A Stochastic Resource-Sharing Network for Electric Vehicle Charging

We consider a distribution grid used to charge electric vehicles such that voltage drops stay bounded. We model this as a class of resource-sharing networks, known as bandwidth-sharing networks in the communication network literature. We focus on resource-sharing networks that are driven by a class of greedy control rules that can be implemented in a decentralized fashion. For a large number of such control rules, we can characterize the performance of the system by a fluid approximation. This leads to a set of dynamic equations that take into account the stochastic behavior of EVs. We show that the invariant point of these equations is unique and can be computed by solving a specific ACOPF problem, which admits an exact convex relaxation. We illustrate our findings with a case study using the SCE 47-bus network and several special cases that allow for explicit computations.Comment: 13 pages, 8 figure

Bounds and Approximations for Stochastic Fluid Networks

The success of modern networked systems has led to an increased reliance and greater demand of their services. To ensure that the next generation of networks meet these demands, it is critical that the behaviour and performance of these networks can be reliably predicted prior to deployment. Analytical modeling is an important step in the design phase to achieve both a qualitative and quantitative understanding of the system. This thesis contributes towards understanding the behaviour of such systems by providing new results for two fluid network models: The stochastic fluid network model and the flow level model. The stochastic fluid network model is a simple but powerful modeling paradigm. Unfortunately, except for simple cases, the steady state distribution which is vital for many performance calculations, can not be computed analytically. A common technique to alleviate this problem is to use the so-called Heavy Traffic Approximation (HTA) to obtain a tractable approximation of the workload process, for which the steady state distribution can be computed. Though this begs the question: Does the steady-state distribution from the HTA correspond to the steady-state distribution of the original network model? It is shown that the answer to this question is yes. Additionally, new results for this model concerning the sample-path properties of the workload are obtained. File transfers compose much of the traffic of the current Internet. They typically use the transmission control protocol (TCP) and adapt their transmission rate to the available bandwidth. When congestion occurs, users experience delays, packet losses and low transfer rates. Thus it is essential to use congestion control algorithms that minimize the probability of occurrence of such congestion periods. Flow level models hide the complex underlying packet-level mechanisms and simply represent congestion control algorithms as bandwidth sharing policies between flows. Balanced Fairness is a key bandwidth sharing policy that is efficient, tractable and insensitive. Unlike the stochastic fluid network model, an analytical formula for the steady-state distribution is known. Unfortunately, performance calculations for realistic systems are extremely time consuming. Efficient and tight approximations for performance calculations involving congestion are obtained

