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 time-limited discipline is more laborious. We follow the line of proof as introduced by Fricker and Ja\"{i}bi 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.