On Rice's formula for stationary multivariate piecewise smooth processes

TL;DR

This paper extends Rice's formula to multivariate stationary processes that move along vector fields, relating crossing intensities of surfaces to the process distribution, with applications in queueing and stress models.

Contribution

It derives a multivariate version of Rice's formula for stationary piecewise smooth processes, linking crossing intensities to initial distributions.

Findings

Derived a multivariate Rice's formula for continuous surface crossings.

Applied the formula to queueing network models.

Provided examples illustrating the theoretical results.

Abstract

Let $X=\{X_t: t\ge 0\}$ be a stationary piecewise continuous $\R^d$-valued process that moves between jumps along the integral curves of a given continuous vector field, and let $S\subset\R^d$ be a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings of $S$ by $X$ to the distribution of $X_0$. Our result is illustrated by examples relating to queueing networks and stress release network models.