A key distinction between today's and tomorrow's wireless networks is the appetite for reliability to enable emerging mission-critical services such as ultra-reliable low-latency communication (URLLC) and hyper-reliable low-latency communication (HRLLC), the staple mission-critical services in IMT-2020 (5G) and IMT-2023 (6G), for which reliable and resilient communication is a must. However, achieving ultra-reliable communication is challenging because of these services' stringent reliability and latency requirements and the stochastic nature of wireless networks. A natural way of increasing reliability and reducing latency is to provision additional network resources to compensate for uncertainty in wireless networks caused by fading, interference, mobility, and time-varying network load, among others. Thus, an important step to enable mission-critical services is to identify and quantify what it takes to support ultra-reliable communication in mobile networks -- a process often referred to as dimensioning. This dissertation focuses on resource dimensioning, notably spectrum, for ultra-reliable wireless communication. This dissertation proposes a set of methods for spectrum dimensioning based on concepts from risk analysis, extreme value theory, and meta distributions. These methods reveal that each ``nine'' in reliability (e.g., five-nines in 99.999%) roughly translates into an order of magnitude increase in the required bandwidth. In ultra-reliability regimes, the required bandwidth can be in the order of tens of gigahertz, far beyond what is typically available in today's networks, making it challenging to provision resources for ultra-reliable communication. Accordingly, this dissertation also investigates alternative approaches to provide resources to enable ultra-reliable communication services in mobile networks. Particularly, this dissertation considers multi-operator network sharing and multi-connectivity as alternatives to make additional network resources available to enhance network reliability and proposes multi-operator connectivity sharing, which combines multi-operator network sharing with multi-connectivity. Our studies, based on simulations, real-world data analysis, and mathematical models, suggest that multi-operator connectivity sharing -- in which mobiles multi-connect to base stations of operators in a sharing arrangement -- can reduce the required bandwidth significantly because underlying operators tend to exhibit characteristics attractive to reliability, such as complementary coverage during periods of impaired connectivity, facilitating the support for ultra-reliable communication in future mobile networks. / Doctor of Philosophy / A key distinction between today's and tomorrow's wireless networks is the appetite for reliability to enable emerging mission-critical services in 5G and 6G, for which ultra-reliable communication is a must. However, achieving ultra-reliable communication is challenging because of these services' stringent reliability and latency requirements and the stochastic nature of wireless networks. Reliability often comes at the cost of additional network resources to compensate for uncertainty in wireless networks. Thus, an important step to enable ultra-reliable communication is to identify and quantify what it takes to support mission-critical services in mobile networks -- a process often denoted as dimensioning. This dissertation focuses on spectrum dimensioning and proposes a set of methods to identify suitable spectrum bands and required bandwidth for ultra-reliable communication. These methods reveal that the spectrum needs for ultra-reliable communication can be beyond what is typically available in today's networks, making it challenging to provide adequate resources to support ultra-reliable communication services in mobile networks. Alternatively, we propose multi-operator connectivity sharing: mobiles simultaneously connect to multiple base stations of different operators. Our studies suggest that multi-operator connectivity sharing can reduce the spectrum needs in ultra-reliability regimes significantly, being an attractive alternative to enable ultra-reliable communication in future mobile networks.

