A stochastic approach in modeling cooperative line networks

We consider a quasi-stationary Markov chain as a model for a decode and forward wireless multi-hop cooperative transmission system that forms successive Opportunistic Large Arrays (OLAs). This paper treats a linear network topology, where the nodes form a one-dimensional horizontal grid with equal spacing. In this OLA approach, all nodes are intended to decode and relay. We derive the transition probability matrix of the Markov chain based on the hypoexponential distribution of the received power at a given time instant assuming that all the nodes have equal transmit power and the channel has Rayleigh fading and path loss with an arbitrary exponent. The Perron-Frobenius eigenvalue and the corresponding eigenvector of the sub-stochastic matrix indicates the signal-to-noise ratio (SNR) margin that enables a given hop distance.