Operadores de Wiener-Hopf, factorizações e teoria da realização

Mestrado em MatemáticaNa presente dissertação é feito um estudo dos operadores de Wiener-Hopf abstractos, em espaços de Banach, e de algumas técnicas de factorização associadas. Estas factorizações permitem deduzir diversas propriedades relativas às eventuais soluções de um largo conjunto de equações integrais e, em especial, das equações integrais de Wiener-Hopf, que foi o ponto de partida de Norbert Wiener e Eberhard Hopf na descoberta dos processos de factorização. Estes processos vêm, ainda, permitir a ligação do estudo das equações integrais de Wiener-Hopf com o estudo das realizações racionais e, assim estabelecer uma ligação entre as factorizações de Wiener-Hopf e a teoria da realização. De facto, se uma equação integral de Wiener-Hopf admitir símbolo racional, este pode tomar a forma de uma realização racional e, assim, pode-se proceder a factorizações conhecidas no seio da teoria da realização, de modo a se alcançar a solução. Neste âmbito, tanto perspectivas gerais como exemplos concretos são apresentados na presente dissertação que não contém resultados matemáticos originais.In the present dissertation, a study of Wiener-Hopf abstract operators, in Banach spaces, and of some associated factorization techniques is performed. These factorizations allow the inference of several properties concerning eventual solutions of an extensive set of integral equations and, especially, of the Wiener-Hopf integral equations, which was the starting point for Norbert Wiener and Eberhard Hopf in the discovery of the factorization processes. Additionally, these processes allow the connection between the study of Wiener-Hopf integral equations and the study of rational realizations establishing, in this manner, a connection between Wiener-Hopf factorizations and the realization theory. In fact, if a certain Wiener-Hopf integral equation admits a rational symbol, the latter can be presented by a rational realization and, hence, one can proceed with the factorizations known in the context of the realization theory and, therefore, attain the solution. In this context, general perspectives, as well as, concrete examples are presented in the current dissertation, which does not contain original mathematical results

Signal reconstruction in structures with two channels

Doutoramento em Engenharia ElectrotécnicaEm sistemas ATM e transmissões em tempo real através de redes IP, os dados são transmitidos em pacotes de informação. Os pacotes perdidos ou muito atrasados levam à perda de informação em posições conhecidas (apagamentos). Contudo, em algumas situações as posições dos erros não são conhecidas e, portanto, a detecção dos erros tem que ser realizada usando um polinómio conhecido. A detecção e correcção de erros são estudadas para sinais digitais em códigos DFT em dois canais que apresentam muito melhor estabilidade que os respectivos códigos DFT num único canal. Para a estrutura de dois canais, um canal processa um código DFT normal, quanto que o outro canal inclui uma permutação, a razão principal para a melhoria na estabilidade. A permutação introduz aleatoriedade e é esta aleatoriedade que é responsável pela boa estabilidade destes códigos. O estudo dos códigos aleatórios vêm confirmar esta afirmação. Para sinais analógicos, foca-se a amostragem funcional e derivativa, onde um canal processa amostras do sinal e o outro processa amostras da derivada do sinal. A expansão sobreamostrada é apresentada e a recuperação de apagamentos é estudada. Neste caso, a estabilidade para a esturtura em dois canais quando a perda de amostras afecta ambos os canais é, em geral, muito pobre. Adicionalmente, a reconstrução de sinais tanto analógicos como digitais é tratada para o modelo do conversor integrate-and-fire. A reconstrução faz uso dos tempos de acção e de valores limites inerentes ao modelo e é viável por meio de um método iterativo baseado em projecções em conjuntos convexos (POCS).In ATM as in real time transmissions over IP networks, the data are transmitted packet by packet. Lost or highly delayed packets lead to lost information in known locations (erasures). However, in some situations the error locations are not known and, therefore, error detection must be performed using a known polynomial. Error detection and correction are studied for digital signals in two-channel DFT codes which presents a much better stability than their single channel counterparts. For the two-channel structure, one channel processes an ordinary DFT code, while the other channel includes an interleaver, the main reason for the improvement in stability. The interleaver introduces randomness and it is this randomness that is responsible for the good stability of these codes. The study of random codes helps confirm this statement. For analogical signals, the focus is given to function and derivative sampling, where one channel processes samples of the signal and the other processes samples of the derivative of the signal. The oversampled expansion is presented and erasure recovery is studied. In this case, the stability of the twochannel structure when sample loss affects both channels is, in general, very poor. Additionally, the reconstruction of analogical as well as digital signals is dealt with for the integrate-and-fire converter model. The reconstruction makes use of the firing times and the threshold values inherent to the model and is viable by means of an iterative method based on projections onto convex sets (POCS)

