abstract: This dissertation examines six different models in the field of econophysics using interacting particle systems as the basis of exploration. In each model examined, the underlying structure is a graph G = (V , E ), where each x ∈ V represents an individual who is characterized by the number of coins in her possession at time t. At each time step t, an edge (x, y) ∈ E is chosen at random, resulting in an exchange of coins between individuals x and y according to the rules of the model. Random variables ξt, and ξt(x) keep track of the current configuration and number of coins individual x has at time t respectively. Of particular interest is the distribution of coins in the long run. Considered first are the uniform reshuffling model, immediate exchange model and model with saving propensity. For each of these models, the number of coins an individual can have is nonnegative and the total number of coins in the system is conserved for all time. It is shown here that the distribution of coins converges to the exponential distribution, gamma distribution and a pseudo gamma distribution respectively. The next two models introduce debt, however, the total number of coins again remains fixed. It is shown here that when there is an individual debt limit, the number of coins per individual converges to a shifted exponential distribution. Alternatively, when a collective debt limit is imposed on the whole population, a heuristic argument is given supporting the conjecture that the distribution of coins converges to an asymmetric Laplace distribution. The final model considered focuses on the effect of cooperation on a population. Unlike the previous models discussed here, the total number of coins in the system at any given time is not bounded and the process evolves in continuous time rather than in discrete time. For this model, death of an individual will occur if they run out of coins. It is shown here that the survival probability for the population is impacted by the level of cooperation along with how productive the population is as whole. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

Abstract not available

In a multi-server system, probability distributions and loss probabilities for customers arriving with different priority categories are studied. Customers arrive in independent Poisson streams and their service times are exponentially distributed, with different rates for different priorities. The non-queuing customers will be lost if the capacity is fully occupied. In these systems, particularly for higher priority customers, the reduction of the loss probabilities is essential to guarantee the quality of the service. Four different policies for high and low priorities were introduced utilizing the fixed capacity of the system, producing different loss probabilities. The same policies were introduced in the case of a low priority being placed in the queue when the system is fully occupied. An application to the Intensive Care and Coronary Care Unit in Campbelltown Public Hospital in Sydney was introduced. This application analyses the admission and discharge by using queuing theory to develop a model which predicts the proportion of patients from each category that would be prematurely transferred as a function of the size of the unit, number of categories, mean arrival rates, and length of stay. / Master of Science (Hons)

In the last 30 to 40 years, many researchers have combined to build the knowledge base of theory and solution techniques that can be applied to the case of differential equations which include the effects of noise. This class of ``noisy'' differential equations is now known as stochastic differential equations (SDEs). Markov diffusion processes are included within the field of SDEs through the drift and diffusion components of the Itô form of an SDE. When these drift and diffusion components are moderately smooth functions, then the processes' transition probability densities satisfy the Fokker-Planck-Kolmogorov (FPK) equation -- an ordinary partial differential equation (PDE). Thus there is a mathematical inter-relationship that allows solutions of SDEs to be determined from the solution of a noise free differential equation which has been extensively studied since the 1920s. The main numerical solution technique employed to solve the FPK equation is the classical Finite Element Method (FEM). The FEM is of particular importance to engineers when used to solve FPK systems that describe noisy oscillators. The FEM is a powerful tool but is limited in that it is cumbersome when applied to multidimensional systems and can lead to large and complex matrix systems with their inherent solution and storage problems. I show in this thesis that the stochastic Taylor series (TS) based time discretisation approach to the solution of SDEs is an efficient and accurate technique that provides transition and steady state solutions to the associated FPK equation. The TS approach to the solution of SDEs has certain advantages over the classical techniques. These advantages include their ability to effectively tackle stiff systems, their simplicity of derivation and their ease of implementation and re-use. Unlike the FEM approach, which is difficult to apply in even only two dimensions, the simplicity of the TS approach is independant of the dimension of the system under investigation. Their main disadvantage, that of requiring a large number of simulations and the associated CPU requirements, is countered by their underlying structure which makes them perfectly suited for use on the now prevalent parallel or distributed processing systems. In summary, l will compare the TS solution of SDEs to the solution of the associated FPK equations using the classical FEM technique. One, two and three dimensional FPK systems that describe noisy oscillators have been chosen for the analysis. As higher dimensional FPK systems are rarely mentioned in the literature, the TS approach will be extended to essentially infinite dimensional systems through the solution of stochastic PDEs. In making these comparisons, the advantages of modern computing tools such as computer algebra systems and simulation software, when used as an adjunct to the solution of SDEs or their associated FPK equations, are demonstrated.

by A. S. Willsky, N. R. Sandell. / Grants AFOSR-77-3281B and ONR-N00014-76-C-0346. / Bibliography: leaf 34.

John N. Tsitsiklis. / "February, 1982." "OSP No. 87049." / Bibliography: p. 39-41. / National Science Foundation Grant DAR78-17826

David J. Sakrison. / "July 10, 1962." "This report is based on a thesis submitted to the Department of Electrical Engineering, M.I.T., May 13, 1961 ..." / Bibliography: p. 74. / Army Signal Corps Contract DA 36-039-sc-78108 Department of the Army Task 3-99-20-001 and Project 3-99-00-000. Army Signal Corps Contract DA-SIG-36-039-61-G14.

J.Y. Hayase. / "May 17, 1957." "This report is based on a thesis submitted to the Department of Electrical Engineering, M.I.T., May 20, 1957, in partial fulfillment of the requirements for the degree of Electrical Engineer." / Bibliography: p. 64-65. / U.S. Army Signal Corps Contract No. DA36-039-sc-64637 Dept. of the Army Task 3-99-06-108 Project 3-99-00-100

by Michael Athans and Sanjoy K. Mitter. / Includes bibliographical references. / Research supported by Air Force Office of Scientific Research (AFSC), Research Grant AFOSR 77-3281. Covers time period, March 1, 1977 to February 28, 1978.

M.H.A. Davis. / Bibliography: leaves 30-33. / "January, 1979." / U.S. Air Force Office of Sponsored Research Grant AFOSR 77-3281 Department of Energy Contract EX-76-A-01-2295