On the Speedup Required to Achieve 100% Throughput for Multicast Traffic over Crossbar Switches

We show an isomorphism between packet scheduling in crossbar switches and circuit switching in three-stage Clos networks, using the concept of rate quantization. We use the analogy for a crossbar switch of size n×n to construct a simple packet scheduler of complexity O(n logn) based on maximal matching. We show that, with this simple scheduler, a speedup of O(logn/ log logn) is necessary to support 100% throughput for any admissible multicast traffic. If fanout splitting of multicast packets is not allowed, we show that an extra speedup of 2 is necessary, even when the arrival rates are within the admissible region for mere unicast traffic. Also we revisit some problems in unicast switch scheduling. We illustrate that the analogy provides useful perspectives and we give a simple proof for a well known result.