The minimum cost D-geodiverse anycast routing with optimal selection of anycast nodes

Consider a geographical network with associated link costs. In anycast routing, network nodes are partitioned into two sets - the source nodes and the anycast (destination) nodes - and the traffic of each source node is routed towards the anycast node providing the minimum routing cost path. By considering a given geographical distance parameter D, we define an anycast routing solution as D-geodiverse when for each source node there are two routing paths, each one towards a different anycast node, such that the geographical distance between the two paths is at least D. Such a solution has the property that any disaster with a coverage diameter below D affecting one routing path (but without involving neither the source node nor its entire set of outgoing links) cannot affect the other path, enhancing in this way the network robustness to natural disasters. The selection of the anycast nodes has an impact both on the feasibility and cost of a D- geodiverse anycast routing solution. Therefore, for a desired number of anycast nodes R, we define the minimum cost D- geodiverse anycast problem (MCD-GAP) aiming to identify a set of R anycast nodes that obtain a minimum cost routing solution. The problem is defined based on integer linear programming and is extended to consider the existence of vulnerability regions in the network, i.e., by imposing the geographical distance D only between network elements belonging to the same region. We present computational results showing the tradeoff between D and R in the optimal solutions obtained with and without vulnerability regions.This paper is based upon work from COST Action CA15127 ("Resilient communication services protecting end user applications from disaster-based failures ‒ RECODIS") supported by COST Association. The work was financially supported by FCT, Portugal, under the projects CENTRO- 01-0145-FEDER-029312 and UID/EEA/50008/2013 and through the postdoc grant SFRH/BPD/ 111503/2015.publishe

Assessment of connectivity-based resilience to attacks against multiple nodes in SDNs

In Software Defined Networks (SDNs), the control plane of a network is decoupled from its data plane. For scalability and robustness, the logically centralized control plane is implemented by physically placing different controllers throughout the network. The determination of the number and placement of controllers is known as the Controller Placement Problem (CPP). In the regular (i.e., failure-free) state, the control plane must guarantee a given maximum delay between every switch and its primary controller and a given maximum delay between every pair of controllers. In general, these delay bounds allow multiple solutions and, so, other goals can be used to determine the best CPP solution. In this paper, we assess the connectivity-based resilience to malicious attacks against multiple network nodes of the CPP solutions obtained with three different aims: the regular state delay optimization without any concern about attacks, the regular state delay optimization taking into consideration the worst-case attacks and the resilience optimization to attacks against multiple nodes. We assess the CPP solutions considering attacks of targeted nature (when the attacker has complete knowledge of the data plane) and attacks of non-targeted nature (i.e., random and epidemic attacks). We present computational results providing an analysis of the CPP solutions to the different types of attacks. The main conclusion is that the connectivity-based resilience between the different CPP solutions strongly depends on the network topology, the regular state delay bounds and the type of attacks. Finally, we provide insights on how SDN operators can consider the conducted assessment when deciding the controller placements in their networks.publishe

Link load balancing optimization of telecommunication networks: a column generation based heuristic approach

This paper deals with optimal load balancing in telecommunication networks. For a capacitated telecommunications network with single path routing and an estimated traffic demand matrix, we wish to determine the routing paths aiming at min-max optimization of link loads. To solve this problem, we propose a column (path) generation based heuristic. In the first step, we use column generation to solve a linear programming relaxation of the basic problem (obtaining a lower bound and a set of paths). In the second step, we apply a multi-start local search heuristic with path-relinking to the search space defined by the paths found in the first step. In order to assess the merits of this approach, we also implemented a search heuristic which is equivalent to the second step of the proposed one but with no constraints on the set of paths that can be used. Through a set of computational results, we show that the proposed heuristic is efficient in obtaining near optimal routing solutions within short running times. Moreover, the comparison of the two heuristics show that constraining the search space to the columns given by column generation gives better results since this solution space contains good quality solutions and, due to its size, enables to find them in short running times

