Stochastic approximation algorithms encompass a class of iterative schemes that converge to a sought value through a series of successive approximations. Such algorithms converge even when the observations are erroneous. Errors in observations may arise due to the stochastic nature of the problem at hand or due to extraneous noise. In other words, stochastic approximation algorithms are self-correcting schemes, in that the errors are wiped out in the limit and the algorithms still converge to the sought values. The rst stochastic approximation algorithm was developed by Robbins and Monro in 1951 to solve the root- nding problem. In 1977 Ljung showed that the asymptotic behavior of a stochastic approximation algorithm can be studied by associating a deterministic ODE, called the associated ODE, and studying it's asymptotic behavior instead. This is commonly referred to as the ODE method. In 1996 Bena•m and Bena•m and Hirsch [1] [2] used the dynamical systems approach in order to develop a framework to analyze generalized stochastic approximation algorithms, given by the following recursion: xn+1 = xn + a(n) [h(xn) + Mn+1] ; (1) where xn 2 Rd for all n; h : Rd ! Rd is Lipschitz continuous; fa(n)gn 0 is the given step-size sequence; fMn+1gn 0 is the Martingale difference noise. The assumptions of [1] later became the `standard assumptions for convergence'. One bottleneck in deploying this framework is the requirement on stability (almost sure boundedness) of the iterates. In 1999 Borkar and Meyn developed a unified set of assumptions that guaranteed both stability and convergence of stochastic approximations. However, the aforementioned frameworks did not account for scenarios with set-valued mean fields. In 2005 Bena•m, Hofbauer and Sorin [3] showed that the dynamical systems approach to stochastic approximations can be extended to scenarios with set-valued mean- fields. Again, stability of the fiterates was assumed. Note that stochastic approximation algorithms with set-valued mean- fields are also called stochastic recursive inclusions (SRIs). The Borkar-Meyn theorem for SRIs [10] As stated earlier, in many applications stability of the iterates is a hard assumption to verify. In Chapter 2 of the thesis, we present an extension of the original theorem of Borkar and Meyn to include SRIs. Specifically, we present two different (yet related) easily-verifiable sets of assumptions for both stability and convergence of SRIs. A SRI is given by the following recursion in Rd: xn+1 = xn + a(n) [yn + Mn+1] ; (2) where 8 n yn 2 H(xn) and H : Rd ! fsubsets of Rdg is a given Marchaud map. As a corollary to one of our main results, a natural generalization of the original Borkar and Meyn theorem is seen to follow. We also present two applications of our framework. First, we use our framework to provide a solution to the `approximate drift problem'. This problem can be stated as follows. When an experimenter runs a traditional stochastic approximation algorithm such as (1), the exact value of the drift h cannot be accurately calculated at every stage. In other words, the recursion run by the experimenter is given by (2), where yn is an approximation of h(xn) at stage n. A natural question arises: Do the errors due to approximations accumulate and wreak havoc with the long-term behavior (convergence) of the algorithm? Using our framework, we show the following: Suppose a stochastic approximation algorithm without errors can be guaranteed to be stable, then it's `approximate version' with errors is also stable, provided the errors are bounded at every stage. For the second application, we use our framework to relax the stability assumptions involved in the original Borkar-Meyn theorem, hence making the framework more applicable. It may be noted that the contents of Chapter 2 are based on [10]. Analysis of gradient descent methods with non-diminishing, bounded errors [9] Let us consider a continuously differentiable function f. Suppose we are interested in nding a minimizer of f, then a gradient descent (GD) scheme may be employed to nd a local minimum. Such a scheme is given by the following recursion in Rd: xn+1 = xn a(n)rf(xn): (3) GD is an important implementation tool for many machine learning algorithms, such as the backpropagation algorithm to train neural networks. For the sake of convenience, experimenters often employ gradient estimators such as Kiefer-Wolfowitz estimator, simultaneous perturbation stochastic approximation, etc. These estimators provide an estimate of the gradient rf(xn) at stage n. Since these estimators only provide an approximation of the true gradient, the experimenter is essentially running the recursion given by (2), where yn is a `gradient estimate' at stage n. Such gradient methods with errors have been previously studied by Bertsekas and Tsitsiklis [5]. However, the assumptions involved are rather restrictive and hard to verify. In particular, the gradient-errors are required to vanish asymptotically at a prescribed rate. This may not hold true in many scenarios. In Chapter 3 of the thesis, the results of [5] are extended to GD with bounded, non-diminishing errors, given by the following recursion in Rd: xn+1 = xn a(n) [rf(xn) + (n)] ; (4) where k (n)k for some fixed > 0. As stated earlier, previous literature required k (n)k ! 0, as n ! 