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Title <Provide the name of the SWIP in the below box. Don’t use bold letters.>
Stochastic models for macro cellular base station locations used for system evaluation
Study Item/Work Item
Submitted For
TSDSI-SG1-SI11-V1.0.0-20150807
Information/Discussion/Decision
Supporters:
Supporter Name
Radha Krishna Ganti
Kiran Kuchi
Klutto Milleth
Email ID
rganti@ee.iitm.ac.in
kkuchi@iith.ac.in
Klutto.milleth@cewit.org.in
Description <Provide the Description in the below box.>
With the recent surge in volume and density of base-stations (BS) being deployed in India, cellular network
geometries are no longer the regular lattice structures that have been in use to model such networks. Cellular
networks have started to resemble more a random network, than a planned network with hexagonal
boundaries. One such instance is the spatial arrangement of BSs in an urban scenario as depicted in Figure
0-1: Base-Station Map in 4G Urban Network, Source - [1]. The behavior of such a network, in terms of mobile
coverage and throughput, is dependent on the spatial structure that is formed by BS. This is primarily because
the signal-to-interference-plus-noise (SINR) ratio at any location in a given network is dependent on the
geometry of neighboring interferers (or transmitters) about that location.
Figure 0-1: Base-Station Map in 4G Urban Network, Source - [1]
Currently, hexagonal model for BS (Macro) locations is the most dominant model for performance evaluation
both in industry and standardization. While, such hexagonal lattices model very planned networks, it is very
rare that such planned networks are possible. This is particularly true in countries like India, where the BS
placement depends on factors (feasibility, density) that cannot be controlled by the operators. Besides the
fact that the hexagonal models do not represent the true geometry of today’s cellular networks, analytical
framework to quantify performance measures of the network using such a model is extremely hard to derive.
Figure 0-2: Base-Station Location Map, London, UK, Source - OfCom
This necessitates the need for us to relook at the models for BS locations that are used to validate
ideas/algorithms during standardizations process. The goal is to come up with statistical models for BS
locations that most represent Indian scenarios. This is akin to using standard fading models for simulation.
There has been lot of work in the academia to develop such models for BS locations that resemble the
practice and also tractable analytically. In the next few sections, we introduce a few such models that are
popular (not exhaustive).
The network is modeled with BS as well as mobile users (MU) being distributed as spatial point processes.
One such well established point process is the Poisson point process (PPP), which models the distribution of
BS as a Poisson random process in space. Such a model is defined by an intensity function of the process. For
instance, if a BS PPP ΦBS defined on two-dimensional space Ɍ2, has intensity λ(x) at a point x ϵ Ɍ2, then for
any closed set A ϵ Ɍ2,
𝑃[ΦBS (𝐴) = 𝑘] =
(∫ 𝜆(𝑥) 𝑑𝑥)𝑘 e− ∫ 𝜆(𝑥) 𝑑𝑥
k!
(1)
Where P [ΦBS (A) = k] denotes the probability of k points being inside the area A and the intensity integral is
evaluated over the closed area A. Equation (1) is same as a Poisson distributed random variable, the
exception being that the distribution is taken over space and the intensity parameter λ(x) is a function of the
Summary <Provide the Summary in the below box.>
location in the space. This means to say that the density of the points spread over the Euclidean space is not
constant. If we take a constant density case, equation (1) gets modified as:
𝑃[ΦBS (𝐴) = 𝑘] =
(𝜆|𝐴|)𝑘 e−𝜆|𝐴|
k!
(2)
Where |A| denotes the volume of area A and λ denotes the constant density of the point process. Such a
process is called a homogeneous PPP and is a well studied [1] [2], [3] tool to analyze performance of wireless
networks. An introduction to modeling a cellular network using a homogeneous PPP is given in this proposal.
A single tier BS network is modeled using homogeneous PPP as an example. Coverage and rate parameters
are analyzed of this network using tools from stochastic geometry. Let λ be the constant density of the BS
network. Figure 0-3: BS PPP and Square Lattice Model is a snapshot of the PPP which matches the actual BS
deployment of Figure 0-1: Base-Station Map in 4G Urban Network, Source - [1] much more than the regular square
lattice (or hexagonal) structure. Here, the square lattice model takes a grid structure with two tiers of
interference consisting of in all 24 interferers.
