Stability of a spatial polling system with greedy myopic service

TL;DR

This paper analyzes a spatial queueing system on a circle with a greedy, myopic server, establishing conditions for stability and ergodicity, and exploring steady-state behavior through simulations and approximations.

Contribution

It introduces a measure-valued process model for the system and proves its positive recurrence and geometric ergodicity under natural conditions, independent of scan range.

Findings

System is positive recurrent under natural stability conditions.

System exhibits geometric ergodicity with light-tailed interpolling times.

Steady-state behavior analyzed via simulations and heuristics.

Abstract

This paper studies a spatial queueing system on a circle, polled at random locations by a myopic server that can only observe customers in a bounded neighborhood. The server operates according to a greedy policy, always serving the nearest customer in its neighborhood, and leaving the system unchanged at polling instants where the neighborhood is empty. This system is modeled as a measure-valued random process, which is shown to be positive recurrent under a natural stability condition that does not depend on the server's scan range. When the interpolling times are light-tailed, the stable system is shown to be geometrically ergodic. The steady-state behavior of the system is briefly discussed using numerical simulations and a heuristic light-traffic approximation.