For a general attractive Probabilistic Cellular Automata on S-Zd, we prove that the (time-) convergence towards equilibrium of this Markovian parallel dynamics, exponentially fast in the uniform norm, is equivalent to a condition (A). This condition means the exponential decay of the influence from the boundary for the invariant measures of the system restricted to finite boxes. For a class of reversible PCA dynamics on {1,+1}(Zd), wit a naturally associated Gibbsian potential rho, we prove that a (spatial-) weak mixing condition (WM) for rho implies the validity of the assumption (A); thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to rho hods. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition.

Production lines with unreliable machines have received a great amount of attention in the literature. Especially, two-station systems have mostly been studied because such systems are easier to handle when compared to the longer lines. In literature, longer lines are usually evaluated by a decomposition algorithm, whereby the long line is partitioned into chunks of two-station lines. Decomposition algorithms require intermediate buffer storages of capacity at least two or three. The trends in modern manufacturing practices, on the other hand, such as the Toyota Production System, dictate that intermediate storages be eliminated. Our work studies multi-station lines with no intermediate storage. We develop software to automate the generation of transition probability matrices to allow the analysis of system behavior. The algorithm allows the use of software packages to handle computations and to solve for exact solutions. Long-run behavior is obtained via the algorithm developed in the computational environment MATLAB. The purpose is to analyze the system performance measures such as starvation and blockage times of stations, production rate and work-in-process. In addition, the production rate and the work-in-process measures over failure and repair probabilities are curve-fit to establish simple and useful empirical formulas for lines consisting three, four and five identical stations. Numerical analyses show that the proposed algorithm is effective for exact solutions and the suggested formulas are valid for approximate solutions.

<p>The theory of Gibbs measures belongs to the borderlandbetween statistical mechanics and probability theory. In thiscontext, the physical phenomenon of phase transitioncorresponds to the mathematical concept of non-uniqueness for acertain type of probability measures.</p><p>The most studied model in statistical mechanics is thecelebrated Ising model. The Potts model is a natural extensionof the Ising model, and the beach model, which appears in adifferent mathematical context, is in certain respectsanalogous to the Ising model. The two main parts of this thesisdeal with the Potts model and the beach model,respectively.</p><p>For the<i>q</i>-state Potts model on an infinite lattice, there are<i>q</i>+1 basic Gibbs measures: one wired-boundary measure foreach state and one free-boundary measure. For infinite trees,we construct "new" invariant Gibbs measures that are not convexcombinations of the basic measures above. To do this, we use anextended version of the random-cluster model together withcoupling techniques. Furthermore, we investigate the rootmagnetization as a function of the inverse temperature.Critical exponents to this function for different parametercombinations are computed.</p><p>The beach model, which was introduced by Burton and Steif,has many features in common with the Ising model. We generalizesome results for the Ising model to the beach model, such asthe connection between phase transition and a certain agreementpercolation event. We go on to study a<i>q</i>-state variant of the beach model. Using randomclustermodel methods again we obtain some results on where in theparameter space this model exhibits phase transition. Finallywe study the beach model on regular infinite trees as well.Critical values are estimated with iterative numerical methods.In different parameter regions we see indications of both firstand second order phase transition.</p><p><b>Keywords and phrases:</b>Potts model, beach model,percolation, randomcluster model, Gibbs measure, coupling,Markov chains on infinite trees, critical exponent.</p>

Managing interference is the main technical challenge in wireless networks. Multiple input multiple output (MIMO) methods are key components to overcome the interference bottleneck and deliver higher data rates. The most efficient MIMO techniques require channel state information (CSI). In practice, this information is inaccurate due to errors in CSI acquisition, as well as mobility and delay. CSI inaccuracy reduces the performance gains provided by MIMO. When compounded with uncoordinated intercell interference, the degradation in MIMO performance is accentuated. This dissertation investigates the impact of CSI inaccuracy on the performance of increasingly complex interference limited networks, starting with a single interferer scenario, continuing to a heterogeneous network with a femtocell overlay, and finishing with a clustered multicell coordination model for randomly deployed transmitting nodes. First, this dissertation analyzes limited feedback beamforming and precoded spatial multiplexing over temporally correlated channels. Assuming uncoordinated interference from one dominant interferer, using Markov chain convergence theory, the gain in the average successful throughput at the mobile user is shown to decrease exponentially with the feedback delay. The decay rate is amplified when the user is interference limited. Interference cancellation methods at the receiver are shown to mitigate the effect of interference. This work motivates the need for practical MIMO designs to overcome the adverse effects of interference. Second, limited feedback beamforming is analyzed on the downlink of a more realistic heterogeneous cellular network. Future generation cellular networks are expected to be heterogeneous, consisting of a mixture of macro base stations and low power nodes, to support the increasing user traffic capacity and reliability demand. Interference in heterogeneous environments cannot be coordinated using traditional interference mitigation techniques due to the on demand and random deployment of low power nodes such as femtocells. Using tools from stochastic geometry, the outage and average achievable rate of limited feedback MIMO is computed with same-tier and cross-tier interference, and feedback delay. A hybrid fixed and random network deployment model is used to analyze the performance in a fixed cell of interest. The maximum density of transmitting femtocells is derived as a function of the feedback rate and delay. The detrimental effect of same-tier interference is quantified, as the mobile user moves from the cell-center to the cell-edge. The third part of this dissertation considers limited coordination between randomly deployed transmitters. Building on the established degrading effect of uncoordinated interference on practical MIMO methods, and the analytical tractability of random deployment models, interference coordination is analyzed. Using multiple antennas at the transmitter for interference nulling in ad hoc networks is first shown to achieve MIMO gains using single antenna receivers. Clustered coordination is then investigated for cellular systems with randomly deployed base stations. As full coordination in the network is not feasible, a random clustering model is proposed where base stations located in the same cluster coordinate. The average achievable rate can be optimized as a function of the number of antennas to maximize the coordination gains. For multicell limited feedback, adaptive partitioning of feedback bits as a function of the signal and interference strength is proposed to minimize the loss in rate due to finite rate feedback. / text

