1
Modeling a Spatially Correlated Cellular
Network with Strong Repulsion
arXiv:1701.02261v1 [cs.IT] 9 Jan 2017
Chang-Sik Choi, Jae Oh Woo, and Jeffrey G. Andrews
Abstract
We propose a new cellular network model that captures both strong repulsion and no correlation
between base stations. The base station locations are modeled as the superposition of two independent
stationary point processes: a random shifted grid with intensity λg for the repulsive base stations and an
independent Poisson point process with intensity λp for the uncorrelated base stations. Assuming that
users are associated with base stations that provide the strongest average receive signal power, we obtain
the probability that a typical user is associated with either a repulsive base station or an uncorrelated
base station. Assuming Rayleigh fading channels, we derive the expression for the coverage probability
of the typical user. The following three observations can be made. First, the association and coverage
probability of the typical user are fully characterized by two system parameters: the intensity ratio ρλ
and the transmit power ratio η. Second, in user association, a bias toward the repulsive base stations
exists. Finally, the coverage expression is a monotonically increasing function with respect to the number
of repulsive base stations, or the inverse of the intensity ratio ρ−1
λ .
I. I NTRODUCTION
A. Motivation and Related Work
The topological distribution of base stations is a first order effect in terms of the signal-tointerference-and-noise ratio (SINR) of a cellular network. According to studies in [1]–[3] that
investigated the topology of cellular networks, base stations (BSs) are usually spread out from
Chang-sik Choi, Jae Oh Woo, and Jeffrey G. Andrews are with the Wireless and Networking and Communication Group,
Department of Electrical and Computer Engineering, The University of Texas at Austin, Texas, 78701, USA. (e-mail:
[email protected], [email protected], [email protected])
Last revised: January 10, 2017
2
one another, which is referred to as repulsion. Studies [1]–[3] provided further evidence of
the repulsions by numerically comparing the SINRs of their repulsive models with the SINRs
obtained from actual deployment data. However, those works did not analytically derive the
SINR expressions, primarily because it is very difficult to do so for repulsive point processes.
B. Background: From Poisson to more general models
The modeling and analysis of BS locations using Poisson point process (PPP) has become
popular in the last five years because the independence of the PPP helps allow computable –
and in some cases quite simple – SINR distribution expressions [4]–[11]. The PPP-based models
often yield a simple expression for the interference [12], [13]. In [4] the coverage probability of
the typical user – meaning the probability that the SINR is above a certain value – was derived in
closed-form, and was extended in [5]–[8] to multi-tier heterogeneous cellular networks (HCNs).
The benefits of HCNs were studied further in [9]–[11]. However, the PPP ignores the repulsive
nature of topology observed in cellular base stations.
In order to mathematically represent the repulsion between base stations, [14]–[17] considered repulsive point processes [18, Chapter 5.] and attempted to derive the SINR expressions
analytically. Several specific point processes can be introduced. For example, [15] considered
the Ginibre point process, [16] provided tractable but complicated analysis for the more general
determinantal point process (of which Ginibre is a special case), and [17] used the Matérn
hard-core point process, which introduces an exclusion region around each BS. However, the
results in [14]–[17] did not explicitly expose how the SINR scales in terms of the level of spatial
repulsion.
The goal of this paper is to introduce and analyze a novel tractable model that captures both
repulsion and randomness in a flexible way.
C. Contributions
This paper proposes a model comprising a superposition of two independent point processes.
The first is a standard PPP with density λp . The second is a random shifted grid with intensity
λg . Two realizations of such point processes along with their association regions (Voronoi
tessellation) can be seen in Figs. 1 and 2. Possible real-world examples where such a model
could be applicable include:
3
•
Vehicular networks where fixed road side units (nearly in a grid) and randomly located
vehicles (a PPP) can transmit information to another vehicle.
•
Device-to-device (D2D) networks with fixed base stations and randomly scattered D2D
devices. A given mobile receiver could receive transmissions from either one.
•
HCNs with repulsively (nearly grid) macro base stations are overlaid with more randomly
scattered small cells.
This paper describes a scalable framework to analyze the these types of cellular networks
using stochastic geometry. However, our framework and numerical results are not limited to
this case. The results are applicable to any wireless network where a repulsive structure exists.
Our theoretical contributions are as follows.
Analytical framework capturing the repulsion between base stations. We propose to
model cellular networks as a combination of two extreme sub-structures repulsive base stations
of transmit power pg are modeled as a random shifted grid with intensity λg ; the uncorrelated
base stations with transmit power pp are modeled into a PPP with intensity λp . We measure the
relative repulsion with the intensity ratio ρλ =
λp
.
λg
Association probability and the signal-to-interference ratio (SIR) coverage probability.
Assuming the typical user is associated with the base station that provides the strongest average
receive signal power, we derive the association probability. Then, we derive the SIR coverage
probability of the typical user, assuming Rayleigh fading channels. The SIR is nearly equivalent to
the SINR for sufficiently dense networks. We show that the association and coverage expressions
are related to the system parameter as functions of intensity ratio ρλ and the power ratio η =
pp
.
pg
Novel observations under the proposed model. First, we show a bias toward the repulsive
base stations in user association. For instance, when ρλ = η = 1, the typical user at the origin
is more likely to be associated with a repulsive base station than an uncorrelated base station.
Secondly, we observe a scale-invariant property of the coverage expression: it is fixed as long as
ρλ does not change. In fact, numerical analysis shows that the coverage probability monotonically
increases with respect to ρ−1
λ . The result suggests that one might increase (or decrease) the
coverage probability of users by adding the repulsive base stations (or by adding the uncorrelated
base stations).
The rest of this paper is organized as follows. Section II discusses our system model and defines
performance metrics. Section III acquires the nearest distance to the proposed point process
and computes the association probability of the typical user. Section IV derives the coverage
4
probability of the typical user. Section V introduces the bounds for the coverage probability and
Section VI presents numerical and simulation results. Section VII concludes the paper.
II. S YSTEM M ODEL
In this section, our system model is depicted. This section explains the characterization of
repulsive cellular networks using two stationary point processes. This section also discusses the
propagation model and the association principle of users in the network.
A. Spatial Modeling of Base Stations
To model the locations of repulsive base stations, one would consider a deterministic square
grid or hexagonal grid that has been considered by practicing engineers [19], [20]. However, the
notion of spatial average or typicality is yet unclear for the deterministic lattice. Furthermore,
under the framework of stochastic geometry, the typicality is obtained by assuming stationary
point processes. For our purposes, the deterministic lattice is not suited for modeling typical
repulsive base stations. Therefore, we adopt a random shifted grid. Specifically, the random
shifted grid is a standard square lattice shifted by a single uniform random variable. The random
locations of the repulsive base stations are given by
X
δs·k+U ,
Φg :=
(1)
k∈Z2
2
where s denotes the width of squares and U represents uniform random variable on − 2s , 2s .
The above equation shows that all points of the square grid are shifted by a random variable U.
The random shifted grid given by (1) is a stationary point process because the joint distribution
{Φg (B1 + x), Φg (B2 + x), . . . , Φg (Bk + x)},
(2)
does not depend on the location of x ∈ R2 for any finite Borel set Bk ∈ B(R2 ) [21, Definition
3.2.1.]. Since it is stationary point process, it admits the intensity parameter λg defined by the
average number of points in a unit area and we have λg = 1/s2 .
On the other hand, we assume an independent homogeneous Poisson point process to represent
uncorrelated base stations.
Φp :=
X
δXi .
i∈N
We use λp to denote the intensity parameter of Φp .
(3)
5
Since the two point processes are stationary, the superposition of the point processes also
yields a stationary point process given by
Φ := Φg ∪ Φp =
X
δs·k+U + δXi .
(4)
k∈Z2 ,i∈N
We produce a stationary point process representing all base stations in repulsive cellular
networks. It can be easily shown that the intensity parameter of the proposed point process is
λg + λp . In addition, we define a key network parameter the intensity ratio ρλ that captures the
repulsion by intensities, i.e.
ρλ :=
λp
.
λg
(5)
Since the intensity of a point process describes the mean number of its points in a set of volume
one, the intensity ratio describes the mean number of the Poisson point process divided by
the mean number of the random shifted grid. It is important to note that we can characterize
the repulsion of base stations by a function of only intensities; we establish a scalable model
for a spatially correlated network. For instance, if the intensity ratio is zero, the proposed
network is composed of the repulsive base stations almost everywhere. On the other hand, if the
intensity ratio is infinity, the proposed network is composed of the uncorrelated base stations
almost everywhere. Figure 1 and 2 demonstrate the proposed point processes and their Voronoi
tessellations where intensity ratios are ρλ = 0.6 or 1.6, respectively.
In addition to the proposed point process Φ, we consider another independent Poisson point
process Φr with intensity parameter λr to describe the locations of users. We assume that all
base stations have full-buffered queues and always transmit1.
B. Signal Propagation and Receive Power
We describe the received signal power P at Y as PX HX /l(kX − Y k), where PX indicates
the transmit power of the base station at X, HX indicates fading, and l(kX − Y k) indicates
large-scale path loss. We do not consider shadowing. This paper assumes that the small scale
fading HX is an independent and identically distributed (i.i.d) exponential random variable with
unit mean. We also assume
l(kX − Y k) := kX − Y kα ,
1
This assumption is justified by assuming λu ≫ λp + λg [4]
(6)
6
5
Random shifted grid
Poisson point process
Voronoi boundary
4
3
2
1
0
-1
-2
-3
-4
-5
-5
-4
-3
-2
-1
0
1
2
3
4
5
Fig. 1. This figure represents Φg (square) and Φp (triangle) when ρλ is 0.6. The dashed lines indicate the Voronoi boundaries
created by Φg ∪ Φp
where α > 2. The transmit power of base station at X is given by