Combined control and data plane robustness of SDN networks against malicious node attacks

In the context of software-defined networking (SDN), we address a variant of the controller placement problem (CPP), which takes into account the network robustness at both control and data plane layers. For given maximum values of switch-controller and controller-controller delays at the regular state (i.e., when the network is fully operational), the aim is to maximize the network robustness against a set of failure states, each state defined as a possible malicious attack to multiple network nodes. We assume that the attacker knows the data plane topology and, therefore, can adopt either one of three commonly considered node centrality attacks (based on the node degree, closeness or betweenness centralities), or an attack to the nodes which are the optimal solution of the critical node detection (CND) problem. We propose a set of robustness metrics which are used to obtain the optimal solutions for the robust CPP variant. We present a set of computational results comparing the average delays and robustness values of the robust CPP solutions against those minimizing only the average switch-controller and controller-controller delays. Moreover, the impact of using the CND based attack in the robustness evaluation of CPP solutions is also assessed in the computational results.publishe

Controller Placement and Availability Link Upgrade Problem in SDN Networks

In SDN networks, the problem of how many controllers and where to place them, has been extensively studied. This is known as the controller placement problem, and has been addressed mainly considering the delays between the SDN switches and controllers. Although the delays between switches and controllers and the intercontroller delays are key aspects, a less addressed issue is the availability of the control paths (the routing paths between the switches and their controllers). In this paper, the problem regarding controller placements with QoS requirements, both in terms of delays and availability of the control paths, is addressed. To guarantee the availability requirements, a set of links is selected to have upgraded availability. An exact method and a heuristic method are proposed for solving the problem. Computational results show that the heuristic method provides near-optimal solutions within reasonable runtimes, when the exact method becomes computationally expensive

Joint optimization of primary and backup controller placement and availability link upgrade in SDN networks

In Software-Defined Networking (SDN), the control and data planes are decoupled, leading to a more programmable and efficient network management. In this paper, the controller placement problem in SDN is addressed, jointly with the problem of exploring a high-availability tree subgraph, in order to support delay and availability requirements between the switches and the controllers. We consider that each switch connects to a primary and to a backup controller. We formulate the joint optimization model as an integer linear programming model (ILP), and propose a heuristic method when the exact model becomes impractical. Furthermore, we compare two ILP formulations, and we also compare the controller redundancy solutions with those considering path redundancy alone.Fundação para a Ciência e a Tecnologia (FCT) under the project grant UIDB/00308/2020. FEDER Funds and National Funds through FCT under the project CENTRO-01-0145-FEDER-029312. COST Action CA15127 (“Resilient communication services protecting end-user applications from disaster-based failures – RECODIS")

Software-Defined Network Design driven by Availability Requirements

In Software Defined Networking (SDN), the controller locations are mainly constrained by delays between switches and controllers, and between the controllers themselves. In addition to the delay requirements, the availability of the connections between switches and controllers is also a key issue for control plane performance. Here, we explore the idea of having a spanning tree substructure called the spine, whose links can be upgraded to have high availability in order to support availability requirements for the control paths (routing paths between switches and controllers). We formulate an optimization model of the joint controller placement and spine design problem for SDN networks, under delay, availability and path redundancy requirements. Numerical results are presented showing the trade-offs between the number of controllers, delay requirements and upgrade cost

SDN Controller Placement With Availability Upgrade Under Delay and Geodiversity Constraints

An inherent problem in Software-Defined Networking (SDN) is the Controller Placement Problem, which addresses how many controllers to deploy in the network, and where to place them. Several variants of this problem have been addressed and researched to find the placements that adapt best to different contexts. In this paper, we address a more complex variant of this problem, to satisfy QoS requirements and to offer robustness against disaster-based failures. We address the joint optimization problem of controller placement and finding a tree subgraph which can be upgraded to have enhanced availability, in order to satisfy delay and availability constraints. Additionally, we consider geodiversity constraints as a way to enhance robustness to disaster-based failures.This work is funded by ERDF Funds through the Centre’s Regional Operational Program and by National Funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P. under the project CENTRO-01-0145-FEDER-029312. This work was also partially supported by FCT under project grant UIDB/00308/2020