La virtualisation de réseau permet de créer des réseaux logiques, dits virtuels sur un réseau physique partagé dit substrat. Pour ce faire, le problème d’allocation des ressources aux réseaux virtuels doit être résolu efficacement. Appelé VNE (Virtual Network Embedding), ce problème consiste à faire correspondre à chaque nœud virtuel un nœud substrat d’un côté, et de l’autre, à tout lien virtuel un ou plusieurs chemins substrat, de manière à optimiser un objectif tout en satisfaisant un ensemble de contraintes. Les ressources de calcul des nœuds et les ressources de bande passante des liens sont souvent optimisées dans un seul réseau substrat. Dans le contexte multi-domaine où la connaissance de l’information de routage est incomplète, l’optimisation des ressources de nœuds et de liens est difficile et souvent impossible à atteindre. Par ailleurs, pour assurer la continuité de service même après une panne, le VNE doit être réalisé de manière à faire face aux pannes. Dans cette thèse, nous étudions le problème d’allocation de ressources (VNE) sous diverses exigences. Pour offrir la virtualisation dans le contexte de réseau substrat multi-domaines, nous proposons une méthode de mappage conjoint des liens inter-domaines et intra-domaines. Avec une information réduite et limitées annoncées par les domaines, notre méthode est capable de mapper simultanément les liens intra-domaines et les liens inter-domaines afin d’optimiser les ressources. De plus, pour améliorer la robustesse des réseaux virtuels, nous proposons un algorithme d’évitement des pannes qui minimise la probabilité de panne des réseaux virtuels. Des solutions exactes et heuristiques sont proposées et détaillées pour des liens à bande passante infinie ou limitée. En outre, nous combinons l’algorithme d’évitement des pannes avec la protection pour proposer un VNE robuste et résistant aux pannes. Avec cette nouvelle approche, les liens protégeables puis les liens les moins vulnérables sont prioritairement sélectionnés pour le mappage des liens. Pour déterminer les liens protégeables, nous proposons une heuristique qui utilise l’algorithme du maxflow afin de vérifier etdedéterminerlesliensprotégeablesàl’étapedumappagedesliensprimaires. Encasd’insuffisance de ressources pour protéger tous les liens primaires, notre approche sélectionne les liens réduisant la probabilité de panne. / Network virtualization allows to create logical or virtual networks on top of a shared physical or substrate network. The resource allocation problem is an important issue in network virtualization. It corresponds to a well known problem called virtual network embedding (VNE). VNE consists in mapping each virtual node to one substrate node and each virtual link to one or several substrate paths in a way that the objective is optimized and the constraints verified. The objective often corresponds to the optimization of the node computational resources and link bandwidth whereas the constraints generally include geographic location of nodes, CPU, bandwidth, etc. In the multi-domain context where the knowledge of routing information is incomplete, the optimization of node and link resources are difficult and often impossible to achieve. Moreover, to ensure service continuity even upon failure, VNE should cope with failures by selecting the best and resilient mappings. Inthisthesis,westudytheVNEresourceallocationproblemunderdifferentrequirements. To embed a virtual network on multi-domain substrate network, we propose a joint peering and intra domain link mapping method. With reduced and limited information disclosed by the domains, our downsizing algorithm maps the intra domain and peering links in the same stage so that the resource utilization is optimized. To enhance the reliability of virtual networks, we propose a failure avoidance approach that minimizes the failure probability of virtual networks. Exact and heuristic solutions are proposed and detailed for the infinite and limited bandwidth link models. Moreover, we combine the failure avoidance with the failure protection in our novel protection-level-aware survivable VNE in order to improve the reliability. With this last approach, the protectable then the less vulnerable links are first selected for link mapping. To determine the protectable links, we propose a maxflow based heuristic that checks for the existence of backup paths during the primary mapping stage. In case of insufficient backup resources, the failure probability is reduced.