pg if X ∈ Φg .
PX :=

pp if X ∈ Φp .
Lower case alphabets are used for the constants. The transmit ratio η is given by pp /pg .
C. User Association and Interference Modeling
This paper assumes a practical association principle. A user is associated with the base station
that provides the strongest average received power among all the base stations. Furthermore, we
add a typical user at the origin using the stationarity of the proposed point process and marks
on it. Since we consider the typical user at the origin, the associated (or tagged) base station of
the typical user is given by
P Xi
P Xi H Xi
≡ arg max
.
X := arg max E
α
kXi k
kXi kα
Xi ∈Φ
Xi ∈Φ
⋆
(7)
7
5
Random shifted grid
Poisson point process
Voronoi boundary
4
3
2
1
0
-1
-2
-3
-4
-5
-5
-4
-3
-2
-1
0
1
2
3
4
5
Fig. 2. This figure represents Φg (square) and Φp (triangle) when ρλ is 1.6. The dashed lines indicate the Voronoi boundaries
created by Φg and Φp . The Voronoi tessellation matches to the user association map if pg = pp . Users in each Voronoi polygon
are associated with the base station located in its center.
See Figure 1 and 2 for visualization representations of user association in R2 when pg = pp ;
which corresponds to the Voronoi tessellation created by Φ. We remark that our association does
not incorporate Rayleigh fading. As a result, the received signal power from the tagged base
station is given by
S := PX ⋆ HX ⋆ kX ⋆ k−α ,
(8)
and the interference is given by
I :=
X
Xi ∈Φ\{X ⋆ }
PXi HXi kXi k−α .
(9)
As a result, we have SINR at the typical user given by
PX ⋆ HX ⋆ kX ⋆ k−α
S
P
=
SINR :=
−α
I +N
Xi ∈Φ\{X ⋆ } PXi HXi kXi k
where N indicates the noise power at the typical user.
(10)
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D. Performance Metric: SIR
In this paper, we use the coverage probability to capture the reliability of the packet transmissions. Provided that noise power is minuscule compared to the signal or interference powers
in practice, we use signal-to-interference (SIR) that have no noise term in (10) to measure the
reliability of the proposed network. We consider slotted time and in each time slot, we declare
successful transmissions if the SIR at the typical user is greater than a threshold value T . The
coverage probability of the typical user is defined as
pcov := P(SIR ≥ T ).
(11)
Another rationale behind using the SIR instead of SINR is that the metric emphasizes the impact
of topological structure by canceling out the deterministic term N.
E. Repulsion and Pair Correlation Function
We discuss the pair correlation function of the our proposed model. It is known that the pair
correlation function is a very useful measure to explain the repulsion. It basically measures the
correlation between two arbitrary points in a point process Ξ.
Definition 1: The pair correlation function is given by
κ(x, y) :=
λ(x, y)
,
λ(x)λ(y)
(12)
where λ(x, y) is the factorial second moment density of Φ, and λ(x) and λ(y) are intensities of
Ξ.
Proposition 1: The pair correlation function of our proposed point process Φ is
κ(x, y)=1 −
1
(1 + ρλ )2
almost everywhere (a.e.)
(13)
Proof: First, it suffices to consider points in {(x, y) : x 6= y ∈ R2 } since the set {(x, y) :
x = y ∈ R2 } is measure zero and we can assume λ(x, x) = 0. To find the factorial second
moment density function λ(x, y), we need to consider four sub-cases where each point can be
from either the random shifted grid or the Poisson point process. By the stationarity, we see that
λ(x, y) = λ2p + λp λg + λg λp + λg 1{x ∼g y} a.e.,
(14)
where 1{x ∼g y} denotes an indicator function that takes one if both x, y are points of the
random shifted grid. Since the event {x ∼g y} is only applied to countably many points, (13)
follows immediately.
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In general, the pair correlation function explains the repulsion between two points and takes
a value between zero and one. If κ(x, y) = 0, it shows a strong repulsion and if κ(x, y) = 1, it
shows no repulsion. If 0 < κ(x, y) < 1, it shows a moderate repulsion. Since we can adjust ρλ
to control κ lying between zero and one, our proposed point process is able to capture repulsion
of any stationary point process network model that has a pair correlation function.
To establish connection with a point process having no pair correlation function, we can define
an average pair correlation κ̂ between two points. i.e. For any increasing convex averaging sets
An ,
1
κ̂ := lim 2
n→∞ λ ν2 (An )2
Z
κ(x, y)λ(x, y)dxdy,
(15)
An ×An
where ν2 (·) is Lebesgue measure in R2 . Given the average pair correlation 0 ≤ κ̂ ≤ 1, one
can easily fit into our proposed network using the average κ̂. Then our proposed point process
minimizes the quadratic variation cost υ below among all point processes having the same
average pair correlation rate:
1
υ := lim 2
n→∞ λ Vol(An )2
III. D ISTANCE
TO THE
Z
An ×An
[κ(x, y) − κ̂]2 λ(x, y)dxdy.
N EAREST BASE S TATION
AND
(16)
A SSOCIATION P ROBABILITY
This section analyzes the distribution of the nearest base station of the proposed point process,
i.e., the distance distribution from the origin to the nearest point of the proposed point process.
Using the distribution, we can derive the association probability. The distances to the nearest
point of Φg and Φp are given by
Rg := inf kXi k,
(17)
Rp := inf kXi k,
(18)
Xi ∈Φg
Xi ∈Φp
The cumulative distribution function (CDF) and probability distribution function (PDF) of Rp
are well-known and are given by
P(Rp ≤ r) = 1 − exp(−πλp r 2 ),
(19)
fRp (r) = 2πλp r exp(−πλp r 2 ).
(20)
Now, we discuss the distribution of the nearest distance to the Φg , Rg .
10
Lemma 1: The PDF of Rg is


