Reliability assessment for load‐sharing systems with exponential components using statistical expansion as a correction

Among recent system models, one specific type of system is generally used to model the dependence among components. Components are connected parallel in such systems as they fail one by one and are supposed to share the system work load. The model is thus referred to as the load‐sharing system model. Despite the availability of extensive reliability assessment methods for different systems, load‐sharing systems have not received enough attention from the scholars who have studied reliability assessment so far. Load‐sharing systems are generally designed for high levels of reliability. Therefore, tests for such systems can be expensive and time consuming. Limitation on resources always leads to small test sample sizes. This increases the difficulties associated with obtaining an accurate and robust system reliability assessment result. This paper proposes a novel assessment method for a certain type of load‐sharing system with components following exponential lifetime distributions. Based on the parameter estimation of the system reliability model, we introduce the Winterbottom‐Cornish‐Fisher asymptotic expansion method for implementing a correction of normal approximation. We demonstrate the accuracy of our method through a series of examples and simulation studies.     

An optimal switching approach toward cost‐effective control of a stand‐alone photovoltaic panel system under stochastic environment

The operation of a stand‐alone photovoltaic (PV) system ultimately aims for the optimization of its energy storage. We present a mathematical model for cost‐effective control of a stand‐alone system based on a PV panel equipped with an angle adjustment device. The model is based on viscosity solutions to partial differential equations, which serve as a new and mathematically rigorous tool for modeling, analyzing, and controlling PV systems. We formulate a stochastic optimal switching problem of the panel angle, which is here a binary variable to be dynamically controlled under stochastic weather condition. The stochasticity comes from cloud cover dynamics, which is modeled with a nonlinear stochastic differential equation. In finding the optimal control policy of the panel angle, switching the angle is subject to impulsive cost and reduces to solving a system of Hamilton‐Jacobi‐Bellman quasi‐variational inequalities (HJBQVIs). We show that the stochastic differential equation is well posed and that the HJBQVIs admit a unique viscosity solution. In addition, a finite‐difference scheme is proposed for the numerical discretization of HJBQVIs. A demonstrative computational example of the HJBQVIs, with emphasis on a stand‐alone experimental system, is finally presented with practical implications for its cost‐effective operation.     

Finite‐time stability of fractional‐order stochastic singular systems with time delay and white noise

In this article, the finite‐time stochastic stability of fractional‐order singular systems with time delay and white noise is investigated. First the existence and uniqueness of solution for the considered system is derived using the basic fractional calculus theory. Then based on the Gronwall's approach and stochastic analysis technique, the sufficient condition for the finite‐time stability criterion is developed. Finally, a numerical example is presented to verify the obtained theory. © 2016 Wiley Periodicals, Inc. Complexity 21: 370–379, 2016     

Modelling a general standby system and evaluation of its performance

Redundancy or standby is a technique that has been widely applied to improving system reliability and availability in system design. In this paper, a general method for modelling standby system is proposed and system performance measures are derived. It is shown that the proposed general standby system includes the cases of cold, hot and warm standby systems with units of exponential distribution, which were studied in the literature, as special cases. An optimal allocation problem for a standby system is also discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

Pricing VIX options in a stochastic vol‐of‐vol model

This paper proposes and makes a study of a new model for volatility index option pricing. Factors such as mean‐reversion, jumps, and stochastic volatility are taken into consideration. In particular, the positive volatility skew is addressed by the jump and the stochastic volatility of volatility. Daily calibration is used to check whether the model fits market prices and generates positive volatility skews. Overall, the results show that the mean‐reverting logarithmic jump and stochastic volatility model (called MRLRJSV in the paper) serves as the best model in all the required aspects. Copyright © 2015 John Wiley & Sons, Ltd.     

Ergodicity and threshold behaviors of a predator‐prey model in stochastic chemostat driven by regime switching

This paper deals with a stochastic predator‐prey model in chemostat which is driven by Markov regime switching. For the asymptotic behaviors of this stochastic system, we establish the sufficient conditions for the existence of the stationary distribution. Then, we investigate, respectively, the extinction of the prey and predator populations. We explore the new critical numbers between survival and extinction for species of the dual‐threshold chemostat model. Numerical simulations are accomplished to confirm our analytical conclusions.     

