Throughputs in stochastic free-choice nets, existence, computations and optimizations

International audienceIn this paper, live and bounded free-choice Petri nets with stochastic firing times are considered. Several classical routing policies, namely the race policy, Bernoulli routings, and periodic routings, are compared in terms of the throughputs of the transitions. First, under general i.i.d. assumptions on the firing times, the existence of the throughput for the three policies is established. We also show that the ratio between the throughputs of two transitions depend only on the asymptotic frequencies of the routings, and not on the routing policy. On the other hand, the total throughput depends on the policy, and is higher for the race policy than for Bernoulli routings. Second, we show how to compute the throughput for exponentially distributed free-choice nets under the three policies. This is done by using Markov processes over appropriate state spaces. We use this to compare the performance of periodic and Bernoulli routings. Finally, we derive optimal policies under several information structures, namely, the optimal pre-allocation, the optimal allocation, and the optimal non-anticipative policy

Approximation methods for stochastic petri nets

Stochastic Marked Graphs are a concurrent decision free formalism provided with a powerful synchronization mechanism generalizing conventional Fork Join Queueing Networks. In some particular cases the analysis of the throughput can be done analytically. Otherwise the analysis suffers from the classical state explosion problem. Embedded in the divide and conquer paradigm, approximation techniques are introduced for the analysis of stochastic marked graphs and Macroplace/Macrotransition-nets (MPMT-nets), a new subclass introduced herein. MPMT-nets are a subclass of Petri nets that allow limited choice, concurrency and sharing of resources. The modeling power of MPMT is much larger than that of marked graphs, e.g., MPMT-nets can model manufacturing flow lines with unreliable machines and dataflow graphs where choice and synchronization occur. The basic idea leads to the notion of a cut to split the original net system into two subnets. The cuts lead to two aggregated net systems where one of the subnets is reduced to a single transition. A further reduction leads to a basic skeleton. The generalization of the idea leads to multiple cuts, where single cuts can be applied recursively leading to a hierarchical decomposition. Based on the decomposition, a response time approximation technique for the performance analysis is introduced. Also, delay equivalence, which has previously been introduced in the context of marked graphs by Woodside et al., Marie's method and flow equivalent aggregation are applied to the aggregated net systems. The experimental results show that response time approximation converges quickly and shows reasonable accuracy in most cases. The convergence of Marie's method and flow equivalent aggregation are applied to the aggregated net systems. The experimental results show that response time approximation converges quickly and shows reasonable accuracy in most cases. The convergence of Marie's is slower, but the accuracy is generally better. Delay equivalence often fails to converge, while flow equivalent aggregation can lead to potentially bad results if a strong dependence of the mean completion time on the interarrival process exists

Nested Fork-Join Queuing Networks and Their Application to Mobility Airfield Operations Analysis

A single-chain nested fork-join queuing network (FJQN) model of mobility airfield ground processing is proposed. In order to analyze the queuing network model, advances on two fronts are made. First, a general technique for decomposing nested FJQNs with probabilistic forks is proposed, which consists of incorporating feedback loops into the embedded Markov chain of the synchronization station, then using Marie\u27s Method to decompose the network. Numerical studies show this strategy to be effective, with less than two percent relative error in the approximate performance measures in most realistic cases. The second contribution is the identification of a quick, efficient method for solving for the stationary probabilities of the λn/Ck/r/N queue. Unpreconditioned Conjugate Gradient Squared is shown to be the method of choice in the context of decomposition using Marie\u27s Method, thus broadening the class of networks where the method is of practical use. The mobility airfield model is analyzed using the strategies described above, and accurate approximations of airfield performance measures are obtained in a fraction of the time needed for a simulation study. The proposed airfield modeling approach is especially effective for quick-look studies and sensitivity analysis

Formal Methods for Schedulings of Latency-Insensitive Designs

LID ( Latency-Insensitive Design) theory was invented to deal with SoC timing closure issues, by allowing arbitrary fixed integer latencies on long global wires. Latencies are coped with using a resynchronization protocol that performs dynamic scheduling of data transportation. Functional behaviour is preserved. This dynamic scheduling is implemented using specific synchronous hardware elements: Relay-Stations (RS) and Shell-Wrappers (SW). Our first goal is to provide a formal modeling of RS and SW, that can then be formally verified. As turns out, resulting behaviour is k-periodic, thus amenable to static scheduling. Our second goal is to provide formal hardware modeling here also. It initially performs Throughput Equalization, adding integer latencies wherever possible; residual cases require introduction of Fractional Registers (FRs) at specific locations. Benchmark results are presented, run on our KPassa tool implementation