2πrs−2
fRg (r) =

2πrs−2 −
if 0 ≤ r ≤ 2s ,
√
8r
arccos( 2rs ) if 2s < r ≤ 2s 2,
s2
p
where the constant s for the width of square lattice is s = 1/ λg .
(21)
Proof: Denoting a ball of radius r centered at x in R2 by B(x, r), the nearest distance Rg
is represented by
(a)
Rg = inf{R > 0|Φg (B(0, R)) 6= ∅} = inf{R > 0| ∪Xi ∈Φg B(Xi , R) ∩ 0 6= ∅}.
(22)
where (a) is obtained by using the stationarity of Φg . Let us define
s s s s
S0 := − ,
× − ,
.
2 2
2 2
Then, the CDF of Rg is given by
P(Rg < r) = P(inf{R > 0| ∪Xi ∈Φg B(Xi , R) ∩ 0 6= ∅} < r)
ν2 (S0 ∩ B(0, r))
ν2 (S0 )


πr 2

if 0 ≤ r ≤ 2s ,

 s2 
q

s
π−4 arccos( 2r
2
)
r2
= 2 s2
− 2 + 4rs2 − 1 if 2s < r ≤ √s2 ,
2





1
if r > √s2
(b)
=
(23)
(24)
where (b) follows from the fact that CDF of the first touch distance to a stationary point process
is equivalent to the volume fraction of the stationary point process [18, Section 6.3.1.]. By taking
the derivative of expression given in (24) with respect to r, we obtain the PDF.
Corollary 1: The nearest distance distribution of the proposed point process is given by
P(R ≤ r) = 1 − (1 − P(Rp ≤ r))(1 − P(Rg ≤ r)),
(25)
where P(Rg < r) and P(Rp < r) are (19) and (24), respectively.
The expressions given in (19), (24) are used to calculate the association probability of the
typical user. Throughout this paper, we write Ag and Ap to denote the event that the typical
user is associated with either Φg or Φp , respectively.
Theorem 1: The probability that the typical user is associated with either Φp or Φp is
Z √2 Z √2
√
π 2
2
2
P(Ap ) =
(1 − r )f (r) dr +
r arccos(1/r) − r − 1 f (r) dr,
4
0
1
P(Ag ) = 1 − P(Ap ),
(26)
(27)
11
where the probability distribution function f (r) in (26) is
ρλ η β 2
ρλ η β
r exp −π
r .
f (r) = 2π
4
4
(28)
Proof: If the typical user is associated with the nearest base station of the Poisson point
process at distance Rp , it implies that received power from the nearest Poisson point process is
greater than the received power from the nearest random shifted grid. The event Ap is described
by
Ap :=
whereas the event Ag is described by
Ag :=
pg
pp
> α
α
Rp
Rg
,
(29)
pp
pg
< α
α
Rp
Rg
.
(30)
Note that Acp = Ag , Acg = Ap , and Ap ∪ Ag is the universal set. Using the event, we can describe
the association probability P(Ap ) as follows.
pg
pp
> α
P(Ap ) = E 1
Rpα
Rg
pg pp
(a)
= E E 1 α > α Rp = r
r
Rg
Z ∞
=
P Rg ≥ rη −1/α fRp (r) dr
0
(b)
Z
(c)
Z
=
∞
0
=
ν2 (S0 \ B(0, rη −1/α ))
2
2πλp re−πλp r dr
ν2 (S0 )
s 1/α
η
2
0
2
(s2 − πr 2 η −2/α )2πλp re−πλp r dr
√s η 1/α
2
π
2
s2 − 4rs sin(θ) − 2r 2
− 2θ 2πλp re−πλp r dr
s 1/α
2
η
2
√
√
Z 2
Z 2
√
π 2
(d)
=
(1 − r )f (r) dr +
r 2 arccos(1/r) − r 2 − 1 f (r) dr,
4
0
1
+
Z
(31)
where (a) is obtained by conditioning on Rp , (b) is obtained by using the expression given in
(23), (c) is given by denoting θ = arccos(s/2r), and (d) follows from the change of variables.
Note that the expressions given in (26) or (27) are functions of the intensity ratio ρ and power
ratio η. We explicitly quantify the impact of repulsion and transmit powers on the association.
In the following, we derive the upper and lower bounds for the association probability.
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Proposition 2: The association probability that the typical user is associated with Φp is lower
and upper bounded by
P(Ap ) ≥ 1 +
P(Ap ) ≤ 1 +
where the variables β, γ, ρ are
π/2 − 1
e−πρ/2 − 1
β − γ,
+√ √
ρ
ρ( 2 − 1)
e−πρ/2 − 1
π/2 − 1
+√ √
β,
ρ
ρ( 2 − 1)
ρ = ρλ η 2/α ,
p
p
β = erf( πρ/2) − erf( πρ/4),
γ = 0.0956(e−πρ/4 − e−πρ/2 ),
and erf(z) :=
√2
π
Rz
0
(32)
(33)
(34)
(35)
(36)
2
e−t dt.
Proof: We develop the bounds by obtaining lower and upper bounds for the second integration in (26). We consider
for 1 ≤ r ≤
√
g(r) := r 2 arccos(1/r) −
√
r 2 − 1,
2. One can easily check that g(r) is a differentiable function with respect to r
and that g(r) is a convex function since its second derivative with respect to r is greater than
√
zero for the interval (1, 2). Then by fully exploiting the differentiability and the convexity of
g(r), we suppose two linear bounds in the following.
√
g( 2) − g(1)
√
g(r) ≥
(r − 1.19) + 0.1662,
(37)
2−1
√
g( 2) − g(1)
√
g(r) ≤
(r − 1),
(38)
2−1
√
for 1 ≤ r ≤ 2. We obtain the lower bound by applying the mean value theorem on the
√
interval (1, 2) with approximating the tangential line numerically. We obtain the upper bound
by finding the smallest linear function greater than g(r). Replacing g(r) of (31) with (37) and
(38), we acquire the lower and upper bounds, respectively.
For the moment, we assume pg = pp in order to emphasize the impact of topology onto the
user association. If λp = λg , then P(Ap ) = 0.36, i.e., approximately two-fifths of users are
associated with Φp . In other words, users associated with Φg is about fifty percent greater than
users associated with Φp on average. In other words, our repulsive model yields the following.
λp = λg =⇒ P(Ap ) ≤ P(Ag ).
(39)
Probability of being associated with Poisson point process
13
0.7
0.6
0.5
Bias toward Φ g @ ρ λ =1
0.4
0.36
∆(P(Ap ))=0.14
0.3
λ g =λ p ; E[Φ g ]=E[Φ p ]
0.2
Analytic expression
An analytic upper bound
An analytic lower bound
Simulation
Poisson point processes
0.1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Intensity ratio
Fig. 3. The probability that the typical user is associated with the Poisson point process. The transmit powers are assumed to
be the same: pg = pp . The curve with square marks describes P(Âp ) given in (40)
This contrasts to the association behavior of the typical user where the cellular network is
modeled as Poisson point processes. Specifically, suppose we replace the random shifted grid