New approaches in statistical modeling

Diese kumulative Dissertation befasst sich mit der statistischen Modellierung von räumlichen Netzwerkdaten, sowie von Daten zur Pandemie des SARS-CoV-2-Virus. Statistische Modellierung kann im übertragenden Sinne als ein großer "Werkzeugkasten'' verstanden werden, mit dem man Phänomene der realen Welt durch eine geeignete mathematische Formalisierung approximiert. Die in dieser Arbeit verwendeten Modelle beruhen in erster Linie auf Regression, wobei die Schwerpunkte auf der Glättung mit penalisierten Splines unter Einbeziehung von zufälligen Effekten liegen. Im Allgemeinen bestehen die Vorteile von Regressions- und statistischen Modellen darin, dass sie interpretierbare Modellergebnisse liefern und Vorhersagen über unbeobachtete Zustände erlauben. Gleichzeitig ist eine Beurteilung der zugrunde liegenden Unsicherheit der Schätzungen möglich. Diese drei Schlüsselaspekte des statistischen Modellierens spielen eine entscheidende Rolle in den fünf Beiträgen dieser kumulativen Dissertation. Die ersten drei Artikel befassen sich mit statistischen Modellen und ihrer Anwendung auf Daten, die auf Netzwerken beobachtet werden. Netzwerke sind Strukturen, die aus durch Kanten verbundene Knoten bestehen. Während Netzwerke in natürlicher Weise abstrakte Beziehungen wie soziale Netzwerke oder ein Netzwerk von Geschäftspartnern darstellen können, liegt der Schwerpunkt in dieser Arbeit auf Netzwerken mit einer räumlichen Interpretation. Im ersten Artikel wird ein neues Modell entwickelt, welches erlaubt, statistische Rückschlüsse auf unbeobachtete Fahrten in Bike-Sharing-Netzwerken zu ziehen. Dabei stellen die Fahrradstationen die Eckpunkte des Netzwerks dar, und die Wege zwischen den Fahrradstationen entsprechen den Kanten. Der darauf folgende Artikel behandelt räumliche Netzwerke und die Schätzung der Intensität von stochastischen Prozessen, deren Realisierungen in räumlichen Netzwerken beobachtet werden. Die Methodik erlaubt auch die Einbeziehung von Kovariablen bei der Schätzung der Intensität. Diese Art der Modellierung ist neu und mit den aktuellen, auf Kerndichteschätzung basierenden Methoden, nicht möglich. Um die Methode frei zugänglich zu machen, wurde ein \textbf{R}-Paket implementiert. Der letzte Beitrag im Bereich der Netzwerke befasst sich mit der Vorhersage der Belegung von Parkplätzen, die entlang eines Straßennetzes verteilt sind. In diesem Zusammenhang wird die Netzwerkstruktur genutzt, um räumliche Abhängigkeiten zu modellieren. Darüber hinaus basieren die Vorhersagen auf einem Semi-Markov-Modell, um die nicht-exponentielle Dauer der einzelnen Zustände zu berücksichtigen. Die Übergangsintensitäten werden mit Hilfe von Überlebenszeitmodellen geschätzt. Der zweite Teil dieser Dissertation befasst sich mit der Pandemie des SARS-CoV-2-Virus, das die Krankheit COVID-19 verursacht. Das deutsche Robert-Koch-Institut (RKI) stellt täglich Daten zu COVID-19-Infektionen und Todesfällen im Zusammenhang mit COVID-19 zur Verfügung, mit zusätzlichen Angaben zu Region, Geschlecht und Alter der Infizierten. Aus verschiedenen Gründen geben die Rohdaten keinen ausreichenden Aufschluss über den Schweregrad der Pandemie, weswegen statistische Modelle auf die Daten angewandt werden. Ein Beitrag befasst sich mit der Vorhersage tödlicher Infektionen auf regionaler Ebene unter Berücksichtigung der lokalen Bevölkerungsstruktur. Damit ist das Modell in der Lage, auch eine regionalspezifische Beurteilung der Schwere der Pandemie vorzunehmen. In einem zweiten Beitrag werden die tödlich endenden Infektionen mit der Anzahl der registrierten Infektionen zueinander in Beziehung gesetzt, um die Veränderung der Fallentdeckungsrate im Laufe der Zeit zu quantifizieren. Darüber hinaus ermöglicht die Methode, den Verlauf der tatsächlichen Zahl der Infektionen zu schätzen, während die gemeldeten Infektionszahlen durch verschiedene Teststrategien beeinflusst sind.This cumulative dissertation is concerned with statistical modeling of data observed on geometric networks and data related to the pandemic of the SARS-CoV-2 virus. Statistical modeling in its broadest sense encompasses a large "toolbox'' to approximate real-world phenomena in a mathematically formalized manner. Models used in this work are primarily regression-based, with an emphasis on penalized spline smoothing and the inclusion of random effects to control for latent heterogeneities. In general, the benefits of regression and statistical models include creating interpretable model results and making predictions about unobserved states while adequately communicating the underlying uncertainty. These three key aspects of statistical modeling play a crucial role in the five contributions of this cumulative dissertation. The first three articles cover statistical models and their application to data observed on networks, i.e. structures consisting of vertices connected by a set of edges. While networks serve as a natural device to represent abstract relationships such as social networks or a network of commercial partners, the focus here is on spatial networks. The first article develops a new model to draw statistical inference about unobserved trips in bike-sharing networks. Here, bike stations represent the network's vertices, and the paths between the bike stations correspond to the edges. The consecutive article treats spatial networks, focusing on estimating stochastic processes' intensity functions with realizations observed on spatial networks. The methodology also allows fitting the intensity with covariates, which is novel and not feasible with the current state-of-the-art methods based on kernel smoothing. To make the methodology freely available, an \textbf{R} package has been implemented. The last contribution in the field of networks covers the prediction of on-street parking occupancy, where parking lots are distributed along a street network. In this context, the network structure is utilized to model spatial dependencies. Moreover, predictions are based on a semi-Markov model to account for non-exponential duration times in each state and the transition intensities are estimated employing time to event models. The second part of this dissertation deals with the pandemic of the SARS-CoV-2 virus, which causes the disease COVID-19. The German Robert Koch Institute (RKI) daily provides data concerning COVID-19 infections and deaths related to COVID-19 with information on the infected's region, gender, and age. For several reasons, the raw data do not indicate the seriousness of the pandemic sufficiently well, which is why statistical models are used to get a clearer picture of the pandemic. One contribution is concerned with nowcasting fatal infections on a regional level while accounting for the local population structure. Thus, the model is capable of evaluating the region-specific seriousness of the pandemic. A second paper relates infections ending fatally to registered infections aiming at quantifying the change of the case detection ratio over time. Furthermore, the method allows assessing the relative course of the actual number of infections while testing strategies influence the reported numbers

