Vodafone Chair Mobile Communications Systems, Prof. Dr.-Ing. Dr. h.c. G. Fettweis
When Base Stations Meet Terminals,
And Some Results Beyond
Gerhard P. Fettweis – Vodafone Chair Professor, TU Dresden
with
Vinay Suryaprakash, Ana Belen Martinez, Ines Riedel, Michael Grieger,
and many more
Introduction
Numerous works have demonstrated the benefits of using spatial point processes to model
wireless networks and the use of a homogeneous Poisson process
to model base stations is
22
validated in [Andrews et al. 2011].
Base stations: big dots. Mobile users: little dots.
Actual BS locations in a 4G Urban Network
20
20
15
10
10
5
5
Y coordinate (km)
15
0
−5
−10
−10
−15
−15
−20
−20
0
−5
−20
−20
−15
−10
−5
0
5
10
15
−15
20
−10
−5
0
5
X coordinate (km)
10
15
20
(a)
Poisson
distributed
baseassociated
stations
mobiles.
A 40base
×station
40
km
view ofbyaa current
base
station
de- flat urban area, wi
Fig. 1. Poisson distributed base
stations
and mobiles,
with each mobile
the×nearest
cell
boundaries
are shown
Fig. 3. with
Aand
40
40 kmBS.
viewThe
of(b)
a current
deployment
major service
provider
in a relatively
by a
boundaries corresponding to a ployment
Voronoi tessellation.
major service provider in a relatively
flat urban area.
nd form a Voronoi tessellation.
Figure 1: A comparison from [Andrews et al. 2011].
Base stations: big dots. Mobiles: little dots.
Coverage probability for α = 4
1
May 20, 2015
5
Gerhard Fettweis , Vinay Suryaprakash
0.9
Grid N=8, SNR=10
Grid N=24, SNR=10
Grid N=24, No Noise
Slide 3
Some Limits of Stochastic Geometry
Downtown of a
major EU city center
 sectorization!
5/21/2015
Gerhard Fettweis
Slide 2
Dresden Field Trial:
(Uplink) Coordinated Multipoint Works!
5/21/2015
Gerhard Fettweis
Slide 3
Sectorization & CoMP
Old World:
isolated sectors
5/21/2015
New World:
overlapping sectos
Gerhard Fettweis
Slide 4
CoMP & Sectorization
N:
DPC:
WF:
NC:
HPBW:
5/21/2015
#sectors
CoMP with dirty paper coding
CoMP with Wiener filtering
no cooperation
half power beam width
Gerhard Fettweis
Slide 5
6-Fold Sectors: Dresden Field Trial Setup
6-fold sectorization
with overlapping antennas for 120° sectors!
5/21/2015
Gerhard Fettweis
Slide 6
6-Fold Versus 3-Fold Sectorization
C: CoMP cluster size
5/21/2015
Gerhard Fettweis
Slide 7
CoMP Outcome
Good News:
Overlapping 6-fold sectors!
 Sectors do not play a role
 Each site can be treated as omni
Bad News:
Now intra-cluster interference occurs,
even for sector-wise orthogonal
signaling, as e.g. OFDMA
5/21/2015
Gerhard Fettweis
Slide 8
Vodafone Chair Mobile Communications Systems, Prof. Dr.-Ing. Dr. h.c. G. Fettweis
When Base Stations Meet Mobile Terminals, and
Some Results Beyond
Gerhard Fettweis
Vinay Suryaprakash
Objectives
Develop models to understand the behavior of interference in homogeneous and
heterogeneous networks while taking load or network traffic into account.
Compute Key Performance Indicators (KPIs), i.e. probability of coverage and spectral
efficiency (spatially averaged rate), for these networks.
Ensure that the expressions obtained for the KPIs are easy to use in other optimization
problems such as those dealing with energy efficiency or deployment cost.
If not, find suitable approximations.
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 2
Model 1:
A simple extension of [Andrews et al. 2011].
− Incorporates the average number of users connected to a base station while deriving
expressions for probability of coverage and spatially averaged rate.
− Relevant publications: [Suryaprakash et al. 2012a], [Suryaprakash et al. 2012b].
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 4
Framework for Model 1
Macro base stations are modeled by a homogeneous Poisson process Φb ⊂ R2 , intensity
λb > 0.
Users are modeled by a homogeneous Poisson process Φu ⊂ R2 , intensity λu > λb .
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 5
Framework for Model 1
Macro base stations are modeled by a homogeneous Poisson process Φb ⊂ R2 , intensity
λb > 0.
Users are modeled by a homogeneous Poisson process Φu ⊂ R2 , intensity λu > λb .
Cell definition:
Cxi ,Φb = z ∈ R2 : SINRz ≥ T
=⇒ Cxi ,Φb = z ∈ R2 : L (z, xi ) ≥ T (IΦb (z) + W )
where
Threshold – T .
Noise – W .
P
Interference – IΦb (z) =
L(z, xj ).
xj
Receive Power – L (z, xi ) =
May 20, 2015
PTx h
;
l(|z−xi |)
h – fading, l (|z − xi |) – pathloss.
Gerhard Fettweis , Vinay Suryaprakash
Slide 5
Model 1: Interference Limited Scenario
Theorem
For a path loss exponent β > 2 and a threshold T , the probability of coverage
is
β−2
,
pc (λb , λu , T , β) =
2 −T λu
2λu T 2 F1 1, β−2
β ;2− β ; λb
(β − 2) +
λb
where 2 F1 (·) is the Gaussian hypergeometric function. If β = 4, the spatially
averaged rate is given by
Z∞
R̄Φb (λb , λu ) =
0
May 20, 2015
2y
h
[1 + y tan−1 (y )] y 2 +
Gerhard Fettweis , Vinay Suryaprakash
λu
λb
i dy .
Slide 6
Model 1: Scenario with Interference and Noise
Theorem
For β = 4 and a threshold T , the probability of coverage is given by