Φg with an independent Poisson point process Φp′ with intensity λp′ . The association probability
of the typical user in the Poisson network was derived in [6, Lemma 2.] and is
P(Âp ) =
λg
λp
, P(Aˆp′ ) =
.
λp ′ + λp
λp ′ + λp
(40)
Therefore, if λp = λp′ , one half of users are associated with Φp .
λp = λp′ =⇒ P(Âp ) = P(Aˆp′ ).
(41)
Figure 3 expresses P(Ap ) and its upper and lower bounds along with numerically obtained
simulation results. The figure also delineates P(Âp ) in (40) for comparison. Figure 3 shows
no intersection between (26) and (40), respectively. In other words, compared to the Poisson
models, we show a bias toward the Φg in user association.
Remark 1: Understanding the reasons for the association bias is critical. The bias in user
association is originated from a geometrical and probabilistic property of the proposed network.
In the proposed network, a randomly located user is always able to find a base station at a
14
p
p
distance less than 1/(2 λg ); In other words, P(R > 1/(2 λg )) = 0. In mathematical terms,
the void probability of the proposed point process has a finite support. This contrasts to the
Poisson point process whose void probability has infinite support P(R > r) = exp(−2πλr)
where intensity is denoted by λ. As a result, the spatial structure of the random shifted grid
produces the bias. It is important to mention the finite support of proposed model is consistent
with most practical wireless communications where transmitters are located at distances less
than infinity.
In addition to characterizing the role of repulsive base stations in user association, Theorem
1 and Proposition 2 provide a practical insight on the deployment design of base stations. For
instance, heterogeneous cellular networks are configured to have offloading gains: the macro
base stations having many users distribute their users to another base stations with fewer users.
The offloading is designed to alleviate heavy loaded queue of the macro base stations. Given
that the locations of macro base stations show repulsion [1]–[3], the association probability of
this paper proves the existence of biasing toward repulsive macro base stations. It means that the
macro base stations would be congested heavier and the offloading affects the network greater
than a Poisson model can predict [9].
IV. C OVERAGE P ROBABILITY
As we discussed in previous section, the typical user is associated with either Φg or Φp with
probability P(Ag ) and P(Ap ), respectively. Since the coverage probability is complex to analyze
directly, we use the association probability to describe the coverage probability as
pcov = P (SIR > T, Ag ) + P (SIR > T, Ap ) .
(42)
In fact, the coverage probability intersected with the association means the coverage probability
of the typical user when the tagged transmitter is associated with either the random shifted grid
or the Poisson point process. To understand the coverage probability in detail, let us suppose two
Poisson point processes denoted by Φp and Φp′ . Simple expression for the coverage probability
is readily obtained since
pcov = P (SIR > T, Ap′ ) + P (SIR > T, Ap )
(43)
and for P(SIR > T, Ap′ ) and P(SIR > T, Ap ), the association does not affect the intensity of
Poisson point processes forming the interference seen at the typical user thanks to the strong
Markov property of Poisson point process.
15
Fig. 4. The Figure depicts the random shifted grid (inverse triangles) and Poisson point process (triangles) seen at the typical
user at the origin (star). we assume pg = pp and {Ap }. The exclusion ball of radius Rp is created around the typical user. In
addition, the correlated structure of the random shifted grid produces the duplicate exclusion circles around all grid points in
R2 . The perfect dependency between points of Φg causes the infinitely repeated circles.
In contrast, our spatial model considers Poisson point process and random shifted grid which
does not satisfy the strong Markov property; and then, for each association event, we need to
consider the conditional point processes such that there are no points inside a certain exclusion
area. We introduce notations such as Φ!∼r
or Φ!∼r
to indicate the point process are conditioned by
g
p
the fact that no points of Φg or Φp exist inside the ball of radius r centered at zero, Φg (B(0, r)) ≡
0 or Φp (B(0, r)) ≡ 0. Since we assume the nearest maximum receive power association, given
that the typical user is associated with Φg at distance kUk, the radius of the exclusion ball for
the Poisson point process is kUkη 1/α . On the other hand, given that the typical user is associated
with Φp at the distance R, the radius of the exclusion ball for the random shifted grid is Rη −1/α .
Now we are ready to work on the coverage probability intersected with the event of association.
16
First, the coverage probability of the typical user when it is associated with random shifted grid
is given by
P (SIR > T, Ag )


= P

P ⋆ H ⋆ /kX ⋆ kα

PX X
> T, Rp > kX ⋆ kη 1/α 
α
PXi HXi /kXi k
Xi ∈Φ\{X ⋆ }

P ⋆ H ⋆ /kX ⋆ kα
(a)
d
 

PX X
= EU  P 
> T, Rp > kX ⋆ kη 1/α X ⋆ = U 
α
PXi HXi /kXi k
Xi ∈Φ\{X ⋆ }




PU HU /kUkα
(b)


1/α 
1/α 
P
R
>
kUkη
R
>
kUkη
>
T
= EU P·|U  P

 (44)
p
p
·|U
PXi HXi /kXi kα
Xi ∈Φ\{U }




 
(c)



= EU 
P
·|U


P


pg HU /kUkα
 P·|U (Rp > kUkη 1/α )
>
T


PXi HXi /kXi kα
(45)
!∼kU kη 1/α
Xi ∈Φg \{U }∪Φp
where (a) is given by conditioning on the random variable U, (b) is obtained by Bayes rule
using the notation P·|U := P(·|U), and (c) follows from the fact that given Rp > kUkη 1/α , the
Poisson point process has no points inside the ball of radius kUkη 1/α . In addition,



P·|U 

P

pg HU /kUkα

>
T

PXi HXi /kXi kα
(46)
!∼kU kη 1/α
Xi ∈Φg \{U }∪Φp
 




(d)