On the Support of Massive Machine-to-Machine Traffic in Heterogeneous Networks and Fifth-Generation Cellular Networks

The widespread availability of many emerging services enabled by the Internet of Things (IoT) paradigm passes through the capability to provide long-range connectivity to a massive number of things, overcoming the well-known issues of ad-hoc, short-range networks. This scenario entails a lot of challenges, ranging from the concerns about the radio access network efficiency to the threats about the security of IoT networks. In this thesis, we will focus on wireless communication standards for long-range IoT as well as on fundamental research outcomes about IoT networks. After investigating how Machine-Type Communication (MTC) is supported nowadays, we will provide innovative solutions that i) satisfy the requirements in terms of scalability and latency, ii) employ a combination of licensed and license-free frequency bands, and iii) assure energy-efficiency and security

Proceedings of the 36th International Workshop Statistical Modelling July 18-22, 2022 - Trieste, Italy

The 36th International Workshop on Statistical Modelling (IWSM) is the first one held in presence after a two year hiatus due to the COVID-19 pandemic. This edition was quite lively, with 60 oral presentations and 53 posters, covering a vast variety of topics. As usual, the extended abstracts of the papers are collected in the IWSM proceedings, but unlike the previous workshops, this year the proceedings will be not printed on paper, but it is only online. The workshop proudly maintains its almost unique feature of scheduling one plenary session for the whole week. This choice has always contributed to the stimulating atmosphere of the conference, combined with its informal character, encouraging the exchange of ideas and cross-fertilization among different areas as a distinguished tradition of the workshop, student participation has been strongly encouraged. This IWSM edition is particularly successful in this respect, as testified by the large number of students included in the program

Sojourn time asymptotics in a parking lot network.

For a two-class two-node bandwidth sharing network called parking lot network we investigate the tail behavior of the queue length and sojourn time under light-tailed assumptions. These results extend previous results in the literature obtained for a single-node network. Explicit conditions are given that indicate whether congestion at the second node influences the large deviations behavior or not. To overcome the complexities that arise when moving away from the single node case, we rely on recent results on overloaded bandwidth sharing networks obtained by Borst et al. (2009), and a comparison with the modified proportional fairness discipline, as introduced by Massoulié (Ann Appl Probab 17: 809-839, 2007). Specifically, our results include upper bounds for the distribution of the number of flows in the network, finiteness of the moment generating function of the workload, and large-deviations asymptotics for the sojourn time assuming flow size distributions having a bounded hazard rate. © 2011 Springer-Verlag