1, at a `prescribed rate'. Sufficient conditions are presented for both stability and convergence of (4). In other words, the conditions presented in Chapter 3 ensure that the errors `do not accumulate' and wreak havoc with the stability or convergence of GD. Further, we show that (4) converges to a small neighborhood of the minimum set, which in turn depends on the error-bound . To the best of our knowledge this is the first time that GD with bounded non-diminishing errors has been analyzed. As an application, we use our framework to present a simplified implementation of simultaneous perturbation stochastic approximation (SPSA), a popular gradient descent method introduced by Spall [13]. Traditional convergence-analysis of SPSA involves assumptions that `couple' the `sensitivity parameters' of SPSA and the step-sizes. These assumptions restrict the choice of step-sizes available to the experimenter. In the context of machine learning, the learning rate may be adversely affected. We present an implementation of SPSA using `constant sensitivity parameters', thereby `decoupling' the step-sizes and sensitivity parameters. Further, we show that SPSA with constant sensitivity parameters can be analyzed using our framework. Finally, we present experimental results to support our theory. It may be noted that contents of Chapter 3 are based on [9]. b(n) a(n) Stochastic recursive inclusions with two timescales [12] There are many scenarios wherein the traditional single timescale framework cannot be used to analyze the algorithm at hand. Consider for example, the adaptive heuristic critic approach to reinforcement learning, which requires a stationary value iteration (for a fixed policy) to be executed between two policy iterations. To analyze such schemes Borkar [6] introduced the two timescale framework, along with a set of sufficient conditions which guarantee their convergence. Perkins and Leslie [8] extended the framework of Borkar to include set-valued mean- fields. However, the assumptions involved were still very restrictive and not easily verifiable. In Chapter 4 of the thesis, we present a generalization of the aforementioned frameworks. The framework presented is more general when compared to the frameworks of [6] and [8], and the assumptions involved are easily verifiable. A SRI with two timescales is given by the following coupled iteration: xn+1 = xn + a(n) un + Mn1+1 ; (5) yn+1 = yn + b(n) vn + Mn2+1 ; (6) where xn 2 R d and yn 2 R k for all n 0; un 2 h(xn; yn) and vn 2 g(xn; yn) for all n 0, where h : Rd Rk ! fsubsets of Rdg and g : Rd Rk ! fsubsets of Rkg are two given Marchaud maps; fa(n)gn 0 and fb(n)gn 0 are the step-size sequences satisfying ! 0 as n ! 1; fMn1+1gn 0 and fMn2+1 gn 0 constitute the Martingale noise terms. Our main contribution is in the weakening of the key assumption that `couples' the behavior of the x and y iterates. As an application of our framework we analyze the two timescale algorithm which solves the `constrained Lagrangian dual optimization problem'. The problem can be stated as thus: Given two functions f : Rd ! R and g : Rd ! Rk, we want to minimize f(x) subject to the condition that g(x) 0. This problem can be stated in the following primal form: inf sup f(x) + T g(x) : (7) 2R 2R0 x d k Under strong duality, solving the above equation is equivalent to solving it's dual: sup inf f(x) + T g(x) : (8) 2Rk x2Rd 0 The corresponding two timescale algorithm to solve the dual is given by: xn+1 = xn a(n) rx f(xn) + nT g(xn) + Mn2+1 ; (9) n+1 = n + b(n) f(xn) + nT g(xn) + Mn1+1 : r We use our framework to show that (9) converges to a solution of the dual given by (8). Further, as a consequence of our framework, the class of objective and constraint functions, for which (9) can be analyzed, is greatly enlarged. It may be noted that the contents of Chapter 4 are based on [12]. Stochastic approximation driven by `controlled Markov' process and temporal difference learning [11] In the field of reinforcement learning, one encounters stochastic approximation algorithms that are driven by Markov processes. The groundwork for analyzing the long-term behavior of such algorithms was laid by Benveniste et. al. [4]. Borkar [7] extended the results of [4] to include algorithms driven by `controlled Markov' processes i.e., algorithms where the `state process' was in turn driven by a time varying `control' process. Another important extension was that multiple stationary distributions were allowed, see [7] for details. The convergence analysis of [7] assumed that the iterates were stable. In reinforcement learning applications, stability is a hard assumption to verify. Hence, the stability assumption poses a bottleneck when deploying the aforementioned framework for the analysis of reinforcement algorithms. In Chapter 5 of the thesis we present sufficient conditions for both stability and convergence of stochastic approximations driven by `controlled Markov' processes. As an application of our framework, sufficient conditions for stability of temporal difference (T D) learning algorithm, an important policy-evaluation method, are presented that are compatible with existing conditions for convergence. The conditions are weakened two-fold in that (a) the Markov process is no longer required to evolve in a finite state space and (b) the state process is not required to be ergodic under a given stationary policy. It may be noted that the contents of Chapter 5 are based on [11].