Selbstorganisierende Strukturen und Metriken komplexer Netzwerke

In nature, society and technology many disordered systems exist, that show emergent behaviour, where the interactions of numerous microscopic agents result in macroscopic, systemic properties, that may not be present on the microscopic scale. Examples include phase transitions in magnetism and percolation, for example in porous unordered media, biological, and social systems. Also technological systems that are explicitly designed to function without central control instances, like their prime example the Internet, or virtual networks, like the World Wide Web, which is defined by the hyperlinks from one web page to another, exhibit emergent properties. The study of the common network characteristics found in previously seemingly unrelated fields of science and the urge to explain their emergence, form a scientific field in its own right, the science of complex networks. In this field, methodologies from physics, leading to simplification and generalization by abstraction, help to shift the focus from the implementation's details on the microscopic level to the macroscopic, coarse grained system level. By describing the macroscopic properties that emerge from microscopic interactions, statistical physics, in particular stochastic and computational methods, has proven to be a valuable tool in the investigation of such systems. The mathematical framework for the description of networks is graph theory, in hindsight founded by Euler in 1736 and an active area of research since then. In recent years, applied graph theory flourished through the advent of large scale data sets, made accessible by the use of computers. A paradigm for microscopic interactions among entities that locally optimize their behaviour to increase their own benefit is game theory, the mathematical framework of decision finding. With first applications in economics e.g. Neumann (1944), game theory is an approved field of mathematics. However, game theoretic behaviour is also found in natural systems, e.g. populations of the bacterium Escherichia coli, as described by Kerr (2002). In the present work, a combination of graph theory and game theory is used to model the interactions of selfish agents that form networks. Following brief introductions to graph theory and game theory, the present work approaches the interplay of local self-organizing rules with network properties and topology from three perspectives. To investigate the dynamics of topology reshaping, coupling of the so called iterated prisoners' dilemma (IPD) to the network structure is proposed and studied in Chapter 4. In dependence of a free parameter in the payoff matrix, the reorganization dynamics result in various emergent network structures. The resulting topologies exhibit an increase in performance, measured by a variance of closeness, of a factor 1.2 to 1.9, depending in the chosen free parameter. Presented in Chapter 5, the second approach puts the focus on a static network structure and studies the cooperativity of the system, measured by the fixation probability. Heterogeneous strategies to distribute incentives for cooperation among the players are proposed. These strategies allow to enhance the cooperative behaviour, while requiring fewer total investments. Putting the emphasis on communication networks in Chapters 6 and 7, the third approach investigates the use of routing metrics to increase the performance of data packet transport networks. Algorithms for the iterative determination of such metrics are demonstrated and investigated. The most successful of these algorithms, the hybrid metric, is able to increase the throughput capacity of a network by a factor of 7. During the investigation of the iterative weight assignments a simple, static weight assignment, the so called logKiKj metric, is found. In contrast to the algorithmic metrics, it results in vanishing computational costs, yet it is able to increase the performance by a factor of 5.In Natur, Gesellschaft und Technik existiert eine Vielzahl ungeordneter Systeme, für die die Emergenz makroskopischer Eigenschaften aus mikroskopischen Wechselwirkungen charakteristisch sind. Beispiele für emergente Eigenschaften in physikalischen Systemen sind Phasenübergänge, wie sie etwa in der Perkolation auftreten. Weitere bedeutende Beispiele sind komplexe technologische Systeme, insbesondere solche, bei deren Entwicklung eine hohe Ausfallsicherheit ohne zentrale Kontrollinstanz eine wichtige Rolle spielt. Ein archetypisches Beispiel eines komplexen, selbstorganisierten Systems, gesteuert durch eigennützig handelnde Einheiten, sind Kommunikationsnetzwerke, insbesondere das Internet. Motiviert durch die immense Bedeutung solcher Kommunikationsnetze für die heutige moderne Gesellschaft untersucht die vorliegende Arbeit Wege zur Optimierung dieser Netze. Hier helfen Methoden der Physik, wie Generalisierung und Reduktion auf grundlegende Eigenschaften, die Aufmerksamkeit von implementationsspezifischen Details der mikroskopischen Dynamik auf makroskopische Folgen zu lenken. Insbesondere die Konzepte und Methoden der Statistischen Physik erweisen sich im Umgang mit komplexen Netzwerken als nützlich. Der mathematische Rahmen zur Beschreibung von Netzwerken ist die Graphentheorie, in deren Formalismus vernetzte Strukturen als Menge von Knoten dargestellt werden, welche durch Kanten miteinander verbunden sind. Ein mathematischer Formalismus zur Beschreibung von eigenständig handelnden Entitäten und deren Wechselwirkung ist die Spieltheorie. Diese beschreibt das Verhalten und die Entscheidungsfindung von Agenten bzw. Spielern, die eigenständig und eigennützig ihr Verhalten, charakterisiert durch ihre Strategie, optimieren. Die vorliegende Arbeit nutzt die Verknüpfung dieser beiden Formalismen um Interaktionen vernetzter eigenständiger Entitäten zu modellieren, und die daraus resultierenden emergenten Eigenschaften der Netzstruktur zu untersuchen. Das Konzept der Reorganisation von Netzwerken wird durch Kopplung der Netzstruktur an die Spieldynamik des Iterated Prisoners' Dilemma untersucht. Dies erlaubt eine Beeinflussung der Reorganisationsdynamik durch kontinuierliche Änderung der Spielparameter und ermöglicht damit eine Optimierung der Spieldynamik in Bezug auf Eigenschaften der emergenten Netzstruktur. Die vorgestellte Art der selbstorganisierenden Netzwerkoptimierung wird exemplarisch anhand einer Quantifizierung der Netzwerkperformanz demonstriert. In Abhängigkeit des gewählten Spielparameters wird im Vergleich zu Zufallsgraphen eine Erhöhung der Performanz um den Faktor 1.2 bis 1.9 erreicht. Eine weitere Herangehensweise zur Untersuchung von Spielen auf Netzwerken wird verfolgt, indem die Kooperativität von Prisoners' Dilemma Spielern auf dem Netz, quantifiziert durch die sogenannte Fixation Probability, untersucht wird. Hier werden Strategien zur Verteilung individueller Kooperationsanreize untersucht. Die vorgeschlagenen Strategien resultieren relativ zur globalen homogenen Verteilung derselben Summe der Anreize zu einer um den Faktor 5 erhöhten Kooperativität. An die spieltheoretischen Betrachtungen anschließend liegt das Hauptaugenmerk der folgenden Betrachtungen auf Kommunikationsnetzwerken. Zusätzlich zu durch vorangegangene Arbeiten vorgeschlagenen Metriken werden drei weitere Metriken eingeführt, von denen sich zwei, die Hybrid Metrik und die logKiKj Metrik, als äußerst erfolgreich im Sinne einer Optimierung der Throughput Capacity erweisen, was durch ausführliche numerische Simulationen belegt wird. Die Vorteile der hier eingeführten Metriken liegen im Fall der Hybrid Metrik in der unter den verglichenen Metriken besten resultierenden Performanz für Netze mit mehr als 3000 Knoten, mit einer mittleren Steigerung um den Faktor 7 im Vergleich zur Performanz ohne Metrik. Für Netze mit bis zu 3000 Knoten erreicht die sogenannte Extremal Metrik zwar eine leicht höhere Performanz, sie ist jedoch wegen ihrer sehr hohen numerischen Anforderungen für größere Netze nicht anwendbar. Im Falle der logKiKj Metrik ist die numerische Komplexität vernachlässigbar, dieser Vorteil an vermindertem numerischen Aufwand wird jedoch durch eine leichte Reduktion des Performanzgewinns erkauft, nichtsdestotrotz bewirkt auch diese Metrik eine mittlere Performanzsteigerung um den Faktor 5 und erreicht damit die Größenordnung der Hybrid Metrik

