Self-stabilizing TDMA Algorithms for Wireless Ad-hoc Networks without External Reference

Time division multiple access (TDMA) is a method for sharing communication media. In wireless communications, TDMA algorithms often divide the radio time into timeslots of uniform size, $\xi$, and then combine them into frames of uniform size, $\tau$. We consider TDMA algorithms that allocate at least one timeslot in every frame to every node. Given a maximal node degree, $\delta$, and no access to external references for collision detection, time or position, we consider the problem of collision-free self-stabilizing TDMA algorithms that use constant frame size. We demonstrate that this problem has no solution when the frame size is $\tau < \max\{2\delta,\chi_2\}$, where $\chi_2$ is the chromatic number for distance-$2$ vertex coloring. As a complement to this lower bound, we focus on proving the existence of collision-free self-stabilizing TDMA algorithms that use constant frame size of $\tau$. We consider basic settings (no hardware support for collision detection and no prior clock synchronization), and the collision of concurrent transmissions from transmitters that are at most two hops apart. In the context of self-stabilizing systems that have no external reference, we are the first to study this problem (to the best of our knowledge), and use simulations to show convergence even with computation time uncertainties.