= E·|U 1







α
pg HU /kUk
P
>T 


PXi HXi /kXi kα


1/α
!∼kU kη
(47)
Xi ∈Φg \{U }∪Φp
α −1
= E·|U P H > T kUk pg
IΦg \{U } + I !∼kU kη1/α
Φp
(e)
α −1
= E·|U exp −T kUk pg
IΦg \{U } + I !∼kU kη1/α
Φp
= LIΦg \{U } T kUkα p−1
LI
g
!∼kU kη 1/α
Φp
T kUkα p−1
,
g
(48)
(49)
(50)
where (d) is given by P(A) = E[1{A}] where 1{·} denotes an indicator function and (e) is
obtained by using the distribution of random variable H. Note that in (50) the L· (ξ) is defined
17
to indicate the conditional expectation of the functional on the the conditional point process.
Inserting (50) into (45), the coverage probability becomes
α −1
α −1
1/α
P(SIR > T, Ag ) = EU LIΦg \{U } T kUk pg LI !∼kU kη1/α T kUk pg P·|U (Rp > kUkη )
Φp
(51)
where the probability P·|U and distribution of U are
2 η 2/α
P·|U (Rp > kUkη 1/α ) = e−πλp kU k
fU (u) =
,
1
.
ν2 (S0 )
Then, deriving the coverage probability boils down to finding the conditional expectation of
functional on the conditional point process LIΦg \{U } (·) and LI
!∼kU kη 1/α
Φp
(·).
Similarly, we derive the coverage probability of the typical user when it is associated with
Poisson point process.
P (SIR > T, Ap )


= P

P ⋆ H ⋆ /kX ⋆ kα

PX X
> T, Rg > kX ⋆ kη −1/α 
α
PXi HXi /kXi k
(52)
Xi ∈Φ\{X ⋆ }

P ⋆ H ⋆ /kX ⋆ kα
d

 
PX X
> T, Rg > kX ⋆ kη −1/α kX ⋆ k = R
= ER  P 
α
PXi HXi /kXi k
Xi ∈Φ\{X ⋆ }




 



P
= ER 
·|R


P


pg HR /kRkα
 P·|R (Rg > Rη −1/α )
>
T


PXi HXi /kXi kα
Xi ∈Φ!∼R
∪Φp!∼Rη
p
(53)
(54)
−1/α
where we use the technique given above. Then the coverage probability becomes
h
i
α −1
−1/α
P(SIR > T, Ap ) = ER LIΦ \Rη−1/α T Rα p−1
T
R
p
L
P
(R
>
Rη
)
, (55)
I
g
·|R
g
g
Φ!∼R
g
p
where the probability P·|R and distribution R are
P·|R (Rg > Rη
−1/α
ν2 (S0 \ B(0, Rη −1/α ))
,
)=
ν2 (S0 )
2
FR (r) = 2πλp re−πλp r .
and similarly, deriving the coverage probability boils down to finding the conditional expectation
of the functional on the conditional point process LI
!∼Rη −1/α
Φg
(·) and LIΦ!∼R (·).
p
18
In the next step, we derive the conditional expectation of functional on the conditional point
process. In the following, we call the Laplace transforms because of its resemblance to the
original Laplace transform.
Lemma 2: The Laplace transforms of interference from Poisson point processes are
Z ∞
pp ξu1−α
du ,
LIΦ∼R (ξ) = exp −2πλp
−α
p
R 1 + pp ξu
Z ∞
pp ξu1−α
LI ∼kU kη1/α (ξ) = exp −2πλp
du .
−α
Φp
kU kη1/α 1 + pp ξu
(56)
(57)
Proof: Since we assume homogeneous Poisson point process, applying the radius of exclusion ball and the strong Markov property produces the results.
Lemma 3: The Laplace transforms of interference from random shifted grids are
Z
Y
1
PU ′ (du′ ),
LI !∼Rη−1/α (ξ) =
′
−α
1
+
p
ξku
+
s(z
,
z
)k
−1/α
Φg
g
1
2
S0 \B(0,Rη
)
2
(58)
(z1 ,z2 )∈Z
LIΦg \kU k (ξ) =
Y
(z1 ,z2 )∈Z2 \(0,0)
1
,
1 + pg ξkU + s(z1 , z2 )k−α
(59)
d
where U ′ = uniform[S0 \ B(0, rη −1/α )].
Proof: We have
−s P
p HkXi k−α
!∼Rη −1/α g
Xi ∈Φg
LI !∼Rη−1/α (ξ) := E·|R e
Φg
−s P
p HkXi k−α
!∼Rη −1/α g
Xi ∈Φg
|Φg
= EΦg |R EH|R e
(a)
(b)
Y
= EΦg |R
Xi ∈Φg!∼Rη
1
1 + pg ξkXi k−α
−1/α
2
(c)
= EU ′ |R
=
Z
Z
Y
(z1 ,z2 )
1 + pg
ξ||u′
2
Z
Y
S0 \B(0,Rη−1/α ) (z ,z )
1 2
1
+ s(z1 , z2 )||−α
1
PU ′ (dv),
1 + pg ξkv + s(z1 , z2 )k−α
(60)
where (a) is obtained by conditioning on the locations of the random shifted grid, (b) is obtained
by the Laplace transform of an exponential random variable H with unit mean, (c) follows from
the fact that all points of random shifted grid is measurable with respect to the uniform random
variable U ′ that take a value uniformly in S0 \B(0, Rη −1/α ). We denote the set S0 \B(0, Rη −1/α )
into S0′
19
On the other hand, the Laplace transform of the interference given that the typical user is
associated with the nearest shifted grid is
i
h P
−s X ∈Φg \{U } pg H/kXi kα
i
LIΦg \kU k (ξ) := E·|U e
ii
h h P
−s X ∈Φg \{u} pg HkXi k−α
i
= E·|U E e
2
Z
Y
=
−α
(z1 ,z2 )6=(0,0)
2
=
Z
Y
(z1 ,z2 )6=(0,0)
E·|U [e−sHpg kXi k ]
1
.
1 + pg ξkU + s(z1 , z2 )k−α
(61)
Note that given that the typical user is associated with kUk, the nearest base station of the random
shifted grid does not comprise the interference; thus, the nearest point Z2 = (0, 0) should be
removed. In addition, the random shifted grid is measurable with respect to the associated base
station kUk.
Since we have the Laplace transform of the interference, the coverage probability of the typical
user is easily obtained.
Theorem 2: The coverage probability of the typical user is

Z
Z2
R∞
Y
T ηkukα v 1−α
2
2/α
dv
−2πλ
p
kukη 1/α 1+T ηkukα v −α

e−πλp kuk η e
pcov =
S0
+
Z
0
∞
(z1 ,z2 )6=(0,0)
ν2 (S0′ ) −2πλp Rr∞
ν2 (S0 )
e
T r α v 1−α
1+T r α v −α
dv