Soit un réseau comprenant des lignes de communication qui échouent indépendamment, dans lequel tous ou certains sites, appelés terminaux, doivent être capables de communiquer entre eux. Dans le modèle stochastique statique classique le réseau est représenté par un graphe probabiliste dont les arêtes sont présentes selon des probabilités connues. La mesure de fiabilité classique (CLR) est la probabilité que les terminaux appartiennent à la même composante connexe. Dans plusieurs contextes il est utile d'imposer la condition plus forte que la distance entre deux terminaux quelconques soit bornée supérieurement par un paramètre d. La probabilité que ça se produise est connue comme la fiabilité diamètre-bornée (DCR). Il s'agit d'une extension de la CLR. Les deux problèmes appartiennent à la classe NP-difficile de complexité; le calcul exact n'est possible que pour les instances de taille limitée ou topologies spécifiques. Dans cette thèse, nous contribuons des résultats concernant le problème du calcul et l'estimation de la DCR. Nous étudions la complexité de calcul de cas particuliers, paramétré par le nombre de terminaux, nœuds et le paramètre d. Nous passons en revue des méthodes pour le calcul exact et étudions des topologies particulières pour lesquelles le calcul de la DCR a une complexité polynomiale. Nous introduisons des résultats de base sur le comportement asymptotique de la DCR lorsque le réseau se développe comme un graphe aléatoire. Nous discutons sur l'impact de la contrainte de diamètre dans l'utilisation des techniques de Monte Carlo, et adaptons et testons une famille de méthodes basées sur le conditionnement de l'espace d'échantillonnage en utilisant des structures nommées d-pathsets et d-cutsets. Nous définissons une famille de mesures de performabilité qui généralise la DCR, développons une méthode de Monte Carlo pour l'estimer, et présentons des résultats expérimentaux sur la performance de ces techniques Monte Carlo par rapport é l'approche naïve. Finalement, nous proposons une nouvelle technique qui combine la simulation Monte Carlo et l'interpolation polynomiale pour les mesures de fiabilité. / Consider a communication network whose links fail independently and a set of sites named terminals that must communicate. In the classical stochastic static model the network is represented by a probabilistic graph whose edges occur with known probabilities. The classical reliability (CLR) metric is the probability that the terminals belong to a same connected component. In several contexts it makes sense to impose the stronger condition that the distance between any two terminals does not exceed a parameter d. The probability that this holds is known as the diameter-constrained reliability (DCR). It is an extension of the CLR. Both problems belong to the NP-hard complexity class; they can be solved exactly only for limited-size instances or specific network topologies. In this thesis we contribute a number of results regarding the problem of DCR computation and estimation. We study the computational complexity of particular cases parameterized by the number of terminals, nodes and the parameter d. We survey methods for exact computation and study particular topologies for which computing the DCR has polynomial complexity. We give basic results on the asymptotic behavior of the DCR when the network grows as a random graph. We discuss the impact that the diameter constraint has in the use of Monte Carlo techniques. We adapt and test a family of methods based on conditioning the sampling space using structures named d-pathsets and d-cutsets. We define a family of performability measures that generalizes the DCR, develop a Monte Carlo method for estimating it, and present numerical evidence of how these techniques perform when compared to crude Monte Carlo. Finally we introduce a technique that combines Monte Carlo simulation and polynomial interpolation for reliability metrics.

This dissertation studies two important problems in the field of biomass supply chain network. In the first part of the dissertation, we study the pre-disaster planning problem that seeks to strengthen the links between the multi-modal facilities of a biomass supply chain network. A mixed-integer nonlinear programming model is developed to determine the optimal locations for multi-modal facilities and bio-refineries, offer suggestions on reliability improvement at vulnerable links, production at bio-refineries, and make transportation decision under both normal and disrupted scenarios. The aim is to assist investors in determining which links’ reliability can be improved under specific budget limitations so that the biouel supply chain network can prevent possible losses when transportation links are disrupted because of natural disasters. We used states Mississippi and Alabama as a testing ground for our model. As part of numerical experimentation, some realistic hurricane scenarios are presented to determine the potential impact that pre-investing may have on improving the bio-mass supply chain network’s reliability on vulnerable transportation links considering limited budget availability. In the second part of the dissertation, we study the impact of feedstock supply uncertainty on the design and management of an inbound biomass coiring supply chain network. A two-stage stochastic mixed integer linear programming model is developed to determine the optimal use of multi-modal facilities, biomass storage and processing plants, and shipment routes for delivering biomass to coal plants under feedstock supply uncertainty while considering congestion into account. To represent a more realistic case, we generated a scenario tree based on the prediction errors obtained from historical and forecasted feedstock supply availability. We linearized the nonlinear problem and solved with high quality and in a time efficient manner by using a hybrid decomposition algorithm that connects a Constraint generation algorithm with Sample average approximation algorithm and enhanced Progressive hedging algorithm. We used states Mississippi and Alabama as a testing ground for our study and conducted thorough computational experiments to test our model and to draw managerial insights.

