A Quasi-Stationary Markov Chain Model of a Cooperative Multi-Hop Linear Network

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. Therefore, the method has potential application as a high-reliability and low-latency approach for broadcasting in a line-shaped network, or unicasting along a pre-designated route. 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 state is represented as a ternary word, which indicates which nodes have decoded in the present hop, in a previous hop, or have not yet decoded. 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.