Search results for: neuro-fuzzy networks

Authors: Kevin Fong-Rey Liu, Ken Yeh, Cheng-Wu Chen, Han-Hsi Liang

Abstract:

The sanitary sewerage connection rate becomes an important indicator of advanced cities. Following the construction of sanitary sewerages, the maintenance and management systems are required for keeping pipelines and facilities functioning well. These maintenance tasks often require sewer workers to enter the manholes and the pipelines, which are confined spaces short of natural ventilation and full of hazardous substances. Working in sewers could be easily exposed to a risk of adverse health effects. This paper proposes the use of Bayesian belief networks (BBN) as a higher level of noncarcinogenic health risk assessment of sewer workers. On the basis of the epidemiological studies, the actual hospital attendance records and expert experiences, the BBN is capable of capturing the probabilistic relationships between the hazardous substances in sewers and their adverse health effects, and accordingly inferring the morbidity and mortality of the adverse health effects. The provision of the morbidity and mortality rates of the related diseases is more informative and can alleviate the drawbacks of conventional methods.

Keywords: Bayesian belief networks, sanitary sewerage, healthrisk assessment, hazard quotient, target organ-specific hazard index.

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Authors: Ankit Agrawal, Ankush Mittal

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A gene network gives the knowledge of the regulatory relationships among the genes. Each gene has its activators and inhibitors that regulate its expression positively and negatively respectively. Genes themselves are believed to act as activators and inhibitors of other genes. They can even activate one set of genes and inhibit another set. Identifying gene networks is one of the most crucial and challenging problems in Bioinformatics. Most work done so far either assumes that there is no time delay in gene regulation or there is a constant time delay. We here propose a Dynamic Time- Lagged Correlation Based Method (DTCBM) to learn the gene networks, which uses time-lagged correlation to find the potential gene interactions, and then uses a post-processing stage to remove false gene interactions to common parents, and finally uses dynamic correlation thresholds for each gene to construct the gene network. DTCBM finds correlation between gene expression signals shifted in time, and therefore takes into consideration the multi time delay relationships among the genes. The implementation of our method is done in MATLAB and experimental results on Saccharomyces cerevisiae gene expression data and comparison with other methods indicate that it has a better performance.

Keywords: Activators, correlation, dynamic time-lagged correlation based method, inhibitors, multi-time delay gene network.

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Authors: J. Fernandez de Canete, S. Gonzalez-Perez, P. del Saz-Orozco

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Neural networks offer an alternative approach both for identification and control of nonlinear processes in process engineering. The lack of software tools for the design of controllers based on neural network models is particularly pronounced in this field. SIMULINK is properly a widely used graphical code development environment which allows system-level developers to perform rapid prototyping and testing. Such graphical based programming environment involves block-based code development and offers a more intuitive approach to modeling and control task in a great variety of engineering disciplines. In this paper a SIMULINK based Neural Tool has been developed for analysis and design of multivariable neural based control systems. This tool has been applied to the control of a high purity distillation column including non linear hydrodynamic effects. The proposed control scheme offers an optimal response for both theoretical and practical challenges posed in process control task, in particular when both, the quality improvement of distillation products and the operation efficiency in economical terms are considered.

Keywords: Distillation, neural networks, software tools, identification, control.

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Authors: M.K.Jeya Kumar

Abstract:

Many advanced Routing protocols for wireless sensor networks have been implemented for the effective routing of data. Energy awareness is an essential design issue and almost all of these routing protocols are considered as energy efficient and its ultimate objective is to maximize the whole network lifetime. However, the introductions of video and imaging sensors have posed additional challenges. Transmission of video and imaging data requires both energy and QoS aware routing in order to ensure efficient usage of the sensors and effective access to the gathered measurements. In this paper, the performance of the energy-aware QoS routing Protocol are analyzed in different performance metrics like average lifetime of a node, average delay per packet and network throughput. The parameters considered in this study are end-to-end delay, real time data generation/capture rates, packet drop probability and buffer size. The network throughput for realtime and non-realtime data was also has been analyzed. The simulation has been done in NS2 simulation environment and the simulation results were analyzed with respect to different metrics.

Keywords: Cluster nodes, end-to-end delay, QoS routing, routing protocols, sensor networks, least-cost-path.

