Capacity Scaling Laws of Cognitive Networks: Dense Ad Hoc Primary Network Case

In this paper, we study the capacity of cognitive networks. We focus on the network model consisting of two overlapping ad hoc networks, called the primary ad hoc network (PaN) and secondary ad hoc network (SaN), respectively. PaN and SaN operate on the same space and spectrum. For PaN (or SaN resp.) we assume that primary (or secondary resp.) nodes are placed according to a Poisson point process of intensity n (or m resp.) over a unit square region. We randomly choose ns (or ms resp.) nodes as the sources of multicast sessions in PaN (or SaN resp.), and for each primary source v (or secondary source v), we pick uniformly at random nd primary nodes (or md secondary nodes) as the destinations of v (or v). Above all, we assume that PaN can adopt the optimal protocol in terms of the throughput. Our main work is to design the multicast strategy for SaN by which it can achieve the optimal throughput, without any negative impact on the throughput for PaN in order sense. Depending on nd and n, we choose the optimal strategy for PaN from two ones called percolation strategy and connectivity strategy, respectively. Subsequently, we design the corresponding throughput-optimal strategy for SaN. We further derive the regimes for n, nd, m and md where the throughput for PaN and SaN can simultaneously be achieved of the upper bound of their capacities asymptotically. Specifically, we show that (1) when n = o( m (log m)2 ), nd = O( n (log n)3 ), and md = O( m (log m)3 ), PaN and SaN can simultaneously achieve the optimal throughput, i.e., the upper bounds of the capacity (Θ( √ n · nd) and Θ(√m ·md), respectively); (2) when nd = Ω( n log n ) and md = Ω( m log m ), PaN and SaN also simultaneously achieve the optimal throughput, i.e., Θ(1) for both networks. Unicast capacity (or broadcast capacity) for the cognitive network can be derived by our results as special cases by letting nd = 1 and md = 1 (or nd = n− 1 and md = m− 1).