Estimating the probability that a sum of random variables (RVs) exceeds a given threshold is a well-known challenging problem. Closed-form expressions for the sum distribution do not generally exist, which has led to an increasing interest in simulation approaches. A crude Monte Carlo (MC) simulation is the standard technique for the estimation of this type of probability. However, this approach is computationally expensive, especially when dealing with rare events. Variance reduction techniques are alternative approaches that can improve the computational efficiency of naive MC simulations. We propose an Importance Sampling (IS) simulation technique based on the well-known hazard rate twisting approach, that presents the advantage of being asymptotically optimal for any arbitrary RVs. The wide scope of applicability of the proposed method is mainly due to our particular way of selecting the twisting parameter. It is worth observing that this interesting feature is rarely satisfied by variance reduction algorithms whose performances were only proven under some restrictive assumptions. It comes along with a good efficiency, illustrated by some selected simulation results comparing the performance of our method with that of an algorithm based on a conditional MC technique.

In this paper, we investigate the simultaneous wireless information and power transfer (SWIPT) for the two-hop Multiple-Input Multiple-Output (MIMO) Amplify-and-Forward (AF) relay communication systems with the multiantenna energy harvesting relay. We derive the optimal source and relay covariance matrices to characterize the achievable region between the sourcedestination rate and the harvested energy at the relay, namely Rate-Energy (R-E) region. In this context, we consider the ideal scenario where the energy harvester (EH) receiver and the information decoder (ID) receiver at the relay can simultaneously decode the information and harvest the energy at the relay. Then, we consider more practical schemes which are the power splitting (PS) and the time switching (TS) which separate the EH and ID transfer over the power domain and the time domain, respectively.