Z

S′
2
Z
Y
1+
(z1 ,z2 )
η−1 T r α
||v+s(z1 ,z2 )||α
1
1+
T kukα
||u+s(z1 ,z2 )||α


 PU (du)
PU ′ (dv) fR (r) dr
(62)
Proof: To obtain P(SIR > T, Ag ), we insert the expressions (57) and (59) evaluated
at ξ = T kukα p−1
into the expression in (51). Then use the fact that P(Rg > rη −1/α ) :=
g
ν2 (S0 \B(0,rη−1/α ))
ν2 (S0 )
=
ν2 (S ′ )
.
ν2 (S0 )
To derive P(SIR > T, Ap ), we insert (56) and (58) evaluated at
2 η 2/α
1/α
ξ = T r α p−1
) = e−πλp kuk
p into the expression in (55). Then use the fact that P(Rp > kukη
.
The summation of P(SIR > T, Ag ) and P(SIR > T, Ap ) produces the coverage probability of
the typical user at the origin.
The above expression for the coverage probability explicitly describes a chance that the typical
user is being covered, i.e., the SIR of the typical user is greater than the threshold T . Since
the above coverage expression is a function of the network parameters including λg , λp , pg ,
pp , and α, we know the behavior of coverage probability with respect to those parameters in
20
the proposed repulsive cellular network. The influence of repulsive structure is clarified in the
following proposition.
Proposition 3: The coverage probability of the typical user at the origin can be rewritten into
Z
R
Y
T ηkûkα w1−α
1
2 2/α
−2πρλ ∞ 1/α
dw
kûkη
1+T ηsα kûkα w−α
e
PÛ (dû)
pcov =
e−πρλ kûk η
T kûkα
1
+
Sˆ0
α
kû+(z1 ,z2 )k
(z ,z )∈Z2
1
2
6=(0,0)
+
Z
0
∞
Z
T r̂ α w1−α
ν2 (Sˆ0′ ) −2πρλ Rr̂∞ 1+T
α w−α dw
r̂
e
ν2 (Sˆ0 )
Ŝ0 \B(r̂η−1/α )
2
Z
Y
PÛ ′ (dû′ )
1+
(z1 ,z2 )
η−1 T r̂ α
kû+(z1 ,z2 )kα
fRˆp (r̂) dr̂,
(63)
′
where Û ′ = uniform[Sˆ0 ] , Sˆ0′ := Sˆ0 \ B(0, r̂η −1/α ), Û = uniform[Sˆ0 ] where Sˆ0 := [−1/2, 1/2] ×
2
[−1/2, 1/2], and fRˆp (r̂) := 2πρλ r̂e−πρλ r̂ . Note that the coverage expression given in (62)
becomes a function of only the intensity ratio ρλ ,the power ratio η, and α.
Proof: Given in Appendix A
Remark 2: The coverage probability of the typical user is explicitly given in Theorem 2 and
Proposition 3 as a function of intensities and transmit powers. In fact, Proposition 3 capture
the repulsive structure of the proposed network into the single variable ρλ and compactly
describes the coverage probability using it. Since ρλ is defined as λg /λp or E[Φg (A)]/ E[Φp (A)],
Proposition 3 concisely shows the influence of the repulsive base stations.
Remark 3: The coverage expression given in Proposition 3 can find its use as a one way
of predicting the coverage probability of any repulsive networks mentioned in Section II-E. In
those applications, κ or κ̂ can be computed numerically or analytically. Then, using the simple
relationship between κ and ρλ that we provided in (13), those networks are projected onto our
proposed network and the coverage probability is obtained by using (63) with η = 1. Therefore,
the expression in (63) fundamentally address the influence of repulsive structure in any type of
wireless networks that adapt the nearest receiver association principle.
V. B OUNDS
FOR THE
C OVERAGE P ROBABILITY
In this section, we provide the lower and upper bounds for the coverage probability. First, we
derive the upper and lower bounds for the Laplace transforms in (51) and (55).
21
Lemma 4: For an arbitrary square truncation window denoted by W in R2 that satisfies
W ⊇ S0 , the Laplace transform of interference from random shifted grid in (51) yields
P
− z
LIΦg \{U } (ξ) ≥ e

ξpg
2
α
1 ,z2 ∈Z \(0,0) kU +s(z1 ,z2 )k
Y
LIΦg \{U } (ξ) ≤ 
,
(64)

