On the Modeling of Randomized Distributed Cooperation for Linear Multi-Hop Networks
A one-dimensional cooperative network is modeled stochastically, such that the nodes are randomly placed according to a Bernoulli process. A discrete time quasi-stationary Markov chain model is considered to characterize the multi-hop transmissions and its transition probability matrix has been derived. By the Perron-Frobenious theorem, the eigen-decomposition of the matrix gives useful information about the coverage of the network and Signal-to-Noise Ratio (SNR) margin that is required for obtaining a given quality of service or packet delivery ratio. An SNR penalty for the random placement of nodes, compared to regular placement, is quantified.