Modeling of a Cooperative One-Dimensional Multi-Hop Network Using Quasi-Stationary Markov Chains

We consider the irreducible discrete time Markov chain, with one absorbing state, as a potential candidate to model a wireless multi-hop transmission system that does cooperative transmission at every hop. This paper describes the modeling for a special geometry where the nodes are aligned on a one-dimensional horizontal grid with equal spacing and such that the cooperating clusters are adjacent. This model can be considered a precursor to a model for an Opportunistic Large Array broadcast for the finite density case. Assuming all the nodes have equal transmit power, the successive transmissions can be modeled as a Markov chain in discrete time. We derive the transition probability matrix of the Markov chain based on the hypoexponential distribution of the received power at a given time instant. The Perron-Frobenius eigenvalue of that sub-stochastic matrix is used in formulating a bound on how far transmissions can reach with a particular relay transmit power.