Sojourn time tails in processor-sharing systems

The processor-sharing discipline was originally introduced as a modeling abstraction for the design and performance analysis of the processing unit of a computer system. Under the processor-sharing discipline, all active tasks are assumed to be processed simultaneously, receiving an equal share of the server capacity. Various extensions of the basic egalitarian discipline have been developed in order to capture scenarios with heterogeneous service shares and network settings. Over the past several years, the processor-sharing discipline has received renewed attention as a powerful tool in modeling and analyzing dynamic bandwidth sharing among elastic transfers in communication networks like the Internet. The sojourn time of a customer, i.e. the amount of time a customer spends in the system from his arrival until his service completion, is the most important performance measure for processor-sharing systems. In this monograph we study various asymptotic properties of the sojourn time distribution. The advantage of considering the asymptotic behavior is that the analysis often provides insight into the typical scenario for a long sojourn time to occur. Moreover, the resulting asymptotic formulas can be used for approximate analysis, providing accurate estimates in situations where numerical procedures become unreliable. In order to analyze the sojourn time asymptotics, we apply several probabilistic and analytic techniques, such as Laplace transforms, branching arguments, large-deviations methods and fluid limits. The main focus in this thesis is on the PS queue where the service time has a light-tailed distribution. This case has received relatively little attention compared to the case of heavy-tailed distributions. Exact asymptotics (of highly uncommon and interesting form) were only available for the M/M/1 queue and were obtained by analytical methods that did not provide insight into the nature of the underlying rare event. In Chapter 2 we analyze the asymptotic behavior of the sojourn time distribution in the classical single-node PS queue. We derive exact tail asymptotics for the sojourn time distribution in the queue with Poisson arrivals and deterministic service times. The proof involves a geometric random sum representation of the sojourn time, and a connection with Yule processes. Numerical experiments demonstrate a remarkable accuracy of the asymptotic approximation. In Chapter 3 we consider the M/G/1 queue, and investigate the tail behavior of the sojourn time distribution for a request of a given length. An exponential asymptote is proved for general service times in two special cases: when the traffic load is sufficiently high and when the request length is sufficiently small. Using the branching process technique, we derive exact asymptotics of exponential type for the sojourn time in the M/M/1 queue. We study the accuracy of the exponential asymptote using numerical methods. In Chapter 4 we study the GI/GI/1 queue operating under a PS discipline with stochastically varying service rate. The focus is on logarithmic estimates of the tail of the sojourn time distribution, under the assumption that the service time distribution has a light tail. The analysis in this chapter relies predominantly on large-deviations techniques. Furthermore, we extend our results to a similar system operating under the discriminatory processor-sharing discipline. In Chapters 5 and 6 we analyze the behavior of alpha-fair bandwidth-sharing networks which can be regarded as generalizations of a processor-sharing discipline from a single node to a network with several shared links. In Chapter 5 we focus on an overload scenario where the traffic load on one or several of the links exceeds the capacity. In order to characterize the overload behavior, we examine the fluid limit, which emerges from a suitably scaled version of the number of flows of the various classes. We derive a functional equation characterizing the fluid limit. We show that any strictly positive solution must be unique, which in particular implies the convergence of the scaled number of flows to the fluid limit for nonzero initial states when the traffic load is sufficiently high. In addition, we establish the uniqueness of the fluid limit for networks with a tree topology. For the case of a zero initial state and zero-degree homogeneous rate allocation functions, we show that there exists a uniquely determined linear solution to the fluid-limit equation, and obtain a fixed-point equation for the corresponding asymptotic growth rates. The results are illustrated for parking lot, linear and star networks as important special cases. We briefly discuss extensions to models with user impatience. In Chapter 6 we derive the asymptotics for the sojourn time distribution in a specific type of bandwidth-sharing network: a parking lot network. Such networks can be practically useful in modeling access networks consisting of several multiplexing stages. Using large-deviations techniques and the fluid-limit results from Chapter 5, we obtain the logarithmic asymptote under the assumption that flow sizes have a light-tailed distribution. In addition, we derive stochastic bounds for the number of flows and the workload in the system