Exact asymptotics for ki-limited exponential polling models

In this paper we find exact asymptotic expressions for the event that the total queue length is large for a ki-limited exponential polling model with equal service rates and two classes of customer. It is found that this behaviour divides into two very different regimes, depending on the arrival rates to the system. Using these exact asymptotic expressions, we provide heuristics for choosing the ki values to provide a given level of quality of service to one class while giving best effort to the other class.