Exit probabilities of constrained simple random walks

Download

index.pdf

Date

2018

Author

Ünlü, Kamil Demirberk

Metadata

Show full item record

Item Usage Stats

198
views

129
downloads

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 thesis 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. We provide numerical examples showing the effectiveness of the computed approximations and indicate possible applications of our results in finance and insurance.

Subject Keywords

Random walks (Mathematics)., Queuing theory., Risk (Insurance)., Financial risk.

URI

http://etd.lib.metu.edu.tr/upload/12622554/index.pdf
https://hdl.handle.net/11511/27774

Collections

Graduate School of Applied Mathematics, Thesis

Suggestions

OpenMETU
Core

APPROXIMATION OF EXCESSIVE BACKLOG PROBABILITIES OF TWO TANDEM QUEUES Sezer, Ali Devin (2018-09-01) Let X be the constrained random walk on Z(+)(2) having increments (1, 0), (-1, 1), and (0, -1) with respective probabilities A lambda,mu 1, and mu 2 representing the lengths of two tandem queues. We assume that X is stable and mu 1 not equal mu 2. Let tau(n) be the first time when the sum of the components of X equals n. Let Y be the constrained random walk on Z x Z(+) having increments (-1, 0), (1, 1), and (0, -1) with probabilities lambda, mu(1), and mu(2). Let tau be the first time that the components of...
Exit probabilities of markov modulated constrained random walks Başoğlu Kabran, Fatma; Sezer, Ali Devin; Department of Financial Mathematics (2018) Let X be the constrained random walk on Z2+ with increments (0, 0), (1, 0), (−1, 1), (0, −1) whose jump probabilities are determined by the state of a finite state Markov chain M. X represents the lengths of two queues of customers (or packets, tasks, etc.) waiting for service from two servers working in tandem; the arrival of customers occur with rate λ(Mk), service takes place at rates μ1(Mk), and μ2(Mk) where Mk denotes the current state of the Markov chain M. We assume that the average arrival rate is l...
REGULARITY OF QUOTIENTS OF DRINFELD MODULAR SCHEMES Kondo, Satoshi; Yasuda, Seidai (Mathematical Sciences Publishers, 2020-02-01) Let A be the coordinate ring of a projective smooth curve over a finite field minus a closed point. For a nontrivial ideal I subset of A, Drinfeld defined the notion of structure of level I on a Drinfeld module.
Bound states of the Klein-Gordon equation for Woods-Saxon potential with position dependent mass Arda, Altug; Sever, Ramazan (World Scientific Pub Co Pte Lt, 2008-05-01) The effective mass Klein-Gordon equation in one dimension for the Woods-Saxon potential is solved by using the Nikiforov-Uvarov method. Energy eigenvalues and the corresponding eigenfunctions are computed. Results are also given for the constant mass case.
Invariant subspaces for Banach space operators with an annular spectral set Yavuz, Onur (2008-01-01) Consider an annulus Omega = {z epsilon C : r(0) 0 such that parallel to p(T)parallel to <= K sup{vertical bar p(lambda)vertical bar : vertical bar lambda vertical bar <= 1} and parallel to p(r(0)T(-1))parallel to <= K sup{vertical bar p(lambda)vertical bar : vertical bar lambda vertical bar <= 1} for all polynomials p. Then there exists a nontrivial common invariant subspace for T* and T*(-1).

Citation Formats

K. D. Ünlü, “Exit probabilities of constrained simple random walks,” Ph.D. - Doctoral Program, Middle East Technical University, 2018.