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# Eindhoven Stochastics Seminar

## May 26, 2015, 00:00

**Philip Whiting**

**Compute and Forward for Wireless Networks – Scheduling and Capacity in Heterogenous Networks **

The first part of the talk presents high SINR asymptotics for Compute and Forward Networks (COFN). For this talk, a COFN is a network with a source node where information (in the form of messages) is produced, intermediate nodes which acts as relays arranged as $K$ columns, and after a succession of these, a sink node where the information (messages) are decoded. The salient characteristic of such networks is that the relay nodes compute (with arbitrary reliability) functions of the messages which are inverted at the sink node to obtain the messages which were sent. The maximum rate at which this can be done, we might call the computational capacity. It is shown that the maximum number of Degrees of Freedom i.e. maximum capacity (relative to the usual Shannon formula) can be achieved, through a certain scheme outlined in the talk. This is shown under the assumption that there is perfect knowledge of the channel state at each stage of the network. Exploiting this fact functions of the original messages are encoded as monomials at each transmitter and then sent on to the next stage, under the scheme. The scheme is completely impractical but other alternate more practical schemes are being investigated. These are based on lattice coding. In the second part of the talk, a definition will be given for Capacity in Heterogeneous Wireless Networks. It is shown that fixed schemes can attain this capacity, provided the network is offered stationary offered traffic. Adaptive schemes offer the prospect of superior delay performance and I will discuss two of these. The first supposes that the so called Cell Range Expansion (CRE)is fixed but then determines the Almost Blanking Subframe (ABS) fraction according to how mobiles are distributed amongst the various pico cells and the macro. The second scheme adapts both the CRE and the ABS and can be shown to be throughput optimal with the limit stationary distribution having geometric moments.