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Motivated by applications in multimedia streaming and in energy systems, we study duality relations in fi nite queues. Dual of a queue is de fined to be a queue in which the arrival and service processes are interchanged. In other words, dual of the G1/G2/1/K queue is the G2/G1/1/K queue, a queue in which the inter-arrival times have the same distribution as the service times of the primal queue and vice versa. Similarly, dual of a fluid flow queue with cumulative input C(t) and available processing S(t) is a fluid queue with cumulative input S(t) and available processing C(t). We are primarily interested in finding relations between the overflow and underflow of the primal and dual queues. Then, using existing results in the literature regarding the probability of loss and the stationary probability of queue being full, we can obtain estimates on the probability of starvation and the probability of the queue being empty. The probability of starvation corresponds to the probability that a queue becomes empty, i.e., the end of a busy period. We study the relations between arrival and departure Palm distributions and their relations to stationary distributions. We consider both the case of point process inputs as well as fluid inputs. We obtain inequalities between the probability of the queue being empty and the probability of the queue being full for both the time stationary and Palm distributions by interchanging arrival and service processes. In the fluid queue case, we show that there is an equality between arrival and departure distributions that leads to an equality between the probability of starvation in the primal queue and the probability of overflow in the dual queue. The techniques are based on monotonicity arguments and coupling. The usefulness of the bounds is illustrated via numerical results.

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Thesis (M.S.)--Oregon State University, 2011. / Printout. Includes bibliographical references (leaves 70-73). Also available on the World Wide Web.

Mean-field games (MFG) are models of large populations of rational agents who seek to optimize an objective function that takes into account their state variables and the distribution of the state variable of the remaining agents. MFG with congestion model problems where the agents’ motion is hampered in high-density regions. First, we study radial solutions for first- and second-order stationary MFG with congestion on Rd. The radial case, which is one of the simplest non one-dimensional MFG, is relatively tractable. As we observe, the Fokker-Planck equation is integrable with respect to one of the unknowns. Consequently, we obtain a single equation substituting this solution into the Hamilton-Jacobi equation. For the first-order case, we derive explicit formulas; for the elliptic case, we study a variational formulation of the resulting equation. For the first case, we use our approach to compute numerical approximations to the solutions of the corresponding MFG systems. Next, we consider second-order stationary MFG with congestion and prove the existence of stationary solutions. Because moving in congested areas is difficult, agents prefer to move in non-congested areas. As a consequence, the model becomes singular near the zero density. The existence of stationary solutions was previously obtained for MFG with quadratic Hamiltonians thanks to a very particular identity. Here, we develop robust estimates that give the existence of a solution for general subquadratic Hamiltonians. Additionally, we study first-order stationary MFG with congestion with quadratic or power-like Hamiltonians. Using explicit examples, we illustrate two key difficulties: the lack of classical solutions and the existence of areas with vanishing densities. Our main contribution is a new variational formulation for MFG with congestion. With this formulation, we prove the existence and uniqueness of solutions. Finally, we devise a discretization that is combined with optimization algorithms to numerically solve various MFG with congestion.

We consider the two-dimensional Navier-Stokes equation on the (possibly) asymmetric torus, D_δ = [0,2𝜋δ] × [0,2𝜋], both with and without stochastic forcing. Absent external force, the vorticity is known to reach a rest state of zero. There exists at least three so called "quasi-stationary states" which attract nearby solutions at rates faster than the global decay rate. The system evolves toward one of these three qualitatively different transient states for long times while the system overall tends toward the final rest state. We develop a finite-dimensional model of the associated deterministic vorticity equation to show how the selection of the dominant quasi-stationary state depends on the aspect ratio of the domain, given by δ. This is followed by formal analysis of the problem as a perturbation from the symmetric domain. Once the selection mechanism for the deterministic model is characterized, stochastic forcing is added to the reduced system. Numerical analysis shows the dominant quasi-stationary state is consistent with what is seen in the deterministic setting. Finally through multiscale averaging methods, the leading order dynamics of the stochastically forced finite-dimensional model for δ close to one is studied. As a result we formally obtain leading order asymptotics of statistics of interest, including the selection mechanism.

