Fibonacci sequence and cascaded directed relay networks with time-division-duplex constraint

Consider K-hop cascaded directed relay networks with time-division duplex (TDD), where the source, the relays, and the destination form a directed chain, and each node only receives the transmissions from its upstream node in the chain. We quantitatively investigate the effect of TDD constraint, which obviously prohibits the use of some network states and reduces the whole transmission rate. Firstly we define a new S sequence based on the famous Fibonacci sequence, which is proven to exactly characterize the amount of all feasible network states. We further find Fibonacci number F_K is equal to the amount of new feasible network states when hop K is added and activated from K - 1 hop networks. Recursive construction method to get all feasible states is also provided, along with some important properties. Then we formulate the scheduling problem into a linear program and find the maximum achievable rate with decode-and-forward relay strategy is r* = min {C₁C₂/C₁+C₂, C₂C₃/C₂+C₃,..., C_K-1C_K/C_K-1+C_k}, where C_k is the capacity of hop k. An example network is uWto demonstrate the scheduling process.