A major current challenge consists in extending formal methods in order to handle infinite-state systems. Infiniteness stems from the fact that the system operates on unbounded data structure such as stacks, queues, clocks, integers; as well as parameterization. Systems with unbounded data structure are natural models for reasoning about communication protocols, concurrent programs, real-time systems, etc. While parameterized systems are more suitable if the system consists of an arbitrary number of identical processes which is the case for cache coherence protocols, distributed algorithms and so forth. In this thesis, we consider model checking problems for certain fundamental classes of probabilistic infinite-state systems, as well as the verification of safety properties in parameterized systems. First, we consider probabilistic systems with unbounded data structures. In particular, we study probabilistic extensions of Lossy Channel Systems (PLCS), Vector addition Systems with States (PVASS) and Noisy Turing Machine (PNTM). We show how we can describe the semantics of such models by infinite-state Markov chains; and then define certain abstract properties, which allow model checking several qualitative and quantitative problems. Then, we consider parameterized systems and provide a method which allows checking safety for several classes that differ in the topologies (linear or tree) and the semantics (atomic or non-atomic). The method is based on deriving an over-approximation which allows the use of a symbolic backward reachability scheme. For each class, the over-approximation we define guarantees monotonicity of the induced approximate transition system with respect to an appropriate order. This property is convenient in the sense that it preserves upward closedness when computing sets of predecessors.

A Markov chain is introduced to the major steps of the process of learning a subject matter by a group of students in the classroom, in order to obtain a mathematical representation of the above process. A classroom experiment for learning mathematics is also presented illustrating the applicability of our results in practice.

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Σκοπός της παρούσας διπλωματικής εργασίας είναι η μελέτη της θεωρίας της Near Complete Decomposability (NCD) και η εφαρμογή της στην ανάλυση της απόδοσης ενός υπολογιστικού συστήματος, του οποίου η μοντελοποίηση με παραδοσιακές τεχνικές οδηγεί σε απαγορευτικά μεγάλο χώρο κατάστασης. Αρχικά, παραθέτουμε τα βασικά σημεία της θεωρίας όπως αυτή θεμελιώνεται μαθηματικά από τον Courtois στην κλασική του μονογραφία (Courtois, 1977), ενώ στη συνέχεια προβαίνουμε στη μοντελοποίηση ενός υποθετικού σταθμού εργασίας κάποιου πολυεπεξεργαστικού συστήματος, στο οποίο εκτελούνται ανά πάσα στιγμή το πολύ Κ έργα. Ο σταθμός εργασίας που μελετάμε διαθέτει buffer πεπερασμένου μεγέθους και είναι επιφορτισμένος με τη συγκέντρωση και το συνδυασμό των επιμέρους υποέργων κάθε έργου και την αποθήκευση του στη μνήμη. Οι κλασικές τεχνικές μοντελοποίησης του buffer οδηγούν σε ένα μοντέλο με πολύ μεγάλο χώρο κατάστασης. Ωστόσο εμείς μοντελοποιούμε μία συναθροιστική εκδοχή του αρχικού μοντέλου, η οποία υπό αρκετά ρεαλιστικές συνθήκες χαίρει της NCD ιδιότητας. Την ιδιότητα αυτή του μοντέλου μας τη δικαιολογούμε τόσο διαισθητικά, όσο και μαθηματικά. Επίσης, επιβεβαιώνουμε πως το NCD μοντέλο πετυχαίνει υψηλής ποιότητας εκτίμηση των πιθανοτήτων μόνιμης κατάστασης και μίας σειράς άλλων χρήσιμων μετρικών, με σημαντικά μικρότερο υπολογιστικό κόστος σε σχέση με το αρχικό μοντέλο, εκτελώντας μία σειρά μετρήσεων στο περιβάλλον Matlab. Παράλληλα, η αξιοποίηση του NCD μοντέλου αυξάνει σημαντικά την ικανότητά μας να ερμηνεύσουμε τη δυναμική συμπεριφορά του συστήματος καθώς αυτό οδεύει προς μια κατάσταση στατιστικής ισορροπίας. Τέλος, επιχειρούμε μία σειρά από “educated guesses” για πιθανές κλάσεις συστημάτων τα οποία θα μπορούσαν να αναλυθούν με μεθοδολογία αντίστοιχη με αυτήν που ακολουθήσαμε εμείς στο παρόν κείμενο. / The purpose of this diploma dissertation is, on one hand the brief study of the theory of Near Complete Decomposability (NCD), and on the other hand the application of NCD in the analysis of a system, the modeling of which leads to a prohibitively large state space. First, we point out the fundamental mathematical principles of NCD as established by Courtois in his classic monograph (Courtois, 1977). Then, we proceed to the modeling of a hypothetical service station (R) of a multiprocessing computer system, which executes at most K jobs simultaneously. R has a finite buffer and its duty is to combine the arriving tasks into a single job and store it to memory. The usual modeling techniques applied to this “task buffer”, lead to a model with extremely large state space. So, we construct a lumped model instead, which enjoys the property of NCD. We prove this, using intuitive arguments as well as mathematical ones. Then, we confirm that the NCD model achieves a reliable estimation of the steady state probability vector and other important metrics, with significantly reduced computational complexity in comparison with the initial model. Furthermore, the exploitation of the NCD model increases significantly our ability to understand the dynamics of our system and to interpret aspects of its transient behavior towards statistical equilibrium. Finally, we make a number of “educated guesses” about possible classes of systems that could be analyzed using the same kind of techniques we used in this dissertation.

