Supplementary Variable Method Research Articles

Bayesian inference for reliability of K-out-of-N: G repairable retrial systems with mixed standbys, switching failure and reboot delay

Existing reliability studies on K-out-of-N: G repairable retrial systems with mixed standbys mainly consider the case where the standby switching is completely reliable and the repair time follows an exponential distribution, and most of them directly analyze the system reliability indices based on probability theory. Given the existence of such phenomenon as standby switching failure in engineering, this paper proposes a novel reliability model of K-out-of-N: G repairable retrial system with switching failure, and performs statistical inference on the reliability to realize the modeling and evaluation of the system. The considered system consists of mixed standbys and a repairer with vacation. When an operating component fails, the warm standby one is switched to the operating one immediately, and the probability of switching failure during the switchover process is p. In the case where the repair time of failed components follows a general distribution, based on the supplementary variable method and recursive algorithm, the steps for calculating the system steady-state availability are presented. In the case of unknown parameters for repair time and retrial time, the Bayesian estimator of the steady-state availability under the squared error loss function and highest posterior density (HPD) confidence interval are obtained. In numerical simulation, the system reliability inference is realized by using the Monte Carlo (MC) method, and the validity of the inference method is analyzed.

A new class of high-order supplementary variable methods for the Klein–Gordon–Zakharov system

In this paper, based on the paradigm of supplementary variable method (SVM), we reformulate the Klein–Gordon–Zakharov system into equivalent optimization problem subject to PDE constraints, and then present a novel class of high-order energy-preserving numerical algorithms to solve it numerically. The optimization model is discretized by applying the Gauss collocation method as well as the prediction–correction technique in time and the sine pseudo-spectral method in space. Experimental results and some comparisons between this method and other reported ones are provided to favor the suggested method in the overall performance.

Conformal structure-preserving SVM methods for the nonlinear Schrödinger equation with weakly linear damping term

In this paper, by applying the supplementary variable method (SVM), some high-order, conformal structure-preserving, linearized algorithms are developed for the damped nonlinear Schrödinger equation. We derive the well-determined SVM systems with the conformal properties and they are then equivalent to nonlinear equality constrained optimization problems for computation. The deduced optimization models are discretized by using the Gauss type Runge-Kutta method and the prediction-correction technique in time as well as the Fourier pseudo-spectral method in space. Numerical results and some comparisons between this method and other reported methods are given to favor the suggested method in the overall performance. It is worthwhile to emphasize that the numerical strategy in this work could be extended to other conservative or dissipative system for designing high-order structure-preserving algorithms.

Bi-objective optimization and reliability analysis in unreliable M/G/1 retrial queues with rational customers, reserved idle times and setup times

ABSTRACT Energy consumption arising from no-loads and failures is a common and challenging problem in the design of service systems. To analyze the impact of no-load consumption and failure consumption on the energy consumption simultaneously, this paper considers an M/G/1 retrial queue with reserved idle times, setup times and server breakdowns. First, the steady-state condition and performance measures of the system are obtained by the embedded Markov chain and the supplementary variable method. Then, the reliability indicators of the system are analyzed, and an algorithm for solving the system reliability R ( t ) is constructed based on the Monte Carlo method. In order to better optimize the system, the customer’s equilibrium joining strategy is derived, and the socially optimal joining strategy is obtained by the particle swarm optimization (PSO) algorithm. From the decision-maker viewpoint, a bi-objective optimization model is constructed to minimize the expected energy consumption and the expected waiting time of retrial customers, and solved by the non-dominated ranking genetic algorithm II (NSGA-II). Finally, we illustrate an application of the proposed analytical approach in the cloud computing data center, which makes it possible to achieve an appropriate balance between energy consumption and quality of service (QoS).

Analysis of Influential Factors of Tourism Culture Development under the Orientation of Territorial Tourism

Abstract Tourism development is one of the main means of productive cultural protection, and culture is an important factor in the attractiveness of tourist attractions. With the development of the concept of regional tourism, China’s tourism and culture have moved to a new stage. In this paper, we screen the indicators from the perspective of industrial income, the number of institutions, service teams, etc., and use the entropy weight method with relevant data to measure the level of tourism and cultural development and tourism priority. Then, the influence factors of tourism and culture development are discussed by combining the least squares regression model and geographically weighted regression model. Finally, to ensure the reliability of the conclusions, the sample reconstruction method and the supplementary variable method are used to conduct the robustness test and endogeneity test. According to the regression results, the regression coefficients for market demand and government support are positive and pass the 5% significance level test. Government support and market demand are currently the main influencing factors for the development of national tourism and culture.

