A Continuum Model for Route Optimization in Large-Scale Inhomogeneous Multi-Hop Wireless Networks

Multi-hop route optimization in large-scale inhomogeneous networks is typically NP-hard, for most problem formulations, requiring the application of heuristics which, despite their relatively low processing complexity, find suboptimal solutions. Where optimal solutions can be determined by Lagrangian based constrained optimization techniques for example, the processing complexity typically scales like O(N³), N being the number of relays employed. Here, we propose an alternative approach to route optimization by considering the limit of infinite relay node density to develop a continuum model, which yields an optimized equivalent continuous relay path. The model is carefully constructed to maintain a constant connection density even though the node density scales without bound. This leads to a formulation for minimizing the end-to-end outage probability that can be solved using methods from the calculus of variations. With the continuum model, we show that the processing complexity scales linearly with the number of points that sample the continuous path, which can be lower than the number of relay nodes in a large scale network. We demonstrate the effectiveness of this new approach and its potential by considering a network subjected to point sources of interference.