A Polling Model with Reneging at Polling Instants

TL;DR

This paper models a cyclic polling system with customer reneging at polling instants, analyzing its impact on waiting times and queue lengths using a novel approach with customer subtypes and generalized Little's law.

Contribution

It introduces a new analysis method for polling systems with customer reneging at polling instants, considering varying arrival rates and customer subtypes.

Findings

Reneging probabilities depend on queue and server location.

Waiting time distributions are derived using a generalized Little's law.

Queue length distributions are obtained by conditioning on system state.

Abstract

In this paper we consider a single-server, cyclic polling system with switch-over times and Poisson arrivals. The service disciplines that are discussed, are exhaustive and gated service. The novel contribution of the present paper is that we consider the reneging of customers at polling instants. In more detail, whenever the server starts or ends a visit to a queue, some of the customers waiting in each queue leave the system before having received service. The probability that a certain customer leaves the queue, depends on the queue in which the customer is waiting, and on the location of the server. We show that this system can be analysed by introducing customer subtypes, depending on their arrival periods, and keeping track of the moment when they abandon the system. In order to determine waiting time distributions, we regard the system as a polling model with varying arrival rates, and apply a generalised version of the distributional form of Little's law. The marginal queue length distribution can be found by conditioning on the state of the system (position of the server, and whether it is serving or switching).