Fast Mixing of Parallel Glauber Dynamics and Low-Delay CSMA Scheduling

Glauber dynamics is a powerful tool to generate randomized, approximate solutions to combinatorially difficult problems. It has been used to analyze and design distributed CSMA (Carrier Sense Multiple Access) scheduling algorithms for multi-hop wireless networks. In this paper we derive bounds on the mixing time of a generalization of Glauber dynamics where multiple links are allowed to update their states in parallel and the fugacity of each link can be different. The results can be used to prove that the average queue length (and hence, the delay) under the parallel Glauber dynamics based CSMA grows polynomially in the number of links for wireless networks with bounded-degree interference graphs when the arrival rate lies in a fraction of the capacity region. We also show that in specific network topologies, the low-delay capacity region can be further improved.