pc
where K =
2
λb , λu , T , PTx , σW
π
2
q
PTx λu
2
λ b T σW
n√
π 3/2
=
λb
2
λu λb T tan−1
s
q
PTx λu
Erfc [K ] exp K 2 ,
2
λ b T σW
T λu
λb
o
+ λb . From which, the
approximated closed form expression for the spatially averaged rate is obtained
as
2
R̄Φb (λb , λu , PTx , σW
)
May 20, 2015
π 5/2
≈
2
s
s
"
#
4 2
λu λb PTx
π 2 λu PTx
π λu PTx
Erfc
exp
.
2
2
2
4
σW
σW
16σW
Gerhard Fettweis , Vinay Suryaprakash
Slide 7
Applications of results obtained using Model 1
Additional power density required
versus a unit increase in user demands.
Comparison of energy management
strategies.
160
Using sleep modes, PS = 0.5W
7
140
6
120
Power saved (W/km2)
Additional power density required
8
5
4
3
2
Using bandwidth variation
100
80
60
40
1
20
0
3−4 Mbps
4−5 Mbps
5−6 Mbps
6−7 Mbps
7−8 Mbps
Unit increase in the average rate provided
May 20, 2015
0
0
1
2
3
4
Gerhard Fettweis , Vinay Suryaprakash
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Hours of the day (t)
Slide 8
Motivation for further exploration
[Andrews et al. 2011] derives tractable expressions for the probability of coverage and
spatially averaged rate in homogeneous networks.
⇒ Model 1 only extends this by introducing a dependence between the interference
and user as well as base station intensities.
[Dhillon et al. 2012] derives similar expressions for heterogeneous networks with many
different types of base stations.
However, these works assume that the point processes used to model each network
component are independent of one another.
In reality, base stations are deployed where ever a large number of users tend to be present
and smaller base stations are deployed when the macro (main) base station is unable to
satisfy user demands.
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 9
Motivation for further exploration
[Andrews et al. 2011] derives tractable expressions for the probability of coverage and
spatially averaged rate in homogeneous networks.
⇒ Model 1 only extends this by introducing a dependence between the interference
and user as well as base station intensities.
[Dhillon et al. 2012] derives similar expressions for heterogeneous networks with many
different types of base stations.
However, these works assume that the point processes used to model each network
component are independent of one another.
In reality, base stations are deployed where ever a large number of users tend to be present
and smaller base stations are deployed when the macro (main) base station is unable to
satisfy user demands.
Today, we present our efforts in bridging this gap.
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 9
Model 2:
Homogeneous networks using a Neyman-Scott Process
− In this model, users are clustered around base stations based on a particular distribution.
− Relevant publication: [Suryaprakash et al. 2013].
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 10
Framework for Model 2
Macro base stations (or cluster centers) are modeled by a stationary Poisson process
Φc ⊂ R2 with intensity λc > 0.
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 11
Framework for Model 2
Macro base stations (or cluster centers) are modeled by a stationary Poisson process
Φc ⊂ R2 with intensity λc > 0.
◦ Conditioned on Φc , the users (or cluster members) are modeled by an inhomogeneous
Poisson process Φu ⊂ R2 with intensity function
X
ρ(y ) = λu
f (y − x), y ∈ R2 ,
x∈Φc
where λu > 0 is a parameter and f is a continuous density function.
◦ Note that Φu (not conditioned on Φc ) is stationary with intensity λc λu .
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 11
Framework for Model 2
Macro base stations (or cluster centers) are modeled by a stationary Poisson process
Φc ⊂ R2 with intensity λc > 0.
◦ Conditioned on Φc , the users (or cluster members) are modeled by an inhomogeneous
Poisson process Φu ⊂ R2 with intensity function
X
ρ(y ) = λu
f (y − x), y ∈ R2 ,
x∈Φc
where λu > 0 is a parameter and f is a continuous density function.
◦ Note that Φu (not conditioned on Φc ) is stationary with intensity λc λu .
The interference at a given location z ∈ R2 is given by
IΦ (z) =
X
L z, xj , with L(z, xj ) =
xj ∈Φc
h
l xj − z
for unit transmit power per user.
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 11
Illustration of Model 2
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 2: Cluster members generated using a
zero-mean radially symmetric Gaussian density f
with variance 0.05.
May 20, 2015
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3: Cluster members generated using a
zero-mean radially symmetric Gaussian density f
with variance 0.5.
Gerhard Fettweis , Vinay Suryaprakash
Slide 12
Model 2: Coverage in Homogeneous Networks
Theorem
For a given distance ‘r ’ between the user and base station, pathloss exponent ‘β’, and threshold
‘T ’, the conditional probability of coverage in a homogeneous network with users clustered
around base stations is