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

Flow-level performance analysis of data networks using processor sharing models

Most telecommunication systems are dynamic in nature. The state of the network changes constantly as new transmissions appear and depart. In order to capture the behavior of such systems and to realistically evaluate their performance, it is essential to use dynamic models in the analysis. In this thesis, we model and analyze networks carrying elastic data traffic at flow level using stochastic queueing systems. We develop performance analysis methodology, as well as model and analyze example systems. The exact analysis of stochastic models is difficult and usually becomes computationally intractable when the size of the network increases, and hence efficient approximative methods are needed. In this thesis, we use two performance approximation methods. Value extrapolation is a novel approximative method developed during this work and based on the theory of Markov decision processes. It can be used to approximate the performance measures of Markov processes. When applied to queueing systems, value extrapolation makes possible heavy state space truncation while providing accurate results without significant computational penalties. Balanced fairness is a capacity allocation scheme recently introduced by Bonald and Proutière that simplifies performance analysis and requires less restrictive assumptions about the traffic than other capacity allocation schemes. We introduce an approximation method based on balanced fairness and the Monte Carlo method for evaluating large sums that can be used to estimate the performance of systems of moderate size with low or medium loads. The performance analysis methods are applied in two settings: load balancing in fixed networks and the analysis of wireless networks. The aim of load balancing is to divide the traffic load efficiently between the network resources in order to improve the performance. On the basis of the insensitivity results of Bonald and Proutière, we study both packet- and flow-level balancing in fixed data networks. We also study load balancing between multiple parallel discriminatory processor sharing queues and compare different balancing policies. In the final part of the thesis, we analyze the performance of wireless networks carrying elastic data traffic. Wireless networks are gaining more and more popularity, as their advantages, such as easier deployment and mobility, outweigh their downsides. First, we discuss a simple cellular network with link adaptation consisting of two base stations and customers located on a line between them. We model the system and analyze the performance using different capacity allocation policies. Wireless multihop networks are analyzed using two different MAC schemes. On the basis of earlier work by Penttinen et al., we analyze the performance of networks using the STDMA MAC protocol. We also study multihop networks with random access, assuming that the transmission probabilities can be adapted upon flow arrivals and departures. We compare the throughput behavior of flow-optimized random access against the throughput obtained by optimal scheduling assuming balanced fairness capacity allocation

Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications

Coding; Communications; Engineering; Networks; Information Theory; Algorithm

A measurement-based approach to service modeling and bandwidth estimation in IEEE 802.11 wireless networks

[no abstract