I. Nedaiborshch, K. Nikolaev, and A. Vladimirov
 

Lipschitz Continuity and Unique Solvability of Fluid Models of Queueing Networks

 

Deterministic fluid models imitate the long-term behavior of stochastic queueing networks. We address the problem of unique solvability of these models under arbitrary time-dependent workloads. This property is stronger than the stability under a constant workload; in particular, it ensures the "smooth" behavior of the network under supernominal inflows. Sufficient conditions of unique solvability may be expressed in terms of Lipschitz constants of open networks in various norms. We find precise Lipschitz constants of a single multiclass server station under FIFO, generalized processor sharing, and priority service disciplines. The best possible Lipschitz constant for a two-class server is proved to be given by a queue-equalizing discipline. For proofs we use the technique of hysteresis operators, in particular, polyhedral Skorokhod problems.