Retrial G-queue with precipitous breakdown and non-tenacious customers

We perform an analysis on an M/G/1 system that has bulk arrival Poisson process as well as instantaneous service, Precipitous breakdown and random repair, Gqueue, and operation under an MBV (Modified Bernoulli vacation) policy. The positive outcome of a customer retrial is lost when an Anti-positive customer arrives during a positive (good) customer?s assistance period. If a new customer enters and notices that the server is currently undergoing repairs or on vacation, they may renege or balk from the system. To obtain a wide variety of additional outcomes, we use the details provided by the supplementary variable method regarding the rates at which different events occur and it is possible to determine the probability-generating functions of queue length distributions as well as the explicit formulations of important performance metrics. The numerical data confirm the analytical findings about the major performance metrics.

A Mathematical Approach to Ullmann Reaction: ANFIS Computing of a Chemical Retrial Queue Model under Hybrid Vacation

Herein, we developed the first report on the chemical retrial queue and hybrid vacation process for the significant Ullmann coupling for enhanced high-throughput reaction discovery. The problem of controlling the waiting time for chemical retrials is addressed here. We incorporated the supplementary variables method (SVT) and generated the steady-state probability generating function (PGF) for system size and orbit size. Important cases are outlined, and a few key metrics are utilized to evaluate system performance. Additionally, the impact of changing certain system parameters has also been analyzed via numerical examples. One particularly interesting aspect of this research is the comparison of neuro-fuzzy technique results with validation results utilizing the "adaptive neuro-fuzzy inference system" (ANFIS).

A single server Non-Markovian with non-compulsory re-service and balking under Modified Bernoulli Vacation

In this paper, a single-server queue with Modified Bernoulli Vacation(MBV) has been considered. The server provides two kinds of services, Compulsory Service (CR) and Non-Compulsory Re-service (NCR). Any new customer who arrives and finds the server engaged, on breakdown, or on vacation will be placed on an orbit; or else if he finds the server free, he enters the service immediately. The server takes a vacation as soon as the orbit is empty at the usual service completion moment. The dissatisfied customer may re-enter the orbit after usual service is complete to receive another service. We have used SVM (Supplementary Variable Method) to derive PGF (Probability Generating Function) for the system. In order to demonstrate the influence that the system parameters have, numerical examples are discussed.

Reliability analysis of a two-dimensional linear consecutive-(r,s)-out-of-(m,n): F repairable system

Most of the existing studies on the two-dimensional consecutive system have focused on the non-repairable case. Nevertheless, considering the repairable case of components in engineering, it is necessary to study the reliability of the two-dimensional consecutive repairable system. Owing to the more complex spatial structure of the two-dimensional consecutive system, the analysis methods of the one-dimensional consecutive repairable system are not applicable to the two-dimensional case. In this work, we give a general method to analyze the reliability of a two-dimensional linear consecutive-(r,s)-out-of-(m,n): F repairable system. Firstly, to calculate the number of different cases of the system in a certain state, a key issue in the analysis of the consecutive repairable system, we propose the sliding window search method and inverse search method, whose computational complexities are compared and correctness is verified. Next, under the assumption that the repair time follows a general distribution, we analyze the reliability of the two-dimensional linear consecutive-(r,s)-out-of-(m,n): F repairable system using the supplementary variable method. Finally, we use an example of a wireless sensor network to analyze the effects of system parameters on system reliability indices, such as steady-state availability, steady-state failure frequency, and MTTFF.

Effect of reduction method on the performance a software defined network system using Gumbel Hougaard family copula distribution

This study examines a system that consists of three subsystems 1, 2 and 3. These three subsystems are each linked together in series. Subsystem 1’s three units are wired in series, subsystem 2’s three units are wired in parallel and operating under the 2-out-of-3: G; policy, and subsystem 3’s three units are wired in parallel and operating under the 1-out-of-3: G; policy. Units and subsystems failure rates are constant and follow an exponential distribution. The repair rate follow two types of distributions, namely general and Gumbel Hougaard family copula distribution. The system was studied using Laplace transforms and supplementary variable methods. For specific values of the failure and repair rates, availability, reliability, mean time to failure (MTTF), and cost analysis have been assessed. As a means to improve the system’s overall effectiveness and availability, a reduction strategy is utilized. Tables and graphs are used to display computed results.

