Uncertain Renewal Process Research Articles

Reliability modeling and analysis of uncertain competing failure systems

The competing failure system is quite common in real life, such as automotive system and aerospace system. Two distinct failure modes usually coexist in this type of system: gradual internal degradation and sudden external shock, where the occurrence of any failure mode can lead to the system failure. The system failure such as aircraft accident may cause a terrible loss of human life and property. Hence, it is worth reducing the rate of system failure by improving the system reliability. While, reliability model is the foundation of any research of reliability and epistemic uncertainty exists in the process of modeling and estimating. Therefore, we introduce uncertainty theory to investigate the competing failure system with epistemic uncertainty. The internal degradation and external shock of the system are characterized by an uncertain differential equation and an uncertain renewal process, respectively. The system reliability is defined as the uncertain measure that neither internal degradation nor external shock surpass corresponding thresholds. In addition, some reliability formulas are derived for the systems subjected to extreme shock, cumulative shock, running shock and δ-shock, respectively. Finally, a case study with respect to micro-electro-mechanical system is given to show how the proposed model is implemented.

Maintenance Policy with Lifetime Reduction Based on Uncertain Renewal Process

This paper focus on a (N, T) preventive maintenance (PM) policy with lifetime reduction discount rate. In our consideration, the lifetime of component is assumed to be an uncertain variable due to the absence of historical operational data, an uncertain (N, T) PM model with the lifetime reduction in a certain proportion is proposed based on uncertain renewal process. Accordingly, the uncertain model, which minimizes the expected maintenance cost rate, is formulated to find the optimal replacement cycle length T∗ and the number N∗ of preventive maintenance. Finally, a numerical example with sensitivity analysis of parameters is provided to illustrate the proposed model, the results imply that the parameters of cost and lifetime can significantly affect the optimal solutions N∗ and T∗, which can provide a useful reference and guidance for aircraft maintenance decision.

Optimal harvesting strategy based on uncertain logistic population model

The variations of population size with respect to time are often described by means of differential equations. This paper assumes the population size follows an uncertain logistic population equation, and calculates its uncertainty distribution and α-paths. The first hitting time that the population size reaches a pre-set level is investigated, which forms an uncertain renewal process, based on which a harvesting strategy is designed. With the help of fundamental theorem of uncertain renewal processes, the optimal harvesting strategy problem is transformed to a traditional optimization problem involving two variables which could be easily solved analytically or numerically.

Uncertain renewal process with general rewards

The renewal reward process describes the cumulative reward related to the renewals of a system. Usually, the rewards at the renewal times are assumed to be positive. Within the framework of uncertainty theory, this assumption is removed in this article such that the rewards can take both positive and negative values. The uncertainty distribution is derived, and the renewal reward theorems are proved for the general uncertain renewal reward process. In addition, the first hitting time that such a process reaches a pre-set level is investigated.

Uncertain insurance risk process with single premium and multiple classes of claims

Traditionally an insurance risk process is considered under the framework of probability theory with a prerequisite that the estimated distribution function is close enough to the real frequency. However, due to economic or technological reasons, sometimes data are unavailable or difficult to obtain, such as when we consider a new insurance product or insurance for valuable weapons. Under these situations, reimbursement policies are based on experts’ belief degree, which has a much wider range than the real frequency. As a result, we should employ uncertain insurance risk models to better deal with human uncertainty in running an insurance company. Noticing the fact that an insurance company pays for different kinds of risks and the uncertainty of the customer’s arrivals and payments, we investigate an uncertain insurance risk process with multiple classes of claims where the premium process follows an uncertain renewal process. Then we derive expressions for the ruin index and the uncertainty distribution of the ruin time. Some numerical examples and a real data example are performed to capture more insights.

Uncertain Stochastic Option Pricing in the Presence of Uncertain Jumps

In this paper, a new differential equation, driven by aleatory and epistemic forms of uncertainty, is introduced and applied to describe the dynamics of a stock price process. This novel class of differential equations is called uncertain stochastic differential equations(USDES) with uncertain jumps. The existence and uniqueness theorem for this class of differential equations is proposed and proved. An appropriate version of the chain rule is derived and applied to solve some examples of USDES with uncertain jumps. The differential equation discussed is applied in an American call option pricing problem. In this problem, it is assumed that the evolution of the stock price is driven by a Brownian motion, the Liu canonical process and an uncertain renewal process. MATLAB is employed for implementing the derived option pricing model. Results show that option prices from the proposed call option pricing formula increase as the jump size increases. As compared to the proposed call option pricing formula, the Black-Scholes overprices options for a certain range of strike prices and under-prices the same options for another range of exercise prices when the jump size is zero.

Stability in mean for uncertain differential equation with jumps

An uncertain differential equation with jumps is a type of uncertain differential equation driven by Liu process and uncertain renewal process, which is used to model discontinuous systems. Up to now, the stability in measure and almost sure stability for such an equation have been studied. The above two types of stability cannot be applied to all cases, so this paper aims at presenting a concept of stability in mean for an uncertain differential equation with jumps as a supplement. Most important of all, a stability theorem is given for an uncertain differential equation with jumps being stable in mean. And some examples are proposed to show how to use the theorem to judge whether the uncertain differential equation with jumps is stable in mean.

Almost sure stability for uncertain differential equation with jumps

Uncertain differential equation with jumps, as a crucial tool to deal with a discontinuous uncertain system, is a type of differential equation driven by both canonical Liu process and uncertain renewal process. So far, a concept of stability in measure for an uncertain differential equation with jumps has been proposed. As a supplement, this paper proposes a concept of almost sure stability for an uncertain differential equation with jumps. A sufficient condition is derived for an uncertain differential equation with jumps being stable almost surely. As a corollary, a sufficient condition is also given for a linear uncertain differential equation with jumps being stable almost surely.

Uncertain differential equation with jumps

Uncertain differential equation is a type of differential equation driven by a canonical Liu process. Traditional uncertain differential equation can only deal with a continuous dynamic uncertain system. To model the sharp drifts embedded in an uncertain dynamic system, this paper proposes a type of uncertain differential equation driven by a canonical Liu process and an uncertain renewal process, which is called an uncertain differential equation with jumps. A sufficient condition for the proposed equation having a unique solution is first given, then a concept of stability for the proposed equation is provided in the sense of uncertain measure, and its sufficient condition is also derived.

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.