1
.
1 + ξpg kU + s(z1 , z2 )k−α
2
(z1 ,z2 )∈Z ∩W
(65)
The Laplace transform of interference from random shifted grid given in (55) yields
2π
−
(ξ) ≥ e
Z
LI !∼Rη−1/α (ξ) ≤
LI
!∼Rη −1/α
Φg
S0′
Φg
R∞
Rη −1/α
Y
(
v log 1+ξpg v −α
) dv
ν2 (S ′ )
0
2
(z1 ,z2 )∈Z ∩W
,
(66)
1
PU ′ (dv).
1 + ξpg kv + s(z1 , z2 )k−α
(67)
where Z2 ∩W indicates the set of integer pairs in R2 ∩W .
Proof: Given in Appendix B
Remark 4: As the size of window kW k increases, both upper bounds given in (65) and (67)
become tighter in the expense of a marginal computational complexity.
Theorem 3: The coverage probability of the typical user is
Z
R
PZ2 \(0,0)
T kukα
2 2/α −
−2πλp ∞ 1/α
kukη
e−πλp kuk η e z1 ,z2 ku+s(z1,z2 )kα e
pcov ≥
ηT kukα v 1−α
1+ηT kukα v −α
dv
PU (du)
S0
+
Z
∞
0
ν2 (S0′ ) −
e
ν2 (S0 )
2π
(
R∞
u log 1+η −1 T r α u−α
rη −1/α
ν2 (S ′ )
0
) du
−2πλp
R∞
r
T r α u1−α
1+T r α u−α
du
fR (r) dr.
(68)
The coverage probability of the typical user is
pcov ≤
+
Z
0
−πλp kuk2 η2/α e
e
S0
Z ∞
−2πλp
ηT kukα v 1−α
kukη 1/α 1+ηT kukα v −α
R∞
1+T
Z
1−α
T rα v
ν2 (S0′ ) −2πλp Rr∞ 1+T
dv
r α v −α
e
ν2 (S0 )
S′
dv
PU (du)
1
PU ′ (du) fRp (r) dr.
(1 + η −1 T r α kuk−α )
(69)
Proof: The expressions (64) and (65) evaluated at ξ = T kUkp−1
g yield the lower and upper
bounds for the P(SIR > T, Ag ); the expressions (66) and (67) evaluated at ξ = T Rp p−1
p produces
lower and upper bounds for the P(SIR > T, Ap ).
The expressions in (68) and (69) are given by adapting the simplest window W := S0 . They
are much simpler than the original coverage expression in (62) and are numerically computable.
The upper bounds and lower bounds are numerically validated in Section VI.
22
VI. N UMERICAL A NALYSIS : S YSTEM L EVEL S IMULATIONS
In this section, we show the coverage probability of the typical user obtained from simulations,
verify the bounds for the coverage probability, and discuss the behavior of typical transmission
affected by repulsive structure. We perform Monte Carlo system level simulation with the number
of iterations N.
Figure 5 and Figure6 showed the simulation results and analytical lower (Figure 5) and upper
(Figure 6) bounds given by Theorem 3. In general, both upper and lower bounds are very tight.
We consider window W = [−3s/2, 3s/2] × [−3s/2, 3s/2]. It is important to mention that the
bounds become tighter as the window size increases. The figures show that changing the intensity
ratio shifts the coverage curves. The influence of changing the intensity ratio is greater in α = 4
than α = 3. Almost all users in repulsive networks are affected by topological changes.
To clarify the impact of repulsive structure, we introduce extra simulation constraint that
nullifies the edge effects. Specifically, for a simulation window W , we maintain the mean number
of base stations to be a constant for any intensity ratio.
E[Φ(W )] := C(≥ 100)
(70)
By doing so, the topological influence of repulsive structure is emphasized. Figure 7 and 8 are
numerically obtained under this constraint. If ρλ = 3 and C = 100 then, λp = 75 and λg = 25.
By doing so, the number of base stations are maintained.
In Figure 7, the SIR coverage probability of the typical user is delineated with respect to the
inverse of the intensity ratio. We assume α = 4 and pg = pp . The figure demonstrates that at
−1
α = 4 and ρ−1
λ = 0.01, about 57% of users have SIRs greater than 0 dB. However, at ρλ = 1.7
about 66% of users have SIRs greater than 0dB. This monotonic increasing property is observed
for any α > 2. Furthermore, with no rigorous proof, when η = 1, we observe that the first order
derivatives of coverage curves are maximized at ρλ ≈ 1. In other words, if pg = pp , the coverage
probability is the most sensitive if the number of repulsive base stations and the number of
random base stations are about the same. In the same vein, Figure 8 describes the coverage
probability with respect to the power ratio η. The figure demonstrates that at ρλ = 1, α = 3 and
η −1 = 0.1, about 30 percent of users have SIR greater than 1. At η −1 = 6.3, 50 percent of users
have SIR greater than 1. The monotonic increasing property is observed for α > 2.
Remark 5: We find the coverage probability is a monotonically increasing function with respect
to the inverse of intensity ratio and/or the inverse of power ratio. The behavior of the typical user
23
1
α=4 ρ =4 (Sim)
λ
α=4 ρ =4 (LB)
0.9
λ
α=3, ρ =4 (Sim)
Coverage probability of the typical user
λ
α=3, ρ =4 (LB)
0.8
λ
α=4,ρ =0.25 (Sim)
λ
α=4,ρ =0.25 (LB)
λ
0.7
α=3,ρ =0.25 (Sim)
λ
α=3,ρ =0.25 (LB)
λ
0.6
0.5
0.4
0.3
0.2
0.1
0
-20
-15
-10
-5
0
5
10
15
20
Threshold (dB)
Fig. 5. This figure represents the coverage probability obtained by simulations (Sim) and their corresponding analytic lower
bounds (LB) given in (68). We consider α = 3 and α = 4 when η = 1. We consider ρλ : 0.25 and 4. The lower bound in
Theorem 3 is very tight for α = 3, 4. The truncation window is defined as W = [−3s/2, 3s/2]2
in the proposed model sharply contrasts to the behavior of the typical user in Poisson networks;
the coverage probability of the typical user in Poisson networks was derived in [4] and it proves
that SIR does not respond to the intensity parameter λ; see [4, Theorem 1]. In other words, the
Poisson models predict that SIR does not scale with respect to intensity. However, we find that if
repulsive base stations are considered, the SIR scales monotonically with respect to ρ−1 ; in other
words, the reliability of any network with repulsive network elements is able to be improved (or
be diminished) by allowing (or denying) repulsive transmitters.
VII. C ONCLUSIONS
In this paper, we developed a scalable modeling technique representing the repulsive base
stations and uncorrelated base stations. This paper derived the association probability, the Laplace
transform of interference, and the coverage probability of the typical user. Our findings can be
summarized into three points. First, the typical user is more likely to be associated with a
repulsive base station rather than an uncorrelated base station. Second, the expressions for the
24
1
α=4,ρ =2 (Sim)
λ
α=4,ρ =2 (UB)
0.9
λ
α=3,ρ =2 (Sim)
Coverage probability of the typical user
λ
α=3,ρ λ =2 (UB)
0.8
α=4,ρ =0.5 (Sim)
λ
α=4,ρ =0.5 (UB)
λ
0.7
α=3,ρ =0.5 (Sim)
λ
α=3,ρ λ =0.5 (UB)
0.6
0.5
0.4
0.3
0.2
0.1
0
-20
-15
-10
-5
0
5
10
15
20
Threshold (dB)
Fig. 6. This figure represents the coverage probability obtained by simulations (Sim) and their corresponding analytic upper
bounds (UB) given in (69). We consider α = 3 and α = 4 when η = 1. We consider ρλ : 0.5 and 2. The upper bound in
Theorem 3 is very tight for α = 3, 4. The truncation window is defined as W = [−3s/2, 3s/2]2
association and the coverage requires only two parameters, intensity ratio and power ratio. Third,
the coverage probability scales with respect to the number of repulsive base stations.
One can extend these results in many ways including the following directions. The repulsive
base stations can be modeled into a hexagonal shifted grid. The Nakagami fading channels or a
dual slope path loss function



akX − Y kα1



l(kX − Y k) = bkX − Y kα2




0
if 0 ≤ kX − Y k ≤ d1
if d1 ≤ kX − Y k ≤ d2
if D2 ≤ kX − Y k
where a, b, α1 , α2 ≥ 0. This assumption can be applied to understand the repulsion in millimeter
wave networks or any dense wireless networks. In addition, the numerical observation made in
Section VI can be rigorously proved by showing that the expression given in (62) is a monotonic
increasing function with respect to the inverse of intensity ratio ρ−1
λ ; its first order derivative
with respect to the intensity ratio is always less than zero.
25
1
0.9
0.8
α=2
α=2.5
α=3
α=3.5
α=4
P(SIR>1)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10 -2
10 -1
10 0
Inverse of intensity ratio
Fig. 7. This figure represents the percentage of user that have SIR greater than one with respect to inverse of the intensity
ratio. We assume pg = pp and X, Y axes are log scaled. For α > 2, the coverage probability is monotonically increasing with
respect to the inverse of the intensity ratio.
A PPENDIX A
P ROOF
OF
P ROPOSITION 3
Therefore, we need to show that P(SIR > T, Ag ) and P(SIR > T, Ap ) are functions of
intensity ratio and power ratio.
First, P(SIR > T, Ag ) can be given by
Z
R
Y
T η||u||α v 1−α
1
2 2/α
dv
−2πλp ∞ 1/α
1+T ηkukα v −α
kukη
PU (du)
e−πλp kuk η
e
T kukα
1
+
S0
α
2
||u+s(z1 ,z2 )||
(z1 ,z2 )∈Z \(0,0)
Z
α
α 1−α
R
Y
1
2 2/α
−2πλp ∞ 1/α s T ηkαûk vα −α dv
u=sû
skûkη
1+T ηs kûk v
=
e−πρλ kûk η
e
PÛ (dû)
T kûkα
1
+
Sˆ0
α
2
kû+(z1 ,z2 )k
(z1 ,z2 )∈Z \(0,0)
Z
R
Y
T ηkûkα w1−α
1
dw
−2πρλ ∞ 1/α
v=sw
−πρλ kûk2 η2/α
1+T
ηsα kûkα w−α
k
ûkη
PÛ (dû),
=
e
e
T kûkα
1 + kû+(z1 ,z2 )kα
Sˆ0
(z ,z )∈Z2 \(0,0)
1
2
where Û = Uniform([− 21 , 12 ] × [− 21 , 21 ]) and then PÛ (du) = 1 · dx dy.
26
1
α=2.5
α=3
α=3.5
α=4
0.9
0.8
0.7
P(SIR>1)
0.6
0.5
0.4
0.3
0.2
0.1
0
10 -1
10 0
10 1
Inverse of power ratio
Fig. 8. This figure represents the percentage of users that have SIR greater than one with respect to the inverse of power ratio.
We assume λg = λp . For given α > 2 and T = 1, the coverage probability of the typical user is monotonically increasing with
respect to the inverse of the power ratio.
On the other hand, P(SIR > T, Ap ) can be given by

