Exit Probabilities of Constrained Simple Random Walks
Tarih:
Başlangıç Zamanı: Starting Time 01:00 pm ~ 02:00 pm
Konum: A 423
Abstract
Consider a nearest neighbor stable two dimensional random walk X constrained to remain on the positive orthant. X is assumed stable, i.e., its average increment points toward the origin. X represents the lengths of two queues (or two stacks in computer science applications) working in parallel. The probability pn that the sum of the components of this random walk reaches a high level n before the random walk returns to the origin is a natural performance measure, representing the probability of a buffer overflow in a busy cycle. The stability of the walk implies that pn decays exponentially in n. Let Y be the same constrained random walk as X, but constrained only on its second component and the jump probabilities on its first component reversed. The present study shows that one can approximate pn with the probability that components
of Y ever equal each other, with exponentially decaying relative error, if X starts from an initial point with nonzero first component. We further construct a class of Y-harmonic functions from single and conjugate points on a characteristic surface, with which the latter probability can be either computed perfectly in some cases, or approximated with bounded relative error in general.
Keywords: Approximation of probabilities of rare events, exit probabilities, constrained random walks, queueing systems, shared memory management, large deviations
Bio
Dr. Kamil Demirberk Ünlü was born in 1986. He graduated from TED Ankara College. He earned his B.S. degree in Business Administration from Cankaya University in 2009 with a double major degree in International Trade in 2010. He received his M.S. and Ph.D. degrees from the Department of Financial Mathematics, Institute of Applied Mathematics, Middle East Technical University in 2012 and 2018, respectively. His research interests include applied probability, data science, time series analysis and especially machine learning and deep learning applied to time series.