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Authors: Shaimaa M. Mohamed, Haitham S. Hamza, Imane A. Saroit

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Many wireless sensor network applications require K-coverage of the monitored area. In this paper, we propose a scalable harmony search based algorithm in terms of execution time, K-Coverage Enhancement Algorithm (KCEA), it attempts to enhance initial coverage, and achieve the required K-coverage degree for a specific application efficiently. Simulation results show that the proposed algorithm achieves coverage improvement of 5.34% compared to K-Coverage Rate Deployment (K-CRD), which achieves 1.31% when deploying one additional sensor. Moreover, the proposed algorithm is more time efficient.

Keywords: Wireless Sensor Networks (WSN), Harmony Search Algorithms, K-Coverage, Mobile WSN.

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Authors: Ahmed Abdolkhalig, Rastko Zivanovic

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This paper evaluates the Total Vector Error of an estimated Phasor as define in IEEE C37.118 standard within different medium access in Local Area Networks (LAN). Three different LAN models (CSMA/CD, CSMA/AMP and Switched Ethernet) are evaluated. The Total Vector Error of the estimated Phasor has been evaluated for the effect of Nodes Number under the standardized network Band-width values defined in IEC 61850-9-2 communication standard (i.e. 0.1, 1 and 10 Gbps).

Keywords: Phasor, Local Area Network, Total Vector Error, IEEE C37.118, IEC 61850.

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Authors: Christian H. Sanabria-Montaña, Rodrigo Huerta-Quintanilla

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A lattice network is a special type of network in which all nodes have the same number of links, and its boundary conditions are periodic. The most basic lattice network is the ring, a one-dimensional network with periodic border conditions. In contrast, the Cartesian product of d rings forms a d-dimensional lattice network. An analytical expression currently exists for the clustering coefficient in this type of network, but the theoretical value is valid only up to certain connectivity value; in other words, the analytical expression is incomplete. Here we obtain analytically the clustering coefficient expression in d-dimensional lattice networks for any link density. Our analytical results show that the clustering coefficient for a lattice network with density of links that tend to 1, leads to the value of the clustering coefficient of a fully connected network. We developed a model on criminology in which the generalized clustering coefficient expression is applied. The model states that delinquents learn the know-how of crime business by sharing knowledge, directly or indirectly, with their friends of the gang. This generalization shed light on the network properties, which is important to develop new models in different fields where network structure plays an important role in the system dynamic, such as criminology, evolutionary game theory, econophysics, among others.

Keywords: Clustering coefficient, criminology, generalized, regular network d-dimensional.

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Authors: Zixin Liu, Jianjun Jiao Wanping Bai

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In this paper, the issue of pth moment exponential stability of stochastic recurrent neural network with distributed time delays is investigated. By using the method of variation parameters, inequality techniques, and stochastic analysis, some sufficient conditions ensuring pth moment exponential stability are obtained. The method used in this paper does not resort to any Lyapunov function, and the results derived in this paper generalize some earlier criteria reported in the literature. One numerical example is given to illustrate the main results.

Keywords: Stochastic recurrent neural networks, pth moment exponential stability, distributed time delays.

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Authors: M. Tarafdar Haque, A. Taheri

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Application of neural networks in execution of programmed pulse width modulation (PPWM) of a voltage source inverter (VSI) is studied in this paper. Using the proposed method it is possible to cancel out the desired harmonics in output of VSI in addition to control the magnitude of fundamental harmonic, contineously. By checking the non-trained values and a performance index, the most appropriate neural network is proposed. It is shown that neural networks may solve the custom difficulties of practical utilization of PPWM such as large size of memory, complex digital circuits and controlling the magnitude of output voltage in a discrete manner.

Keywords: Neural Network, Inverter, PPWM.

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Authors: Dimitra Alexiou

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In an urban area the location allocation of emergency services mobile units, such as ambulances, police patrol cars must be designed so as to achieve a prompt response to demand locations. In this paper the partition of a given urban network into distinct sub-networks is performed such that the vertices in each component are close and simultaneously the sums of the corresponding population in the sub-networks are almost uniform. The objective here is to position appropriately in each sub-network a mobile emergency unit in order to reduce the response time to the demands. A mathematical model in framework of graph theory is developed. In order to clarify the corresponding method a relevant numerical example is presented on a small network.

Keywords: Distances, Emergency Service, Graph Partition, location.

