1

Cross-Mode Interference Characterization in
Cellular Networks with Voronoi Guard Regions

arXiv:1702.05018v3 [] 20 Oct 2017

Stelios Stefanatos, Antonis G. Gotsis, and Angeliki Alexiou

Abstract—Advances in cellular networks such as device-todevice communications and full-duplex radios, as well as the inherent elimination of intra-cell interference achieved by networkcontrolled multiple access schemes, motivates the investigation
of the cross-mode interference properties under a guard region
corresponding to the Voronoi cell of an access point (AP). By
modeling the positions of interfering APs and user equipments
(UEs) as Poisson distributed, analytical expressions for the
statistics of the cross-mode interference generated by either APs
or UEs are obtained based on appropriately defined density
functions. The considered system model and analysis are general
enough to capture many operational scenarios of practical interest, including conventional downlink/uplink transmissions with
nearest AP association, as well as transmissions where not both
communicating nodes lie within the same cell. Analysis provides
insights on the level of protection offered by a Voronoi guard
region and its dependence on type of interference and receiver
position. Numerical examples demonstrate the validity/accuracy
of the analysis in obtaining the system coverage probability for
operational scenarios of practical interest.
Index Terms—Stochastic geometry, cellular networks, guard
region, D2D communications, full-duplex radios, interference.

I. I NTRODUCTION
Characterization of the interference experienced by the
receivers of a wireless network is of critical importance for
system analysis and design [1]. This is especially the case
for the future cellular network, whose envisioned fundamental
changes in its architecture, technology, and operation will have
significant impact on the interference footprint [2]. Interference characterization under these new system features is of
the utmost importance in order to understand their potential
merits as well as their ability to co-exist.
Towards increasing the spatial frequency reuse, two of the
most prominent techniques/features considered for the future
cellular network are device-to-device (D2D) communications
[3] and full-duplex (FD) radios [4]. Although promising,
application of these techniques introduces additional, crossmode interference. For example, an uplink transmission is
no longer affected only by interfering uplink transmissions
but also by interfering D2D and/or downlink transmissions.
Although it is reasonable to expect that the current practice of
eliminating intra-cell interference by employing coordinated
transmissions per cell will also hold in the future [5], the
continually increasing density of APs and user equipments
(UEs) suggest that inter-cell interference will be the major
limiting factor of D2D- and/or FD-enabled cellular networks,
rendering its statistical characterization critical.
The authors are with the Department of Digital Systems, University of
Piraeus, Piraeus, Greece. Email: {sstefanatos, agotsis, alexiou}@unipi.gr.

A. Previous Work
Stochastic geometry is by now a well accepted framework
for analytically modeling interference in large-scale wireless
networks [6]. Under this framework, most of the numerous
works on D2D-enabled cellular networks (without FD radios)
consider the interfering D2D nodes as uniformly distributed
on the plane, i.e., there is no spatial coordination of D2D
transmissions (see, e.g., [7], [8], [9], [10]). Building on the
approach of [11], various works consider the benefits of
employing spatially coordinated D2D transmissions where, for
each D2D link in the system, a circular guard region (zone)
is established, centered at either the receiver or transmitter,
within which no interfering transmissions are performed [12],
[13]. However, when the D2D links are network controlled
[5], a more natural and easier to establish guard region is the
(Voronoi) cell of a coordinating AP. Under a non-regular AP
deployment [14], this approach results in a random polygon
guard region, which makes the interference characterization a
much more challenging task.
Interference characterization for this type of guard region
has only been partially investigated in [15], [16], [17] and only
for the case of conventional uplink transmissions with nearest
AP association and one active UE per cell (with no crossmode interference). As the positions of the interfering UEs
are distributed as a Voronoi perturbed lattice process (VPLP)
in this case [18], [19], which is analytically intractable, an
approximation based on a Poisson point process (PPP) model
with a heuristically proposed equivalent density is employed.
This approach of approximating a complicated system model
by a simpler one with appropriate parameters (in this case,
by a PPP of a given density) was also used in [20] for the
characterization of downlink systems (with no cross-mode
interference as well). Reference [19] provides a rigorous
characterization of the equivalent density of the UE-generated
interference both from the “viewpoint” of an AP as well as
its associated UE, with the latter case of interest in case
of cross-mode interference. The analysis reveals significant
differences in the densities corresponding to these two cases
suggesting that the equivalent density is strongly dependent on
the considered receiver position. Interference characterization
for the case of arbitrary receiver position that may potentially
lie outside the Voronoi guard region as, e.g., in the case of
cross-cell D2D links [21], has not been investigated in the
literature, let alone under cross-mode interference conditions.
Investigation of interference statistics under a cell guard
region is also missing in the (much smaller) literature on FDenabled cellular networks, which typically assumes no spatial

