Stochastic Analysis of Homogeneous Wireless Networks Assisted by Intelligent Reflecting Surfaces

In this paper, we study the impact of the existence of multiple IRSs in a homogeneous wireless network, in which all BSs, users (U), and IRSs are spatially distributed by an independent homogeneous PPP, with density $\lambda_{{\rm BS}}\rm{[BS/m^2]}$, $\lambda_{{\rm U}}\rm{[U/m^2]}$, and $\lambda_{{\rm IRS}}\rm{[IRS/m^2]}$, respectively. We utilize a uniformly random serving strategy for BS and IRS to create stochastic symmetry in the network. We analyze the performance of the network and study the effect of the existence of the IRS on the network performance. To this end, for a typical user in the system, we derive analytical upper and lower bounds on the expectation of the power (second statistical moment) of the desired signal and the interference caused by BSs and other users. After that, we obtain analytical upper bounds on the decay of the probability of the power of the desired signal and the interference for the typical user (which results in a lower bound for the cumulative distribution function (CDF)). Moreover, we derive upper bounds on the decay of the probability of the capacity of one typical user, which results in a lower bound for the outage probability. In the numerical results, we observe that the numerical calculation of the power of the desired signal and the interference is near the derived lower bounds and we show that the increment of the parameter ${(\lambda_{\rm IRS})}$ causes increment in powers of both the desired and interference signals. We also observe that the increment of the parameter ${\lambda_{\rm IRS}}$ causes the decrement of outage probability.