Network adiabatic theorem

The popularity of Aloha -like algorithms for resolution of contention between multiple entities accessing common resources is due to their extreme simplicity and distributed nature. Example applications of such algorithms include Ethernet and recently emerging wireless multi-access networks. Despite a long and exciting history of more than four decades, the question of designing an algorithm that is essentially as simple and distributed as Aloha while being efficient has remained unresolved. In this paper, we resolve this question successfully for a network of queues where contention is modeled through independent-set constraints over the network graph. The work by Tassiulas and Ephremides (1992) suggests that an algorithm that schedules queues so that the summation of `weight' of scheduled queues is maximized, subject to constraints, is efficient. However, implementing such an algorithm using Aloha-like mechanism has remained a mystery. We design such an algorithm building upon a Metropolis-Hastings sampling mechanism along with selection of `weight' as an appropriate function of the queue-size. The key ingredient in establishing the efficiency of the algorithm is a novel adiabatic -like theorem for the underlying queueing network, which may be of general interest in the context of dynamical systems.