Non-stationary service curves : model and estimation method with application to cellular sleep scheduling

In today’s computer networks, short-lived flows are predominant. Consequently, transient start-up effects such as the connection establishment in cellular networks have a significant impact on the performance. Although various solutions are derived in the fields of queuing theory, available bandwidths, and network calculus, the focus is, e.g., about the mean wake-up times, estimates of the available bandwidth, which consist either out of a single value or a stationary function and steady-state solutions for backlog and delay. Contrary, the analysis during transient phases presents fundamental challenges that have only been partially solved and is therefore understood to a much lesser extent. To better comprehend systems with transient characteristics and to explain their behavior, this thesis contributes a concept of non-stationary service curves that belong to the framework of stochastic network calculus. Thereby, we derive models of sleep scheduling including time-variant performance bounds for backlog and delay. We investigate the impact of arrival rates and different duration of wake-up times, where the metrics of interest are the transient overshoot and relaxation time. We compare a time-variant and a time-invariant description of the service with an exact solution. To avoid probabilistic and maybe unpredictable effects from random services, we first choose a deterministic description of the service and present results that illustrate that only the time-variant service curve can follow the progression of the exact solution. In contrast, the time-invariant service curve remains in the worst-case value. Since in real cellular networks, it is well known that the service and sleep scheduling procedure is random, we extend the theory to the stochastic case and derive a model with a non-stationary service curve based on regenerative processes. Further, the estimation of cellular network’s capacity/ available bandwidth from measurements is an important topic that attracts research, and several works exist that obtain an estimate from measurements. Assuming a system without any knowledge about its internals, we investigate existing measurement methods such as the prevalent rate scanning and the burst response method. We find fundamental limitations to estimate the service accurately in a time-variant way, which can be explained by the non-convexity of transient services and their super-additive network processes. In order to overcome these limitations, we derive a novel two-phase probing technique. In the first step, the shape of a minimal probe is identified, which we then use to obtain an accurate estimate of the unknown service. To demonstrate the minimal probing method’s applicability, we perform a comprehensive measurement campaign in cellular networks with sleep scheduling (2G, 3G, and 4G). Here, we observe significant transient backlogs and delay overshoots that persist for long relaxation times by sending constant-bit-rate traffic, which matches the findings from our theoretical model. Contrary, the minimal probing method shows another strength: sending the minimal probe eliminates the transient overshoots and relaxation times

Spatial point pattern analysis with application to confocal microscopy data

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Extended formulations for convex envelopes

In this work we derive explicit descriptions for the convex envelope of nonlinear functions that are component-wise concave on a subset of the variables and convex on the other variables. These functions account for more than 30% of all nonlinearities in common benchmark libraries. To overcome the combinatorial difficulties in deriving the convex envelope description given by the component-wise concave part of the functions, we consider an extended formulation of the convex envelope based on the Reformulation-Linearization-Technique introduced by Sherali and Adams(SIAM J Discret Math 3(3):411-430, 1990). Computational results are reported showing that the extended formulation strategy is a useful tool in global optimization

Relaxation refinement for mixed-integer nonlinear programs with applications in engineering

