Nadhir Ben Rached

A fast simulation method for the Log-normal sum distribution using a hazard rate twisting technique

The probability density function of the sum of Log-normally distributed random variables (RVs) is a well-known challenging problem. For instance, an analytical closed-form expression of the Log-normal sum distribution does not exist and is still an open problem. A crude Monte Carlo (MC) simulation is of course an alternative approach. However, this technique is computationally expensive especially when dealing with rare events (i.e. events with very small probabilities). Importance Sampling (IS) is a method that improves the computational efficiency of MC simulations. In this paper, we develop an efficient IS method for the estimation of the Complementary Cumulative Distribution Function (CCDF) of the sum of independent and not identically distributed Log-normal RVs. This technique is based on constructing a sampling distribution via twisting the hazard rate of the original probability measure. Our main result is that the estimation of the CCDF is asymptotically optimal using the proposed IS hazard rate twisting technique. We also offer some selected simulation results illustrating the considerable computational gain of the IS method compared to the naive MC simulation approach.

An Improved Hazard Rate Twisting Approach for the Statistic of the Sum of Subexponential Variates

In this letter, we present an improved hazard rate twisting technique for the estimation of the probability that a sum of independent but not necessarily identically distributed subexponential Random Variables (RVs) exceeds a given threshold. Instead of twisting all the components in the summation, we propose to twist only the RVs which have the biggest impact on the right-tail of the sum distribution and keep the other RVs unchanged. A minmax approach is performed to determine the optimal twisting parameter which leads to an asymptotic optimality criterion. Moreover, we show through some selected simulation results that our proposed approach results in a variance reduction compared to the technique where all the components are twisted.

Unified Importance Sampling Schemes for Efficient Simulation of Outage Capacity Over Generalized Fading Channels

The outage capacity (OC) is among the most important performance metrics of communication systems operating over fading channels. Of interest in the present paper is the evaluation of the OC at the output of the Equal Gain Combining (EGC) and the Maximum Ratio Combining (MRC) receivers. In this case, it can be seen that this problem turns out to be that of computing the Cumulative Distribution Function (CDF) for the sum of independent random variables. Since finding a closed-form expression for the CDF of the sum distribution is out of reach for a wide class of commonly used distributions, methods based on Monte Carlo (MC) simulations take pride of price. In order to allow for the estimation of the operating range of small outage probabilities, it is of paramount importance to develop fast and efficient estimation methods as naive MC simulations would require high computational complexity. In this line, we propose in this work two unified, yet efficient, hazard rate twisting Importance Sampling (IS) based approaches that efficiently estimate the OC of MRC or EGC diversity techniques over generalized independent fading channels. The first estimator is shown to possess the asymptotic optimality criterion and applies for arbitrary fading models, whereas the second one achieves the well-desired bounded relative error property for the majority of the well-known fading variates. Moreover, the second estimator is shown to achieve the asymptotic optimality property under the particular Log-normal environment. Some selected simulation results are finally provided in order to illustrate the substantial computational gain achieved by the proposed IS schemes over naive MC simulations.

An Efficient Simulation Scheme of the Outage Probability with Co-Channel Interference

The outage probability (OP) of the signal-to-interference-plus-noise ratio (SINR) is an important metric used to evaluate the performance of wireless communication systems operating over fading channels. One major difficulty toward assessing the OP is that, in most of the realistic scenarios, closed-form expressions cannot be derived. This is for instance the case of Log-normal fading environments, in which evaluating the OP of the SINR amounts to computing the probability that a sum of correlated Log-normal variates exceeds a given threshold. Since such a probability is not known to admit a closed-form expression, it has thus far been evaluated by several approximation techniques, the accuracies of which are unfortunately not guaranteed in the interesting region of small outage probabilities. For these regions, simulation techniques based on variance reduction algorithms can represent a good alternative, being well-recognized to be quick and highly accurate for estimating rare event probabilities. This constitutes the major motivation behind our work. More specifically, we propose an efficient importance sampling approach which is based on a covariance matrix scaling technique and illustrate its computational gain over naive Monte Carlo simulations through some selected simulation results.