Cette thèse traite de quatre problèmes dans le contexte des grands systèmes distribués. Ce travail est motivé par les questions soulevées par l'expansion du Cloud Computing et des technologies associées. Le présent travail étudie l'efficacité de différents algorithmes d'allocation de ressources dans ce cadre. Les méthodes utilisées impliquent une analyse mathématique de plusieurs modèles stochastiques associés à ces réseaux. Le chapitre 1 fournit une introduction au sujet, ainsi qu'une présentation des principaux outils mathématiques utilisés dans les chapitres suivants. Le chapitre 2 présente un mécanisme de contrôle de congestion dans les services de Video on Demand fournissant des fichiers encodés dans diverses résolutions. On propose une politique selon laquelle le serveur ne livre la vidéo qu'à un débit minimal lorsque le taux d'occupation du serveur est supérieur à un certain seuil. La performance du système dans le cadre de cette politique est ensuite évaluée en fonction des taux de rejet et de dégradation. Les chapitres 3, 4 et 5 explorent les problèmes liés aux schémas de coopération entre centres de données (CD) situés à la périphérie du réseau. Dans le premier cas, on analyse une politique dans le contexte des services de cloud multi-ressources. Dans le second cas, les demandes arrivant à un CD encombré sont transmises à un CD voisin avec une probabilité donnée. Au troisième, les requêtes bloquées dans un CD sont transmises systématiquement à une autre où une politique de réservation (trunk) est introduite tel qu'une requête redirigée est acceptée seulement s'il y a un certain nombre minimum de serveurs libres dans ce CD. / This PhD thesis investigates four problems in the context of Large Distributed Systems. This work is motivated by the questions arising with the expansion of Cloud Computing and related technologies. The present work investigates the efficiency of different resource allocation algorithms in this framework. The methods used involve a mathematical analysis of several stochastic models associated to these networks. Chapter 1 provides an introduction to the subject in general, as well as a presentation of the main mathematical tools used throughout the subsequent chapters. Chapter 2 presents a congestion control mechanism in Video on Demand services delivering files encoded in various resolutions. We propose a policy under which the server delivers the video only at minimal bit rate when the occupancy rate of the server is above a certain threshold. The performance of the system under this policy is then evaluated based on both the rejection and degradation rates. Chapters 3, 4 and 5 explore problems related to cooperation schemes between data centres on the edge of the network. In the first setting, we analyse a policy in the context of multi-resource cloud services. In second case, requests that arrive at a congested data centre are forwarded to a neighbouring data centre with some given probability. In the third case, requests blocked at one data centre are forwarded systematically to another where a trunk reservation policy is introduced such that a redirected request is accepted only if there are a certain minimum number of free servers at this data centre.

