Wireless networks equipped with CSMA are scheduled in a fully distributed manner. A disadvantage of such distributed control in multihop networks is the hidden node problem, which causes the effect of stealing, in which a downstream node steals the channel from an upstream node with probability $p$. Aziz, Starobinski, and Thiran [IEEE/ACM Trans. Networking, to appear] have recently shown that the $N$-hop model with stealing is stable only in the case $N=3$ and $p\in(0,1]$. This 3-hop model can be represented as a random walk in the quarter plane. We derive various asymptotic expressions for the stationary large buffer probabilities of the 3-hop model that capture the effect of $p$. We mainly rely on the ray method, and we show how this method is related to the compensation approach and singularity analysis.
Guillemin, F., Knessl, C., & Leeuwaarden, van, J. S. H. (2011). Wireless multihop networks with stealing : large buffer asymptotics via the ray method. SIAM Journal on Applied Mathematics, 71(4), 1220-1240. https://doi.org/10.1137/100808629