On the generalization of the hazard rate twisting-based simulation approach

Estimating the probability that a sum of random variables (RVs) exceeds a given threshold is a well-known challenging problem. A naive Monte Carlo 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. An alternative approach is represented by the use of variance reduction techniques, known for their efficiency in requiring less computations for achieving the same accuracy requirement. Most of these methods have thus far been proposed to deal with specific settings under which the RVs belong to particular classes of distributions. In this paper, we propose a generalization of the well-known hazard rate twisting Importance Sampling-based approach that presents the advantage of being logarithmic efficient for arbitrary sums of 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 the proposed method with some existing techniques.

A Time-Varied Probabilistic ON/OFF Switching Algorithm for Cellular Networks

In this letter, we develop a time-varied probabilistic on/off switching planning method for cellular networks to reduce their energy consumption. It consists in a risk-aware optimization approach that takes into consideration the randomness of the user profile associated with each base station (BS). The proposed approach jointly determines 1) the instants of time at which the current active BS configuration must be updated due to an increase or decrease of the network traffic load and 2) the set of minimum BSs to be activated to serve the networks’ subscribers. Probabilistic metrics modeling the traffic profile variation are developed to trigger this dynamic on/off switching operation. Selected simulation results are then performed to validate the proposed algorithm for different system parameters.

A Unified Moment-Based Approach for the Evaluation of the Outage Probability With Noise and Interference

In this paper, we develop a novel moment-based approach for the evaluation of the outage probability (OP) in a generalized fading environment with interference and noise. Our method is based on the derivation of a power series expansion of OP of the signal-to-interference-plus-noise ratio. It does not necessitate stringent requirements, the only major ones being the existence of a power series expansion of the cumulative distribution function of the desired user power and the knowledge of the cross moments of the interferers’ powers. The newly derived formula is shown to be applicable for most of the well-practical fading models of the desired user under some assumptions on the parameters of the powers’ distributions. A further advantage of our method is that it is applicable irrespective of the nature of the fading models of the interfering powers, the only requirement being the perfect knowledge of their cross moments. In order to illustrate the wide scope of applicability of our technique, we present a convergence study of the provided formula for the Generalized Gamma and the Rice fading models. Moreover, we show that our analysis has direct bearing on recent multi-channel applications using selection diversity techniques. Finally, we assess by simulations the accuracy of the proposed formula for various fading environments.

Energy Management Optimization for Cellular Networks Under Renewable Energy Generation Uncertainty

The integration of renewable energy (RE) as an alternative power source for cellular networks has been deeply investigated in the literature. However, RE generation is often assumed to be deterministic; an impractical assumption for realistic scenarios. In this paper, an efficient energy procurement strategy for cellular networks powered simultaneously by the smart grid (SG) and locally deployed RE sources characterized by uncertain processes is proposed. For a one-day operation cycle, the mobile operator aims to reduce its total energy cost by optimizing the amounts of energy to be procured from the local RE sources and SG at each time period. Additionally, it aims to determine the amount of extra generated RE to be sold back to SG. A chance constrained optimization is first proposed to deal with the RE generation uncertainty. Then, two convex approximation approaches: 1) Chernoff and 2) Chebyshev methods, characterized by different levels of knowledge about the RE generation, are developed to determine the energy procurement strategy for different risk levels. In addition, their performances are analyzed for various daily scenarios through selected simulation results. It is shown that the higher complex Chernoff method outperforms the Chebyshev one for different risk levels set by the operator.

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.

Optimized Energy Procurement for Cellular Networks with Uncertain Renewable Energy Generation

Renewable energy (RE) is an emerging solution for reducing carbon dioxide (CO2) emissions from cellular networks. One of the challenges of using RE sources is to handle its inherent uncertainty. In this paper, a RE powered cellular network is investigated. For a one-day operation cycle, the cellular network aims to reduce energy procurement costs from the smart grid by optimizing the amounts of energy procured from their locally deployed RE sources as well as from the smart grid. In addition to that, it aims to determine the extra amount of energy to be sold to the electrical grid at each time period. Chance constrained optimization is first proposed to deal with the randomness in the RE generation. Then, to make the optimization problem tractable, two well- know convex approximation methods, namely; Chernoff and Chebyshev based-approaches, are analyzed in details. Numerical results investigate the optimized energy procurement for various daily scenarios and compare between the performances of the employed convex approximation approaches.