Hitting probabilities of constrained random walks representing tandem networks

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 $i\neq j.$