Fluid networks are useful tools for analyzing complex manufacturing environments especially in semiconductor wafer fabrication. The makespan of a fluid network is defined as the time to drain the system, when there is fluid present in the buffers initially. Based on this definition, the question of determining the allocation of resources so as to minimize the makespan of a fluid network is known as the makespan problem. In the deterministic version of the makespan problem, it is assumed that the parameters of the system, such as incoming rates, service rates and initial inventory, are known deterministically. The deterministic version of the makespan problem for reentrant lines and multiclass fluid networks has been investigated in the literature and an analytical solution for the problem is well-known. In this work, we provide another formulation for the deterministic makespan problem and prove that the problem can be solved for each station separately. Optimal solutions for the deterministic makespan problem have been used as a guide to develop heuristics methods to solve makespan scheduling problem in the job-shop context in the literature. This provides one motivation for further investigation of the fluid makespan problem. In this work our main focus is solving the makespan problem when the problem parameters are uncertain. This uncertainty may be caused by various factors such as the unpredictability of the arrival process or randomness in machine availability due to failures. In the presence of parameter uncertainty, the decision maker's goal is to optimally allocate the capacity in order to minimize the expected value of the makespan. We assume that the decision maker has distributional information about the parameters at the time of decision making. We consider two decision making schemes. In the first scheme, the controller sets the allocations before observing the parameters. After the initial allocations are set, they cannot be changed. In the second scheme, the controller is allowed a recourse action after a data collection process. It is shown that in terms of obtaining the optimal control, both schemes differ considerably from the deterministic version of the problem. We formulate both schemes using stochastic programming techniques. The first scheme is easier to analyze since the resulting model is convex. Unfortunately, under the second decision scheme, the objective function is non-convex. We develop a branch-and-bound methodology to solve the resulting stochastic non-convex program. Finally, we identify some special cases where the stochastic problem is analytically solvable. This work uses stochastic programming techniques to formulate and solve a problem arising in queueing networks. Stochastic programming and queueing systems are two major areas of Operations Research that deal with decision making under uncertainty. To the best of our knowledge, this dissertation is one of the first works that brings these two major areas together.

We consider a general stochastic differential delay equation (SDDE) with multiplicative colored noise. We study the limit as the time delays and the correlation times of the noises go to zero at the same rate. First, we derive the limiting equation for the equation obtained by Taylor expanding the SDDE to first order in the time delays. The limiting equation contains a noise-induced drift term that depends on the ratios of the time delays to the correlation times of the noises. We prove that, under appropriate assumptions, the solution of the equation obtained by the Taylor expansion converges to the solution of this limiting equation in probability with respect to the sup norm over compact time intervals. Next, we derive the limiting equation for the SDDE and prove a similar convergence result regarding convergence of the solution of the SDDE to the solution of this limiting equation. We see that the limiting equation corresponding to the equation obtained by the Taylor expansion is an approximation of the limiting equation corresponding to the SDDE. Finally, we study the effects of time delay on a particular model of active Brownian motion.

No description available.

Stochastic fluctuations can have profound impacts on engineered systems. Nonetheless, we can achieve significant benefits such as cost reduction based upon expanding our fundamental knowledge of stochastic systems. The primary goal of this thesis is to contribute to our understanding by developing and analyzing stochastic models for specific types of engineered systems. The knowledge gained can help management to optimize decision making under uncertainty. This thesis has three parts. In Part I, we study many-server queues that model large-scale service systems such as call centers. We focus on the positive recurrence of piecewise Ornstein-Uhlenbeck (OU) processes and the validity of using these processes to predict the steady-state performance of the corresponding many-server queues. In Part II, we investigate diffusion processes constrained to the positive orthant under infinitesimal changes in the drift. This sensitivity analysis on the drift helps us understand how changes in service capacities at individual stations in a stochastic network would affect the steady-state queue-length distributions. In Part III, we study the trading mechanism known as limit order book. We are motivated by a desire to better understand the interplay among order flow rates, liquidity fluctuation, and optimal executions. The goal is to characterize the temporal evolution of order book shape on the “macroscopic” time scale.

Response and stability of vessel rolling motion with strongly nonlinear softening stiffness will be studied in this dissertation using the methods of stochastic dynamical systems. As one of the most classic stability failure modes of vessel dynamics, large amplitude rolling motion in random beam waves has been studied in the past decades by many different research groups. Due to the strongly nonlinear softening stiffness and the stochastic excitation, there is still no general approach to predict the large amplitude rolling response and capsizing phenomena. We studied the rolling problem respectively using the shaping filter technique, stochastic averaging of the energy envelope and the stochastic Melnikov function. The shaping filter technique introduces some additional Gaussian filter variables to transform Gaussian white noise to colored noise in order to satisfy the Markov properties. In addition, we developed an automatic cumulant neglect tool to predict the response of the high dimensional dynamical system with higher order neglect. However, if the system has any jump phenomena, the cumulant neglect method may fail to predict the true response. The stochastic averaging of the energy envelope and the Melnikov function both have been applied to the rolling problem before, it is our first attempt to apply both approaches to the same vessel and compare their efficiency and capability. The inverse of the mean first passage time based on Markov theory and rate of phase space flux based on the stochastic Melnikov function are defined as two different, but analogous capsizing criteria. The effects of linear and nonlinear damping and wave characteristic frequency are studied to compare these two criteria. Further investigation of the relationship between the Markov and Melnikov based method is needed to explain the difference and similarity between the two capsizing criteria.

In this thesis we use techniques from white noise analysis to study solutions of semilinear stochastic differential equations in a Hilbert space H: {dX[subscript]t = (AX[subscript]t + F(t,X[subscript]t)) dt + ơ(t,X[subscript]t) δB[subscript]t, t∈ (0,T], X[subscript]0 = ξ, where A is a generator of either a C[subscript]0-semigroup or an n-times integrated semigroup, and B is a cylindrical Wiener process. We then consider applications to forward interest rate models, such as in the Heath-Jarrow-Morton framework. We also reformulate a phenomenological model of the forward rate. / Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Science, 2009

In this thesis we use techniques from white noise analysis to study solutions of semilinear stochastic differential equations in a Hilbert space H: {dX[subscript]t = (AX[subscript]t + F(t,X[subscript]t)) dt + ơ(t,X[subscript]t) δB[subscript]t, t∈ (0,T], X[subscript]0 = ξ, where A is a generator of either a C[subscript]0-semigroup or an n-times integrated semigroup, and B is a cylindrical Wiener process. We then consider applications to forward interest rate models, such as in the Heath-Jarrow-Morton framework. We also reformulate a phenomenological model of the forward rate. / Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Science, 2009

Diss. Uppsala : Univ., 2004.

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

Thesis (Ph. D.)--Northeastern University, 2008. / Title from title page (viewed March 25, 2009). Graduate School of Arts and Sciences, Dept. of Physics. Includes bibliographical references (p. 292-298).