Power law condition for stability of Poisson hail

TL;DR

This paper establishes a power law condition on the tail distributions of jobs in the Poisson hail model to guarantee stability at low arrival rates, demonstrating near-optimality of this condition.

Contribution

It introduces a power law tail condition for the Poisson hail model that ensures stability and proves its near-optimality.

Findings

Power law tail condition guarantees stability as rate tends to zero

The condition is shown to be nearly optimal in a weak sense

Provides insights into tail behavior and stability in Poisson hail models

Abstract

We consider the Poisson hail model introduced by Baccelli and Foss. We give a power law condition for the tails (spatial and temporal) of the distribution of jobs to ensure stability as the rate parameter $\lambda $ tends to zero. We then show that in a weak sense it is optimal.