Lösungsstrategien für Gemischt-Ganzzahlige Nichtlineare Programme (MINLPs) basieren häufig auf einer konvexen Relaxierung der zulässigen Menge. Diese Relaxierung wird benutzt um untere Schranken zu ermitteln und um die Qualität lokaler Lösungen zu beurteilen. In dieser Thesis diskutieren wir verschiedene Ansätze um geeignete Relaxierungen zu konstruieren und zu verbessern. Außerdem analysieren wir diese in Hinblick auf Strenge und Qualität der resultierenden unteren Schranken. Dabei betrachten wir sowohl allgemeine MINLPs als auch spezifische Probleme, die sich aus der Anwendung ergeben. Wir entwickeln ein Schnittebenenverfahren für die konvexe Hülle der zulässigen Menge von relativ allgemeinen MINLPs. Es basiert auf der simultanen Betrachtung von Nebenbedingungen und auf einem konvexen Optimierungsproblem. Dieses Separationsproblem ist nicht-differenzierbar und benötigt die konvexe Einhüllende von Linearkombinationen der Nebenbedingungen. Wir analysieren seine Struktur und Glätte ausführlich und diskutieren passende Lösungsansätze. Außerdem entwickeln wir Approximationen der konvexen Einhüllenden und ein ensprechendes approximatives Separationsproblem. Dieses führt zu schwächeren Resultaten aber zu einer höheren Anwendbarkeit. Das obige Schnittebenenverfahren wird außerdem auf eine Menge von Nebenbedingungen angewendet, die aus bivariaten quadratischen Absolutwertfunktionen besteht. Wir präsentieren allgemeine analytische Hilfsmittel und Konzepte und bestimmen die konvexe Einhüllende für diese Funktionen unter gewissen Voraussetzungen. Diese Klasse von Funktionen wird auch bei der Modellierung von Gasnetzwerken verwendet, was es uns erlaubt den Einfluss des Schnittebenenverfahrens auf Probleme aus der Anwendung zu untersuchen. Schließlich betrachten wir noch ein Beispiel eines optimalen Designproblems aus dem Bereich des Chemieingenieurwesens. Für das Modell einer Destillationskolonne bieten wir eine Reformulierung an und beweisen monotones Verhalten von bestimmten Folgen relevanter Variablen. Reformulierung und Monotonie werden benutzt um die Formulierung der zugehörigen zulässigen Menge zu verbessern. Insbesondere entwickeln wir eine problemspezifische Bound-Tightening-Strategie. Unsere Ergebnisse werden an einigen Testinstanzen computergestützt evaluiert.Solution strategies for Mixed-Integer Nonlinear Programs (MINLPs) often rely on a convex relaxation of the feasible set. This relaxation is used to derive lower bounds and to evaluate the quality of local solutions. In this thesis, we discuss different approaches of constructing and improving suitable relaxations. We further analyze these relaxations with respect to tightness and quality of the resulting lower bounds. This is done for general MINLPs as well as for specific problems arising from certain real world applications. We develop a cutting plane method for the convex hull of the feasible set of relatively general MINLPs. It is based on simultaneous considerations of the involved constraints and on solving a convex optimization problem. This underlying separation problem is non-differentiable and requires the convex envelope of linear combinations of the constraint functions. We analyze its structure and smoothness in detail, and discuss suitable solution approaches. Furthermore, we introduce approximation strategies for the convex envelope and discuss the resulting approximate version of the separation problem. This approximate version leads to weaker results but to a greater applicability. The proposed cutting plane approach is further applied to constraint sets consisting of bivariate quadratic absolute value functions. We present general analytic tools and concepts, and derive the convex envelope of the considered functions under certain assumptions. This type of functions also emerges from the modeling of gas networks, which allows us to computationally evaluate the impact of our cutting plane approach on a real world application. Finally, we consider an example of optimal design problems in chemical engineering. For a distillation column model, we introduce a suitable reformulation and prove monotonic behavior of several sequences of relevant variables. Reformulation and monotonicity are used to improve the formulation of the respective feasible set. In particular, we develop a problem specific bound tightening strategy. Our results are computationally evaluated on multiple test instances

Toward quantitative limited-angle ultrasound reflection tomography to inform abdominal HIFU treatment planning

High-Intensity Focused Ultrasound (HIFU) is a treatment modality for solid cancers of the liver and pancreas which is non-invasive and free from many of the side-effects of radiotherapy and chemotherapy. The safety and efficacy of abdominal HIFU treatment is dependent on the ability to bring the therapeutic sound waves to a small focal ”lesion” of known and controllable location within the patient anatomy. To achieve this, pre-treatment planning typically includes a numerical simulation of the therapeutic ultrasound beam, in which anatomical compartment locations are derived from computed tomography or magnetic resonance images. In such planning simulations, acoustic properties such as density and speed-of-sound are assumed for the relevant tissues which are rarely, if ever, determined specifically for the patient. These properties are known to vary between patients and disease states of tissues, and to influence the intensity and location of the HIFU lesion. The subject of this thesis is the problem of non-invasive patient-specific measurement of acoustic tissue properties. The appropriate method, also, of establishing spatial correspondence between physical ultrasound transducers and modeled (imaged) anatomy via multimodal image reg-istration is also investigated; this is of relevance both to acoustic tissue property estimation and to the guidance of HIFU delivery itself. First, the principle of a method is demonstrated with which acoustic properties can be recovered for several tissues simultaneously using reflection ultrasound, given accurate knowledge of the physical locations of tissue compartments. Second, the method is developed to allow for some inaccuracy in this knowledge commensurate with the inaccuracy typical in abdominal multimodal image registration. Third, several current multimodal image registration techniques, and two novel modifications, are compared for accuracy and robustness. In conclusion, relevant acoustic tissue properties can, in principle, be estimated using reflected ultrasound data that could be acquired using diagnostic imaging transducers in a clinical setting