 
 
Z 
Z






 
 
µ
1 − exp −λu1 −
 dx ×
p (λc , λu , f (·), T , β | r ) = exp −λc
f
(y
)dy

 
 
µl(r )T



µ + l(x+y )


R2
R2
Z



Z



exp 
−λu 1 −
R2
May 20, 2015
µ
µ+
R2
Gerhard Fettweis , Vinay Suryaprakash
µl(r )T
l(x−y )


f (x)dx 
 f (y )dy .
Slide 13
Model 2: Coverage in Homogeneous Networks
Conditional probability of coverage vs.
distance
Conditional probability of coverage vs.
threshold
λc = 1/km2, σ2 = 0.5, β = 4, r = 0.3 km
λc = 1/km2, T = −9dB, β = 4, σ2 = 0.5
0.9
0.9
λu = 5/km2
0.8
λu = 10/km2
0.7
λu = 15/km2
0.6
λu = 20/km
0.5
λu = 40/km2
0.4
λu = 50/km2
Probability of coverage ( p (λc, λu, f(⋅), T, β | r) )
Probability of coverage ( p( λc, λu, f(⋅), T, β | r) )
1
2
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
λu = 5/km2
0.8
λu = 10/km2
0.7
λu = 15/km2
0.6
λu = 20/km2
0.5
λu = 40/km2
0.4
λu = 50/km2
0.3
0.2
0.1
0
−15
−12
Distance between user and base station (r km)
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
−9
−6
−3
0
3
6
9
12
15
Threshold (T dB)
Slide 14
Model 2: Coverage in Homogeneous Networks
Conditional probability of coverage vs.
cluster variance
Conditional probability of coverage vs.
cluster variance
λc = 1/km2, T = −9dB, β = 4, r = 0.3 km
Probability of coverage ( p ( λc, λu, f(⋅), T, β | r) )
Probability of coverage ( p (λc, λu, f(⋅), T, β | r) )
λc = 1/km2, T = −9dB, β = 4, r = 0.03 km
0.9
0.8
0.7
0.6
0.5
λu = 5
0.4
λu = 10
λu = 15
0.3
λu = 20
0.2
λu = 40
0.1
0
λu = 50
0
0.2
0.4
0.6
0.8
0.8
λu = 10
0.7
λu = 15
0.6
λu = 20
λu = 40
0.5
λu = 50
0.4
0.3
0.2
0.1
0
1.0
λu = 5
0
Figure 4: User to base station distance, r = 0.03 km.
May 20, 2015
0.2
0.4
0.6
0.8
1.0
Variance of the cluster distribution (σ2)
Variance in cluster distribution (σ2)
Figure 5: User to base station distance, r = 0.3 km.
Gerhard Fettweis , Vinay Suryaprakash
Slide 15
Model 2: Coverage in Homogeneous Networks
Theorem
The probability of coverage in a homogeneous network with users clustered around base
stations, pathloss exponent ‘β’, and threshold ‘T ’ is




 


Z








 
µ






1
−
−λ
1
−
exp
p (λc , λu , f (·), T , β) u
exp −λc
f
(y
)dy
dx
u



 
µl(r )T



µ + l(x+y )