2

coordination for the UE transmissions (see e.g., [22], [23],
[24]).
B. Contributions
This paper considers a stochastic geometry framework for
modeling the cross-mode interference power experienced at
an arbitrary receiver position due to transmissions by APs
or UEs and under the spatial protection of a Voronoi guard
region. Modeling the positions of interfererers (APs or UEs)
as a Poisson point process, the statistical characterization of
the cross-mode interference is pursued via computation of its
Laplace transform. The main contributions of the paper are
the following.
• Consideration of a general system model, which allows
for a unified analysis analysis of cross-mode interference
statistics. By appropriate choice of the system model
parameters, the interference characterization is applicable
to multiple operational scenarios, including conventional
downlink/uplink communications with nearest AP association and (cross-cell) D2D links where the transmitterreceiver pair does not necessarily lie in a single cell.
• Exact statistical characterization of AP-generated crossmode interference, applicable, e.g., in the case where
interference experienced at an AP is due to transmissions
by other FD-enabled APs. An equivalent interferer density is given in a simple closed form, allowing for an
intuitive understanding of the interfernece properties and
its dependence on the position of the receiver relative
to the position of the AP establishing the Voronoi guard
region.
• Determination of a lower bound of the Laplace transform in case of UE-generated cross-mode interference,
applicable, e.g., in the case where a UE experiences
interference due to other FD-enabled or D2D-operating
UEs. The properties of the corresponding equivalent
density of interferers is studied in detail, providing insights for various operational scenarios, including the
standard uplink communication scenario (with no crossmode interfernece) where a rigorous justification of why
the heuristic approaches previously proposed in [16], [17]
are accurate.
Simulated examples indicate the accuracy of the analytical
results also for cases where the positions of interferering UEs
are VPLP distributed, suggesting their use as a basis for
determination of performance as well as design of optimal
resource allocation algorithms for future, D2D- and/or FDenabled cellular networks.
C. Notation
The origin of the two-dimensional plane R2 will be denoted
as o. The Euclidean norm of x ∈ R2 is denoted as |x| with
operator | · | also used to denote the absolute value of a scalar
or the Lebesgue measure (area) of a bounded subset of R2 .
The polar form representation of x ∈ R2 will be denoted as
(|x|, ∠x) or |x|∠x, where ∠x is the principal branch of the
angular coordinate of x taking values in [−π, π). The open ball

in R2 , centered at x ∈ R2 and of radius R > 0, is denoted
as B(x, R) , {y ∈ R2 : |y − x| < R}, whereas its boundary
is denoted as C(x, R) , {y ∈ R2 : |y − x| = R}. I(·) is the
indicator (0 − 1) operator, P(·) is the probability measure,
and E(·) is the expectation operator. Functions arccos(·) :
[−1, 1] → [0, π] and arcsin(·) : [−1, 1] → [−π/2, π/2]
are the principal branches of the inverse cosine and sine,
respectively. The Laplace transform of a random variable z
equals Lz (s) , E(e−sz ), for all s ∈ R for which the
expectation exists.
II. S YSTEM M ODEL
A large-scale model of a cellular network with APs and
UEs positioned over R2 is considered. The positions of APs
are modeled as a realization of a homogeneous PPP (HPPP)
Φa ⊂ R2 of density λa > 0. All the communication scenarios
considered in this paper involve three nodes, namely,
• a, so called, typical node, which, without loss of generality (w.l.o.g.), will be assumed that its position coincides
with the origin o. The typical node is an AP if o ∈ Φa
or a UE, otherwise;
∗
• the closest AP to the typical node, located at x
,
arg minx∈Φa |x| (x∗ = o in case o ∈ Φa , i.e., the typical
node is an AP);
• a (typical) receiver located at an arbitrary position xR ∈
R2 . Typical node is an AP if xR ∈ Φa or a UE, otherwise.
The case xR = o is also allowed, meaning that the typical
node is also the typical receiver.
Let V ∗ denote the Voronoi cell of the AP at x∗ , generated by
the Poisson-Voronoi tessellation of the plane from Φa , i.e.,