Autonomous vehicles is soon a reality in the every-day life. Though before it is used commercially the vehicles need to be proven safe. The current standard for functional safety on roads, ISO 26262, does not include autonomous vehicles at the moment, which is why in this project an approach using semi-Markov models is used to assess safety. A semi-Markov process is a stochastic process modelled by a state space model where the transitions between the states of the model can be arbitrarily distributed. The approach is realized as a MATLAB tool where the user can use a steady-state based analysis called a Loss and Risk based measure of safety to assess safety. The tool works and can assess safety of semi-Markov systems as long as they are irreducible and positive recurrent. For systems that fulfill these properties, it is possible to draw conclusions about the safety of the system through a risk analysis and also about which autonomous driving level the system is in through a sensitivity analysis. The developed tool, or the approach with the semi-Markov model, might be a good complement to ISO 26262.

Este trabalho apresenta um modelo analítico para a análise de vazão (throughput) da camada MAC do IEEE 802.11. A proposta se diferencia da maioria dos desenvolvimentos realizados na área por apresentar uma modelagem de ambientes com estações não-saturadas, levando em consideração o perfil do tráfego gerado pelas camadas superiores ao nível de enlace. Dessa maneira, o modelo permite uma análise do desempenho de aplicações reais sobre redes sem fio, a partir de resultados provenientes de um modelo matemático, enquanto diversos outros trabalhos avaliam somente situações de saturação das estações. Simulações utilizando Network Simulator 2 e comparações com resultados obtidos em trabalhos análogos desenvolvidos por outros pesquisadores validam o modelo proposto. Cenários com variações na quantidade de estações e carga oferecida à rede são avaliados utilizando o modelo teórico e comparados aos resultados simulados, provendo uma análise do comportamento dos limites de vazão do sistema. As estruturas do IEEE 802.11 MAC foram estudadas a fim de embasar o desenvolvimento do modelo proposto. Também foi realizada uma pesquisa a respeito de Cadeias de Markov, que serve como estrutura básica para a descrição do IEEE 802.11 DCF. / This work provides an analytical model to compute the IEEE 802.11 DCF throughput. The proposed analysis differs from most of the related work by presenting scenario modeling with unsaturated stations, accounting for the traffic characteristics generated by the upper layers to the data link layer. As a consequence, the proposed model allows for an accurate analysis of real applications over wireless networks, based on mathematical model results, while several other researches evaluate only saturate stations situations. Simulations ran using Network Simulator 2 and comparisons with results obtained on similar works developed by other researchers are used to validate the proposed model. Scenarios with different number of stations and offered loads are evaluated using the theoretical model and compared to the simulated results, providing an analysis on the behavior of the maximum throughput provided by the network. A study was made on the structures and mechanisms of IEEE 802.11 MAC in order to support the development of the proposed model. A research on Markov Chains was also done, since it is the basic structure used to describe the IEEE 802.11 DCF.