Fundamental Limits on Synchronizing Clocks Over Networks

We characterize what is feasible concerning clock synchronization in wireline or wireless networks. We consider a net work of n nodes, equipped with affine clocks relative to a designated clock that exchange packets subject to link delays. Determining all unknown parameters, i.e., skews and offsets of all the clocks as well as the delays of all the communication links, is impossible. All nodal skews, as well as all round-trip delays between every pair of nodes, can be determined correctly. Also, every transmitting node can predict precisely the time indicated by the receiver's clock at which it receives the packet. However, the vector of unknown link delays and clock offsets can only be determined up to an (n - 1)-dimensional subspace, with each degree of freedom corresponding to the offset of one of the (n - 1) clocks. Invoking causality, that packets cannot be received before they are transmitted, the uncertainty set can be reduced to a polyhedron. We also investigate structured models for link delays as the sum of a transmitter-dependent delay, a receiver-dependent delay, and a known propagation delay, and identify conditions which permit a unique solution, and conditions under which the number of the residual degrees of freedom is independent of the network size. For receiver-receiver synchronization, where only receipt times are available, but no time-stamping is done by the sender, all nodal skews can still be determined, but delay differences between neighboring communication links with a common sender can only be characterized up to an affine transformation of the (n - 1) un known offsets. Moreover, causality does not help reduce the uncertainty set.