Some Properties of a Nonlinear Model of a System for Synchronizing Digital Transmission Networks*
Irwin W. Sandberg · Bell System Technical Journal · 1969
J. R. Pierce has recently proposed a system for synchronizing an arbitrary number of geographically separated oscillators, and, under the assumption of zero transmission delays between stations, has shown that a certain linear model of the system is stable in the sense that all of the station frequencies approach a common final value as t → ∞. The purpose of this paper is to report on some results concerning the dynamic behavior of a nonlinear version of an important special case of Pierce's model. The nonlinear model takes into account transmission delays. It is proved under certain very general conditions that the nonlinear model possesses the stability property required of a synchronization system. More explicitly, it is proved that the model is stable for all nonnegative values of the delays. The results show that the model possesses some additional fundamental properties of engineering interest, and they provide an analytical basis for using a computer for further studies. In particular, a complete solution to the problem of determining the final frequency of the system and the final value of the content of an arbitrary buffer is presented, in the sense that it is shown that these quantities can be determined by solving a certain set of nonlinear equations which is proved to possess a unique solution.