The snowblower problem
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 Levy snowfall and arbitrary snowblower delays, and study it by analyzing an underlying stochastic Poincare map governing the system's evolution. The log-Laplace transform and mean of the Poincare map are computed, convergence to steady state ( equilibrium) is proved, and the system's equilibrium behavior is explored.
snowblower problem snowblowing systems continuous polling systems levy snowfall poincare map polling on the circle