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Authors: Siddhartha Chauhan, Nitin Kumar Kotania

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Traditional Wireless Sensor Networks (WSNs) generally use static sinks to collect data from the sensor nodes via multiple forwarding. Therefore, network suffers with some problems like long message relay time, bottle neck problem which reduces the performance of the network.

Many approaches have been proposed to prevent this problem with the help of mobile sink to collect the data from the sensor nodes, but these approaches still suffer from the buffer overflow problem due to limited memory size of sensor nodes. This paper proposes an energy efficient scheme for data gathering which overcomes the buffer overflow problem. The proposed scheme creates virtual grid structure of heterogeneous nodes. Scheme has been designed for sensor nodes having variable sensing rate. Every node finds out its buffer overflow time and on the basis of this cluster heads are elected. A controlled traversing approach is used by the proposed scheme in order to transmit data to sink. The effectiveness of the proposed scheme is verified by simulation.

Keywords: Buffer overflow problem, Mobile sink, Virtual grid, Wireless sensor networks.

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Authors: Truong Quang Dang Khoa, Masahiro Nakagawa

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Chaos and fractals are novel fields of physics and mathematics showing up a new way of universe viewpoint and creating many ideas to solve several present problems. In this paper, a novel algorithm based on the chaotic sequence generator with the highest ability to adapt and reach the global optima is proposed. The adaptive ability of proposal algorithm is flexible in 2 steps. The first one is a breadth-first search and the second one is a depth-first search. The proposal algorithm is examined by 2 functions, the Camel function and the Schaffer function. Furthermore, the proposal algorithm is applied to optimize training Multilayer Neural Networks.

Keywords: learning and evolutionary computing, Chaos Optimization Algorithm, Artificial Neural Networks, nonlinear optimization, intelligent computational technologies.

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Authors: Bingquan Shen

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Convolutional Neural Networks (CNN) have demonstrated their effectiveness in synthesizing 3D views of object instances at various viewpoints. Given the problem where one have limited viewpoints of a particular object for classification, we present a pose normalization architecture to transform the object to existing viewpoints in the training dataset before classification to yield better classification performance. We have demonstrated that this Pose Normalization Network (PNN) can capture the style of the target object and is able to re-render it to a desired viewpoint. Moreover, we have shown that the PNN improves the classification result for the 3D chairs dataset and ShapeNet airplanes dataset when given only images at limited viewpoint, as compared to a CNN baseline.

Keywords: Convolutional neural networks, object classification, pose normalization, viewpoint invariant.

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Authors: Jiann-Ming Wu, Hsiao-Chang Chen, Chun-Chang Wu, Pei-Hsun Hsu

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This work explores blind image deconvolution by recursive function approximation based on supervised learning of neural networks, under the assumption that a degraded image is linear convolution of an original source image through a linear shift-invariant (LSI) blurring matrix. Supervised learning of neural networks of radial basis functions (RBF) is employed to construct an embedded recursive function within a blurring image, try to extract non-deterministic component of an original source image, and use them to estimate hyper parameters of a linear image degradation model. Based on the estimated blurring matrix, reconstruction of an original source image from a blurred image is further resolved by an annealed Hopfield neural network. By numerical simulations, the proposed novel method is shown effective for faithful estimation of an unknown blurring matrix and restoration of an original source image.

Keywords: Blind image deconvolution, linear shift-invariant(LSI), linear image degradation model, radial basis functions (rbf), recursive function, annealed Hopfield neural networks.

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Authors: V. Agrawal, A. Sharma

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High Strength Concrete (HSC) is defined as concrete that meets special combination of performance and uniformity requirements that cannot be achieved routinely using conventional constituents and normal mixing, placing, and curing procedures. It is a highly complex material, which makes modeling its behavior a very difficult task. This paper aimed to show possible applicability of Neural Networks (NN) to predict the slump in High Strength Concrete (HSC). Neural Network models is constructed, trained and tested using the available test data of 349 different concrete mix designs of High Strength Concrete (HSC) gathered from a particular Ready Mix Concrete (RMC) batching plant. The most versatile Neural Network model is selected to predict the slump in concrete. The data used in the Neural Network models are arranged in a format of eight input parameters that cover the Cement, Fly Ash, Sand, Coarse Aggregate (10 mm), Coarse Aggregate (20 mm), Water, Super-Plasticizer and Water/Binder ratio. Furthermore, to test the accuracy for predicting slump in concrete, the final selected model is further used to test the data of 40 different concrete mix designs of High Strength Concrete (HSC) taken from the other batching plant. The results are compared on the basis of error function (or performance function).

