Adversarial queuing theory with setups

Abstract

We look at routing and scheduling problems on Kelly type networks where the injection process is under the control of an adversary. The novelty of the model we consider is that the adversary injects requests of distinct types. Resources are subject to switch-over delays or setups when they begin servicing a new request class. In this new setting, we study the behavior of sensible policies as introduced by Dai and Jennings [J. Dai, O. Jennings, Stabilizing queueing networks with setups, Math. Oper. Res. (2004) 891 922]. We first show that the model is robust in the sense that under some mild conditions universal stability of work conserving packet routing protocols is preserved for natural variants of the underlying model. Also, the model's equivalence to so called token networks is established. We adapt to the multi-type request and setup setting, standard arguments for proving stability. Nevertheless, we provide counterexamples that show that for several reasonable adaptations of contention resolution protocols to the multi-type case, stability results do not carry over from the single-type scenario. This motivates us to explore fluid model based arguments that could be used for proving stability for a given network. Specifically we show analogues of results obtained by Gamarnik [D. Gamarnik, Stability of adversarial queues via fluid model, in: Proc. of the 39th Annual Symposium on Foundations of Computer Science, 1998, pp. 60-70] but in the multi-type request with setups scenario.