Delay-throughput tradeoff for supportive two-tier networks

Consider a static wireless network that has two tiers with different priorities: a primary tier vs. a secondary tier. The primary tier consists of randomly distributed legacy nodes of density n, which have an absolute priority to access the spectrum. The secondary tier consists of randomly distributed cognitive nodes of density m = n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">beta</sup> with beta ges 2, which can only access the spectrum opportunistically to limit the interference to the primary tier. By allowing the secondary tier to route the packets for the primary tier, we show that the primary tier can achieve a throughput scaling of lambda <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (n) = Theta(1/log n) per node and a delay-throughput tradeoff of D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (n) = Theta (radic(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">beta</sup> log nlambda <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (n))) for lambda <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (n) = O (1/log n), while the secondary tier still achieves the same optimal delay-throughput tradeoff as a stand-alone network.