizon ruin probability based on importance sampling via a suitable change of probability measure is also provided; illustrative numerical examples are also provided.     

A renewal jump-diffusion process with threshold dividend strategy

In this paper, we consider a jump-diffusion risk process with the threshold dividend strategy. Both the distributions of the inter-arrival times and the claims are assumed to be in the class of phase-type distributions. The expected discounted dividend function and the Laplace transform of the ruin time are discussed. Motivated by Asmussen [S. Asmussen, Stationary distributions for fluid flow models with or without Brownian noise, Stochastic Models 11 (1) (1995) 21–49], instead of studying the original process, we study the constructed fluid flow process and their closed-form formulas are obtained in terms of matrix expression. Finally, numerical results are provided to illustrate the computation.     

Semi-Markov model of a renewal process with switching

A semi-Markov process with a discrete-continuous phase space is applied to describe a renewal process with switching. Formulas are derived for the stationary distribution of the embedded Markov chain and the stationary characteristics of the system.Sevastopol' Instrument-Building Institute. Translated from Dinamicheskie Sistemy, No. 10, pp. 63–68, 1992.     

Replacement process decomposition for discounted Markov renewal programming

An iterative decomposition method is presented for computing the values in an infinite-horizon discounted Markov renewal program (DMRP). The states are partitioned intoM groups, with each iteration involving disaggregation of one group at a time, with the otherM–1 groups being collapsed intoM–1 singletons using the replacement process method. Each disaggregation also looks like a DMRP and can be performed by policy-iteration, value-iteration or linear programming. Anticipated benefits from the method include reduced computer time and memory requirements, scale-invariance and greater robustness to the starting point.     

Random fuzzy EOQ model with imperfect quality items

This paper investigates an economic order quantity (EOQ) problem with imperfect quality items, where the percentage of imperfect quality items in each lot is characterized as a random fuzzy variable while the setup cost per lot, the holding cost of each unit item per day, and the inspection cost of each unit item are characterized as fuzzy variables, respectively. In order to maximize the expected long-run average profit, a random fuzzy EOQ model is constructed. Since it is almost impossible to find an analytic method to solve the proposed model, a particle swarm optimization (PSO) algorithm based on the random fuzzy simulation is designed. Finally, the effectiveness of the designed algorithm is illustrated by a numerical example.     

Beyond the Poisson renewal process: A tutorial survey

After sketching the basic principles of renewal theory and recalling the classical Poisson process, we discuss two renewal processes characterized by waiting time laws with the same power asymptotics defined by special functions of Mittag–Leffler and of Wright type. We compare these three processes with each other.     

Random fuzzy differential equations under generalized Lipschitz condition

We present the studies on two kinds of solutions to random fuzzy differential equations (RFDEs). The different types of solutions to RFDEs are generated by the usage of two different concepts of fuzzy derivative in the formulation of a differential problem. Under generalized Lipschitz condition, the existence and uniqueness of both kinds of solutions to RFDEs are obtained. We show that solutions (of the same kind) are close to each other in the case when the data of the equation did not differ much. By an example, we present an application of each type of solutions in a population growth model which is subjected to two kinds of uncertainties: fuzziness and randomness.     

Characterizations of the Poisson process as a renewal process via two conditional moments

Given two independent positive random variables, under some minor conditions, it is known that fromE(X^rX+Y)=a(X+Y)^r andE(X^sX+Y)=b(X+Y)^s, for certain pairs ofr ands, wherea andb are two constants, we can characterizeX andY to have gamma distributions. Inspired by this, in this article we will characterize the Poisson process among the class of renewal processes via two conditional moments. More precisely, let {A(t), t0} be a renewal process, with {S _k, k1} the sequence of arrival times, andF the common distribution function of the inter-arrival times. We prove that for some fixedn andk, kn, ifE(S _k ^r A(t)=n)=at^r andE(S _k ^s A(t)=n)=bt^s, for certain pairs ofr ands, wherea andb are independent oft, then {A(t), t0} has to be a Poisson process. We also give some corresponding results about characterizingFto be geometric whenF is discrete.Support for this research was provided in part by the National Science Council of the Republic of China, Grant No. NSC 81-0208-M110-06.     

A fuzzy process of individuation

ABSTRACT

It is shown that an aspect of the process of individuation may be thought of as a fuzzy set. The process of individuation has been interpreted as a two-valued problem in the history of philosophy. In this work, I intend to show that such a process in its psychosocial aspect is better understood in terms of a fuzzy set, characterized by a continuum membership function. According to this perspective, species and their members present different degrees of individuation. Such degrees are measured from the membership function of the psychosocial process of individuation. Thus, a social analysis is suggested by using this approach in human societies.