Keywords: Artificial Neural Networks, Concrete, prediction ofslump, slump in concrete

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Authors: Alaa Al Hawari, Mohammad Khader, Tarek Zayed, Osama Moselhi

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Ground Penetrating Radar (GPR) is one of the most effective electromagnetic techniques for non-destructive non-invasive subsurface features investigation. Water leak from pipelines is the most common undesirable reason of potable water losses. Rapid detection of such losses is going to enhance the use of the Water Distribution Networks (WDN) and decrease threatens associated with water mains leaks. In this study, GPR approach was developed to detect leaks by implementing an appropriate imaging analyzing strategy based on image refinement, reflection polarity and reflection amplitude that would ease the process of interpreting the collected raw radargram image.

Keywords: Water Networks, Leakage, Water pipelines, Ground Penetrating Radar.

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Authors: Fayez Mostafa Alhamoui, Adnan Hadi Mahdi Al- Helali

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The main issue in designing a wireless sensor network (WSN) is the finding of a proper routing protocol that complies with the several requirements of high reliability, short latency, scalability, low power consumption, and many others. This paper proposes a novel routing algorithm that complies with these design requirements. The new routing protocol divides the WSN into several subnetworks and each sub-network is divided into several clusters. This division is designed to reduce the number of radio transmission and hence decreases the power consumption. The network division may be changed dynamically to adapt with the network changes and allows the realization of the design requirements.

Keywords: Wireless sensor networks, routing protocols, ad hoc topology, cluster, sub-network, WSN design requirements.

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Authors: Xingyuan Qu, Shouming Zhong

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In this paper, the problem of stability criteria of neural networks (NNs) with two-additive time-varying delay compenents is investigated. The relationship between the time-varying delay and its lower and upper bounds is taken into account when estimating the upper bound of the derivative of Lyapunov functional. As a result, some improved delay stability criteria for NNs with two-additive time-varying delay components are proposed. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

Keywords: Delay-dependent stability, time-varying delays, Lyapunov functional, linear matrix inequality (LMI).

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Authors: Daiming Wang

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In this paper, by using Mawhin-s continuation theorem of coincidence degree and a method based on delay differential inequality, some sufficient conditions are obtained for the existence and global exponential stability of periodic solutions of cellular neural networks with distributed delays and impulses on time scales. The results of this paper generalized previously known results.

Keywords: Periodic solutions, global exponential stability, coincidence degree, M-matrix.

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Authors: Abdesselem Dakhli, Wajdi Bellil, Chokri Ben Amar

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DNA Barcode provides good sources of needed information to classify living species. The classification problem has to be supported with reliable methods and algorithms. To analyze species regions or entire genomes, it becomes necessary to use the similarity sequence methods. A large set of sequences can be simultaneously compared using Multiple Sequence Alignment which is known to be NP-complete. However, all the used methods are still computationally very expensive and require significant computational infrastructure. Our goal is to build predictive models that are highly accurate and interpretable. In fact, our method permits to avoid the complex problem of form and structure in different classes of organisms. The empirical data and their classification performances are compared with other methods. Evenly, in this study, we present our system which is consisted of three phases. The first one, is called transformation, is composed of three sub steps; Electron-Ion Interaction Pseudopotential (EIIP) for the codification of DNA Barcodes, Fourier Transform and Power Spectrum Signal Processing. Moreover, the second phase step is an approximation; it is empowered by the use of Multi Library Wavelet Neural Networks (MLWNN). Finally, the third one, is called the classification of DNA Barcodes, is realized by applying the algorithm of hierarchical classification.

Keywords: DNA Barcode, Electron-Ion Interaction Pseudopotential, Multi Library Wavelet Neural Networks.

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Authors: Laurent Enel, Franck Guillem

Abstract:

This paper considers the benefits gained by using an efficient quality of service management such as DiffServ technique to improve the performance of military communications. Low delay and no blockage must be achieved especially for real time tactical data. All traffic flows generated by different applications do not need same bandwidth, same latency, same error ratio and this scalable technique of packet management based on priority levels is analysed. End to end architectures supporting various traffic flows and including lowbandwidth and high-delay HF or SHF military links as well as unprotected Internet sub domains are studied. A tuning of Diffserv parameters is proposed in accordance with different loads of various traffic and different operational situations.