Z
Z
R+
R2
R2

Z

×




Z



exp 
−λu 1 −
R2
R2
µ
µ+
µl(r )T
l(x−y )





f (x)dx 
 f (y )dy  g (r )dr .
where the continuous density function of R (the distance between a user and a base station)
d
g (r ) = − dr
v (r ) and v (r ) is the void probability given by
!# !
R"
R
v (r ) = exp −λc
1 − exp −λm
f (y − x)dy
dx .
R2
May 20, 2015
b(o,r )
Gerhard Fettweis , Vinay Suryaprakash
Slide 16
Model 2: Coverage in Homogeneous Networks
Probability of coverage (different view)
T = −9dB, σ2 = 0.5, β = 4
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0
10
75
Ba8se
May 20, 2015
50
6
statio
4
n inte
25
2
nsity
( λ )0
c
0
Us
er in
ity (
tens
λu)
Probability of coverage ( p( λc, λu, f(⋅), T, β) )
Probability of coverage ( p ( λc, λu, f(⋅), T, β) )
Probability of coverage
T = −9dB, σ2 = 0.5, β = 4
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
25
User in
Gerhard Fettweis , Vinay Suryaprakash
tensity 50
(λ )
u
75
0
2
Base
4
6
inten
station
8
10
sity ( λ c
)
Slide 17
Model 3:
Heterogeneous networks using a stationary Poisson
Cluster Process
− An alternative to [Dhillon et al. 2012] in which there are only two types of base stations
and micro base stations are clustered around macro base stations.
− Relevant publication: [Suryaprakash et al. 2014].
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 18
Framework for Model 3
Base stations are modeled using a stationary Poisson cluster process Φ ⊂ R2 .
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 19
Framework for Model 3
Base stations are modeled using a stationary Poisson cluster process Φ ⊂ R2 .
◦ Macro base stations (or cluster centers) are modeled by a stationary Poisson process
Φc ⊂ R2 with intensity λc > 0.
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 19
Framework for Model 3
Base stations are modeled using a stationary Poisson cluster process Φ ⊂ R2 .
◦ Macro base stations (or cluster centers) are modeled by a stationary Poisson process
Φc ⊂ R2 with intensity λc > 0.
◦ Conditioned on Φc , the micro base stations (or cluster members) are modeled by an
inhomogeneous Poisson process Φm ⊂ R2 with intensity function
X
ρ(y ) = λm
f (y − x), y ∈ R2 ,
x∈Φc
where λm > 0 is a parameter and f is a continuous density function.
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 19
Framework for Model 3
Base stations are modeled using a stationary Poisson cluster process Φ ⊂ R2 .
◦ Macro base stations (or cluster centers) are modeled by a stationary Poisson process
Φc ⊂ R2 with intensity λc > 0.
◦ Conditioned on Φc , the micro base stations (or cluster members) are modeled by an
inhomogeneous Poisson process Φm ⊂ R2 with intensity function
X
ρ(y ) = λm
f (y − x), y ∈ R2 ,
x∈Φc
where λm > 0 is a parameter and f is a continuous density function.
◦ Note that Φm (not conditioned on Φc ) is stationary with intensity λc λm .
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 19
Framework for Model 3
Base stations are modeled using a stationary Poisson cluster process Φ ⊂ R2 .
◦ Macro base stations (or cluster centers) are modeled by a stationary Poisson process
Φc ⊂ R2 with intensity λc > 0.
◦ Conditioned on Φc , the micro base stations (or cluster members) are modeled by an
inhomogeneous Poisson process Φm ⊂ R2 with intensity function
X
ρ(y ) = λm
f (y − x), y ∈ R2 ,
x∈Φc
where λm > 0 is a parameter and f is a continuous density function.
◦ Note that Φm (not conditioned on Φc ) is stationary with intensity λc λm .
Hence, the base stations, i.e., the superposition Φ = Φc ∪ Φm , form a stationary Poisson
cluster process with intensity λ = λc (1 + λm ).
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 19
Framework for Model 3
Base stations are modeled using a stationary Poisson cluster process Φ ⊂ R2 .
◦ Macro base stations (or cluster centers) are modeled by a stationary Poisson process
Φc ⊂ R2 with intensity λc > 0.
◦ Conditioned on Φc , the micro base stations (or cluster members) are modeled by an
inhomogeneous Poisson process Φm ⊂ R2 with intensity function
X
ρ(y ) = λm
f (y − x), y ∈ R2 ,
x∈Φc
where λm > 0 is a parameter and f is a continuous density function.
◦ Note that Φm (not conditioned on Φc ) is stationary with intensity λc λm .
Hence, the base stations, i.e., the superposition Φ = Φc ∪ Φm , form a stationary Poisson
cluster process with intensity λ = λc (1 + λm ).
The interference at a given location z ∈ R2 is given by
IΦ (z) =
X
L z, xj , with L(z, xj ) =
xj ∈Φ
h
l xj − z
for unit transmit power per user.
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 19
Illustration of Model 3
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 6: Cluster members generated using a
zero-mean radially symmetric Gaussian density f
with variance 0.05.
May 20, 2015
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 7: Cluster members generated using a
zero-mean radially symmetric Gaussian density f
with variance 0.5.
Gerhard Fettweis , Vinay Suryaprakash
Slide 20
Model 3: Coverage in Heterogeneous Networks
Theorem
For a given distance ‘r ’ between the user and base station, pathloss exponent ‘β’, and threshold
‘T ’, the conditional probability of coverage in a heterogeneous network with micro base stations
clustered around macro base stations is derived as
p (λc , λm , f (·), T , β | r ) =



