Sebastian S. Szyszkowicz

On the Tails of the Distribution of the Sum of Lognormals

Finding the distribution of the sum of lognormal random variables is an important mathematical problem in wireless communications, as well as in many other fields. While several methods exist to approximate this distribution, their performance tends to deteriorate in both tails. Finding a good overall fit remains an open problem. Other disadvantages of these methods are their complexity and, in some cases, their limitation to particular scenarios. In this paper we examine the sum of independent lognormal random variables with arbitrary parameters. We define the concept of best lognormal fit to a tail and show what it means in terms of convergence. We restate a known result about asymptotes to the higher tail of the distribution. To our knowledge, the lower tail has not yet been studied. We give a simple closed-form expression for an asymptote to the lower tail. We also show that known methods for finding the sum of lognormals use distribution functions that do not have this asymptotic behaviour in the tails. Our results are complementary to the existing knowledge, which together can combine to solve the problem of the sum of lognormals simply and exactly. We support our results by simulations.

Limit theorem on the sum of identically distributed equally and positively correlated joint lognormals

We prove that the distribution of the sum of N identically distributed jointly lognormal random variables, where all pairs have the same strictly positive correlation coefficient, converges to a lognormal with known parameters as N becomes large. We confirm our theorem by simulations and give an application of the theorem.

Analysis of Interference from Large Clusters as Modeled by the Sum of Many Correlated Lognormals

We examine the statistical distribution of the interference produced by a cluster of very many co-channel interferers, e.g., a sensor network, or a city full of active wireless devices and access points. We consider an arbitrary statistical interferer layout and consider the interference as experienced at a given point outside (and not immediately near to) the interferer area. We model the paths as experiencing power law attenuation and lognormal correlated shadowing. It has been shown in literature that adding correlation to the shadowing model can give qualitatively different (and probably more realistic) results. Our results are mostly analytical, with a small amount of numerical integration required. Whereas simulations of very many correlated interferers are very computationally heavy, our methods complexity is independent of the number of interferers, and its precision in fact improves when increasing the number of terms.

Outage in a cellular network overlaid with an ad hoc network: The uplink case

We analyze the uplink outage probability at the randomly, but not necessarily independently nor homogeneously, distributed cellular network receivers caused by an overlaid secondary ad hoc network. Unlike the previous works that assumed exact circular exclusion regions (or guard zones), we incorporate a sensing mechanism to decrease the significant effect of the nearby interferers. We consider Rayleigh-faded links, large-scale power control, and the knowledge of the initial outages in the absence of the secondary network. We derive an upper bound on the outage probability and show it to be tight when the co-channel primary receivers are separated by relatively significant distances, which is inherently the case for the cellular network. Furthermore, we derive a closed-form expression that shows the significant impact of the sensing mechanism on the spectrum sharing gains‥ Additionally, we optimize the decision threshold, which is used by secondary network users to decide whether to transmit or not, in order to maximize the spectrum sharing gains under the given outage constraints.

Aggregate Interference Distribution From Large Wireless Networks With Correlated Shadowing: An Analytical–Numerical–Simulation Approach

As the number and variety of wireless devices sharing spectrum increases, it becomes increasingly important to characterize the sum interference that is produced by a large number of interferers. We show that, in the case of several interferers, the assumption of independent shadowing paths is very inaccurate and must be replaced by an appropriate correlation model. We choose one such model, which has desirable mathematical and physical properties, is tunable, and is particularly well suited for simulation, although our approach can also be used with other correlation models. In addition, we allow a very versatile channel and system model. The simulation cost of such systems quickly grows for large numbers of interferers due to the time and memory constraints of the Monte Carlo simulation algorithm using the classic matrix factorization (e.g., Cholesky) approach. We show how an alternative simulation approach using shadowing fields can significantly reduce the order of the computational cost. In addition, we show how judicious random sample reuse and extrapolation based on a numerical analysis of moments can be used to further simplify the simulation. Through the combination of these three approaches, using a mixture of simulation, numerical, and analytical techniques, we can obtain accurate approximations of the distribution of the total interference power while reducing computational time by factors of more than 1000. We can also make some mathematical statements about the problem, which may be useful for further developments. We argue that our model is complex enough to accommodate a good degree of realism and that our approach is a viable alternative to the pure analysis of such a complex and versatile problem.

Fitting the Modified-Power-Lognormal to the Sum of Independent Lognormals Distribution

We propose a new method for calculating a tight approximation to the distribution of the sum of independent lognormal random variables. We make use of a three-parameter modified-power-lognormal distribution function as the approximating distribution. We use theoretical results from our previous work on the tails of the distribution of the sum of lognormals to match the slope of the modified-power-lognormal function at both tails. This would not have been possible with many of the recently-proposed distribution functions, which do not behave properly in the tails. We then also use moment-matching to find the best curve match. Our method is mostly closed-form, requiring only one simple numerical integral. We compare our method with those in literature in terms of complexity and accuracy. We conclude that our method is more accurate than the simple (closed-form) methods, and much simpler to understand and implement than the more accurate methods which rely heavily on numerical integration.

Automated Placement of Individual Millimeter-Wave Wall-Mounted Base Stations for Line-of-Sight Coverage of Outdoor Urban Areas

In this letter, advanced concepts in polygon computational geometry are combined to automate the placement of thousands of millimeter-wave (mmWave) wall-mounted base stations over two large urban areas, and to gather their line-of-sight coverage statistics; this is the first large-scale fully-automated channel modeling study of mmWave cells over massive open building data. This allows for a more accurate modeling of small-cell mmWave access networks, which are currently considered as a strong candidate technology for providing gigabit rates to dense urban areas.

Separating the Effect of Independent Interference Sources with Rayleigh Faded Signal Link: Outage Analysis and Applications

We show that, for independent interfering sources and a signal link with exponentially distributed received power, the total probability of outage can be decomposed as a simple expression of the outages from the individual interfering sources. We give a mathematical proof of this result, and discuss some immediate implications, showing how it results in important simplifications to statistical outage analysis. We also discuss its application to two active topics of study: spectrum sharing and sum of interference powers (e.g., lognormal) analysis.

Analytical Modeling of Interference in Cellular Fixed Relay Networks

We develop a simple yet accurate analysis of the interference distribution in a cellular system, with particular emphasis on a two-hop fixed relay network, though the analysis may apply to much wider contexts. Similar analyses have already been proposed, but suffer from being too specific in their assumptions, are analytically difficult, consider only the uplink, and are not necessarily validated. We provide a simple closed-form solution for a wide variety of cases and validate all our theoretical curves directly by Monte-Carlo simulations of the exact same models. Our method is flexible for many channel and system parameters, and for arbitrary cellular layouts, thus it can readily be applied to a two-hop relay context

Cell Switch-Off for Networks Deployed With Variable Spatial Regularity

Cell switch-off (CSO) is considered to be a promising approach to reducing the energy consumed by cellular networks. In this letter, we set a new CSO research direction that focuses on saving energy and increasing the performance of a network—deployed with variable amounts of spatial regularity—by switching off some cells so as to maximize the spatial regularity of the remaining active cells. We propose three greedy algorithms for tackling this new problem. Improving the spatial regularity using a greedy algorithm results in either: 1) much extra energy could be saved while maintaining network performance or 2) saving the same amount of energy as the random CSO with better network performance.