<P> If m ≥ n, the Clos network is rearrangeably nonblocking, meaning that an unused input on an ingress switch can always be connected to an unused output on an egress switch, but for this to take place, existing calls may have to be rearranged by assigning them to different centre stage switches in the Clos network . To prove this, it is sufficient to consider m = n, with the Clos network fully utilised; that is, r × n calls in progress . The proof shows how any permutation of these r × n input terminals onto r × n output terminals may be broken down into smaller permutations which may each be implemented by the individual crossbar switches in a Clos network with m = n . </P> <P> The proof uses Hall's marriage theorem which is given this name because it is often explained as follows . Suppose there are r boys and r girls . The theorem states that if every subset of k boys (for each k such that 0 ≤ k ≤ r) between them know k or more girls, then each boy can be paired off with a girl that he knows . It is obvious that this is a necessary condition for pairing to take place; what is surprising is that it is sufficient . </P> <P> In the context of a Clos network, each boy represents an ingress switch, and each girl represents an egress switch . A boy is said to know a girl if the corresponding ingress and egress switches carry the same call . Each set of k boys must know at least k girls because k ingress switches are carrying k × n calls and these cannot be carried by less than k egress switches . Hence each ingress switch can be paired off with an egress switch that carries the same call, via a one - to - one mapping . These r calls can be carried by one middle - stage switch . If this middle - stage switch is now removed from the Clos network, m is reduced by 1, and we are left with a smaller Clos network . The process then repeats itself until m = 1, and every call is assigned to a middle - stage switch . </P> <P> Real telephone switching systems are rarely strict - sense nonblocking for reasons of cost, and they have a small probability of blocking, which may be evaluated by the Lee or Jacobaeus approximations, assuming no rearrangements of existing calls . Here, the potential number of other active calls on each ingress or egress switch is u = n − 1 . </P>

When does a two stage network become a fully connected network