Figure 0-3: BS PPP and Square Lattice Model
Coverage and rate are modeled as functions of the fundamental parameter SINR. The SINR is taken with
respect to a typical user assumed to be located at the origin of the space under consideration. Instead of
taking the typical user around at every point of the network area to derive the average coverage, PPPs
provide an alternate solution. Fixing the typical user at the origin, the spatial distribution of the nodes (BS) is
varied over all possible distributions on the given space to derive the average coverage in the given network.
This property is at the helm of stochastic geometry being able to model practical networks with high
accuracy. The mathematical ease of averaging over Poisson distribution of nodes imparts this model high
analytical tractability as well as quickness of performance quantification.
I.
Analysis
Analysis of the abovementioned cellular network is briefed in this section. Consider a single-tier cellular
network modeled with a PPP Φ of constant intensity measure λ over the Euclidean space Ɍ2. The BSs transmit
at a constant unit power. Association follows the nearest neighbor rule. Small scale fading between any pair
of nodes is assumed to follow Rayleigh distribution and hence the fading power is exponentially distributed
with unit mean. Simple path loss model is assumed with path loss exponent α, such that power received at a
distance x from a node transmitting at unity power is |x|-α, α > 2. This implies that the SINR of the typical user
located at the origin and having a nearest BS at a distance r is given as:
SINR =
ℎ𝑟 −𝛼
(3)
𝑁+𝐼𝑟
Where the N denotes the variance of the zero-mean Gaussian noise at the typical user receiver and Ir denotes
the interference to the typical user by all the other BSs, assuming a frequency reuse 1 network. Small scale
fading is denoted by h, such that |h|2 is an exponential random variable with unit mean and Ir is as:
𝐼𝑟 = ∑𝑖 𝜖 𝛷\𝑏0 𝑔𝑖 𝑅𝑖−𝛼
(4)
The subscript denotes the interference is due to all the BSs of the PPP Φ, but the BS to which the typical user
has associated.
Coverage is modeled as a probability measure, such that it denotes the probability that the typical user
receives an SINR greater than a threshold T, i.e. using (3) we can have probability of coverage given the nearest
BS to the typical user is at distance r is given as:
𝑃[𝑆𝐼𝑁𝑅 > 𝑇|𝑟] = 𝑃[ℎ𝑟 −𝛼 > 𝑇(𝑁 + 𝐼𝑟 )|𝑟]
∞
=> 𝑃[𝑆𝐼𝑁𝑅 > 𝑇] = ∫ 𝑃[𝑆𝐼𝑁𝑅 > 𝑇|𝑟] 𝑓(𝑟)𝑑𝑟
0
Where f(r) denotes the nearest neighbor density function and is given as:
𝑓(𝑟) = 2𝜋𝜆𝑟 𝑒 −𝜆𝜋𝑟
2
(5)
The coverage probability could then be derived as in [1] to be the following expression:
∞
𝑃[𝑆𝐼𝑁𝑅 > 𝑇] = 𝜋𝜆 ∫ 𝑒 −𝜋𝜆𝑣(1+𝑝(𝑇,𝛼))−𝑇𝜎
2 𝑣 𝛼/2
𝑑𝑣
0
When noise is neglected,
𝑝(𝑇, 𝛼) =
2
𝑇𝛼
∞
∫
1
𝛼
𝑑𝑢
(6)
2 1 + 𝑢2
−
𝑇 𝛼
Equation (5) denotes a quickly computable closed form expression for coverage that could be evaluated using
Mathematica or MATLAB. The average rate R’ experienced by the typical user could be derived as:
∞
𝑅 ′ = 𝐸[𝑙𝑜𝑔(1 + 𝑆𝐼𝑁𝑅)] = ∫ (log(1 + 𝑆𝐼𝑁𝑅) > 𝑡)𝑑𝑡
(6)
𝑡=0
Using the fact that log (1+SINR) is a non-negative random variable. The probability of coverage expression of
(5) could be directly plugged in (6) to get the average typical user rate.
II.
Results and Discussion
Following plots show the result of network analysis based on stochastic geometry and the square lattice based
simulation approach and compare them with the results got from the actual BS deployment (as in Figure 0-1:
Base-Station Map in 4G Urban Network, Source - [1]) simulation.
Figure 0-4: Coverage Probability vs. SINR Threshold T. Left - α = 2.5, Right - α = 4
Figure 0-5: Coverage Probability vs. SINR Threshold T. Left - Frequency Reuse factor δ = 2, Right - δ = 4
Notice that the coverage curves got by the PPP analysis closely follows the actual coverage curves. The good
thing being that PPPs give a more conservative bound on coverage in contrast to the lattice based analysis
that hikes up coverage probability.