L'objet de cette thèse est l'étude de certaines propriétés analytiques et asymptotiques des processus de Markov, et de leurs applications à la méthode de Stein. Le point de vue considéré consiste à déployer des inégalités fonctionnelles pour majorer la distance entre lois de probabilité. La première partie porte sur l'étude asymptotique de processus de Markov inhomogènes en temps via des inégalités de type Poincaré, établies par l'analyse spectrale fine de l'opérateur de transition. On se place d'abord dans le cadre du théorème central limite, qui affirme que la somme renormalisée de variables aléatoires converge vers la mesure gaussienne, et l'étude est consacrée à l'obtention d'une borne à la Berry-Esseen permettant de quantifier cette convergence. La distance choisie est une quantité naturelle et encore non étudiée dans ce cadre, la distance du chi-2, complétant ainsi la littérature relative à d'autres distances (Kolmogorov, variation totale, Wasserstein). Toujours dans le contexte non-homogène, on s'intéresse ensuite à un processus peu mélangeant relié à un algorithme stochastique de recherche de médiane. Ce processus évolue par sauts de deux types (droite ou gauche), dont la taille et l'intensité dépendent du temps. Une majoration de la distance de Wasserstein d'ordre 1 entre la loi du processus et la mesure gaussienne est établie dans le cas où celle-ci est invariante sous la dynamique considérée, et étendue à des exemples où seule la normalité asymptotique est vérifiée. La seconde partie s'attache à l'étude des entrelacements entre processus de Markov (homogènes) et gradients, qu'on peut interpréter comme un raffinement du critère de Bakry-Emery, et leur application à la méthode de Stein, qui est un ensemble de techniques permettant de majorer la distance entre deux mesures de probabilité. On prouve l'existence de relations d'entrelacement du second ordre pour les processus de naissance-mort, allant ainsi plus loin que les relations du premier ordre connues. Ces relations sont mises à profit pour construire une méthode originale et universelle d'évaluation des facteurs de Stein relatifs aux mesures de probabilité discrètes, qui forment une composante essentielle de la méthode de Stein-Chen. / The object of this thesis is the study of some analytical and asymptotic properties of Markov processes, and their applications to Stein's method. The point of view consists in the development of functional inequalities in order to obtain upper-bounds on the distance between probability distributions. The first part is devoted to the asymptotic study of time-inhomogeneous Markov processes through Poincaré-like inequalities, established by precise estimates on the spectrum of the transition operator. The first investigation takes place within the framework of the Central Limit Theorem, which states the convergence of the renormalized sum of random variables towards the normal distribution. It results in the statement of a Berry-Esseen bound allowing to quantify this convergence with respect to the chi-2 distance, a natural quantity which had not been investigated in this setting. It therefore extends similar results relative to other distances (Kolmogorov, total variation, Wasserstein). Keeping with the non-homogeneous framework, we consider a weakly mixing process linked to a stochastic algorithm for median approximation. This process evolves by jumps of two sorts (to the right or to the left) with time-dependent size and intensity. An upper-bound on the Wasserstein distance of order 1 between the marginal distribution of the process and the normal distribution is provided when the latter is invariant under the dynamic, and extended to examples where only the asymptotic normality stands. The second part concerns intertwining relations between (homogeneous) Markov processes and gradients, which can be seen as refinment of the Bakry-Emery criterion, and their application to Stein's method, a collection of techniques to estimate the distance between two probability distributions. Second order intertwinings for birth-death processes are stated, going one step further than the existing first order relations. These relations are then exploited to construct an original and universal method of evaluation of discrete Stein's factors, a key component of Stein-Chen's method.

Nessa tese, estudamos o limite hidrodinâmico de um sistema estocástico de neurônios cujas interações são dadas por potenciais de Kac que imitam sinapses elétricas e químicas, e as correntes de vazamento. Esse sistema consiste de $\\ep^$ neurônios imersos em $[0,1)^2$, cada um disparando aleatoriamente de acordo com um processo pontual com taxa que depende tanto do seu potential de membrana como da posição. Quando o neurônio $i$ dispara, seu potential de membrana é resetado para $0$, enquanto que o potencial de membrana do neurônio $j$ é aumentado por um valor positivo $\\ep^2 a(i,j)$, se $i$ influencia $j$. Além disso, entre disparos consecutivos, o sistema segue uma movimento determinístico devido às sinapses elétricas e às correntes de vazamento. As sinapses elétricas estão envolvidas na sincronização do potencial de membrana dos neurônios, enquanto que as correntes de vazamento inibem a atividade de todos os neurônios, atraindo simultaneamente todos os potenciais de membrana para $0$. No principal resultado dessa tese, mostramos que a distribuição empírica dos potenciais de membrana converge, quando o parâmetro $\\ep$ tende à 0 , para uma densidade de probabilidade $ho_t(u,r)$ que satisfaz uma equação diferencial parcial nâo linear do tipo hiperbólica . / We study the hydrodynamic limit of a stochastic system of neurons whose interactions are given by Kac Potentials that mimic chemical and electrical synapses and leak currents. The system consists of $\\ep^$ neurons embedded in $[0,1)^2$, each spiking randomly according to a point process with rate depending on both its membrane potential and position. When neuron $i$ spikes, its membrane potential is reset to $0$ while the membrane potential of $j$ is increased by a positive value $\\ep^2 a(i,j)$, if $i$ influences $j$. Furthermore, between consecutive spikes, the system follows a deterministic motion due both to electrical synapses and leak currents. The electrical synapses are involved in the synchronization of the membrane potentials of the neurons, while the leak currents inhibit the activity of all neurons, attracting simultaneously their membrane potentials to 0. We show that the empirical distribution of the membrane potentials converges, as $\\ep$ vanishes, to a probability density $ho_t(u,r)$ which is proved to obey a nonlinear PDE of Hyperbolic type.