The aim of this study is a preparation of polyaniline stationary phases doped with silver for application in HPLC. Various polyaniline coated stationary phases differing in the addition of AgNO3 were polymerized. Some of them were subjected to Ag sedimentation, in others AgNO3 was added after polymerization. Stationary phases were investigated by scanning electron microscopy, atomic absorption spectroscopy, Raman spectroscopy, and Fourier transformation infrared spectroscopy. Columns filled with prepared stationary phases were compared with silver-free polyaniline column by separating a mixture consisting of caffeine, theobromine and theophylline in three chromatographic modes (NP-pure ACN, HILIC-98/2 (v/v) ACN/water and RP-20/80 (v/v) ACN/water) at a flow rate of 5 µL/min with UV detection at 265 nm and also by separating a mixture of 2'-aminoacetophenone, 3'-aminoacetophenone and 4'-aminoacetophenone in the same chromatographic modes, at a flow rate of 5 µL/min except from RP mode where a flow rate of 10 µL/min was used to accelerate separation. Polyaniline-coated columns doped with silver showed different selectivity in the RP mode of the mixture of caffeine, theobromine and theophylline when compared to polyaniline-coated columns without the addition of silver. Further, the columns were tested for...

Submitted master thesis is by its character focused on development of polyaniline which would possess attributes typical for monolithic stationary phase. Developed polyaniline aims to be used in capillary liquid chromatography. Accompanying target of submitted master thesis is to provide solutions for variety of complications which follow the preparation of polyaniline in its desired form such as consistency in a form of solutions, gels and pastes, extremely fast polymerization reaction, limited solubility of cross-linking agents, solubility of monolithic polyaniline in polar mobile phases and insufficient bond of polyaniline to the inner wall of fused silica capillary tubing. Solutions for this complications are selection of ideal oxidation agent (ammonium persulfate), cross-linking agent (tris(4-aminophenyl)amine) and porogenic agent (methanol). Also, setting ideal ratio between oxidation agent, cross-linking agent and porogenic agent, heat treatment, choice of suitable mobile phase (tetrahydrofuran), adjustment of capillary wall using silanization agent (3-[3-(trimethoxysilyl)propyl]-aniline) and construction of equipment needed for quick filling of capillaries. Described development concluded in the preparation of solid polyaniline monolith which had a steady bond to the inner wall of fused...

Due essentially to the difficulties associated with obtaining explicit forms of stationary marginal distributions of non-linear stationary processes, appropriate characterizations of such processes are worked upon little. After discussing an elaborate motivation behind this thesis and presenting preliminaries in Chapter 1, we characterize, in Chapter 2, the stationary marginal distributions of certain non-linear multivariate stationary processes. To do so, we show that the stationary marginal distributions of these processes belong to specific skew-distribution families, and for a given skew-distribution from the corresponding family, a process, with stationary marginal distribution identical to that given skew-distribution, can be found. While conventional time series analysis greatly depends on the assumption of stationarity, measurements taken from many physical systems, which consist of both periodicity and randomness, often exhibit cyclostationarity (i.e., a periodic structure in their first- and second-order moments). Identifying the hourly global horizontal irradiances (GHIs), collected at a solar monitoring station of Saudi Arabia, as a cyclostationary process and considering the significant impact of that on the energy production in Saudi Arabia, Chapter 3 provides a temporal model of GHIs. Chapter 4 extends the analysis to a spatio-temporal cyclostationary modeling of 45 different solar monitoring stations of the Kingdom. Both the proposed models are shown to produce better forecasts, more realistic simulations, and reliable photovoltaic power estimates in comparison to a classical model that fails to recognize the GHI data as cyclostationary. Chapter 5 extends the notion of cyclostationarity to a novel and flexible class of processes, coined evolving period and amplitude cyclostationary (EPACS) processes, that allows periods and amplitudes of the mean and covariance functions to evolve and, therefore, accommodates a much larger class of processes than the cyclostationary processes. Thereafter, we investigate its properties, provide methodologies for statistical inference, and illustrate the presented methods using a simulation study and a real data example, from the heavens, of the magnitudes of the light emitted from the variable star R Hydrae. Finally, Chapter 6 summarizes the findings of the thesis and discusses its significance and possible future extensions.

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