 
Z µl(r )T







 
µ
l(y )




exp −λc
exp
−λ
f
(y
−
x)dy
1−
.
m

 dx 
µl(r )T
µl(r )T





µ + l(x)
µ + l(y )


Z
R2
May 20, 2015
R2
Gerhard Fettweis , Vinay Suryaprakash
Slide 21
Model 3: Coverage in Heterogeneous Networks
Theorem
The probability of coverage in a heterogeneous network with micro base stations clustered
around macro base stations, pathloss exponent ‘β’, and threshold ‘T ’ is
p (λc , λm , f (·), T , β) =



 
Z µl(r )T







 
µ
l(y )

−λm
 dx 
exp −λc
exp
f
(y
−
x)dy
1−
g (r ) dr ,

 
µl(r )T
µl(r )T





µ + l(x)
µ + l(y )


Z
Z
R2
R2
R+
where the continuous density function of the distance between the user and the base station
d
g (r ) = − dr
v (r ) and the void probability v (r ) is given as

Z

v (r ) = exp 
−λc




/ b(o, r )) exp −λm
1 − 1 (x ∈

 
f (y − x) dy  dx 
.
b(o,r )
R2
May 20, 2015

Z
Gerhard Fettweis , Vinay Suryaprakash
Slide 22
Model 3: Coverage in Heterogeneous Networks
Probability of coverage ( p ( λc,λm, f(⋅), T, β | r) )
Conditional probability of coverage
λc = 1/km2, T = −9dB, β = 4, σ2 = 0.5
1
λm = 1
0.9
λm = 6
0.8
λm = 11
0.7
λm = 21
0.6
λm = 41
0.5
0.4
0.3
0.2
0
0.5
1
1.5
2
2.5
3
Distance between user and base station (r km)
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 23
Model 3: Coverage in Heterogeneous Networks
Probability of coverage
T = −9dB, σ2 = 0.5, β = 4
λc = 1/km , T = −9dB, β = 4, σ = 0.5
2
2
Probablity of coverage ( p ( λ ,λ , f(⋅), T, β) )
1
λm = 1
0.9
λm = 6
0.8
c m
Probability of coverage ( p ( λc,λm, f(⋅), T, β | r) )
Conditional probability of coverage
λm = 11
0.7
λm = 21
0.6
λm = 41
0.5
0.4
0.3
0.2
0
0.5
1
1.5
2
2.5
3
1
0.9
0.8
0.7
0.6
0.5
λc = 0.1/km2
0.4
λc = 0.3/km2
0.3
λc = 0.5/km2
0.2
λc = 0.9/km2
0.1
0
λc = 1.0/km2
1
Distance between user and base station (r km)
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
2
3
4
5
6
7
8
9
10
Micro base station intensity (λm)
Slide 23
Comments about expressions obtained using Models 2
and 3
The expressions, shown in the previous slides, are easily evaluated using commercially
available computational software.
However, they are rather large and cumbersome which prevents easy re-use in other
optimization problems (which need more than the final value obtained by evaluating the
expressions numerically by fixing certain parameters).
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 24
Comments about expressions obtained using Models 2
and 3
The expressions, shown in the previous slides, are easily evaluated using commercially
available computational software.
However, they are rather large and cumbersome which prevents easy re-use in other
optimization problems (which need more than the final value obtained by evaluating the
expressions numerically by fixing certain parameters).
Hence, we investigate other suitable approximations for describing the interference which
could allow easy re-use.
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 24
Models 2 and 3:
May 20, 2015
Asymptotic Behavior of the Interference
Gerhard Fettweis , Vinay Suryaprakash
Slide 25
Asymptotic Behavior of the Interference
Define an estimator S(r ) of the interference, where the distance between the user and base
station pair is r .
More Details
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 26
Asymptotic Behavior of the Interference
Define an estimator S(r ) of the interference, where the distance between the user and base
station pair is r .
Then, the estimator Sn (r ) over eroded sampling windows
T
Wn,r ≡ Wn b(o, r ) =
(Wn + x) can be defined as
x∈b(0,r )
Sn (r ) =
1 X
1W (x)1H(x,r ) (Φ − δx ),
|Wn,r | x∈Φ n,r
where, for a set A, 1A (·) is its indicator function and H(x, r ) are the sets of point
configurations which are not r -close to x.
More Details
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 26
Asymptotic Behavior of the Interference
Define an estimator S(r ) of the interference, where the distance between the user and base
station pair is r .
Then, the estimator Sn (r ) over eroded sampling windows
T
Wn,r ≡ Wn b(o, r ) =
(Wn + x) can be defined as
x∈b(0,r )
Sn (r ) =
1 X
1W (x)1H(x,r ) (Φ − δx ),
|Wn,r | x∈Φ n,r