High-order supplementary variable methods for thermodynamically consistent partial differential equations

In recent years, there has been growing interest in developing numerical methods to retain consistency between the partial differential equation system and its deduced equations (e.g., the thermodynamically consistent model and its energy law) after numerical approximations. A supplementary variable method (SVM) of second-order was recently introduced in [1,2] to preserve thermodynamic consistency for thermodynamically consistent models. In this paper, we present a new class of high-order supplementary variable methods. These methods consist of two parts: firstly, the original model is discretized using a high-order prediction method, and secondly, the SVM reformulated system is approximated by a high-order correction method. The newly developed schemes are rigorously proven to preserve high-order accuracy with increased stability through the use of a stabilizer. Additionally, the schemes inherently maintain thermodynamic consistency for thermodynamically consistent models, retaining the original energy law at the discrete level. To assess the high-order supplementary variable methods, we apply them to two thermodynamically consistent partial differential equation systems: the functionalized Cahn–Hilliard model and the ternary Cahn–Hilliard model. We then provide several benchmark examples to demonstrate the accuracy, stability, and efficiency of the newly developed high-order numerical methods.

Thermodynamically consistent hydrodynamic phase-field computational modeling for fluid-structure interaction with moving contact lines

This paper proposes a novel computational modeling approach to investigate the fluid-structure interactions with moving contact lines. By embracing the generalized Onsager principle, a coupled hydrodynamics and phase field system is introduced that can describe the fluid-structure interactions with moving contact lines in a thermodynamically consistent manner. The resulting partial differential equation (PDE) model consists of the Navier-Stokes equation and a nonlinear Allen-Cahn type equation. Volume conservation is enforced through an additional penalty term. A fully-discrete structure-preserving numerical scheme is proposed by combining several techniques to solve this coupled PDE system effectively and accurately. For the temporal discretization, we utilize the supplementary variable method for preserving the thermodynamic structure and the projection approach for reducing the problem size. Then, we use the finite difference method on the staggered grid for spatial discretization. Furthermore, we have rigorously proved that the proposed numerical scheme based on the second-order backward difference formula respects the original energy stability, i.e., the scheme is energy stable. Additionally, with the aid of the supplementary variable method, the resultant scheme can be transformed into a constrained optimization problem, where the solutions of the supplementary variables are the arguments of the objective function that reaches the optimality. Then the augmented Lagrangian method is introduced in part to bring robustness and efficiency to solving such a constrained optimization problem. Finally, various numerical simulations verify the model's capability and demonstrate the scheme's effectiveness, accuracy, and stability.

A Distributed Projection Neurodynamic Approach for Solving BP Denoising Problem in Sparse Recovery

This paper investigates a novel projection neurodynamic approach for solving the basis pursuit denoising (BPDN) in a distributed manner. First, by using the distributed consensus theorem over undirected graph and supplementary variables method, the distributed version of BPDN is obtained. Second, with the help of projection operators, primal-dual dynamical system and derivative feedback terms, a novel distributed neurodynamic approach is proposed to deal with the distributed version of BPDN for sparse recovery. Moreover, the optimality and convergence properties of the proposed distributed projection neurodynamic approach (DPNA) are analyzed rigorously. Finally, we apply DPNA to sparse signal reconstruction which demonstrates the effectiveness of DPNA through numerical experiments. In addition, inspired by the role of image reconstruction technology in the field of defense against adversarial attack, we use DPNA as a preprocessing method to enhance the robustness of the deep model. Compared with known defense schemes such as JEPG, ComDefend, and OMP, our DPNA is better than them.

Petri net analysis of a queueing inventory system with orbital search by the server

Abstract In this article, a queueing inventory system with finite sources of demands, retrial demands, service time, lead time, ( s , S ) \left(s,S) replenishment policy, and demands search from the orbit was studied. When the lead time is exponentially distributed (resp. lead time is generally distributed), generalized stochastic Petri net (GSPN) (resp. Markov regenerative stochastic Petri net [MRSPN]) is proposed for this inventory system. The quantitative analysis of this stochastic Petri net model was obtained by continuous time Markov chain for the GSPN model (resp. the supplementary variable method for the MRSPN model). The probability distributions are obtained, witch allowed us to compute performance measures and the expected cost rate of the studied system.

Can Digitalization Foster Sustainable Financial Inclusion? Opportunities for Both Banks and Vulnerable Groups

Financial inclusion is a crucial link in achieving the Sustainable Development Goals (SDGs). Strengthening the capability of financial institutions to provide inclusive financial services can help to narrow the inequality gap and increase access to opportunities. Digitalization, with its competitive advantages and rapid growth, may be a powerful tool to foster financial inclusion and sustainable development. This paper examines the effects of bank digitalization on sustainable financial inclusion and explores two underlying incentive mechanisms in banks: profit driven and risk aversion. We construct a basic model and a mechanism model and exploit a nonlinear attempt, heterogeneous estimation as well as supplementary variable and instrument variable methods for a robustness test. The results of the basic model demonstrate that bank digitalization has significant positive effects on financial inclusion and the current financial inclusive effects are sustainable. The mechanism models designed as the mediation effect panel model suggest that digitalization enables banks to expand the business probability frontier of profit-driven behavior and pursuit of credit portfolio diversity in risk aversion behavior, thereby promoting sustainable financial inclusion. As a result of digitalization, vulnerable groups can benefit from sustainable financial inclusion, while financial inclusion feeds back into banks’ sustainable development. This paper conforms to the trend of the development of digitalization and provides theoretical and empirical support for banks to build digitalization and realize sustainable financial inclusion, which contributes to the “triple-win” financial ecology for improving banks’ performance, increasing the rights of vulnerable groups and promoting sustainable development throughout society.