Keywords: Military data networks, Quality of service, Tacticalsystems.

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Authors: Nima Farkhondeh Jahromi, George Papaefthymiou, Lou van der Sluis

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The incorporation of renewable energy sources for the sustainable electricity production is undertaking a more prominent role in electric power systems. Thus, it will be an indispensable incident that the characteristics of future power networks, their prospective stability for instance, get influenced by the imposed features of sustainable energy sources. One of the distinctive attributes of the sustainable energy sources is exhibiting the stochastic behavior. This paper investigates the impacts of this stochastic behavior on the small disturbance rotor angle stability in the upcoming electric power networks. Considering the various types of renewable energy sources and the vast variety of system configurations, the sensitivity analysis can be an efficient breakthrough towards generalizing the effects of new energy sources on the concept of stability. In this paper, the definition of small disturbance angle stability for future power systems and the iterative-stochastic way of its analysis are presented. Also, the effects of system parameters on this type of stability are described by performing a sensitivity analysis for an electric power test system.

Keywords: Power systems stability, Renewable energy sources, Stochastic behavior, Small disturbance rotor angle stability.

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Authors: Yan Yu, Wang Yu, Chen Xintong, Liu Yi, Zhang Yanzhong, Wang Yanji, Chen Xingyu, Zhang Miaocheng, Tong Yi

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Neuromorphic computing based on spiking neural networks (SNNs) has emerged as a promising avenue for building the next generation of intelligent computing systems. Owing to their high-density integration, low power, and outstanding nonlinearity, memristors have attracted emerging attention on achieving SNNs. However, fabricating a low-power and robust memristor-based spiking neuron without extra electrical components is still a challenge for brain-inspired systems. In this work, we demonstrate a TiO2-based threshold switching (TS) memristor to emulate a leaky integrate-and-fire (LIF) neuron without auxiliary circuits, used to realize single layer fully connected (FC) SNNs. Moreover, our TiO2-based resistive switching (RS) memristors realize spiking-time-dependent-plasticity (STDP), originating from the Ag diffusion-based filamentary mechanism. This work demonstrates that TiO2-based memristors may provide an efficient method to construct hardware neuromorphic computing systems.

Keywords: Leaky integrate-and-fire, memristor, spiking neural networks, spiking-time-dependent-plasticity.

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Authors: C. Iraklis, G. Evmiridis, A. Iraklis

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Renewable energy sources and distributed power generation units already have an important role in electrical power generation. A mixture of different technologies penetrating the electrical grid, adds complexity in the management of distribution networks. High penetration of distributed power generation units creates node over-voltages, huge power losses, unreliable power management, reverse power flow and congestion. This paper presents an optimization algorithm capable of reducing congestion and power losses, both described as a function of weighted sum. Two factors that describe congestion are being proposed. An upgraded selective particle swarm optimization algorithm (SPSO) is used as a solution tool focusing on the technique of network reconfiguration. The upgraded SPSO algorithm is achieved with the addition of a heuristic algorithm specializing in reduction of power losses, with several scenarios being tested. Results show significant improvement in minimization of losses and congestion while achieving very small calculation times.

Keywords: Congestion, distribution networks, loss reduction, particle swarm optimization, smart grid.

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Authors: A. Abid, F. M. Abbou, H. T. Ewe

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Optical Bursts Switching (OBS) is a relatively new optical switching paradigm. Contention and burst loss in OBS networks are major concerns. To resolve contentions, an interesting alternative to discarding the entire data burst is to partially drop the burst. Partial burst dropping is based on burst segmentation concept that its implementation is constrained by some technical challenges, besides the complexity added to the algorithms and protocols on both edge and core nodes. In this paper, the burst segmentation concept is investigated, and an implementation scheme is proposed and evaluated. An appropriate dropping policy that effectively manages the size of the segmented data bursts is presented. The dropping policy is further supported by a new control packet format that provides constant transmission overhead.

Keywords: Burst length, Burst Segmentation, Optical BurstSwitching.