where, for a set A, 1A (·) is its indicator function and H(x, r ) are the sets of point
configurations which are not r -close to x.
Create a centered random variable using the estimator over the windows, which is given by
Zn (r ) = |Wn,r |1/2 (Sn (r ) − E [Sn (r )]) .
More Details
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 26
Asymptotic Behavior of the Interference
Theorem
For any radius r ≥ 0, such that lim Var [Zn (r )] = σλ2 (r ) > 0, we have
n→∞
D
Zn (r ) −−−→ N (0, σλ2 (r ))
n→∞
where N (0, σλ2 (r )) is a zero-mean Gaussian distribution with variance σλ2 (r ) and
D denotes convergence in distribution.
Proof
The proof is derived along lines similar to those used in [Heinrich 1988].
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 27
Asymptotic Behavior of the Interference
Theorem
For any radius r ≥ 0, such that lim Var [Zn (r )] = σλ2 (r ) > 0, we have
n→∞
D
Zn (r ) −−−→ N (0, σλ2 (r ))
n→∞
where N (0, σλ2 (r )) is a zero-mean Gaussian distribution with variance σλ2 (r ) and
D denotes convergence in distribution.
Proof
The proof is derived along lines similar to those used in [Heinrich 1988].
Therefore, interference in clustered (and highly correlated) networks can be
approximated by a Gaussian random variable with mean E [Sn (r )] and variance
2 (r ).
Mean
Variance
|Wn,r |σλ
It also implies that the influence of the transmit power, fading, and pathloss can be
observed solely in the mean and variance of the interference.
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 27
Number of instances taking a particular value
Verification of the results for Model 3
120
Interference values
Gaussian fit from theory
100
80
60
40
20
0
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
Normalized interference values (binned)
Figure 8: Histogram for simulated interference values and the Gaussian density (from theory) with the mean
and variance using the equations derived. Note that the mean has been subtracted to center both the
histogram and the theoretic curve.
Plots of mean and variance
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 28
Verification of Expressions
Variance
Mean
0
0
Mean of the interference (normalized)
λc = 2 /km2
λc = 3 /km2
λc = 5 /km2
λc = 6 /km2
−1
10
λc = 8 /km2
2
λc = 10/km
2
λc = 2 /km
2
λc = 3 /km
−2
10
2
λc = 5 /km
2
λc = 6 /km
λc = 8 /km2
λc = 10/km2
−3
10
5
6
7
8
9
10
11
12
13
14
15
Variance of the interference (normalized)
10
10
λc = 2/km2
λc = 3/km2
−1
10
λc = 5/km2
−2
λc = 6/km2
10
λc = 8/km2
−3
10
λc = 10/km2
λc = 2/km2
−4
10
λc = 3/km2
λc = 5/km2
−5
10
λc = 6/km2
−6
10
λc = 8/km2
λc = 10/km2
−7
10
5
6
7
Micro base stations intensity (λm)
8
9
10
11
12
13
14
15
Micro base station intensity (λm)
Back
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 42
Conclusions
Reality:
Base station and terminal distributions are correlated processes
Generic Results:
 intra-cell interference
 generic expressions for HetNets derived
Open HetNet Challenge:
 correlated distribution of mobiles to base stations, and
 correlated distribution of mobiles to micro/small cells
Open Stochastic Geometry Challenge: finding simpler approximations
5/21/2015
Gerhard Fettweis
Slide 9
References
Jeffrey G. Andrews, François Baccelli, Radha Krishna Ganti,
A Tractable Approach to Coverage and Rate in Cellular Networks,
in IEEE Transactions on Communications, vol. 59, pp. 3122 - 3134, 2011.
Dhillon, H.S. and Ganti, R.K. and Baccelli, F. and Andrews, J.G.
Modeling and analysis of K-tier downlink heterogeneous cellular networks
in IEEE Journal on Selected Areas in Communications, vol. 30 pp. 550 - 560, 2012.
Heinrich, L.
Asymptotic behaviour of an empirical nearest-neighbour distance function for stationary poisson
cluster processes
in Mathematische Nachrichten, vol. 136, no. 1, pp. 131 - 148, 1988.
Heinrich, L.
Stable limit theorems for sums of multiply indexed m-dependent random variables
in Mathematische Nachrichten, vol. 127, no. 1, pp. 193 - 210, 1986.
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 30
References
Vinay Suryaprakash, Albrecht Fehske, André Fonseca dos Santos, Gerhard. P. Fettweis,
On the Impact of Sleep Modes and BW Variation on the Energy Consumption of Radio Access
