Impact of Fairness and Heterogeneity on Delays in Large-scale Content Delivery Networks.

We consider multi-class queueing systems where the per class service rates depend on the network state, fairness criterion, and is constrained to be in a symmetric polymatroid capacity region. We develop new comparison results leading to explicit bounds on the mean service time under various fairness criteria and possibly heterogeneous loads. We then study large-scale systems with growing numbers of service classes n (e.g., files), heterogenous servers m and polymatroid capacity resulting from a random bipartite graph modeling service availability (e.g., placement of files across servers). This models, for example, a large scale content delivery network (CDN) supporting parallel servicing of a download request. For an appropriate asymptotic regime, we show that the system's capacity region is uniformly close to a symmetric polymatroid -- i.e., heterogeneity in servers' capacity and file placement disappears. Combining our comparison results and the asymptotic 'symmetry' in large systems, we study performance robustness to heterogeneity in per class loads and fairness criteria. Roughly, if each class can be served by cn = ω(log n) servers, the load per class does not exceed θn = o (min(n/log n, cn)), and average server utilization is bounded by λ < 1, then mean delay satisfies the following bound: E[D(n)] ≤ K θn/cn 1/λ log (1/1--λ}\right), where K is a constant. Thus, large, randomly configured CDNs with a logarithmic number of file copies are robust to substantial load and server heterogeneities for a class of fairness criteria.