Statistical analysis of the best relay location in a random two-way relay network with multi-slope path loss

This paper presents an exact statistical analysis for the best relay location in a random two-way relay network whose channels are subject to multi-slope path loss and Rayleigh fading. In this network, relays are distributed randomly as a two-dimensional Poisson point process, and the best relay is selected according to a max–min criterion. The midpoint between the two sources is adopted as a reference point and the relays are indexed such that the closer to the reference point, the smaller is the index. An exact analytical expression is derived for the probability that the � th nearest relay to the reference point is the best relay. Then, a detailed statistical analysis for the location of the best relay with respect to the reference point is presented. In each case, a low-complexity approximation is also derived and its accuracy is verified by numerical simulation. The proposed analysis demonstrates the counter-intuitive fact that the best relay in a random two-way relay network is quite unlikely to be located on, or very close to the reference point. Numerical results also suggest that the proposed analysis can be used to restrict the search region for finding the best relay.