On the Asymptotic Number of Active Links in a Random Network
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A network of n transmitters and n receivers is considered. We assume that transmitter i aims to send data to its designated destination, receiver i. Communications occur in a single-hop fashion and destination nodes are simple linear receivers without multi-user detection. Therefore, in each time slot every source node can only talk to one other destination node. Thus, there is a total of n communication links. An important question now arises. How many links can be active in such a network so that each of them supports a minimum rate Rmin? This dissertation is devoted to this problem and tries to solve it in two di erent settings, dense and extended networks. In both settings our approach is asymptotic, meaning, we only examine the behaviour of the network when the number of nodes tends to in nity. We are also interested in the events that occur asymptotically almost surely (a.a.s.), i.e., events that have probabilities approaching one as the size of the networks gets large. In the rst part of the thesis, we consider a dense network where fading is the dominant factor a ecting the quality of transmissions. Rayliegh channels are used to model the impact of fading. It is shown that a.a.s. log(n)^2 links can simultaneously maintain Rmin and thus be active. In the second part, an extended network is considered where nodes are distant from each other and thus, a more complete model must take internode distances into account. We will show that in this case, almost all of the links can be active while maintaining the minimum rate.