Networks,
in the proceedings of the 75th IEEE Vehicular Technology Conference (VTC Spring), 2012, 2012.
Vinay Suryaprakash, André Fonseca dos Santos, Albrecht Fehske, Gerhard. P. Fettweis,
Energy Consumption Analysis of Wireless Networks using Stochastic Deployment Models,
in the proceedings of the IEEE Global Communications Conference (GLOBECOM), 2012, 2012.
Vinay Suryaprakash, Gerhard. P. Fettweis,
A stochastic examination of the interference in heterogeneous radio access networks,
in the proceedings of the 11th International Symposium on Modeling Optimization in Mobile, Ad Hoc
Wireless Networks (WiOpt), 2013, pp. 68 - 74, 2013.
Vinay Suryaprakash, Jesper Møller, Gerhard. P. Fettweis,
On the Modeling and Analysis of Heterogeneous Radio Access Networks using a Poisson Cluster
Process,
in the IEEE Transactions on Wireless Communications, 2014.
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 31
Thank You!
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 32
Back up
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 33
Asymptotic Behavior of the Interference
The asymptotic behavior of the interference is studied using an increasing sequence of
compact sampling windows (Wn )n≥1 in Rd and eroded sets
Wn,r ≡ Wn b(o, r ) = {x ∈ Wn : b(x, r ) ⊆ Wn }, which satisfy the Regularity Condition.
Regularity Condition : There exist a sequence of subsets of Rd satisfying
(a) each Wn is convex and compact;
(b) Wn ⊂ Wn+1 ;
(c) sup{r ≥ 0 : B(x, r ) ⊂ Wn for some x} → ∞ as n → ∞.
Define an estimator S(r ) of the interference where the user and base station pair is r .
Then, the estimator Sn (r ) over eroded sampling windows can be defined as
Sn (r ) =
1 X
1W (x)1H(x,r ) (Φ − δx ),
|Wn,r | x∈Φ n,r
where, for a set A, 1A (·) is its indicator function and H(x, r ) are the sets of point
configurations which are not r -close to x.
Create a centered random variable using the estimator over the windows, which is given by
Zn (r ) = |Wn,r |1/2 (Sn (r ) − E [Sn (r )]) .
Back
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 34
Proof of Asymptotic Behavior of the Interference
Introduce a truncated Poisson cluster process, Φρ whose cluster center process is still Φc
but the process of cluster members Φmρ consists of atoms of Φm which are located in the
sphere b(0, ρ), where ρ > r , i.e., Φmρ ({x}) > 0 if Φm ({x}) > 0 and ||x|| ≤ ρ. For A ∈ B0d ,
Snρ (r , A) =
X
1
1A∩Wn,r (x)1H(x,r ) (Φρ − δx )
|Wn,r | x∈Φ
ρ
which implies that the centered random variable can be written as
Znρ (r ) = (|Wn,r |)1/2 Snρ (r , Wn,r ) − E [Snρ (r , Wn,r )] .
Define a set Ez = [z1 − 1, z1 ) × · · · × [zd − 1, zd ) for any z ∈ Un ⊂ Z d where
d
Z = {z = (z1 , · · · , zd ) : zi = 0, ±1, ±2, · · · ; i = 1, · · · , d} and
o
d n
(i)
Un = × 1, 2, · · · , [an ] + 1 . Now, consider a family of random variables
i=1
Xnz (r ) =
Znρ (r )
(|Wn,r |)1/2 (Snρ (r , Ez ) − E [Snρ (r , Ez )])
=
.
Var [Znρ (r )]
Var [Znρ (r )]
Back
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 35
Proof of Asymptotic Behavior of the Interference
This implies Xnz (r ) forms an m-dependent random field.
From [Heinrich 1986], we know that
Znρ (r )
D
−−−−→ N(0, 1),
Var [Znρ (r )] n→∞
since the following conditions are satisfied for every > 0.
X
(i)
P (|Xnz | ≥ ) −−−−→ 0,
n→∞
z∈Un
(ii)
X
2
E Xnz
() ≤ C () < ∞,
z∈Un
(iii) E [Sn ()] −−−−→ a ∈ R and Var [Sn ()] −−−−→ σ 2 ,
n→∞
n→∞
where C () is a positive constant that changes with and σ > 0.
Back
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 36
Proof of Asymptotic Behavior of the Interference
Then, we show that
lim sup Var Zn (r ) − Znρ (r ) = 0, ∀r ≥ 0.
ρ→∞ n≥1
For the functional limit theorem to hold, the tightness of Zn must be proven. This is done
4
by determining bounds on E Zn (t) − Zn (s) , ∀ 0 ≤ s ≤ t ≤ R by means of the fourth- and
second-order cumulants.
From Lemma 2 of [Heinrich 1988], the bounds are given by
4
E Zn (t) − Zn (s) ≤ C1 (t − s)/|Wn,r | + (t − s)2 ,
for a constant C1 > 0.
Hence, proving the theorem stated.
Back
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 37
Mean of the Estimator of the Interference
The mean of the estimator is


