A Polling Model with Smart Customers

TL;DR

This paper analyzes a cyclic polling system with state-dependent arrival rates, deriving queue length and waiting time distributions, and introduces efficient methods and laws applicable under certain conditions.

Contribution

It introduces a novel polling model with server-location-dependent arrival rates and develops new analytical tools for queue length and waiting time distributions.

Findings

Derived joint and marginal queue length distributions.

Established a generalized Little's law for this model.

Demonstrated the model's features through numerical examples.

Abstract

In this paper we consider a single-server, cyclic polling system with switch-over times. A distinguishing feature of the model is that the rates of the Poisson arrival processes at the various queues depend on the server location. For this model we study the joint queue length distribution at polling epochs and at server's departure epochs. We also study the marginal queue length distribution at arrival epochs, as well as at arbitrary epochs (which is not the same in general, since we cannot use the PASTA property). A generalised version of the distributional form of Little's law is applied to the joint queue length distribution at customer's departure epochs in order to find the waiting time distribution for each customer type. We also provide an alternative, more efficient way to determine the mean queue lengths and mean waiting times, using Mean Value Analysis. Furthermore, we show that under certain conditions a Pseudo-Conservation Law for the total amount of work in the system holds. Finally, typical features of the model under consideration are demonstrated in several numerical examples.