Moore and Shannon's reliability polynomial can be used as a global statistic to explore the behavior of diffusive processes on a graph dynamical system representing a finite sized interacting system. It depends on both the network topology and the dynamics of the process and gives the probability that the system has a particular desired property. Due to the complexity involved in evaluating the exact network reliability, the problem has been classified as a NP-hard problem. The estimation of the reliability polynomials for large graphs is feasible using Monte Carlo simulations. However, the number of samples required for an accurate estimate increases with system size. Instead, an adaptive method using Bernstein polynomials as kernel density estimators proves useful. Network reliability has a wide range of applications ranging from epidemiology to statistical physics, depending on the description of the functionality. For example, it serves as a measure to study the sensitivity of the outbreak of an infectious disease on a network to the structure of the network. It can also be used to identify important dynamics-induced contagion clusters in international food trade networks. Further, it is analogous to the partition function of the Ising model which provides insights to the interpolation between the low and high temperature limits. / Ph. D. / The research presented here explores the effects of the structural properties of an interacting system on the outcomes of a diffusive process using Moore-Shannon network reliability. The network reliability is a finite degree polynomial which provides the probability of observing a certain configuration for a diffusive process on networks. Examples of such processes analyzed here are outbreak of an epidemic in a population, spread of an invasive species through international trade of commodities and spread of a perturbation in a physical system with discrete magnetic spins. Network reliability is a novel tool which can be used to compare the efficiency of network models with the observed data, to find important components of the system as well as to estimate the functions of thermodynamic state variables.

Viele Probleme der Kombinatorik, Zahlentheorie, Wahrscheinlichkeitstheorie, Zuverlässigkeitstheorie und Statistik lassen sich durch Anwendung einer einheitlichen Methode lösen, die als Prinzip der Inklusion-Exklusion bekannt ist. Das Prinzip der Inklusion-Exklusion drückt die Indikatorfunktion einer Vereinigung endlich vieler Ereignisse als alternierende Summe der Indikatorfunktionen ihrer Durchschnitte aus. Die vorliegende Schrift befasst sich mit verbesserten Inklusions-Exklusions-Identitäten und verbesserten Bonferroni-Ungleichungen, die voraussetzen, dass die Ereignisfamilie gewissen strukturellen Anforderungen genügt. Solche wohl-strukturierten Ereignisfamilien finden sich u.a. in der schließenden Statistik, der kombinatorischen Zuverlässigkeitstheorie und der chromatischen Graphentheorie. / Many problems in combinatorics, number theory, probability theory , reliability theory and statistics can be solved by applying a unifying method, which is known as the principle of inclusion-exclusion. The principle of inclusion-exclusion expresses the indicator function of a union of finitely many events as an alternating sum of indicator functions of their intersections. This thesis deals with improved inclusion-exclusion identities and improved Bonferroni inequalities that require the family of events to satisfy some structural restrictions. Examples of such well-structured families arise in problems of statistical inference, combinatorial reliability theory and chromatic graph theory.

