On the Efficient Simulation of the Distribution of the Sum of Gamma–Gamma Variates With Application to the Outage Probability Evaluation Over Fading Channels

The Gamma-Gamma distribution has recently emerged in a number of applications ranging from modeling scattering and reverberation in sonar and radar systems to modeling atmospheric turbulence in wireless optical channels. In this respect, assessing the outage probability achieved by some diversity techniques over this kind of channels is of major practical importance. In many circumstances, this is related to the difficult question of analyzing the statistics of a sum of Gamma-Gamma random variables. Answering this question is not a simple matter. This is essentially because outage probabilities encountered in practice are often very small, and hence, the use of classical Monte Carlo methods is not a reasonable choice. This lies behind the main motivation of this paper. In particular, this paper proposes a new approach to estimate the left tail of the sum of Gamma-Gamma variates. More specifically, we propose robust importance sampling schemes that efficiently evaluates the outage probability of diversity receivers over Gamma-Gamma fading channels. The proposed estimators satisfy the well-known bounded relative error criterion for both maximum ratio combining and equal gain combining cases. We show the accuracy and the efficiency of our approach compared with naive Monte Carlo via some selected numerical simulations.

I. Introduction

The Gamma-Gamma distribution has recently emerged in many communication systems. Not only that it generalizes other types of distributions that are used to model the statistics of the fading channels, such as the K and the double Rayleigh distribution for instance, but it also shows a good agreement with the measurements from the experiments conducted in [1] and [2] for example. In radar and sonar systems, it has been used to model scattering [3] and reverbation [4]. It has been broadly accepted as an appropriate model for both line-of sight and non-line-of sight wireless radio-frequency channels [5], [6]. Lately, it started to gain popularity in modeling atmospheric turbulence in wireless optical channels [7], [8]. The Gamma-Gamma distribution is a compound model since it presents the product of two Gamma random variables (RVs) that model the small and large scale fluctuations, respectively [9]. It was shown that this probability density function (PDF) models accurately the atmospheric turbulence in free-space optics (FSO) over a wide range being a good fit for both weak and strong turbulence regimes [10], [11]. A concise review of the use of Gamma-Gamma to model the statistics of the signal fading channels can be found in [10].