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Interference in Cellular Networks: Sum of Lognormals Modeling
, 2007
"... ii We examine the existing methods for evaluating the distribution of the sum of lognormal random variables, focusing on closedform results. We find that there are no results in literature that are both simple and accurate. We then derive a new closedform expression for the lower tail of the distr ..."
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ii We examine the existing methods for evaluating the distribution of the sum of lognormal random variables, focusing on closedform results. We find that there are no results in literature that are both simple and accurate. We then derive a new closedform expression for the lower tail of the distribution, and use it to construct a new method using a powerlognormal distribution. We apply both basic momentmatching and our new method the problem of the total interference power in a cellular system. For both methods, we derive equations that find the interference distribution essentially in closed form, using minimal numerical integration. We apply both methods to the uplink and downlink in systems with and without power control, for various cellular layouts, channel models and user activity probability. We compare distributions obtained by MonteCarlo simulation directly with those obtained by our method, and find very good matches in many useful cases. iii Acknowledgements
Regular Analog/RF Integrated Circuits Design Using Optimization With Recourse Including Ellipsoidal Uncertainty
, 2008
"... Abstract—Long design cycles due to the inability to predict silicon realities are a wellknown problem that plagues analog/RF integrated circuit product development. As this problem worsens for nanoscale IC technologies, the high cost of design and multiple manufacturing spins causes fewer products ..."
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Abstract—Long design cycles due to the inability to predict silicon realities are a wellknown problem that plagues analog/RF integrated circuit product development. As this problem worsens for nanoscale IC technologies, the high cost of design and multiple manufacturing spins causes fewer products to have the volume required to support fullcustom implementation. Design reuse and analog synthesis make analog/RF design more affordable; however, the increasing process variability and lack of modeling accuracy remain extremely challenging for nanoscale analog/RF design. We propose a regular analog/RF IC using metalmask configurability design methodology Optimization with Recourse of Analog Circuits including Layout Extraction (ORACLE), which is a combination of reuse and shareduse by formulating the synthesis problem as an optimization with recourse problem. Using a twostage geometric programming with recourse approach, ORACLE solves for both the globally optimal shared and applicationspecific variables. Furthermore, robust optimization is proposed to treat the design with variability problem, further enhancing the ORACLE methodology by providing yield bound for each configuration of regular designs. The statistical variations of the process parameters are captured by a confidence ellipsoid. We demonstrate ORACLE for regular Low Noise Amplifier designs using metalmask configurability, where a range of applications share common underlying structure and applicationspecific customization is performed using the metalmask layers. Two RF oscillator design examples are shown to achieve robust designs with guaranteed yield bound. Index Terms—Configurable design, optimization with recourse, robustness, statistical optimization. I.
Constantine CaramanisRobust Algorithms for Area and Power Optimization of Digital Integrated Circuits under Variability
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Fitting the Modified–Power–Lognormal to the Sum of Independent Lognormals Distribution
"... Abstract—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 prev ..."
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Abstract—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. Index Terms—sum of lognormals, interference analysis I.
1569004089 1 On the Tails of the Distribution of the Sum of Lognormals
"... Abstract—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 tail areas. Findin ..."
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Abstract—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 tail areas. 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 closedform 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. Index Terms—interference statistics, sum of lognormals, tail distribution. T I.
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"... 1Accurate and fast approximations of momentgenerating functions and their inversion for lognormal and similar distributions∗ ..."
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1Accurate and fast approximations of momentgenerating functions and their inversion for lognormal and similar distributions∗
A Simple Discrete Approximation for the Renewal Function
"... Background: The renewal function is widely useful in the areas of reliability, maintenance and spare component inventory planning. Its calculation relies on the type of the probability density function of component failure times which can be, regarding the region of the component lifetime, modelled ..."
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Background: The renewal function is widely useful in the areas of reliability, maintenance and spare component inventory planning. Its calculation relies on the type of the probability density function of component failure times which can be, regarding the region of the component lifetime, modelled either by the exponential or by one of the peakshaped density functions. For most peakshaped distribution families the closed form of the renewal function is not available. Many approximate solutions can be found in the literature, but calculations are often tedious. Simple formulas are usually obtained for a limited range of functions only. Objectives: We propose a new approach for evaluation of the renewal function by the use of a simple discrete approximation method, applicable to any probability density function. Methods/Approach: The approximation is based on the well known renewal equation. Results: The usefulness is proved through some numerical results using the normal, lognormal, Weibull and gamma density functions. The accuracy is analysed using the normal density function. Conclusions: The approximation proposed enables simple and fairly accurate calculation of the renewal function irrespective of the type of the probability density function. It is especially applicable to the peakshaped density functions when the analytical solution hardly ever exists.
Optimal Approximations for Risk Measures of Sums of Lognormals based on Conditional Expectations
, 2007
"... In this paper we investigate approximations for the distribution function of a sum S of lognormal random variables. These approximations are obtained by considering the conditional expectation E[S  Λ] of S with respect to a conditioning random variable Λ. The choice for Λ is crucial in order to obt ..."
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In this paper we investigate approximations for the distribution function of a sum S of lognormal random variables. These approximations are obtained by considering the conditional expectation E[S  Λ] of S with respect to a conditioning random variable Λ. The choice for Λ is crucial in order to obtain accurate approximations. The different alternatives for Λ that have been proposed in literature to date are ‘global ’ in the sense that Λ is chosen such that the entire distribution of the approximation E[S  Λ] is ‘close ’ to the corresponding distribution of the original sum S. In an actuarial or a financial context one is often only interested in a particular tail of the distribution of S. Therefore in this paper we propose approximations E[S  Λ] which are only locally optimal, in the sense that the relevant tail of the distribution of E[S  Λ] is an accurate approximation for the corresponding tail of the distribution of S. Numerical illustrations reveal that local optimal choices for Λ can improve the quality of the approximations in the relevant tail significantly. We also explore asymptotic properties of the approximations E[S  Λ] and investigate links with results from Asmussen & RoyasNandayapa (2005). Finally, we briefly adress the suboptimality of Asian options from the point of view of risk averse decision makers with a fixed investment horizon.
ModifiedPowerLognormal Approximation to the Sum of Lognormals Distribution
"... We propose a new method for calculating a tight approximation to the distribution of the sum of independent lognormal random variables. We use a threeparameter modifiedpowerlognormal distribution function as the approximating distribution. We use theoretical results from our previous work on the ..."
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We propose a new method for calculating a tight approximation to the distribution of the sum of independent lognormal random variables. We use a threeparameter modifiedpowerlognormal 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 modifiedpowerlognormal function at both tails. This would not have been possible with many of the recentlyproposed distribution functions, which do not behave properly in the tails. We then use momentmatching to find the best curve match. Our method is mostly closedform, 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 (closedform) methods, and much simpler to understand and implement than the more accurate methods which rely heavily on numerical integration. sum of lognormals, interference analysis