V ∗ , y ∈ R2 : |y − x∗ | < |y − x|, for all x ∈ Φa \ {x∗ } .
The goal of this paper is to characterize the interference power
experienced at xR due to transmissions from nodes (APs or
UEs) that lie outside V ∗ , i.e., with V ∗ effectively forming a
spatial guard region within which no interference is generated.
Let Φu ⊂ R2 denote the point process representing the
positions of the UEs in the system other than the typical
node and typical receiver, and IxR ,a and IxR ,u denote the
interference power experienced at xR due to transmissions
by all APs and UEs in the system, respectively. Under the
Voronoi guard region scheme described above, the standard
interference power model is adopted in this paper, namely [1],
X
IxR ,k ,
Pk gx |x − xR |−αk , k ∈ {a, u},
(1)
x∈Φk \V ∗

where gx ≥ 0 is the channel gain of a transmission generated
by a node at x ∈ R2 , assumed to be an independent of x,
exponentially distributed random variable of mean 1 (Rayleigh
fading), αk > 2 is the path loss exponent and Pk > 0 is
the transmit power, which, for simplicity, is assumed fixed
and same for all nodes of the same type. Figure 1 shows an
example realization of V ∗ and the positions of the interferers
for the case xR 6= x∗ 6= o. Note that V ∗ is a random guard
region as it depends positions of the APs, therefore, there is no
guarantee that xR lies within V ∗ , i.e., it holds P(xR ∈ V ∗ ) <

3

TABLE I
S PECIAL C ASES OF S YSTEM M ODEL
V∗
|x∗ |

condition

xR 6= x∗ ,
x∗ 6= o

xR = o,
x∗ 6= o
xR = x∗ ,
x∗ 6= o

Fig. 1. Random realization of the system model. APs and UEs are shown as
triangle and circle markers, respectively. The typical node in this example
is also a UE shown as a square. The shaded polygon area indicates the
Voronoi guard region V ∗ within which no node (indicated by open marker)
transmits. All nodes outside V ∗ (filled markers) generate interference that is
experienced by the receiver of interest whose position (xR ) is indicated by
an open diamond marker. This scenario may correspond to a (cross-cell) D2D
link between typical node and receiver or to the receiver acting as relay aiding
the donwlink or uplink communication of the typical node with its nearest
AP.

1, unless xR is a point on the line segment joining the origin
with x∗ (inclusive).
The above model is general enough so that, by appropriately
choosing xR and/or x∗ , IxR ,a and IxR ,u correspond to many
practical instances of (cross-mode) interference experienced
in D2D/FD-enabled as well as conventional cellular networks.
For example, for the case where x∗ 6= xR 6= o, Eq. (1) may
correspond to the AP- and UE-generated interference power
experienced by the receiver of a, potentially cross-cell, D2D
link between the typical nodes at o and xR . Various other
scenarios of practical interest are captured by the model, some
of which are described in Table I. Note that from the scenarios
identified in Table I, only the standard downlink and uplink
scenarios have been considered previously in the literature
[15], [16], [17], [25], however, without consideration of crossmode interference, i.e., UE-generated and AP-generated interference for downlink and uplink transmissions, respectively.
Reference [19] considers the UE-generated cross-mode interference experienced at the typical node in downlink mode,
however, the results provided cannot not be straightforwardly
generalized to the general case.
For characterization of the performance of the communication link as well as obtaining insights on the effectiveness of
the guard region V ∗ for reducing interference, it is of interest to
describe the statistical properties of the random variables IxR ,a
and IxR ,u . This is the topic of the following section, where
the marginal statistics of IxR ,a and IxR ,u conditioned on x∗
and treating xR as a given, but otherwise free, parameter, are
investigated in detail. Characterization of the joint distribution
of IxR ,a and IxR ,u is left for future work.
Note that the respective unconditional interference statistics
can be simply obtained by averaging over the distribution
of x∗ , which corresponds to a uniformly distributed
∠x∗
√
∗
and a Rayleigh distributed |x | of mean 1/(2 λa ) [27].
However, results conditioned on x∗ are also of importance

xR =
6 o,
x∗ = o
xR = x∗ ,
x∗ = o

scenario
General case. Corresponds to (a) a D2D link when the
typical node sends data to the typical receiver or (b) to a
relay-assisted cellular communication where the typical
receiver acts a relay for the link between the typical
node and it nearest AP (either downlink or uplink).
Typical receiver is not guaranteed to lie within V ∗ .
Typical node is a UE in receive mode and lies within
V ∗ . Represents a standard downlink when the nearest
AP is transmitting data to the typical node/receiver.
Nearest AP to the typical node is in receive mode with
the typical node lying in V ∗ . Represents a standard
uplink when the AP receives data from the typical node.
Typical node is an AP. When the typical node/AP is the
one transmitting to the typical receiver, a non-standard
downlink is established as xR is not necessarily lying
within V ∗ .
Typical node is an AP in receive mode; the position of
the corresponding transmitter is unspecified and may as
well lie outside V ∗ , thus modeling a non-standard uplink.

