Wireless Ad Hoc and Sensor Networks

Distributed Power Control of Wireless Cellular and Peer-to-Peer Networks 213are extremely random and do not offer easy analysis. In wireless networks,channel uncertainties such as path loss, the shadowing, and Rayleighfading can attenuate the power of the signal at the receiver and thus causevariations in the received SIR and, therefore, degrading the performanceof any DPC scheme. It is important to understand these uncertaintiesbefore the development of a DPC scheme.5.4.1.1 Path LossAs presented earlier, if only path loss is considered, the power attenuationis taken to follow the inverse fourth power law (Aein 1973):gij=gdij n(5.44)where g is a constant usually equal to 1 and d ij is the distance betweenthe transmitter of the jth link to the receiver of the ith link, and n is thepath loss exponent. A number of values for n have been proposed fordifferent propagation environments, depending on the characteristics ofthe communication medium. A value of n = 4 is taken in our simulations,which is commonly used to model path loss in an urban environment.Further, without user mobility, is a constant.g ij5.4.1.2 ShadowingHigh buildings, mountains, and other objects block the wireless signals.A blind area is often formed behind a high building or in the middle oftwo buildings. This is often seen especially in large urban areas. Theterm 10 01 . ζ is often used to model the attenuation of the shadowing to thereceived power (Canchi and Akaiwa 1999, Hashem and Sousa 1998),where ς is assumed to be a Gaussian random variable.5.4.1.3 Rayleigh FadingIn mobile radio channels, the Rayleigh distribution is commonly used todescribe the statistical time-varying nature of the received envelope of aflat fading signal, or the envelope of an individual multipath component.The Rayleigh distribution has a probability density function (pdf) givenby (Rappaport 1999)⎧ x ⎛2x ⎞⎪ exp − ( x )px ( ) =⎜ ⎟ 0 ≤ ≤∞22⎨σ⎝ 2σ⎠⎪⎩ 0 ( x < 0)(5.45)