A Network Containing a Periodically Operated Switch Solved by Successive Approximations

One main contributor to the cost of transmission circuits is the transmission medium itself. Thus it is important to share the transmission medium among as many messages as possible. One possible method is the frequency multiplex where each message utilizes a different frequency band of the whole band available in the medium. An alternate method is the time multiplex where each message is assigned a time slot of duration r and has access to that time slot once every T seconds. It is obvious that the economics of the situation requires that T be as small as possible and T as large as possible so that the largest possible number of messages are transmitted over the medium. For this very reason the analysis of periodically switched networks is of special interest in the case where T/T is small. W. R. Bennett 4 has published an exact analysis of this problem without any restrictions either on the network or on the ratio T/T. It is believed, however, that the analysis presented in this paper will, in most practical cases, give the desired answer with a considerable reduction in the amount of calculations. The simplification of the analysis is mainly a result of the assumption that T/T is small. First the successive approximation method of solution will be discussed in general terms. Next it will be shown that the zeroth approximation to the transmission through the network can be obtained from the gain of a block diagram analogous to those used in the analysis of sampled servomechanisms.