The rearrangement process in a two-stage broadcast switching network

This paper considers tbe rearrangement process in the twostage btoadcast switching network presented by F. IC. Hwang and G . W. Richards in IEEE TRANSACTIONS ON COMMUNICATIONS, October 1985. By defining a certain function it is possible to calculate an upper bound on the number bf connections to be moved during a rearrangement. When each inlet channel appears twice, the maximum number of connections to be moved is found. For a special class of inlet assignment patterns in the case where each inlet channel appears three times, the mPxlmum number of connections to be moved is also found. In the general case, an upper bound is given when the number of outlets at each secand-stage sdtch i s kept below a certain bound. I. THE KNOWN PROPERTIES OF THE NETWORK HE network to be considered here (see Fig. 1) is identical T to the one presented in [ 11, and it is described by the three parathetres nl, n2, and M where nl is the number of inlet channels at each first-stage switch, n2 is the number of outlets at each second-stage switch, and M is the number of times each inlet channel appears in the first stage. The number of crosspoints in the network divided by the number of crosspoints in the corresponding rectangular switch is called the reduced number of crosspoints and is given by C&=M(l/nl+ l/nz). (1.1) To minimize C,, the fraction M/nz has to be made as close to zero as possible but the rearrangement requirement puts a lower bound on the fraction. Hall's theorem on a system of distinct representatives [2] ensures that the network is rearrangeable, if and only if, the following condition is fulfilled. The Rearrangement Condition: For any n 5 n2, there are at least n first-stage switches containing appearance of any n inlet channels. To ensure that the n f inlet channels are effectively rotated in the M blocks the following condition is assumed to be fulfilled. The Pair Condition: No pair of inlet channels appears on the same first-stage switch more than once throughout the Mn, first-stage switches. All the inlet assignment patterns presented in [l] fulfill the pair condition, but instead of working with some explicit patterns, it is more advantageous in a general approach just to assume the pair condition to be fulfilled. 11. A N UPPER BOUND ON THE NUMBER OF CONNECTIONS TO BE MOVED DURING A REARRANGEMENT BY MEANS OF THE FUNCTION As it will be seen later, the rearrangement condition as well as an upper bound on the number of connections to be moved Paper approved by the Editor for Communication Switching of the IEEE Communications Society. Manuscript received November 19, 1986; revised July 2, 1987. The author is with the Electromagnetics Institute, Technical University of Denmark, DK-2800 Lyngby, Denmark. IEEE Log Number 8718999. i T L \ . . I I n ! Fig. 1. The network presented in [ 11. Every inlet channel has exactly one appearance in each of the M blocks. during a rearrangement can be given by means of a function whose values are in general unknown. The function will be called and it is defined by means of another function TM,,,(E) := The number of first-stage switches having Let E be a subset of the set of inlet channels. Then, elements from E amongst its inlet channels. (2.1)