The Snowblower Problem

Abstract

A snowblower is circling a closed-loop racetrack, driving clockwise and clearing off snow in a constant snowblowing rate. Both the snowfall and the snowblower's driving speed vary randomly (in both space and time coordinates). The snowblower's motion and the snowload profile on the racetrack are co-dependent and co-evolve, resulting in a coupled stochastic dynamical system of ‘random motion (snowblower) in a random environment (snowload profile)’. Snowblowing systems are closely related to continuous polling systems – or, so-called, polling systems on the circle – which are the continuum limits of ‘standard’ polling systems. Our aim in this manuscript is to introduce a stochastic model that would apply to a wide class of stochastic snowblower-type systems and, simultaneously, generalize the existing models of continuous polling systems. We present a general snowblowing-system model, with arbitrary Lévy snowfall and arbitrary snowblower delays, and study it by analyzing an underlying stochastic Poincaré map governing the system's evolution. The log-Laplace transform and mean of the Poincaré map are computed, convergence to steady state (equilibrium) is proved, and the system's equilibrium behavior is explored.