On achieving throughput in an input-queued switch

We establish some lower bounds on the speedup required to achieve throughput for some classes of switching algorithms in a input-queued switch with virtual output queues (VOQs). We use a weak notion of throughput, which will only strengthen the results, since an algorithm that cannot achieve weak throughput cannot achieve stronger notions of throughput. We focus on priority switching algorithms, i.e., algorithms that assign priorities to VOQs and forward packets of high priority first. We show a lower bound on the speedup for two fairly general classes of priority switching algorithms: input priority switching algorithms and output priority switching algorithms. An input priority scheme prioritizes the VOQs based on the state of the input queues, while an output priority scheme prioritizes the VOQs based on their output ports. We first show that, for output priority switching algorithms, a speedup S ≥ 2 is required to achieve weak throughput. From this, we deduce that both maximal and maximum size matching switching algorithms do not imply weak throughput unless S ≥ 2. The bound of S ≥ 2 is tight in all cases above, based on a result in Dai et al. Finally, we show that a speedup S ≥ 3/2 is required for the class of input priority switching algorithms to achieve weak throughput.