Maximum size matching is unstable for any packet switch

Input-queued packet switches use a matching algorithm to configure a nonblocking switch fabric (e.g., a crossbar). Ideally, the matching algorithm will guarantee 100% throughput for a broad class of traffic, so long as the switch is not oversubscribed. An intuitive choice is the maximum size matching (MSM) algorithm, which maximizes the instantaneous throughput. It was shown (McKeown et al. (1999)) that with MSM the throughput can be less than 100% when N > 3, even with Terms-Instability,benign Bernoulli i.i.d. arrivals. In this letter, we extend this result to N > 2, and hence show it to be true for switches of any size.