Multicast Throughput Order of Network Coding in Wireless Ad-hoc Networks

Abstract

We consider a network with n nodes distributed uniformly in a unit square. We show that, under the protocol model, when n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> = Ω (log(n) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1+α</sup> ) out of the n nodes, each act as source of independent information for a multicast group consisting of m randomly chosen destinations, the per-session capacity in the presence of network coding (NC) has a tight bound of Θ(√n/n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> √mlog(n)) when m = O(n/log(n)) and Θ(1/n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> ) when m = Ω(n/log(n)). In the case of the physical model, we consider n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> = n and show that the per-session capacity under the physical model has a tight bound of Θ(1/√mn) when m = O(n/(log(n)) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ), and Θ(1/n) when m = Ω(n/log(n)). Prior work has shown that these same order bounds are achievable utilizing only traditional store-and-forward methods. Consequently, our work implies that the network coding gain is bounded by a constant for all values of m. For the physical model we have an exception to the above conclusion when m is bounded by O(n/(log(n)) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) and Ω(n/log(n)). In this range, the network coding gain is bounded by O((log(n)) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> ).