This matches with intuition since a PPP could place two BSs arbitrarily close to each other thereby increasing
the interference. The lattice based model avoids this interference quite clearly and hence is an upper bound
on the actual coverage probability. Both the bounds are almost equally accurate, but the ease and tractability
of the PPP framework makes it much more worthwhile to use. The Poisson BS model becomes more accurate
at lower path loss exponents. There are two reasons for this. First, the PPP models distant interference
whereas a 1 or 2 tier grid model does not; and the interference of far-off base stations is more significant for
small α. Second, since a weakness of the Poisson model is the artificially high probability of a nearby and
dominant interfering base station, at lower path loss exponents, perhaps counter-intuitively, such an effect
is less corrupting because a dominant base station contributes a lower fraction of the total interference due
to the slower attenuation of non-dominant interferers.
Though the analysis shown here is for a single-tier network, the same could be easily extended to multi-tier
heterogeneous networks (HetNet). Cardinal work on modeling the downlink performance of a general K-tier
HetNet was done in [4] and is an extension of the analysis shown here.
Though PPPs model a wireless cellular network with reasonable accuracy, the arbitrary closeness of the
associated and an interfering BS make them a little less appropriate. In particular, a real cellular network will
have nodes with certain repulsion amongst them. That means to say two macros can simply not be arbitrarily
close to each other and thus are believed to have repulsion or regularity in their spatial distribution. This is
exactly modeled by Determinantal point processes (DPP) that are described in great detail in [2] and [3].
Following results show the performance of a class of DPP called the Ginibre point processes (β-GPP).
In this proposal, we show the usefulness and accuracy of modeling wireless networks using stochastic
geometry. We show that the hexagonal or regular lattice models for wireless networks are not accurate.
They suffer from two major defects. First, the real networks are no longer symmetrical with square or
hexagonal boundaries but irregular and more random geometries that are highly dependent on the terrain
and demographics of the area. Second, even if analytical models were to be derived for such regular
structures, the models turn out to be way too cumbersome to track and suffer from inaccurate assumptions.
Stochastic geometry not only offers a rich tool of modeling spatial distributions that closely resemble real
networks, but is blessed with the prime virtues of mathematical ease and tractable analysis. PPP is one such
interesting and easy to use option. For more apt distributions we have introduced DPPs that model repulsion
between the BS and are thereby much more accurate than PPPs, while still maintaining analytical tractability.
We therefore propose to come up with statistical models for BS locations in India. Research should focus on
finding the suitability of PPP, DPP models to Indian cellular networks and trying to investigate into models
that suit Indian urban environment. These models would provide with test cases for performance evaluation
in addition to the commonly used hexagonal model.
Figure 0-6 - Left: Urban BS Location Map. Right: GPP fit to Urban BS distribution [2]
and [3].
The fit of DPPs to model the urban BS distribution as shown by Figure 0-6 - Left: Urban BS Location Map. Right:
GPP fit to Urban BS distribution is remarkable. This shows that β-GPP are able to model the repulsion between
the BSs with high accuracy and could be used to model modern cellular networks of practically any form of
distribution.
Impact <Provide the Impact of SWIP in TSDSI in the below box.>
References <Provide the References in the below box.>
[1] J. Andrews, F. Baccelli, and R. K. Ganti, “A tractable approach to coverage and rate in cellular networks,”
Communications, IEEE Transactions on, vol. 59, pp. 3122–3134, November 2011.
[2] N. Deng, W. Zhou, and M. Haenggi, “The Ginibre Point Process as a Model for Wireless Networks with
Repulsion,” IEEE Transactions on Wireless Communications, vol. 14, pp. 107-121, Jan. 2015.
[3] Y. Li, F. Baccelli, H. Dhillon, and J. Andrews, "Statistical Modeling and Probabilistic Analysis of Cellular
Networks with Determinantal Point Processes." IEEE Transactions on Communications, Dec. 2014.
[4] H. Dhillon, R. K. Ganti , F. Baccelli, and J. Andrews, “Modeling and analysis of k-tier downlink
heterogeneous cellular networks,” Selected Areas in Communications, IEEE Journal on, vol. 30, pp. 550–560,
April 2012.
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