Limit theorems for polling models with increasing setups

Multiclass single-server systems with significant setup times (polling models) are common in industry. This article considers asymptotics for polling models with increasing setup times. Two types of polling model are considered, namely (a) polling models with polling tables, exhaustive service, and deterministic setups, and (b) cyclic exhaustive service polling models with general setups under heavy traffic. It is shown that as the mean setup time increases to infinity, the scaled intervisit time for each queue (time between service of that queue) converges in probability to a constant. This, in turn, is shown to imply that scaled steady-state waiting time converges in distribution to either a uniform distribution or a simple discrete random variable multiplied by a uniform random variable as setups tend to infinity. These results lead to considerable insight into the behavior of systems with setups, and conclusions are drawn with respect to previous studies.