An asymptotic property of branching-type overloaded polling networks

Yuejiao Wang,Yuqing Chu,Zaiming Liu,Yingqiu Li

doi:10.1515/math-2019-0116

Abstract

Abstract Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] studied the fluid asymptotics of the joint queue length process for an overloaded cyclic polling system with multigated service discipline by exploiting the connection with multi-type branching processes. In contrast to the heavy traffic behaviors, the cycle time of the overloaded polling system increases by a deterministic times over times under passage to the fluid dynamics and the fluid limit preserves some randomness. The present paper aims to extend the overloaded asymptotics in Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] to the corresponding polling system with general branching-type service disciplines and customer re-routing policy. A unifying overloaded asymptotic property is derived. Due to the exhaustiveness, the property is a natural extension of the classical polling model with multigated service discipline in Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] and provides new exact results that have not been observed before for rerouting policy. Additionally, a stochastic simulation is undertaken for the validation of the fluid limit and the optimization of the gating indexes to minimize the total population is considered as an example to demonstrate the usefulness of the random fluid limit.

Highlights

In this paper, we consider a cyclic N-queue (Q, · · ·, QN, N ≥ ) polling system with general branchingtype service discipline within each queue and customer re-routing policies: after completing service at Qi, a customer is either routed to Qj with probability pi,j or leaves the system with probability pi
The present paper aims to extend the overloaded asymptotics in Remerova et al [Random uid limit of an overloaded polling model, Adv
The property is a natural extension of the classical polling model with multigated service discipline in Remerova et al [Random uid limit of an overloaded polling model, Adv

Summary

Introduction

We consider a cyclic N-queue (Q , · · · , QN , N ≥ ) polling system with general branchingtype service discipline within each queue and customer re-routing policies: after completing service at Qi, a customer is either routed to Qj with probability pi,j or leaves the system with probability pi,. The possibility for re-routing of customers further enhances the already-extensive modeling capabilities of polling models, since in many applications, customers require service at more than one facility of the system. In the vast majority of papers that have appeared on polling models, it is almost invariably assumed that the system is stable and the stable performance measures are concerned. With the advent of the era of Internet+, the study of critically or strictly super-critically loaded polling systems is vigorously pioneered due to the overloaded Internet channel or online shopping orders

Objectives

Conclusion