Heavy traffic analysis of a polling model with retrials and glue periods

TL;DR

This paper analyzes a complex polling model with retrials and glue periods in heavy traffic, deriving asymptotic queue length distributions and providing accurate system load approximations.

Contribution

It introduces a heavy traffic analysis for a polling model with retrials and glue periods, deriving new asymptotic queue length distributions.

Findings

Closed-form heavy-traffic queue length distributions

Accurate approximation of mean customer numbers

Insights into system behavior near capacity

Abstract

We present a heavy traffic analysis of a single-server polling model, with the special features of retrials and glue periods. The combination of these features in a polling model typically occurs in certain optical networking models, and in models where customers have a reservation period just before their service period. Just before the server arrives at a station there is some deterministic glue period. Customers (both new arrivals and retrials) arriving at the station during this glue period will be served during the visit of the server. Customers arriving in any other period leave immediately and will retry after an exponentially distributed time. As this model defies a closed-form expression for the queue length distributions, our main focus is on their heavy-traffic asymptotics, both at embedded time points (beginnings of glue periods, visit periods and switch periods) and at arbitrary time points. We obtain closed-form expressions for the limiting scaled joint queue length distribution in heavy traffic and use these to accurately approximate the mean number of customers in the system under different loads.