# The Distribution of Path Losses for Uniformly Distributed Nodes in a Circle

Authors: Zubin Bharucha; Harald Haas
Journal: Research Letters in Communications
 (2008)

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## Abstract

When simulating a wireless network, users/nodes are usually assumed to be distributed uniformly in space. Path losses between nodes in a simulated network are generally calculated by determining the distance between every pair of nodes and applying a suitable path loss model as a function of this distance (power of distance with an environment-specific path loss exponent) and adding a random component to represent the log-normal shadowing. A network withN nodes consists of N(N−1)/2 path loss values. In order to generate statistically significant results for system-level simulations, Monte Carlo simulations must be performed where the nodes are randomly distributed at the start of every run. This is a time-consuming operation which need not be carried out if the distribution of path losses between the nodes is known. The probability density function (pdf) of the path loss between the centre of a circle and a node distributed uniformly within a the circle is derived in this work.


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## Body

## 1. Introduction

System-level computer simulations of wireless, mobile
networks are commonplace in current research in the wireless communications
field. Users are oftentimes uniformly distributed in hexagonal cells
in a system simulation. In order to find the signal-to-interference-and-noise
ratio (SINR) on any link, path loss information must be known between the two
communicating entities as well as the entity in question and all other
potential interfering entities. Various empirically obtained path loss models exist which
simulate varying propagation environments as shown in [1–4]. In [5], the authors derive the probability density function
(pdf) of the distance between two nodes within communication range of one
another (based on the path loss between them). This is then extended to
calculating the number of communicable nodes in the vicinity of the node in
question. However, to the best of the authors' knowledge, no work has been done
to analytically model the distribution of path losses between uniformly
distributed nodes in a network (irrespective of whether they can sustain
communication between one another or not).Calculating the path loss between the nodes in a
cellular system is usually done by actually distributing the nodes uniformly,
calculating the distances between them and applying an appropriate path loss
model to these distances. This must be repeated many times in order to get a
statistically significant result. This operation is computationally expensive
and simulation runtimes can be drastically reduced if this step could be
skipped. In order to do so, knowledge of the distribution of path losses
between uniformly distributed nodes in a hexagon is essential.In this letter, the pdf of the distribution of path
losses between the centre of a circle and uniformly distributed nodes within it
is derived. This novel derivation can be used to find the distribution of a
whole class of path loss models. A circular scenario is preferred over a
hexagonal one because the derivation of the aforementioned pdf is
straightforward for the circular case. Furthermore, it is also shown in this
paper that the theory derived for a circular scenario agrees with the
simulations for the hexagonal case very closely and is therefore a very good
approximation.The derivation of the pdf is shown in Section2. The
comparisons between theory and simulation and discussions thereof are shown in
Section 3. Section 4 contains concluding remarks.


