Hitting probabilities of constrained random walks representing tandem networks

TL;DR

This paper develops approximation formulas for hitting probabilities of constrained random walks modeling tandem networks, with exponential decay of relative error, using harmonic systems and supermartingale constructions.

Contribution

It introduces a new approach to approximate hitting probabilities in high-dimensional tandem networks using harmonic systems and explicit formulas, with proven exponential error decay.

Findings

Derived approximation formulas for hitting probabilities.

Proved exponential decay of approximation error.

Provided explicit formulas based on harmonic systems.

Abstract

Let $X$ be the constrained random walk on $\mathbb{Z}_+^d$ $d >2$, having increments $e_1$, $-e_i+e_{i+1}$ $i=1,2,3,...,d-1$ and $-e_d$ with probabilities $\lambda$, $\mu_1$, $\mu_2$,...,$\mu_d$, where $\{e_1,e_2,..,e_d\}$ are the standard basis vectors. The process $X$ is assumed stable, i.e., $\lambda < \mu_i$ for all $i=1,2,3,...,d.$ Let $\tau_n$ be the first time the sum of the components of $X$ equals $n$. We derive approximation formulas for the probability ${\mathbb P}_x(\tau_n < \tau_0)$. For $x \in \bigcup_{i=1}^d \Big\{x \in {\mathbb R}^d_+: \sum_{j=1}^{i} x(j)$ $> \left(1 - \frac{\log \lambda/\min \mu_i}{\log \lambda/\mu_i}\right) \Big\}$ and a sequence of initial points $x_n/n \rightarrow x$ we show that the relative error of the approximation decays exponentially in $n$. The approximation formula is of the form ${\mathbb P}_y(\tau < \infty)$ where $\tau$ is the first time the sum of the components of a limit process $Y$ is $0$; $Y$ is the process $X$ as observed from a point on the exit boundary except that it is unconstrained in its first component (in particular $Y$ is an unstable process); $Y$ and ${\mathbb P}_y(\tau< \infty)$ arise naturally as the limit of an affine transformation of $X$ and the probability ${\mathbb P}_x(\tau_n < \tau_0).$ The analysis of the relative error is based on a new construction of supermartingales. We derive an explicit formula for ${\mathbb P}_y(\tau < \infty)$ in terms of the ratios $\lambda/\mu_i$ which is based on the concepts of harmonic systems and their solutions and conjugate points on a characteristic surface associated with the process $Y$; the derivation of the formula assumes $\mu_i \neq \mu_j$ for $i\neq j.$