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Authors: Ritu Vijay, Rekha Govil

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The design of a complete expansion that allows for compact representation of certain relevant classes of signals is a central problem in signal processing applications. Achieving such a representation means knowing the signal features for the purpose of denoising, classification, interpolation and forecasting. Multilayer Neural Networks are relatively a new class of techniques that are mathematically proven to approximate any continuous function arbitrarily well. Radial Basis Function Networks, which make use of Gaussian activation function, are also shown to be a universal approximator. In this age of ever-increasing digitization in the storage, processing, analysis and communication of information, there are numerous examples of applications where one needs to construct a continuously defined function or numerical algorithm to approximate, represent and reconstruct the given discrete data of a signal. Many a times one wishes to manipulate the data in a way that requires information not included explicitly in the data, which is done through interpolation and/or extrapolation. Tidal data are a very perfect example of time series and many statistical techniques have been applied for tidal data analysis and representation. ANN is recent addition to such techniques. In the present paper we describe the time series representation capabilities of a special type of ANN- Radial Basis Function networks and present the results of tidal data representation using RBF. Tidal data analysis & representation is one of the important requirements in marine science for forecasting.

Keywords: ANN, RBF, Tidal Data.

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Authors: A. Greco, N. Mammone, F.C. Morabito, M.Versaci

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This paper is mainly concerned with the application of a novel technique of data interpretation to the characterization and classification of measurements of plasma columns in Tokamak reactors for nuclear fusion applications. The proposed method exploits several concepts derived from soft computing theory. In particular, Artifical Neural Networks have been exploited to classify magnetic variables useful to determine shape and position of the plasma with a reduced computational complexity. The proposed technique is used to analyze simulated databases of plasma equilibria based on ITER geometry configuration. As well as demonstrating the successful recovery of scalar equilibrium parameters, we show that the technique can yield practical advantages compares with earlier methods.

Keywords: Tokamak, sensors, artificial neural network.

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Authors: P.S. Prakash, S. Selvan

Abstract:

Quality of Service (QoS) Routing aims to find path between source and destination satisfying the QoS requirements which efficiently using the network resources and underlying routing algorithm and to fmd low-cost paths that satisfy given QoS constraints. One of the key issues in providing end-to-end QoS guarantees in packet networks is determining feasible path that satisfies a number of QoS constraints. We present a Optimized Multi- Constrained Routing (OMCR) algorithm for the computation of constrained paths for QoS routing in computer networks. OMCR applies distance vector to construct a shortest path for each destination with reference to a given optimization metric, from which a set of feasible paths are derived at each node. OMCR is able to fmd feasible paths as well as optimize the utilization of network resources. OMCR operates with the hop-by-hop, connectionless routing model in IP Internet and does not create any loops while fmding the feasible paths. Nodes running OMCR not necessarily maintaining global view of network state such as topology, resource information and routing updates are sent only to neighboring nodes whereas its counterpart link-state routing method depend on complete network state for constrained path computation and that incurs excessive communication overhead.

Keywords: QoS Routing, Optimization, feasible path, multiple constraints.

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Authors: Hyoup-Sang Yoon

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The quality of video transmitted by mobile ad hoc networks (MANETs) can be influenced by several factors, including protocol layers; parameter settings of each protocol. In this paper, we are concerned with understanding the functional relationship between these influential factors and objective video quality in MANETs. We illustrate a systematic statistical design of experiments (DOE) strategy can be used to analyze MANET parameters and performance. Using a 2^k factorial design, we quantify the main and interactive effects of 7 factors on a response metric (i.e., mean opinion score (MOS) calculated by PSNR with Evalvid package) we then develop a first-order linear regression model between the influential factors and the performance metric.

Keywords: Evalvid, full factorial design, mobile ad hoc networks, ns-2.

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Authors: Kriangkrai Maneerat, Chutima Prommak

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This paper presents an analysis of the localization accuracy of indoor positioning systems using Cramer-s rule via IEEE 802.15.4 wireless sensor networks. The objective is to study the impact of the methods used to convert the received signal strength into the distance that is used to compute the object location in the wireless indoor positioning system. Various methods were tested and the localization accuracy was analyzed. The experimental results show that the method based on the empirical data measured in the non line-of-sight (NLOS) environment yield the highest localization accuracy; with the minimum error distance less than 3 m.

Keywords: Indoor positioning systems, localization accuracy, wireless networks, Cramer's rule.

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