On the Ratio of Exponential and Generalized Gamma Random Variables with Applications to Ad Hoc SISO Networks

A Poisson point process (PPP)-based model for a single-input single-output (SISO) transmission between two randomly located nodes is developed and analyzed. The power received at a node, when a randomly deployed transmitter transmits the message signal in the presence of Rayleigh fading and path loss, is shown to be the ratio of an exponential random variable (RV) and a generalized gamma (GG) RV. The cumulative distribution function (CDF) of the received power is derived, which is used to find the outage probability at the receiver. The study is then further extended to SISO multi-hop links where the coverage probability of the network is calculated. Finally, a transmission model is proposed that saves a significant amount of energy by carefully selecting an intermediate node in a multi-hop network such that the lifetime of the network can be increased. Numerical simulations are presented to validate the theoretical models.