Stability of Two Exponential Time-limited Polling Models

In this article, we consider the stability of two single-server polling models. More specifically, we will state and prove the stability conditions of single-server polling systems operating under the pure and exhaustive exponential time-limited service discipline. These conditions will be proven for the polling system operating under the periodic polling strategy and preemptive service. The stability proof of the pure time-limited discipline is straightforward as stability may be considered for each queue in isolation. The proof for the exhaustive timelimited discipline is more laborious. We follow the line of proof as introduced by Fricker and Jäıbi [1] for a large class of service disciplines. Unfortunately, the preemptive nature of the exhaustive time-limited discipline excludes it from this class and as a result substantial efforts are required to modify the proof as to allow for preemptive disciplines. Finally, the extension of the proofs to the Markovian polling strategy is discussed.