Pulse Coupled Discrete Oscillators dynamics for network scheduling

The dynamics of coupled oscillators, first introduced in mathematical biology, have increasingly become the inspiration for solving wireless scheduling problems. The appeal lies in the fact that coupled oscillators models suggest remarkably simple and scalable policies to enforce temporal events patterns in the absence of central control. Most authors have studied the emergent network behavior of 'desynchronization', i.e. the state in which nodes equally partition a time frame into individual slots. However, less has been said about using these dynamics for the assignment of discrete resources, and the outcome of discrete oscillators dynamics. This problem is important because transmission events in general cannot have arbitrary duration, due to modulation constraints. Our problem has many features in common with 'quantized consensus' problems which will be highlighted in this paper. In particular, in this paper we provide a model for analyzing Pulse Coupled Discrete Oscillators (PCDO) dynamics. As we will describe, the PCDO can naturally nest beneath the Pulse Coupled Oscillators (PCO) synchronization protocol, to attain a common shared slotted time. The PCDO dynamics assign a set of consecutive PCO slots, in an arbitrarily long frame of PCO slots. Our analysis shows that the algorithm converges almost surely and provides a bound on the convergence time of the PCDO dynamics.