A reduced-load equivalence for generalised processor sharing networks with heavy-tailed input flows

We consider networks where traffic is served according to the Generalised Processor Sharing (GPS) principle. GPS-based scheduling algorithms are considered important for providing differentiated quality of service in integrated-services networks. We are interested in the workload of a particular flow~$i$ at the bottleneck node on its path. Flow $i$ is assumed to have long-tailed traffic characteristics. We distinguish between two traffic scenarios, (i) flow~$i$ generates instantaneous traffic bursts and (ii) flow $i$ generates traffic according to an on/off process. In addition, we consider two configurations of feed-forward networks. First we focus on the situation where other flows join the path of flow~$i$. Then we extend the model by adding flows which can branch off at any node, with cross traffic as a special case. We prove that under certain conditions the tail behaviour of the workload distribution of flow~$i$ is equivalent to that in a {em two-node tandem network where flow~$i$ is served in isolation at {em constant rates. These rates only depend on the traffic characteristics of the other flows through their average rates. This means that the results do not rely on any specific assumptions regarding the traffic processes of the other flows. In particular, flow~$i$ is not affected by excessive activity of flows with `heavier-tailed' traffic characteristics. This confirms that GPS has the potential to protect individual flows against extreme behaviour of other flows, while obtaining substantial multiplexing gains.