Asymptotically Optimal Load Balancing Topologies

We consider a system of N servers inter-connected by some underlying graph topology G(N). Tasks with unit-mean exponential processing times arrive at the various servers as independent Poisson processes of rate lambda. Each incoming task is irrevocably assigned to whichever server has the smallest number of tasks among the one where it appears and its neighbors in G(N). The above model arises in the context of load balancing in large-scale cloud networks and data centers, and has been extensively investigated in the case G(N) is a clique. Since the servers are exchangeable in that case, mean-field limits apply, and in particular it has been proved that for any lambda < 1, the fraction of servers with two or more tasks vanishes in the limit as N -> infinity. For an arbitrary graph G(N), mean-field techniques break down, complicating the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph G(N) is said to be N-optimal or root N-optimal when the queue length process on G(N) is equivalent to that on a clique on an N-scale or root N-scale, respectively. We prove that if G(N) is an Erdos-Renyi random graph with average degree d(N), then with high probability it is N-optimal and v N-optimal if d(N) -> infinity and d(N)/(root N log( N)) -> infinity as N -> infinity, respectively. This demonstrates that optimality can be maintained at N-scale and v N-scale while reducing the number of connections by nearly a factor N and root N/log(N) compared to a clique, provided the topology is suitably random. It is further shown that if G(N) contains Theta(N) bounded-degree nodes, then it cannot be N-optimal. In addition, we establish that an arbitrary graph G(N) is N-optimal when its minimum degree is N - o(N), and may not be N-optimal even when its minimum degree is cN + o(N) for any 0 < c < 1/2. Simulation experiments are conducted for various scenarios to corroborate the asymptotic results.