Thesis (MSc (Mathematical Sciences. Computer Science))--University of Stellenbosch, 2009. / ENGLISH ABSTRACT: Since 2002 data tra c has overtaken voice tra c in volume [1]. Telecom / Network operators still generate most of their income carrying voice tra c. There is however a huge revenue potential in delivering reliable guaranteed data services. Network survivability and recovery from network failures are integral to network reliability. Due to the nature of the Internet, recovery from link failures needs to be distributed and dynamic in order to be scalable. Link failure recovery schemes are evaluated in terms of the survivability of the network, the optimal use of network resources, scalability, and the recovery time of such schemes. The need for recovery time to be improved is highlighted by real-time data tra c such as VoIP and video services carried over the Internet. The goal of this thesis is to examine existing link failure recovery schemes and evaluate the need for their extension, and to evaluate the performance of the proposed link failure recovery schemes. i / AFRIKAANSE OPSOMMING: Sedert 2002 het data verkeer die stem verkeer in volume verbygesteek [1]. Telekommunikasie / netwerk operateurs genereer egter steeds die meeste van hul inkomste met stem verkeer. Netwerk oorlewing en die herstel van netwerk mislukkings is integraal tot netwerk stabiliteit. Die samestelling van die Internet noodsaak dat die herstel van skakel mislukkings verspreid en dinamies van natuur moet wees. Die herstel-skema van skakel mislukkings word evalueer in terme van die oorleefbaarheid van die netwerk, die mees e ektiewe benutting van network bronne, aanpasbaarheid, en die herstel tydperk van die skema. Die vinnig moontlikste herstel tydperk word genoodsaak deur oombliklike data verkeer soos VoIP en beeld dienste wat oor die Internet gedra word. The doel van hierdie tesis is om bestaande skakel mislukking herstel skemas te evalueer, en dan verder ondersoek in te stel na hul uitbreiding. Daarna word die voorgestelde skakel mislukking skema se e ektiwiteit gemeet.

L'émergence de la technologie de virtualisation des fonctions réseau (NFV) a suscité un vif intérêt autour de la conception, la gestion et le déploiement de services réseau de manière flexible, automatisée et indépendante du fournisseur. La mise en œuvre de la technologie NFV devrait être une solution profitable pour les fournisseurs de services et les clients. Cependant, ce changement de paradigme, amorcé par NFV, nécessite un abandon progressif des services réseau fournis à travers des équipements dédiés. En contrepartie, un environnement totalement ou partiellement virtualisé est proposé pour instancier dynamiquement et à la demande des modules logiciels appelés fonctions de réseau virtuelles (VNF). Cette évolution soulève un ensemble de défis liés au déploiement et à l'exploitation de services, tels que l'orchestration et la gestion, la résilience des services, le contrôle de la qualité de service (QoS), l’approvisionnement des ressources, etc. En outre, la question centrale à résoudre dans le contexte NFV est la suivante : « comment placer et chaîner effacement des fonctions virtuelles d’un service afin de fournir un niveau de qualité demandé par le client tout en optimisant l'utilisation des ressources par le fournisseur de services ? ”.Ainsi, cette thèse étudie la problématique du placement et du chaînage des VNF en tenant compte de certaines exigences de service telles que le délai de bout en bout, la disponibilité du service et la consommation d'énergie, et propose un ensemble d'algorithmes et de mécanismes visant à optimiser le déploiement des services demandés/fournis. Nos contributions dans cette thèse sont triples. Premièrement, nous proposons deux algorithmes de placement et de chaînage de VNF sensibles au délai de bout-en-bout pour des applications temps-réel. Les algorithmes proposés visent à respecter le délai approprié de bout-en-bout qui dépend du service déployé (exemples : VoIP, Streaming, etc.). Deuxièmement, nous présentons une analyse comparative de la disponibilité des services et nous proposons deux mécanismes de placement et de chaînage de VNF pour garantir un niveau prédéfini de disponibilité. L’objectif est de fournir des services résilients en ajustant avec précision les paramètres du schéma de protection (nombre, type, emplacement et taille des instances VNF) nécessaires pour atteindre ce niveau de disponibilité en dépit des défaillances du réseau. Enfin, nous proposons une architecture générale qui explore la possibilité d’étendre le paradigme de la virtualisation à l’Internet des objets (IoT). À cette fin, nous définissons un mécanisme de placement et de chaînage respectant