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.

💡 Research Summary

The paper investigates a generalized version of Glauber dynamics in which multiple links in a wireless multi‑hop network may update their activity states simultaneously, and each link can be assigned its own fugacity (weight) λi. This “parallel Glauber dynamics” is motivated by the need to accelerate the convergence of distributed CSMA (Carrier Sense Multiple Access) scheduling algorithms, which rely on a Markov chain whose stationary distribution is a weighted independent‑set (or Boltzmann) distribution over the network’s interference graph.

First, the authors formalize the model. The network is represented by an interference graph G(V,E) with N vertices (links). A feasible schedule corresponds to an independent set σ⊆V. Each link i carries a fugacity λi>0 that reflects its traffic demand or priority. In each discrete time slot a subset of at most K links is selected uniformly at random; these links independently decide whether to turn on or off according to the standard Glauber rule, i.e., a link attempts to become active with probability λi/(1+λi) provided none of its neighbors are currently active, otherwise it stays inactive. The resulting process is a Markov chain on the space of independent sets.

The paper proves that this chain satisfies detailed balance with respect to the product‑form stationary distribution π(σ)∝∏i∈σ λi, regardless of the heterogeneity of the λi’s and the parallel update size K, as long as K does not exceed the maximum degree Δ of the interference graph. Consequently the chain is ergodic and converges to π.

The core technical contribution is a rigorous mixing‑time analysis. Using a path‑coupling argument, the authors bound the expected Hamming distance between two coupled chains after one parallel update step. They show that when K=O(Δ) (i.e., the number of simultaneously updated links is proportional to the graph’s maximum degree) the distance contracts by a factor (1−c/Δ) for some constant c that depends only on the ratio maxi λi / mini λi. This yields a mixing‑time bound τ(ε) ≤ C·Δ·log N·log(1/ε), where C is a universal constant. In contrast, the classic single‑site Glauber dynamics has a mixing time of order O(N log N) for the same class of graphs. Thus parallel updates reduce the convergence time from linear‑in‑N to logarithmic‑in‑N, a dramatic improvement for large networks.

Having established fast mixing, the authors translate the result into performance guarantees for CSMA scheduling. In the CSMA protocol, each link runs a back‑off timer that is geometrically distributed with mean proportional to 1/λi; when the timer expires the link attempts to transmit if none of its neighbors are active. The schedule generated by the protocol at any time is exactly a sample from the Glauber chain. Because the chain mixes rapidly, the distribution of schedules quickly approaches π, and the system behaves as if it were operating under the stationary distribution. Using a Lyapunov drift argument, the paper shows that if the arrival rate vector a lies inside a scaled capacity region α·C, where C is the convex hull of all feasible independent sets, then the expected total queue length satisfies E