P‐moment stability under small Gauss type random excitation of stochastic system

In this paper, the stochastic stability under small Gauss type random excitation is investigated theoretically and numerically. When p is larger than 0, the p‐moment stability theorem of stochastic models is proved by Lyapunov method, Ito isometry formula, matrix theory and so on. Then the application of p‐moment such as k‐order moment of the origin and k‐order moment of the center is introduced and analyzed. Finally, p‐moment stability of the power system is verified through the simulation example of a one machine and infinite bus system. Copyright © 2016 John Wiley & Sons, Ltd.     

A fractional‐order model for chronic lymphocytic leukemia and immune system interactions

In this study, a mathematical, fractional‐order model was developed for B cell chronic lymphocytic leukemia, with immune system, and then analyzed. Interactions between B leukemia cells, natural killer cells, cytotoxic T cells, and T‐helper cells are considered to be incorporated into a system consisting of four fractional differential equations. For estimation of the parameters, clinical data of six patients were used. By numerical solution of the system, the interactions between the leukemia cell population and the immune system cell populations for values of α ∈ (0,1) at different times were explained. By determining points of equilibrium and stability of the system were met. Bifurcation analysis showed that use of the fractional‐order model, figure out unpredictable behaviors of the system such as saddle‐node, bistability and hysteresis phenomenon occurred in the system by changing the values of some of the parameters, it was predictable.     

Strong convergence of the split‐step θ‐method for stochastic age‐dependent population equations

In this paper, we constructed the split‐step θ (SSθ)‐method for stochastic age‐dependent population equations. The main aim of this paper is to investigate the convergence of the SS θ‐method for stochastic age‐dependent population equations. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from the theory, and comparative analysis with Euler method is given, the results show the higher accuracy of the SS θ‐method. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal reinsurance–investment problem in a constant elasticity of variance stock market for jump‐diffusion risk model

In this paper, we consider the jump‐diffusion risk model with proportional reinsurance and stock price process following the constant elasticity of variance model. Compared with the geometric Brownian motion model, the advantage of the constant elasticity of variance model is that the volatility has correlation with the risky asset price, and thus, it can explain the empirical bias exhibited by the Black and Scholes model, such as volatility smile. Here, we study the optimal investment–reinsurance problem of maximizing the expected exponential utility of terminal wealth. By using techniques of stochastic control theory, we are able to derive the explicit expressions for the optimal strategy and value function. Numerical examples are presented to show the impact of model parameters on the optimal strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

Estimation of a smooth quantile function under the proportional hazards model

The problem of estimating a smooth quantile function, Q(·), at a fixed point p, 0 < p < 1, is treated under a nonparametric smoothness condition on Q. The asymptotic relative deficiency of the sample quantile based on the maximum likelihood estimate of the survival function under the proportional hazards model with respect to kernel type estimators of the quantile is evaluated. The comparison is based on the mean square errors of the estimators. It is shown that the relative deficiency tends to infinity as the sample size, n, tends to infinity.     

On optimal operating conditions for a data processing system: A stochastic approach

In this paper, we consider the problem of determining optimal operating conditions for a data processing system. The system is burned‐in for a fixed burn‐in time before it is put into field operation and, in field operation, it has a work size and follows an age‐replacement policy. Assuming that the underlying lifetime distribution of the system has a bathtub‐shaped failure rate function, the properties of optimal burn‐in time, optimal work size and optimal age‐replacement policy will be derived. It can be seen that this model is a generalization of those considered in the previous works, and it yields a better optimal operating conditions. This paper presents an analytical method for three‐dimensional optimization problem. An algorithm for determining optimal operating conditions is also given. Copyright © 2007 John Wiley & Sons, Ltd.     

