Optimal load-balancing

This paper is about load-balancing packets across multiple paths inside a switch, or across a network. It is motivated by the recent interest in load-balanced switches. Load-balanced switches provide an appealing alternative to crossbars with centralized schedulers. A load-balanced switch has no scheduler, is particularly amenable to optics, and - most relevant here -guarantees 100% throughput. A uniform mesh is used to load-balance packets uniformly across all 2-hop paths in the switch. In this paper we explore whether this particular method of load-balancing is optimal in the sense that it achieves the highest throughput for a given capacity of interconnect. The method we use allows the load-balanced switch to be compared with ring, torus and hypercube interconnects, too. We prove that for a given interconnect capacity, the load-balancing mesh has the maximum throughput. Perhaps surprisingly, we find that the best mesh is slightly non-uniform, or biased, and has a throughput of N/(2N - 1), where N is the number of nodes.