This paper focuses on the performance analysis of time-varying fading channels, introducing a new general metric called fade duration. Fade duration measures the time during which a signal remains below a specified threshold within a fixed time interval. To model the signal, we utilize established models for the inphase and quadrature components, employing stochastic differential equations (SDEs) to capture the continuous-time statistical properties of the fading channel. We estimate the complementary cumulative distribution function (CCDF) of the fade duration in different fading environments using Monte Carlo simulations and analyze how various system parameters impact its behavior. To enhance the efficiency of our estimates, we leverage importance sampling (IS), a well-known variance-reduction technique, for accurately estimating the tail of the CCDF. The proposed IS scheme involves solving a high-dimensional controlled partial differential equation. To overcome the curse of dimensionality, we use Markovian projection to develop a novel one-dimensional SDE for signal envelope variations, enhancing the computational feasibility of IS. We present numerical results for the CCDF of fade duration in Rayleigh and Rice environments using our proposed IS estimators.

We propose a unified rare-event estimator for the performance evaluation of wireless communication systems. The estimator is derived from the well-known multilevel splitting algorithm. In its original form, the splitting algorithm cannot be applied to the simulation and estimation of time-independent problems, because splitting requires an underlying continuous-time Markov process whose trajectories can be split. We tackle this problem by embedding the static problem of interest within a continuous-time Markov process, so that the target time-independent distribution becomes the distribution of the Markov process at a given time instant. The main feature of the proposed multilevel splitting algorithm is its large scope of applicability. For illustration, we show how the same algorithm can be applied to the problem of estimating the cumulative distribution function (CDF) of sums of random variables (RVs), the CDF of partial sums of ordered RVs, the CDF of ratios of RVs, and the CDF of weighted sums of Poisson RVs. We investigate the computational efficiency of the proposed estimator via a number of simulation studies and find that it compares favorably with existing estimators.

We consider the evaluation of the outage probability (OP) for L-branch equal gain combining diversity receivers operating over fading channels, i.e. equivalently the cumulative distribution function (CDF) of the sum of the L channel envelopes. Generally, closed-form expressions of the OP values are out of reach. Moreover, the use of naive Monte Carlo (MC) simulations is not a good alternative since it is expensive in terms of number of samples when small values of OP are considered. In this paper, we use the concept of importance sampling (IS), being known to yield accurate estimates using few number of simulations runs. The proposed IS scheme is essentially based on sample rejection where the IS probability density function (PDF) is the truncation of the underlying PDF over the L dimensional sphere. It assumes the knowledge of the CDF of the sum of the L channel gains in a closed-form expression. Such an assumption is not restrictive since it holds for various challenging fading models. As an illustration, we apply the proposed estimator to the cases of independent Rayleigh, correlated Rayleigh, and independent and identically distributed Rice fading channels and prove that it achieves the well-desired bounded relative error property. Finally, we validate these theoretical results through some selected experiments.

We consider the problem of evaluating outage probability (OP) values of generalized selection combining diversity receivers over fading channels. This is equivalent to computing the cumulative distribution function (CDF) of the sum of order statistics. Generally, closed-form expressions of the CDF of order statistics are unavailable for many practical distributions. Moreover, the naive Monte Carlo method requires a substantial computational effort when the probability of interest is sufficiently small. In the region of small OP values, we propose instead an efficient, yet universal, importance sampling (IS) estimator that yields a reliable estimate of the CDF with small computing cost. The main feature of the proposed IS estimator is that it has bounded relative error under a certain assumption that is shown to hold for most of the challenging distributions. Moreover, an improvement of this estimator is proposed for the Pareto and the Weibull cases. Finally, the efficiency of the proposed estimators are investigated through various numerical experiments.

We consider the problem of evaluating the cumulative distribution function (CDF) of the sum of order statistics, which serves to compute outage probability (OP) values at the output of generalized selection combining receivers. Generally, closed-form expressions of the CDF of the sum of order statistics are unavailable for many practical distributions. Moreover, the naive Monte Carlo (MC) method requires a substantial computational effort when the probability of interest is sufficiently small. In the region of small OP values, we instead propose two effective variance reduction techniques that yield a reliable estimate of the CDF with small computing cost. The first estimator, which can be viewed as an importance sampling estimator, has bounded relative error under a certain assumption that is shown to hold for most of the challenging distributions. A possible improvement of this estimator is then proposed for the Pareto and the Weibull cases. The second is a conditional MC estimator that achieves the bounded relative error property for the generalized Gamma case and the logarithmic efficiency for the Log-normal case. Finally, the efficiency of these estimators is compared via various numerical simulations.

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.

The outage probability (OP) of the signal-to-interference-plus-noise ratio (SINR) is an important metric that is used to evaluate the performance of wireless systems. One difficulty toward assessing the OP is that, in realistic scenarios, closed-form expressions cannot be derived. This is, for instance, the case of the Log-normal environment, 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 does not admit a closed-form expression, it has thus far been evaluated by several approximation techniques, the accuracies of which are not guaranteed in the region of small OPs. For these regions, simulation techniques based on variance reduction algorithms are a good alternative, being quick and highly accurate for estimating rare event probabilities. This constitutes the major motivation behind this paper. More specifically, we propose a generalized hybrid importance sampling scheme, based on a combination of a mean shifting and a covariance matrix scaling, to evaluate the OP of the SINR in a Log-normal environment. We further our analysis by providing a detailed study of two particular cases. Finally, the performance of these techniques is performed both theoretically and through various simulation results.

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.

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.

In this paper, we develop a generalized moment-based approach for the evaluation of the outage probability (OP) in the presence of co-channel interference and additive white Gaussian noise. The proposed method allows the evaluation of the OP of the signal-to-interference-plus-noise ratio by a power series expansion in the threshold value. Its main advantage is that it does not require a particular distribution for the interference channels. The only necessary ingredients are a power series expansion for the cumulative distribution function of the desired user power and the cross-moments of the interferers' powers. These requirements are easily met in many practical fading models, for which the OP might not be obtained in closed-form expression. For a sake of illustration, we consider the application of our method to the Rician fading environment. Under this setting, we carry out a convergence study of the proposed power series and corroborate the validity of our method for different values of fading parameters and various numbers of co-channel interferers.

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.

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.

The outage capacity (OC) is among the most important performance metrics of communication systems over fading channels. The evaluation of the OC, when Equal Gain Combining (EGC) or Maximum Ratio Combining (MRC) diversity techniques are employed, boils down to computing the Cumulative Distribution Function (CDF) of the sum of channel envelopes (equivalently amplitudes) for EGC or channel gain (equivalently squared enveloped/amplitudes) for MRC. Closed-form expressions of the CDF of the sum of many generalized fading variates are generally unknown and constitute open problems. In this paper, we develop a unified hazard rate twisting Importance Sampling (IS) based approach to efficiently estimate the CDF of the sum of independent arbitrary variates. The proposed IS estimator is shown to achieve an asymptotic optimality criterion, which clearly guarantees its efficiency. Some selected simulation results are also shown to illustrate the substantial computational gain achieved by the proposed IS scheme over crude Monte-Carlo simulations.