A Computational Framework for Efficient Reliability Analysis of Complex Networks

With the growing scale and complexity of modern infrastructure networks comes the challenge of developing efficient and dependable methods for analysing their reliability. Special attention must be given to potential network interdependencies as disregarding these can lead to catastrophic failures. Furthermore, it is of paramount importance to properly treat all uncertainties. The survival signature is a recent development built to effectively analyse complex networks that far exceeds standard techniques in several important areas. Its most distinguishing feature is the complete separation of system structure from probabilistic information. Because of this, it is possible to take into account a variety of component failure phenomena such as dependencies, common causes of failure, and imprecise probabilities without reevaluating the network structure. This cumulative dissertation presents several key improvements to the survival signature ecosystem focused on the structural evaluation of the system as well as the modelling of component failures. A new method is presented in which (inter)-dependencies between components and networks are modelled using vine copulas. Furthermore, aleatory and epistemic uncertainties are included by applying probability boxes and imprecise copulas. By leveraging the large number of available copula families it is possible to account for varying dependent effects. The graph-based design of vine copulas synergizes well with the typical descriptions of network topologies. The proposed method is tested on a challenging scenario using the IEEE reliability test system, demonstrating its usefulness and emphasizing the ability to represent complicated scenarios with a range of dependent failure modes. The numerical effort required to analytically compute the survival signature is prohibitive for large complex systems. This work presents two methods for the approximation of the survival signature. In the first approach system configurations of low interest are excluded using percolation theory, while the remaining parts of the signature are estimated by Monte Carlo simulation. The method is able to accurately approximate the survival signature with very small errors while drastically reducing computational demand. Several simple test systems, as well as two real-world situations, are used to show the accuracy and performance. However, with increasing network size and complexity this technique also reaches its limits. A second method is presented where the numerical demand is further reduced. Here, instead of approximating the whole survival signature only a few strategically selected values are computed using Monte Carlo simulation and used to build a surrogate model based on normalized radial basis functions. The uncertainty resulting from the approximation of the data points is then propagated through an interval predictor model which estimates bounds for the remaining survival signature values. This imprecise model provides bounds on the survival signature and therefore the network reliability. Because a few data points are sufficient to build the interval predictor model it allows for even larger systems to be analysed. With the rising complexity of not just the system but also the individual components themselves comes the need for the components to be modelled as subsystems in a system-of-systems approach. A study is presented, where a previously developed framework for resilience decision-making is adapted to multidimensional scenarios in which the subsystems are represented as survival signatures. The survival signature of the subsystems can be computed ahead of the resilience analysis due to the inherent separation of structural information. This enables efficient analysis in which the failure rates of subsystems for various resilience-enhancing endowments are calculated directly from the survival function without reevaluating the system structure. In addition to the advancements in the field of survival signature, this work also presents a new framework for uncertainty quantification developed as a package in the Julia programming language called UncertaintyQuantification.jl. Julia is a modern high-level dynamic programming language that is ideal for applications such as data analysis and scientific computing. UncertaintyQuantification.jl was built from the ground up to be generalised and versatile while remaining simple to use. The framework is in constant development and its goal is to become a toolbox encompassing state-of-the-art algorithms from all fields of uncertainty quantification and to serve as a valuable tool for both research and industry. UncertaintyQuantification.jl currently includes simulation-based reliability analysis utilising a wide range of sampling schemes, local and global sensitivity analysis, and surrogate modelling methodologies

Learning curves of generic features maps for realistic datasets with a teacher-student model

Teacher-student models provide a framework in which the typical-case performance of high-dimensional supervised learning can be described in closed form. The assumptions of Gaussian i.i.d. input data underlying the canonical teacher-student model may, however, be perceived as too restrictive to capture the behaviour of realistic data sets. In this paper, we introduce a Gaussian covariate generalisation of the model where the teacher and student can act on different spaces, generated with fixed, but generic feature maps. While still solvable in a closed form, this generalization is able to capture the learning curves for a broad range of realistic data sets, thus redeeming the potential of the teacher-student framework. Our contribution is then two-fold: First, we prove a rigorous formula for the asymptotic training loss and generalisation error. Second, we present a number of situations where the learning curve of the model captures the one of a realistic data set learned with kernel regression and classification, with out-of-the-box feature maps such as random projections or scattering transforms, or with pre-learned ones - such as the features learned by training multi-layer neural networks. We discuss both the power and the limitations of the framework.Comment: v3: NeurIPS camera-read