In a wireless communication system the transmitted information is subjected to a number of impairments, among which inter-symbol interference (ISI), thermal noise and fading are the most prevalent. Owing to the dispersive nature of the communication channel, ISI results from the arrival of multiple delayed copies of the transmitted signal at the receiver. Thermal noise is caused by the random fluctuation on electrons in the receiver hardware, while fading is the result of constructive and destructive interference, as well as absorption during transmission. To protect the source information, error-correction coding (ECC) is performed in the transmitter, after which the coded information is interleaved in order to separate the information to be transmitted temporally. Turbo equalization (TE) is a technique whereby equalization (to correct ISI) and decoding (to correct errors) are iteratively performed by iteratively exchanging extrinsic information formed by optimal posterior probabilistic information produced by each algorithm. The extrinsic information determined from the decoder output is used as prior information by the equalizer, and vice versa, allowing for the bit-error rate (BER) performance to be improved with each iteration. Turbo equalization achieves excellent BER performance, but its computational complexity grows exponentially with an increase in channel memory as well as with encoder memory, and can therefore not be used in dispersive channels where the channel memory is large. A number of low complexity equalizers have consequently been developed to replace the maximum a posteriori probability (MAP) equalizer in order to reduce the complexity. Some of the resulting low complexity turbo equalizers achieve performance comparable to that of a conventional turbo equalizer that uses a MAP equalizer. In other cases the low complexity turbo equalizers perform much worse than the corresponding conventional turbo equalizer (CTE) because of suboptimal equalization and the inability of the low complexity equalizers to utilize the extrinsic information effectively as prior information. In this thesis the author develops two novel iterative low complexity turbo equalizers. The turbo equalization problem is modeled on superstructures, where, in the context of this thesis, a superstructure performs the task of the equalizer and the decoder. The resulting low complexity turbo equalizers process all the available information as a whole, so there is no exchange of extrinsic information between different subunits. The first is modeled on a dynamic Bayesian network (DBN) modeling the Turbo Equalization problem as a quasi-directed acyclic graph, by allowing a dominant connection between the observed variables and their corresponding hidden variables, as well as weak connections between the observed variables and past and future hidden variables. The resulting turbo equalizer is named the dynamic Bayesian network turbo equalizer (DBN-TE). The second low complexity turbo equalizer developed in this thesis is modeled on a Hopfield neural network, and is named the Hopfield neural network turbo equalizer (HNN-TE). The HNN-TE is an amalgamation of the HNN maximum likelihood sequence estimation (MLSE) equalizer, developed previously by this author, and an HNN MLSE decoder derived from a single codeword HNN decoder. Both the low complexity turbo equalizers developed in this thesis are able to jointly and iteratively equalize and decode coded, randomly interleaved information transmitted through highly dispersive multipath channels. The performance of both these low complexity turbo equalizers is comparable to that of the conventional turbo equalizer while their computational complexities are superior for channels with long memory. Their performance is also comparable to that of other low complexity turbo equalizers, but their computational complexities are worse. The computational complexity of both the DBN-TE and the HNN-TE is approximately quadratic at best (and cubic at worst) in the transmitted data block length, exponential in the encoder constraint length and approximately independent of the channel memory length. The approximate quadratic complexity of both the DBN-TE and the HNN-TE is mostly due to interleaver mitigation, requiring matrix multiplication, where the matrices have dimensions equal to the data block length, without which turbo equalization using superstructures is impossible for systems employing random interleavers. / Thesis (PhD)--University of Pretoria, 2013. / gm2013 / Electrical, Electronic and Computer Engineering / unrestricted