les contraintes énergétiques pour des services IoT. Notre architecture propose de découpler et de virtualiser les fonctionnalités inhérentes à un objet connecté de l’équipement IoT physique. En étendant NFV au domaine IoT, notre solution ouvre de nouvelles perspectives d’application en supportant de nouveaux cas d’usages. / The emergence of Network Function Virtualization (NFV) technology has aroused keen interest to design, manage and deploy network services in a flexible, automated and vendor-agnostic manner. Implementing NFV technology is expected to be a win-win solution for both service providers and costumers. However, this paradigm shift, sparked by NFV, calls for a progressive abandon of network services that are provided as hardware appliance and rather it proposes a fully or partially virtualized environment that offers software modules called Virtual Network Functions (VNFs). This shift rises a set of challenges related to service deployment and operation such as orchestration and management, service resiliency, Quality of Service (QoS) and resource provisioning among others. Furthermore, the core question that needs to be solved within NFV context is “What is the best way to place and chain VNFs that form a service in order to meet Service Level Agreement requirements (costumer side) while optimizing resource usage (service provider side)?”.This thesis investigates the problem of VNF Placement and Chaining considering service requirements such as end-to-end delay, service availability and energy consumption and proposes a set of algorithms and mechanisms that aim to achieve an optimized deployment of the requested/provided services. Our contributions in this thesis are threefold. First, we propose a delay-aware Placement and Chaining algorithms for delay-sensitive applications over NFV networks. The proposed algorithms aim to meet the appropriate end-to-end delay defined according to the deployed service (VoIP, Streaming, etc.). Second, we provide a comprehensive service availability benchmarking and we propose two availability-aware mechanisms for VNFs chain. The aim is to provide resilient service provisioning by fine-tuning the parameters of the protection scheme (the number, the type, the placement and the size of the spare instances) needed to reach a predefined availability level, despite network failures. Finally, we propose a framework architecture that explores the possibility to extend the virtualization paradigm to Internet of Things (IoT). Toward this end, we define an energy-aware Placement and Chaining for IoT services where inherent IoT functionalities are decoupled from specific dedicated IoT devices and instantiated on-demand. By bringing together NFV and IoT paradigms, this extension opens new perspectives and push toward designing new use cases.

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In this thesis, we will study the concept of k-edge connected and k-connected reliability. There, vertices are modelled as fail-safe and edges fail stochastically independent. For a fixxed k, the network is then considered operational when each pair of vertices has k edge disjoint or internally disjoint paths, respectively, connecting them in the surviving subnetwork. Thus, the property of being operating covers the connectivity of the surviving graph together with some minimum bandwidth. We study essential and irrelevant edges for those reliability measures. Further, we study a splitting approach to transform the reliability of the graph into the probability that subgraphs have a certain connectivity. We also extend an approximation algorithm of Karger from the All-Terminal Unreliability to k-edge connected Unreliability and study the k-edge connected Reliability for some special graph classes, namely graphs with restricted treewidth, edge-transitive graphs and the complete graph.:1 Introduction 2 Monotone Systems 2.1 Monotone Binary Systems 2.2 Monotone Multistate Systems 3 Graphs and Graph Operations 4 Higher Connectivity 4.1 Connectivity Number and Edge-Connectivity Number 4.2 Algorithms 5 Essential and Irrelevant Edges 6 Probabilistic Graphs and Reliability Measures 7 Reductions 8 Splitting 8.1 Some Special Cases for Small Separators/Cuts 8.2 Generalization to Arbitrary Separators 8.3 Constructing the Splitting Classes for 2-ec, 3-ec and 2-vc 8.4 Minimality 8.5 A Lattice-based Approach 9 An Approximation Scheme 9.1 Definition of Approximation Algorithms 9.2 The FPRAS for All-Terminal-Unreliability 9.3 Improved Bound for the Number of alpha-cuts 9.4 Extension to k-edge-connected Unreliability 10 Special Graph Classes 10.1 Graphs with Bounded Treewidth 10.2 Edge-Transitive Graphs 10.3 The Complete Graph 11 Future Research 12 Summary