Fluid model for a data network with alpha-fair bandwidth sharing and general document size distributions : two examples of stability

The design and analysis of congestion control mechanisms for mod- ern data networks such as the Internet is a challenging problem. Mathematical models at various levels have been introduced in an efiort to provide insight to some aspects of this problem. A model introduced and studied by Roberts and Massouli¶ (13) aims to capture the dynamics of document arrivals and departures in a network where bandwidth is shared fairly amongst ∞ows that correspond to continuous transfers of individual elastic documents. With gener- ally distributed interarrival times and document sizes, except for a few special cases, it is an open problem to establish stability of this stochastic ∞ow level model under the nominal condition that the average load on each resource is less than its capacity. As a step towards the study of this model, in a separate work (8), we introduced a measure valued process to describe the dynamic evo- lution of the residual document sizes and proved a ∞uid limit result: under mild assumptions, rescaled measure valued processes corresponding to a sequence of connection level models (with flxed network structure) are tight, and any weak limit point of the sequence is almost surely a solution of a certain ∞uid model. The invariant states for the ∞uid model were also characterized in (8). In this paper, we review the structure of the stochastic ∞ow level model, describe our ∞uid model approximation and then give two interesting examples of network topologies for which stability of the ∞uid model can be established under a nominal condition. The two types of networks are linear networks and tree networks. The result for tree networks is particularly interesting as there the distribution of the number of documents process in steady state is expected to be sensitive to the (non-exponential) document size distribution (2). Future work will be aimed at further analysis of the ∞uid model and at using it for studying stability and heavy tra-c behavior of the stochastic ∞ow level model.