On the Age of Status Updates in Unreliable Multi-Source M/G/1 Queueing Systems

The timeliness of status message delivery in communications networks is subjective to time-varying wireless channel transmissions. In this letter, we investigate the age of information (AoI) of each source in a multi-source M/G/1 queueing update system with active server failures. In particular, we adopt the method of supplementary variables to derive a closed-form expression for the average AoI in terms of system parameters, where the server repair time follows a general distribution and the service time of packets generated by independent sources is a general random variable. Numerical results are provided to validate the effectiveness of the proposed packet serving policy under different parametric settings.

Unreliable retrial queueing system with working vacation

<abstract><p>This paper investigates an unreliable $ M/G(P_{1}, P_{2})/1 $ retrial queueing system with a woking vacation. An arriving customer successfully starts the first phase service with the probability $ \alpha $ or the server fails with the probability $ \bar{\alpha} $. Once failure happens, the serving customer is taken to the orbit. The failed server is taken for repair with some delay. Once the repair is comleted, the server is ready to provide service once again. In this background, we implemented the working vacation scenario. During working vacation, the service will be provided at a slower rate, rather than entirely stopping the service. The supplementary variable method was adopted to find the orbit and system lengths. Additionally, some unique results and numerical evaluations have been presented.</p></abstract>

Performance analysis for Bernoulli feedback queues subject to disasters: a system with batch Poisson arrivals under a multiple vacation policy

ABSTRACT We analyze an M/G/1 system with batch Poisson arrivals and instantaneous Bernoulli feedback, operating under a Multiple Vacation Policy. The system is subject to disasters that occur according to an independent Poisson process and are followed by (random) repair periods with general distribution. The analysis is carried out using the supplementary variable method. The Laplace transform of the time between two consecutive disasters is obtained and the existence of the stationary regime for the system is shown. Besides obtaining the stationary distribution for the number of customers in the system, we use the information regarding the rates of occurrence of various events provided by the supplementary variable solution to obtain a great variety of additional results. These include the Laplace transform of the busy period distribution and the probability that a customer completes service. We indicate areas of application of our model to real-life systems, in particular in Vehicle ad hoc Networks (VANETs), and we use the analytic results obtained to optimize such a system under a Quality of Service constraint. Finally, we analyze a variant of the system subject to disasters even when the server is not busy.

A serial inventory system with lead-time-dependent backordering: A reduced-state approximation

We study a serial inventory system where the external customers may have a maximum time that they would be willing to wait for delivery in cases of stock-out and the demand would be lost if the remaining delivery lead time of the next available item is longer. This lead-time-dependent backordering behavior subsumes the models of partial backordering regardless of the wait that a customer would experience. In the inventory literature, this behavior has only been analyzed in single-location settings. We study this behavior in a multi-stage setting. We consider continuous review policies at all stages facing external Poisson demands. Using the method of supplementary variables, we define the stochastic process representing the inventory system and obtain the expressions for the operating characteristics of the inventory system. Based on the solution structures for the special cases, we propose an approximate solution which rests on replacing the state-dependent purchasing decision of the customer with an averaged-out purchase probability computed using only the age of the oldest item. An extensive numerical study indicates that the proposed approximation performs very well. Our numerical study provides additional insights about the sensitivity and allocation of stock levels across stages.

Analysis of a retrial queueing system with priority service and modified multiple vacations

This paper studies an M/G/1 retrial queueing system with modified multiple vacations, in which a new external arrival may expel the customer being served out of the system and directly starts to be served or join the retrial orbit. Once the server finds the orbit is empty at the end of a service, it will immediately take a random length vacation. If the orbit is still empty when a vacation is finished, the server takes another same vacation. This pattern continues until the orbit is not empty when a vacation is completed or the server has already taken M vacations. By constructing an embedded Markov chain, we provide the sufficient and necessary condition of system stability. The distributions of the orbit size and the system size in steady-state are derived through the supplementary variable method. Then, some system performance measures and the Laplace-Stieltjes transform of sojourn time distribution are obtained. Besides, we perform the cost analysis of the system. Finally, several numerical illustrations are given to investigate the effects of system parameters on the essential system characteristics.