Stochastic Modeling of Cooperative Multi-Hop Strip Networks With Fixed Hop Boundaries

In this paper, a strip-shaped cooperative multi-hop wireless network is modeled stochastically with quasi-stationary Markov chain. The network is considered to be a fixed boundary decode-and-forward Opportunistic Large Array (OLA), where each level is of the same size and contains the same number of nodes placed randomly. The state of the system is represented by the number of nodes that decode the message in the current level. The distribution of the received power at a node is derived to formulate the transition probability matrix. For the distribution of power, a closed-form expression of the distribution of distance between a pair of nodes in disjoint levels is derived. It is seen that the distribution of distance can be well-approximated by the Weibull distribution. The Weibull approximation is then carried forward to find the distribution of the received power at a node assuming all nodes transmit with the same power and the channel has Rayleigh fading and path loss with an arbitrary exponent. The coverage and outage characteristics for various network sizes and path loss exponents are quantified. The signal-to-noise ratio (SNR) margin required for a given network coverage is determined by using the Perron-Frobenius theorem of non-negative matrices. Numerical simulations are performed to validate the theoretical results.