Millimeter wave (mmWave) holds promise as a carrier frequency for fifth generation cellular networks. Because mmWave signals are sensitive to blockage, prior models for cellular networks operated in the ultra high frequency (UHF) band do not apply to analyze mmWave cellular networks directly. Leveraging concepts from stochastic geometry, this paper proposes a general framework to evaluate the coverage and rate performance in mmWave cellular networks. Using a distance-dependent line-of-site (LOS) probability function, the locations of the LOS and non-LOS base stations are modeled as two independent non-homogeneous Poisson point processes, to which different path loss laws are applied. Based on the proposed framework, expressions for the signal-to-noise-and-interference ratio (SINR) and rate coverage probability are derived. The mmWave coverage and rate performance are examined as a function of the antenna geometry and base station density. The case of dense networks is further analyzed by applying a simplified system model, in which the LOS region of a user is approximated as a fixed LOS ball. The results show that dense mmWave networks can achieve comparable coverage and much higher data rates than conventional UHF cellular systems, despite the presence of blockages. The results suggest that the cell size to achieve the optimal SINR scales with the average size of the area that is LOS to a user.

Coverage and Rate Analysis for Millimeter-Wave Cellular Networks

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Positive Operators

Zeros of orthogonal polynomials

We provide a comprehensive overview of mathematical models and analytical techniques for millimeter wave (mmWave) cellular systems. The two fundamental physical differences from conventional sub-6-GHz cellular systems are: 1) vulnerability to blocking and 2) the need for significant directionality at the transmitter and/or receiver, which is achieved through the use of large antenna arrays of small individual elements. We overview and compare models for both of these factors, and present a baseline analytical approach based on stochastic geometry that allows the computation of the statistical distributions of the downlink signal-to-interference-plus-noise ratio (SINR) and also the per link data rate, which depends on the SINR as well as the average load. There are many implications of the models and analysis: 1) mmWave systems are significantly more noise-limited than at sub-6 GHz for most parameter configurations; 2) initial access is much more difficult in mmWave; 3) self-backhauling is more viable than in sub-6-GHz systems, which makes ultra-dense deployments more viable, but this leads to increasingly interference-limited behavior; and 4) in sharp contrast to sub-6-GHz systems cellular operators can mutually benefit by sharing their spectrum licenses despite the uncontrolled interference that results from doing so. We conclude by outlining several important extensions of the baseline model, many of which are promising avenues for future research.

Modeling and Analyzing Millimeter Wave Cellular Systems

THE theory of Fourier integrals arises out of the elegant pair of reciprocal formulae The Laplace Transform By David Vernon Widder. (Princeton Mathematical Series.) Pp. x + 406. (Princeton: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

The Laplace Transform

We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, star-shaped, and superadditive functions which are related to Γ and?.

https://www.ams.org/mcom/1997-66-217/S0025-5718-97-00807-7/S0025-5718-97-00807-7.pdf

On some inequalities for the gamma and psi functions

Let p ¬= 1 be a positive real number. We determine all real numbers α = α(p) and β = β(p) such that the inequalities formula math. formula math. are valid for all x > 0. And, we determine all real numbers a and b such that - log(1 - e -ax ) ≤ √ x ∞ e-t/t ≤ - log(1 - e -bx ) hold for all > 0.

/pdf/on-some-inequalities-for-the-incomplete-gamma-function-2ydct08q1a.pdf

On some inequalities for the incomplete gamma function

Monotonicity theorems and inequalities for the complete elliptic integrals

We prove: (i) Let F n (r) = P n (x)[e - (1 + 1/x) x ] and G n (x) = P n (x)[(1 + 1/x) x + 1 - e], where P n (x) = x n + Σ n - 1 v = 0 c v , x v is a polynomial of degree n > 1 with real coefficients. F n is completely monotonic if and only if n = 1 and c 0 ≥ 11/12; and G n is completely monotonic if and only if n = 1 and c 0 ≥ 1/12. (ii) The functions x → e - (1 + 1/x) x and x → (1 + 1/x) x + 1 - e are Stieltjes transforms and in particular they are completely monotonic. (iii) Let a > 0 and b be real numbers. The function x → (1 + a/x) x + b - e a is completely monotonic if and only if a < 2b. Part (i) extends and complements a recently published result of Sandor and Debnath, while part (iii) generalizes a theorem of Schur.

Some classes of completely monotonic functions

We present various new inequalities involving the logarithmic mean $ L(x,y)=(x-y)/(\log{x}-\log{y}) $
 , the identric mean $ I(x,y)=(1/e)(x^x/y^y)^{1/(x-y)} $
 , and the classical arithmetic and geometric means, $ A(x,y)=(x+y)/2 $
 and $ G(x,y)=\sqrt{xy} $
 . In particular, we prove the following conjecture, which was published in 1986 in this journal. If $ M_r(x,y)= (x^r/2+y^r/2)^{1/r}(r\neq{0}) $
 denotes the power mean of order r, then $ M_c(x,y)(\frac{1}{2}(L(x,y)+I(x,y)) {(x,y>0,\, x\neq{y})} $
 with the best possible parameter $ c=(\log{2})/(1+\log{2}) $
 .

Horst Alzer

Papers

On some inequalities for the gamma and psi functions

On some inequalities for the incomplete gamma function

Monotonicity theorems and inequalities for the complete elliptic integrals

Some classes of completely monotonic functions

Inequalities for Means in Two Variables