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