Hitting Probabilities of Constrained Simple Random Walks in Three Dimensions

Date

2025-1-7

Author

Aktepe İlter, Cansu

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We study the constrained simple random walk in three dimensions modeling the state of a queueing system with three nodes working in parallel. The process is assumed to be stable, i.e., the service rate at each node is greater than the arrival rate. The stability assumption implies that the process follows a repeating cycle, starting anew each time the process hits the origin. Consider the probability pn that the sum of the components of the process equals n before the process hits the origin, which can be thought of as the probability of a buffer overflow in a cycle. The stability assumption implies that pn decays exponentially in n. The goal of the present thesis is to develop approximation formulas for pn. In the literature, this problem is treated for two dimensional simple walks using an affine transformation of the problem. We extend this analysis to three dimensions. As in two dimensions, the affine transformation yields a limit process and a limit hitting probability. We show, for the case of the three dimensional stable constrained simple random walk, the limit probability approximates pn with an exponentially diminishing relative error, assuming that the first component of the initial point of the process is nonzero. We further approximate the limit probability by harmonic functions of the limit process constructed from solutions of harmonic systems associated with the problem. We provide a numerical example and discuss a possible application to finance.

Subject Keywords

constrained simple random walks, rare events, queueing systems, financial modelling, harmonic systems

URI

https://hdl.handle.net/11511/113054

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Graduate School of Applied Mathematics, Thesis