Z ∞
Z
Z2
R ∞ T r α v1−α
′
Y
ν2 (S0 ) −2πλp r 1+T rα v−α dv 
e
ν2 (S0 )
1+
0
S0 \B(0,rη−1/α )
(z1 ,z2 )
u′ =sû′
=
r=sr̂
=
v=sw
=
Z
Z
∞
0
∞
0
Z
0
∞
ν2 (S0′ ) −2πλp Rr∞ T rα vα1−α
dv
1+T r v −α
e
ν2 (S0 )
Z
1
η−1 T r α
||u′ +s(z1 ,z2 )||α
2
−1/α
Sˆ0 \B(0, rη s
Z
Y
) (z ,z )
1 2

PU ′ (du′ ) fR (r) dr
s
1+
s−α η−1 T r α
||û′ +(z1 ,z2 )||α
PÛ ′ (dû′ ) fRp (r) dr
Z
Z2
R ∞ sα T r̂ α v1−α
Y
ν2 (Sˆ0′ )
−πρλ r̂ 2 −2πλp sr̂ 1+sα T r̂ α v−α dv
2πρλ r̂e
e
ν2 (Sˆ0 )
Ŝ0 \B(0,r̂η−1/α ) (z ,z ) 1 +
1 2
R
α 1−α
ν2 (Sˆ0′ )
2 −2πρλ ∞ T r̂ w
dw
r̂ 1+T r̂ α w−α
2πρλ r̂e−πρλ r̂ e
ˆ
ν2 (S0 )
Z
2
Z
Y
Ŝ0 \B(0,r̂η−1/α ) (z ,z )
1 2
1
η−1 T r̂ α
kû+(z1 ,z2 )kα
1
1+
η−1 T r̂ α
kû+(z1 ,z2 )kα
PÛ ′ (dû′ )
PÛ ′ (dû′ ) dr̂,
where Û ′ = Uniform([− 12 , 12 ] × [− 21 , 12 ] \ B(0, r̂η −1/α )). Therefore, the coverage probability of
the typical user is a function of only ρλ and η.
27
A PPENDIX B
P ROOF
OF
L EMMA 4
Firstly, we develop upper and lower bounds for the Laplace transform of the interference
given in (51). Note that the interference is comprised of signals from both random shifted grid
and Poisson point process. The Laplace transform of interference created by the Poisson point
process is well-known and does not require a bound. We need to find the upper and lower bounds
for the interference created by the random shifted grid in (51). We have
P
LIΦg \{U } (ξ) = e
≥ e−
where we use log
1
1+x
(z1 ,z2 )∈Z2 ∩W \(0,0)
P
log
(z1 ,z2 )∈Z2 ∩W \(0,0)
1
1+ξpg kU +s(z1 ,z2 )k−α
ξpg kU +s(z1 ,z2 )k−α
(71)
,
(72)
≥ −x for x ∈ (0, ∞).
In order to find an upper bound for (51), we define a measurable function ϕ(·) such that
Y
1
ϕ(U, ξ) :=
.
(73)
1 + ξpg ||U + s(z1 , z2 )||−α
2
(z1 ,z2 )∈Z
Since the element 1/(1 + ξpg kU + s(z1 , z2 )k−α ) takes a value between zero and one, we have
Y
1
,
(74)
LIΦg \{U } (ξ) = ϕ(U, ξ) ≤
1 + ξpg kUk−α
2
(z1 ,z2 )∈Z ∩W
where the inequality follows by truncating the function ϕ(U, ξ) by finite terms. We denote the
truncation window by W . Applying (72) or (74) into (51), we obtain the lower and the upper
bounds, respectively.
Secondly, we develop lower and upper bounds for the Laplace transform of the interference
created by the random shifted grid. In order to find the lower bound for (55), we have


P
log
!∼Rη −1/α

LI !∼Rη−1/α(ξ) = E·|R eXi ∈Φg
1
1+ξpg kXi k−α
Φg
−
= E·|R e
R
R2
log(1+ξpg kxk−α )Φg!∼Rη
−1/α


(dx)
,
28
where we have the second equality by changing the summation into an integration form. Moreover,
Z
ξpg
!∼Rη−1/α
Φg
(dx)
E·|R exp −
log 1 +
kxkα
R2
Z
ξpg
!∼Rη−1/α
≥ exp − E·|R
log 1 +
Φg
(dx)
kxkα
R2
Z
ξpg
!∼Rη−1/α
E[Φg
(dx)] ,
= exp −
log 1 +
kxkα
R2
(75)
(76)
(77)
where we use Jensen’s inequality and Campbells mean value formula [13]. We denote the
conditional intensity measure of the random shifted grid by M
Φg!∼Rη
−1/α
(dx) and it is
S
1{x ∈
/ Xi ∈s·Z2 B(Xi , Rη −1/α )}
M !∼Rη−1/α (dx) =
dx
Φg
ν2 (S0 \ B(Rη −1/α ))
1{x ∈ R2 \B(0, Rη −1/α )}
dx.
ν2 (S0 \ B(0, Rη −1/α ))
≤
(78)
where 1{A} denote the indicator function for event {A}. In other words, the intensity measure
is upper bounded by the same uniform measure in S0 \ B(0, rη −1/α ) that will nonzero for all
the grid points except the origin. By inserting (78) into (77), we have
2π
LI
!∼rη −1/α
Φg
−
(ξ) ≥ e
ξpg
du
u log 1+ α
u
Rη −1/α
ν2 (S0 \B(0,Rη −1/α ))
R∞
.
In order to find the upper bound for (55), we use the measurable function ϕ(·). Then,
Y
1
ϕ(U ′ , ξ) :=
1 + ξpg ||U ′ + s(z1 , z2 )||−α
2
(79)
(80)
(z1 ,z2 )∈Z
≤
Y
2
(z1 ,z2 )∈Z ∩W
1 + ξpg
kU ′
1
.
+ s(z1 , z2 )k−α
Here we also consider the truncation window W . Then applying ϕ(U ′ ) ≤ ϕtrun (U ′ )
E[ϕ(U)] ≤ E[ϕtrun (U ′ )] yields
Z
LI !∼rη−1/α (ξ) ≤
Φg
S0 \B(0,Rη−1/α )
Y
(z1 ,z2 )∈Z2 ∩W
1
PU ′ (du).
1 + ξpg ku + s(z1 , z2 )k−α
(81)
=⇒
(82)
Applying the expression given in (79) or (82) into (55), we obtain the lower and upper bounds.
29
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