A new two‐stage algorithm for solving power flow tracing

This article proposed a new hybrid algorithm for solving power flow tracing (PFT) through the comparison by other techniques. This proposed hybrid strategy in detail discuses over the achieved results. Both methods use the active and reactive power balance equations at each bus to solve the tracing problem, where the first method considers the proportional sharing assumption and the second one considers the circuit laws to find the relationship between power inflows and outflows through each line, generator, and load connected to each bus of the network. Both algorithms are able to handle loop flow and loss issues in tracing the problem. A mathematical formulation is also introduced to find the share of each unit in provision of each load. These algorithms are employed to find the producer and consumer's shares on the cost of transmission for each line in different case studies. As the results of these studies show, both algorithms can effectively solve the PFT problem. © 2014 Wiley Periodicals, Inc. Complexity 21: 187–194, 2015     

Hopf bifurcation of a diffusive Gause‐type predator‐prey model induced by time fractional‐order derivatives

Since population behaviors possess the characteristic of history memory, we, in this paper, introduce time fractional‐order derivatives into a diffusive Gause‐type predator‐prey model, which is time fractional‐order reaction‐diffusion equations and a generalized form of its corresponding first‐derivative model. For this kind of model, we prove the existence and uniqueness of a global positive solution by using the theory of evolution equations and the comparison principle of time fractional‐order partial differential equations. Besides, we obtain the stability and Hopf bifurcation of the Gause‐type predator‐prey model in the forms of the time fractional‐order ordinary equations and of the time fractional‐order reaction‐diffusion equations, respectively. Our results show that the stable region of the parameters in these 2 models can be enlarged by the time fractional‐order derivatives. Some numerical simulations are made to verify our results.     

A matrix rational Lanczos method for model reduction in large‐scale first‐ and second‐order dynamical systems

In the present paper, we describe an adaptive modified rational global Lanczos algorithm for model‐order reduction problems using multipoint moment matching‐based methods. The major problem of these methods is the selection of some interpolation points. We first propose a modified rational global Lanczos process and then we derive Lanczos‐like equations for the global case. Next, we propose adaptive techniques for choosing the interpolation points. Second‐order dynamical systems are also considered in this paper, and the adaptive modified rational global Lanczos algorithm is applied to an equivalent state space model. Finally, some numerical examples will be given.     

A second‐order finite difference approximation for a mathematical model of erythropoiesis

We present a second‐order finite difference scheme for approximating solutions of a mathematical model of erythropoiesis, which consists of two nonlinear partial differential equations and one nonlinear ordinary differential equation. We show that the scheme achieves second‐order accuracy for smooth solutions. We compare this scheme to a previously developed first‐order method and show that the first order method requires significantly more computational time to provide solutions with similar accuracy. We also compare this numerical scheme with other well‐known second‐order methods and show that it has better capability in approximating discontinuous solutions. Finally, we present an application to recovery after blood loss. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

System availability in a shock model under preventive repair and phase‐type distributions

A device that can fail by shocks or ageing under policy N of maintenance is presented. The interarrival times between shocks follow phase‐type distributions depending on the number of cumulated shocks. The successive shocks deteriorate the system, and some of them can be fatal. After a prefixed number k of nonfatal shocks, the device is preventively repaired. After a fatal shock the device is correctively repaired. Repairs are as good as new, and follow phase‐type distributions. The system is governed by a Markov process whose infinitesimal generator, stationary probability vector, and availability are calculated, obtaining well‐structured expressions due to the use of phase‐type distributions. The availability is optimized in terms of the number k of preventive repairs. Copyright © 2009 John Wiley & Sons, Ltd.     

First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first‐order system in a stress‐velocity formulation for the Stokes problem and a volumetric flux‐hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well‐known Beavers–Joseph–Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss–Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss–Newton method are well posed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1150–1173, 2015     

Optimal investment for the defined-contribution pension with stochastic salary under a CEV model

In this paper, we study the optimal investment strategy of defined-contribution pension with the stochastic salary. The investor is allowed to invest in a risk-free asset and a risky asset whose price process follows a constant elasticity of variance model. The stochastic salary follows a stochastic differential equation, whose instantaneous volatility changes with the risky asset price all the time. The HJB equation associated with the optimal investment problem is established, and the explicit solution of the corresponding optimization problem for the CARA utility function is obtained by applying power transform and variable change technique. Finally, we present a numerical analysis.