## 2. Distribution of Path Losses for Uniformly Distributed Users in a Circle

Path loss is
generally represented as some power of distance,γ,
plus a random variation about this power law due to shadowing [1]. Beyond some close-in
distance d0,
the path loss (in dB) can be written as(1)L=a+10γ⁢log⁡10(dd0)+ξ;d≥d0,where a is an intercept which is the free-space path
loss at distance d0,d is the distance between the two points, and ξ is the shadow fading variation about the
linear relationship in the log domain and is a zero-mean, normally distributed
random variable with standard deviation σ,
that is, 𝒩(0,σ2).
For the sake of convenience, in (1), we make the substitutions b=10γ and X=d/d0.
Thus, the path loss between two points separated by Xd0 meters is written as(2)L=a+blog⁡10(X)+ξ.For a circle of radius R having uniform node distribution, it is known
that the pdf of the distance of any point from the centre, x,
is [6](3)fX(x)=2xR2,x∈[0,R].The angular distribution is
uniform on [0,2π].
Deriving the pdf of L involves the addition of two random variables
and a constant. First we find the pdf of the random variable Y=b⁢log⁡10X using the transformation of random variables.
Thus,(4)fY(y)=2(ln⁡10)102y/bbR2,y∈(−∞,b⁢log⁡10R].Next, we
calculate the pdf of (5)Z=b⁢log⁡10X︷Y+𝒩(0,σ2)︷W.The pdf of W is known to be(6)fW(w)=1σ2πexp⁡(−w22σ2).Since this is an addition of two
independent random variables (Y and W), we can use the convolution integral to find
the pdf of Z,
that is, fZ(z)=fY(y)⋆fW(w) (where ⋆ is the convolution operator):(7)fZ(z)=∫−∞∞fW(w)fY(z−w)dw=2(ln⁡10)2πσbR2∫A∞exp⁡(−w22σ2)102(z−w)/bdw=2(ln⁡10)2πσbR2exp⁡{2b(ln⁡10)z+2σ2(ln⁡10)2b2}×∫A∞exp⁡{−(w+(2σ2(ln⁡10))/b)22σ2}dw,where A=z−blog⁡10R.
Let k=w+(2σ2(ln⁡10))/b.
Using this in (7), then(8)fZ(z)=2(ln⁡10)2πσbR2exp⁡{2b(ln⁡10)z+2σ2(ln⁡10)2b2}×∫B∞exp⁡{−k22σ2}dk,where B=z−b⁢log⁡10R+(2σ2(ln⁡10))/b.
Let l=k/2σ.
Decomposing the integral into the standard form of the error
function, then(9)fZ(z)=2(ln⁡10)2πσbR2exp⁡{2b(ln⁡10)z+2σ2(ln⁡10)2b2}×∫C∞exp⁡{−l2}2σdl,where (10)C=z2σ−b⁢log⁡10R2σ+2σ(ln⁡10)b.Using the definition of the error function as follows:(11)erf(x)=2π∫0xe−t2dt,we get(12)fZ(z)=(ln⁡10)bRexp⁡{2b(ln⁡10)z+2σ2(ln⁡10)2b2}×{1−erf(C)}.Finally, we solve for the pdf
of(13)L=a+b⁢log⁡10X+𝒩(0,σ2)︷Z.This final step translates to a
shift and the pdf of L is given as(14)fL(ℓ)=(ln⁡10)bR2exp⁡{2b(ln⁡10)(ℓ−a)+2σ2(ln⁡10)2b2}×{1−erf(D)};∀ℓ,where(15)D=ℓb−ab−b2log⁡10R+2σ2(ln⁡10)2bσ.Equation (14) gives the pdf of
path loss distribution in a circular scenario of radius R as seen from the centre of the circle. Using
an alternate expression for the error function as shown in [7], (14) can be written
as(16)fL(ℓ)=(ln⁡10)bR2exp⁡{2b(ln⁡10)(ℓ−a)+2σ2(ln⁡10)2b2}×{1−(1−2π∫0π/2exp⁡[−2D22sin⁡2θ]dθ)sgn(D)};∀ℓ,where D holds the same meaning as defined above and sgn(⋅) is the signum function.


## 3. Results

The parameters
specified in the UMTS path loss model [8] are used for the simulations (intercept, a=37 and b=30). The match between theory and simulations is
investigated for varying circle radii and standard deviations for the
log-normal shadowing component. Simulations are also carried out with the nodes
distributed uniformly in hexagons having the same major radii as the circles to
demonstrate the practicality of the derived theory.The simulation is run by uniformly distributing
100,000 nodes in a circle or hexagon of the specified radius and then
calculating the distance between the centre and each point. The histogram of
these distances is then compared against theory. Figure1(a) shows the match
between the pdfs obtained through simulations and those derived in Section 2
for varying scenario radii and log-normal shadowing standard deviation values.
Good agreement between theory and simulation is seen in all cases.
Furthermore, it is observed that the theory for uniformly distributed nodes in
a circle very closely approximates the case when the nodes are uniformly
distributed in a hexagon having the same major radius as the circle. As
expected, the pdf for the hexagonal case shifts to the left.  This is because the major radius of the hexagon is the same as
the radius of the circle, causing the hexagon to fit inside the circle.
Therefore, there are a few areas not covered by nodes in the hexagonal case
which causes the shift. The situation would be reversed if the minor radius of
the hexagon were made equal to the radius of the
circle. Figure 1(b) shows the cumulative distribution functions (cdfs)
associated with the pdfs depicted in Figure 1(a).(a) Pdfs of path loss distribution for uniformly
distributed nodes in circles and hexagons having the same major radii.
Simulation and  theory are in good agreement as scenario radii and standard deviation of
the log-normal component are varied.
(b) The corresponding cdfs.
(a)(b)


## 4. Conclusion

This paper has
presented an analytical derivation of the pdf of path loss distribution between
the centre of a circle and uniformly distributed nodes within it.The derived pdf assists in calculating the
carrier-to-interference ratios, interference thresholds and exclusion regions
inad hoc and sensor networks. In addition, it is envisaged that the
result can be used to develop and assess routing and scheduling algorithms for
such networks without running time-consuming Monte Carlo simulations.



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*Source: 376895-2008-05-08.xml*