X
1
E [Sn (r )] =
E
1Wn,r (x)1H(x,r ) (Φ − δx ) .
|Wn,r |
x∈Φ
It can then be written as


X 
X

ξM (r ) ≡ |Wn,r |E [Sn (r )] = E
1Wn,r (y )1H(y ,r ) (Φ − δy ) .
1Wn,r (x)1H(x,r ) (Φ − δx ) +
x∈Φc
(x)
y ∈Φm
(x)
Using the Slivnyak-Mecke Theorem first for Φ and then for Φm (after conditioning on both
(x)
(x)
Φ − Φm and Φ̃m ), we get
Z
ξM (r ) = λc
h
i
(x)
E 1H(x,r ) (Φ)1H(x,r ) (Φ̃m ) dx +
Wn,r
ZZ
λc λm
(x)
1Wn,r (y ) P (Φ ∈ H(y , r )) P Φ̃m ∈ H(y , r ) f (x − y ) dy dx.
||x−y ||>r
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 38
Mean of the Estimator of the Interference
Thereby, we get


Z

E [Sn (r )] = λc v (r ) vm (r ) + λm

vm (z, r )f (z)dz  ,
kzk>r
by a simple change of variables where


Z 
Z



v (r ) = exp 
/ b(o, r )) exp −λm
1 − 1 (x ∈
−λc


 
f (y − x) dy  dx 
.
b(o,r )
Rd
(x)
and the probability of Φm not being r -close to y is

Z

(x)
P Φm ↑̸ b(y , r ) = exp −λm


f (z − x) dz  = vm (x − y , r ).
kz−y k≤r
Here, vm (r ) = vm (o, r ).
May 20, 2015
Back
Gerhard Fettweis , Vinay Suryaprakash
Slide 39
Variance of the Estimator of the Interference
The variance of the estimator is given by
1
2
σλ
(r ) =
ξCC (r ) + 2 ξCM (r ) + ξCCM (r ) +
2
|Wn,r |
ξCMM (r ) + ξCCMM (r ) + ξM (r ) − {ξM (r )}2 ,
where
Z
ξCC (r ) = λ2c
|Wn,r ∩ (Wn,r + z)| v (z, r ) (um (z, r ))2 dz,
kzk>r
Z
|Wn,r ∩ (Wn,r + z)| v (z, r ) um (z, r ) f (z)dz,
ξCM (r ) = λc λm
kzk>r
ξCCM (r ) = λ2c λm
ZZ
|Wn,r ∩ (Wn,r + z)| v (z, r ) um (z, r ) um (z, w , r ) f (w ) dz dw ,
kz−w k>r
kzk>r
kw k>r
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 40
Variance of the Estimator of the Interference
ξCMM (r ) = λc λ2m
ZZ
|Wn,r ∩ (Wn,r + w − z)| v (w − z, r ) um (w − z, w , r )f (z) f (w ) dz dw ,
kzk>r
kw k>r
kz−w k>r
ξCCMM (r ) = λ2c λ2m
ZZZ
|Wn,r ∩ (Wn,r + w )| v (w , r ) um (w , w − z, r )
kw k>r
kzk>r
kw +z 0 k>r
kz−w k>r
kz 0 k>r
kz+z 0 −w k>r
and
um (w , −z 0 , r ) f (z) f (z 0 ) dw dz dz 0 ,


Z

ξM (r ) = λc |Wn,r | v (r ) vm (r ) + λm

vm (z, r )f (z)dz  .
kzk>r
Back
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 41
Conclusions & Future Work
Conclusions:
- Models for homogeneous and heterogeneous networks have been developed.
- Expressions for the interference and the relevant KPI’s in these scenarios have been
derived.
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 29
Conclusions & Future Work
Conclusions:
- Models for homogeneous and heterogeneous networks have been developed.
- Expressions for the interference and the relevant KPI’s in these scenarios have been
derived.
Future Work:
- Though current insights are useful, finding good approximations for some of the more
unwieldy expressions could help easier reuse in other optimization problems.
- In order to improve upon the models presented in this work, alternatives in which the
degree of heterogeneity as well as the extent of correlation between locations of
different network components are adjustable can be explored.
Investigate triply stochastic point process models wherein users are clustered
around micro base stations, and micro base stations are in turn clustered around
macro base stations.
May 20, 2015
Gerhard Fettweis , Vinay Suryaprakash
Slide 29