Thesis (Ph. D.)--University of Wisconsin--Madison, 1991. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 351-357).

by Ma Yiguang. / "April 2000." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (p. 117-[124]). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Understand the behavior of queueing networks in heavy tra c is very important due to its importance in evaluating the network performance in related applications. However, in many cases, the stationary distributions of such networks are intractable. Based on di usion limits of queueing networks, we can use Re ected Brownian Motion (RBM) processes as reasonable approximations. As such, we are interested in obtaining the stationary distribution of RBM. Unfortunately, these distributions are also in most cases intractable. However, the tail behavior (large deviations) of RBM may give insight into the stationary distribution. Assuming that a large deviations principle holds, we need only solve the corresponding variational problem to obtain the rate function. Our research is mainly focused on how to solve variational problems in the case of rotationally symmetric (RS) data. The contribution of this dissertation primarily consists of three parts. In the rst part we give out the speci c stability condition for the RBM in the octant in the RS vi case. Although the general stability conditions for RBM in the octant has been derived previously, we simplify these conditions for the case we consider. In the second part we prove that there are only two types of possible solutions for the variational problem. In the last part, we provide a simple computational method. Also we give an example under which a spiral path is the optimal solution. / text

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Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

We explore the effect of multiscale structure on weakly interacting diffusions through two main projects. In the first, we consider a collection of weakly interacting diffusion processes moving in a two-scale locally periodic environment. We study the large deviations principle of the empirical distribution of the particles' positions in the combined limit as the number of particles grow to infinity and the time-scale separation parameter goes to zero simultaneously. We make use of weak convergence methods providing a convenient representation for the large deviations rate function, which allow us to characterize the effective controlled mean field dynamics. In addition, we obtain equivalent representations for the large deviations rate function of the form of Dawson-Gartner which hold even in the case where the diffusion matrix depends on the empirical measure and when the particles undergo averaging in addition to the propagation of chaos. In the second, we consider a fully-coupled slow-fast system of McKean-Vlasov SDEs with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy-Problem on the Wasserstein space that are of independent interest.

We study the totally asymmetric exclusion process on the positive integers with a single particle source at the origin. Liggett (1975) has shown that the long term behaviour of this process has a phase transition: If the particle production rate at the source and the initial density are below certain critical values, the stationary measure is a product measure, otherwise the stationary measure is spatially correlated. Following the approach of Derrida et al. (1993) it was shown by Grosskinsky (2004) that these correlations can be described by means of a matrix product representation. In this thesis we derive a large deviation principle with explicit rate function for the particle density in a macroscopic box based on this representation. The novel and rigorous technique we develop for this problem combines spectral theoretical and combinatorial ideas and has the potential to be applicable to other models described by matrix products.

Nesta dissertação de mestrado, vamos apresentar uma prova para os grandes desvios para variáveis aleatórias independentes e identicamente distribuídas com todos os momentos finitos e para a medida empírica de cadeias de Markov com espaço de estados finito e tempo discreto. Além disso, abordaremos os teoremas de Sanov e Gärtner-Ellis. / In this master thesis it is presented a proof of the large deviations for independent and identically distributed random variables with all finite moments and for the empirical measure of Markov chains with finite state space and with discrete time. Moreover, we address the theorems of Sanov and of Gartner-Ellis.

This thesis studies two topics in sequence analysis. In the first part, we investigate the large deviations of the shape of the random RSK Young diagrams, associated with a random word of size n whose letters are independently drawn from an alphabet of size m=m(n). When the letters are drawn uniformly and when both n and m converge together to infinity, m not growing too fast with respect to n, the large deviations of the shape of the Young diagrams are shown to be the same as that of the spectrum of the traceless GUE. Since the length of the top row of the Young diagrams is the length of the longest (weakly) increasing subsequence of the random word, the corresponding large deviations follow. When the letters are drawn with non-uniform probability, a control of both highest probabilities will ensure that the length of the top row of the diagrams satisfies a large deviation principle. In either case, both speeds and rate functions are identified. To complete our study, non-asymptotic concentration bounds for the length of the top row of the diagrams, are obtained for both models. In the second part, we investigate the order of the r-th, 1<= r < +∞, central moment of the length of the longest common subsequence of two independent random words of size n whose letters are identically distributed and independently drawn from a finite alphabet. When all but one of the letters are drawn with small probabilities, which depend on the size of the alphabet, the r-th central moment is shown to be of order n^{r/2}. In particular, when r=2, we get the order of the variance of the longest common subsequence.

The problem of designing scheduling algorithm for multi-channel (e.g., OFDM-based) wireless downlink networks is considered, where the system has a large bandwidth and proportionally large number of users to serve. For this system, while the classical MaxWeight algorithm is known to be throughput-optimal, its buffer-overflow performance is very poor (formally, it is shown that it has zero rate function in our setting). To address this, a class of algorithms called iHLQF (iterated Heaviest matching with Longest Queues First) is proposed. The algorithms in this class are shown to be throughput-optimal for a general class of arrival/channel processes, and also rate-function optimal (i.e., exponentially small buffer overflow probability) for certain arrival/channel processes, where the channel-rates are 0 or 1 packets per timeslot. iHLQF however has higher computational complexity than MaxWeight (n⁴ vs. n² computations per timeslot respectively). To overcome this issue, a new algorithm called SSG (Server-Side Greedy) is proposed. It is shown that SSG is throughput-optimal, results in a much better per-user buffer overflow performance than the MaxWeight algorithm (positive rate function for certain arrival/channel processes), and has a computational complexity (n²) that is comparable to the MaxWeight algorithm. Thus, it provides a nice trade-off between buffer-overflow performance and computational complexity. For multi-rate channel processes, where the channels can serve multiple packets per timeslot, new Markov chain-based coupling arguments are used to derive rate-function positivity results for the SSG algorithm. Finally, an algorithm called DMEQ is proposed and shown to be rate-function optimal for certain multi-rate channel scenarios, whose definition characterizes the sufficient conditions for rate-function optimality in this regime. These results are validated by both analysis and simulations. / text