SINR Outage Evaluation in Cellular Networks: Saddle Point Approximation (SPA) Using Normal Inverse Gaussian (NIG) Distribution

Signal-to-noise-plus-interference ratio (SINR) outage probability is among one of the key performance metrics of a wireless cellular network. In this paper, we propose a semi-analytical method based on saddle point approximation (SPA) technique to calculate the SINR outage of a wireless system whose SINR can be modeled in the form $\frac{\sum{i=1}^M Xi}{\sum{i=1}^N Yi +1}$ where $Xi$ denotes the useful signal power, $Yi$ denotes the power of the interference signal, and $\sum{i=1}^M Xi$, $\sum{i=1}^N Yi$ are independent random variables. Both $M$ and $N$ can also be random variables. The proposed approach is based on the saddle point approximation to cumulative distribution function (CDF) as given by \tit{Wood-Booth-Butler formula}. The approach is applicable whenever the cumulant generating function (CGF) of the received signal and interference exists, and it allows us to tackle distributions with large skewness and kurtosis with higher accuracy. In this regard, we exploit a four parameter \tit{normal-inverse Gaussian} (NIG) distribution as a base distribution. Given that the skewness and kurtosis satisfy a specific condition, NIG-based SPA works reliably. When this condition is violated, we recommend SPA based on normal or symmetric NIG distribution, both special cases of NIG distribution, at the expense of reduced accuracy. For the purpose of demonstration, we apply SPA for the SINR outage evaluation of a typical user experiencing a downlink coordinated multi-point transmission (CoMP) from the base stations (BSs) that are modeled by homogeneous Poisson point process. We characterize the outage of the typical user in scenarios such as (a)~when the number and locations of interferers are random, and (b)~when the fading channels and number of interferers are random. Numerical results are presented to illustrate the accuracy of the proposed set of approximations.