On the Stability of Multiclass Queueing Networks: A Relaxed Sufficient Condition via Limiting Fluid Processes

#### A.L. Stolyar

1995, v.1, №4, 491-512

ABSTRACT

We consider a multiclass queueing network, whose underlying stochastic process is a countable, continuous time Markov chain. Stability of the network is understood as ergodicity of this Markov chain. The message class determines a message route through the network and the mean message service time in each node on its route. Each node may have its own queueing discipline within a wide class, including FCFS, LCFS, Priority and Processor Sharing. We will show that the sequence of scaled (in space and time) underlying stochastic processes converges to a fluid process with sample paths defined as fixed points of a special operator. This convergence together with continuity and similarity properties of the family of sample paths of the fluid process allows us to prove the following result. If each sample path of the fluid process with non-zero initial state is such that the amount of fluid'' in the network falls below its initial value at least once, then the network is stable.

Keywords: Multiclass queueing network,stability,fluid limit