On s’intéresse au problème de contrôle impulsionnel à horizon infini avec facteur d’oubli pour les processus de Markov déterministes par morceaux (PDMP). Dans un premier temps, on modélise l’évolution d’un système opto-électronique par des PDMP. Afin d’optimiser la maintenance du système, on met en place un problème de contrôle impulsionnel tenant compte à la fois du coût de maintenance et du coût lié à l’indisponibilité du matériel auprès du client.On applique ensuite une méthode d’approximation numérique de la fonction valeur associée au problème, faisant intervenir la quantification de PDMP. On discute alors de l’influence des paramètres sur le résultat obtenu. Dans un second temps, on prolonge l’étude théorique du problème de contrôle impulsionnel en construisant de manière explicite une famille de stratégies є-optimales. Cette construction se base sur l’itération d’un opérateur dit de simple-saut-ou-intervention associé au PDMP, dont l’idée repose sur le procédé utilisé par U.S. Gugerli pour la construction de temps d’arrêt є-optimaux. Néanmoins, déterminer la meilleure position après chaque intervention complique significativement la construction de telles stratégies et nécessite l’introduction d’un nouvel opérateur. L’originalité de la construction de stratégies є-optimales présentée ici est d’être explicite, au sens où elle ne nécessite pas la résolution préalable de problèmes complexes. / We are interested in a discounted impulse control problem with infinite horizon forpiecewise deterministic Markov processes (PDMPs). In the first part, we model the evolutionof an optronic system by PDMPs. To optimize the maintenance of this equipment, we study animpulse control problem where both maintenance costs and the unavailability cost for the clientare considered. We next apply a numerical method for the approximation of the value function associated with the impulse control problem, which relies on quantization of PDMPs. The influence of the parameters on the numerical results is discussed. In the second part, we extendthe theoretical study of the impulse control problem by explicitly building a family of є-optimalstrategies. This approach is based on the iteration of a single-jump-or-intervention operator associatedto the PDMP and relies on the theory for optimal stopping of a piecewise-deterministic Markov process by U.S. Gugerli. In the present situation, the main difficulty consists in approximating the best position after the interventions, which is done by introducing a new operator.The originality of the proposed approach is the construction of є-optimal strategies that areexplicit, since they do not require preliminary resolutions of complex problems.

L’objectif de cette thèse est de construire un framework qui représente les incertitudes aléatoires et épistémiques basé sur les approches probabilistes et des théories d’incertain, de comparer les méthodes et de trouver les propres applications sur les grands systèmes avec événement rares. Dans la thèse, une méthode de normalité asymptotique a été proposée avec simulation de Monte Carlo dans les cas binaires ainsi qu'un modèle semi-Markovien dans les cas de systèmes multi-états dynamiques. On a aussi appliqué la théorie d’ensemble aléatoire comme un modèle de base afin d’évaluer la fiabilité et les autres indicateurs de performance dans les systèmes binaires et multi-états avec technique bootstrap. / Our research objective is to build frameworks representing both aleatory and epistemic uncertainties based on probabilistic approach and uncertainty approaches and to compare these methods and find the proper applicatin for these methods in large scale systems with rare event. In this thesis, an asymptotic normality method is proposed with Monte Carlo simulation in case of binary systems as well as semi-Markov model for cases of dynamic multistate system. We also apply random set as a basic model to evaluate system reliability and other performance indices on binary and multistate systems with bootstrap technique.

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Kollektive Migration und Schwarmverhalten sind Beispiele für Selbstorganisation und können in verschiedenen biologischen Systemen beobachtet werden, beispielsweise in Vogel-und Fischschwärmen oder Bakterienpopulationen. Im Zentrum dieser Arbeit steht ein räumlich diskretes und zeitlich stetiges Model, welches das kollektive Migrieren von Individuen mittels eines stochastischen Vielteilchensystems (VTS) beschreibt und analysierbar macht. Das konstruierte Modell ist in keiner Klasse gut untersuchter Vielteilchensysteme enthalten, sodass der größte Teil der Arbeit der Entwicklung von Methoden zur Untersuchung des Langzeitverhaltens bestimmter VTS gewidmet ist. Eine entscheidende Rolle spielen hier Gibbs-Maße, die zu zeitlich invarianten Maßen in Beziehung gesetzt werden. Durch eine Simulationsstudie und die Analyse des Einflusses der Parameter Migrationsgeschwindigkeit, Sensitivität der Individuen und (räumliche) Dichte der Anfangsverteilung können Eigenschaften kollektiver Migration erklärt und Hypothesen für weitere Analysen aufgestellt werden. / Collective migration and swarming behavior are examples of self-organization and can be observed in various biological systems, such as in flocks of birds, schools of fish or populations of bacteria. In the center of this thesis lies a stochastic interacting particle system (IPS), which is a spatially discrete model with a continuous time scale that describes collective migration and which can be treated using analytical methods. The constructed model is not contained in any class of well-understood IPS’s. The largest part of this work is used to develop methods that can be used to study the long-term behavior of certain IPS’s. Thereby Gibbs-Measures play an important role and are related to temporally invariant measures. One can explain the properties of collective migration and propose a hypothesis for further analyses by a simulation study and by analysing the parameters migration velocity, sensitivity of individuals and (spatial) density of the initial distribution.