# Asymptotic Properties of a Branching-type Overloaded Polling Network

## TL;DR

This paper analyzes the long-term behavior of an overloaded cyclic polling network with rerouting, using branching process theory and fluid limits, and demonstrates how to optimize gating indexes to minimize total queue length.

## Contribution

It introduces an asymptotic analysis of a rerouting polling network using branching process theory and fluid limits, enabling optimization of gating indexes.

## Key findings

- Fluid limit resembles that of systems without rerouting
- Optimal gating indexes can be identified for minimizing total population
- Simulation confirms theoretical fluid limit and optimization results

## Abstract

In this paper, we consider an $N$-queue overloaded polling network attended by a single cyclically roving server. Upon the completion of his service, a customer is either routed to another queue or leaves the system. All the switches are instantaneous and random multi-gated service discipline is employed within each queue. With the asymptotic theorem of multi-type branching processes and the exhaustiveness of service discipline, the fluid asymptotic process of the scaled joint queue length process is investigated. The fluid limit is of the similar shape with that of the polling system without rerouting policy, which allows us to optimize the gating indexes. Additionally, a stochastic simulation is undertaken to demonstrate the fluid limit and the optimization of the gating indexes to minimize the total population is considered.