Lower bounds for wide-sense nonblocking Clos network

The 3-stage Clos network is generally considered the most basic multistage interconnecting network (MIN). The nonblocking property of such network has been extensively studied in the past. However, there are only a few lower bound results regarding wide-sense nonblocking. We show that in the classical circuit switching environment, to guarantee wide-sense nonblocking for large r, 2n - 1 center switches are necessary, where r is the number of input switches and n is the number of inlets of each input switch. For the multirate environment, we show that for large r, any 3-stage Clos network needs atleast 3n - 2 center switches to guarantee wide-sense nonblocking. Our proof works even for the two-rate environment.