on their own as they can serve as the mathematical basis
for (optimal) resource allocation algorithms given network
topology information, e.g., decide on whether the typical node
employs D2D or cellular mode given knowledge of x∗ .
For tractability purposes, the following assumption on the
statistics of Φu will be considered throughout the analysis.
Assumption. The positions of the (potentially) interfering
UEs, Φu , is a realization of an HPPP of density λu > 0,
that is independent of the positions of the APs, typical node,
and typical receiver.
Remark: In general, this assumption is not exact since
resource allocation and scheduling decisions over the entire
network affect the distribution of interfering UE positions.
For example, in the conventional uplink scenario with at least
one UE per cell, the transmitting UE positions correspond
to a VPLP process of density equal to λu = λa [17],
[19]. However, as also shown in [17], [19] for the standard
uplink scenario, the HPPP assumption of the UE point process
allows for a tractable, yet accurate approximation of the actual
performance, which, as will be demonstrated in Sec. V, is also
the case for other operational scenarios as well. For generality
purposes, an arbitrary value λu 6= λa is also allowed in
the analysis, which is actually the case, e.g., when one UE
is only allowed to transmit per cell but there exist empty
cells and, therefore, λu < λa (ultra dense networks [26]),
and also serves as an approximate model for the case when
some arbitrary/undefined coordination scheme is employed by
other cells in the system (if at all), that may as well result in
λu > λa .
III. I NTERFERENCE C HARACTERIZATION
Towards obtaining a tractable characterization of the distribution of the (AP or UE) interference power, or, equivalently,
its Laplace transform, the following standard result in PPP
theory is first recalled [27].
P
Lemma 1. Let I , x∈Φ̃ P hx |x − z|−α , z ∈ R2 , P > 0,
with Φ̃ an inhomogeneous PPP of density λ : R2 → [0, ∞),

4

and {hx }x∈Φ i.i.d. exponential random variables of mean 1.
The Laplace transform of I equals
 Z

−α
LI (s) = exp −
λ(x)γ(sP |x − z| )dx
(2)
R2


Z ∞
= exp −2π
λz (r)rγ(sP r−α )dr ,
(3)
0

1
1+t ,

and the second equality holds only
where γ(t) , 1 −
in the case of a circularly-symmetric density function w.r.t.
point z, i.e., λ(z + x) = λz (|x|), for an appropriately defined
function λz : [0, ∞) → [0, ∞).
The above result provides a complete statistical characterization in integral form of the interference experienced at
a position z ∈ R2 due to Poisson distributed interferers.
Of particular interest is the case of a circularly-symmetric
density, which only requires a single integration over the radial
coordinate and has previously led to tractable analysis for
various wireless network models of interest such as mobile ad
hoc [1] and downlink cellular [25]. In the following it will
be shown that even though the interference model of Sec.
II suggests a non circularly-symmetric interference density
due to the random shape of the guard region, the Laplace
transform formula of (3) holds exact for IxR ,a and is a (tight)
lower bound for IxR ,u with appropriately defined circularlysymmetric equivalent density functions.
A. AP-Generated Interference
Note that considering the Voronoi cell of a random AP at
x ∈ Φa as a guard region has no effect on the distribution of
the interfering APs. This is because the Voronoi cell of any
AP is a (deterministic) function of Φa , therefore, it does not
impose any constraints on the realization of Φa , apart from
implying that the AP at x does not produce interference. By
well known properties of HPPPs [27], conditioning on the
position x of the non-interfering AP, has no effect on the
distribution of the interfering APs. That is, the interfering AP
positions are still distributed as an HPPP of density λa as in
the case without imposing any guard region whatsoever.
However, when it is the Voronoi cell of the nearest AP
to the originthat is considered as a guard region, an implicit
guard region is formed. Indeed, the interfering APs, given the
AP at x∗ is not interfering, are effectively distributed as an
inhomogeneous PPP with density [25]
λ̃a (x) = λa I(x ∈
/ B(o, |x∗ |)), x ∈ R2 ,

(4)

i.e., an implicit circular guard zone is introduced around the
typical node (see Fig. 1), since, if this was not the case,
another AP could be positioned at a distance smaller than
|x∗ | from the origin, which is impossible by assumption. This
observation may be used to obtain the Laplace transform of
the AP-generated interference experienced at xR directly from
the formula of (2). However, the following result shows that
the two-dimensional integration can be avoided as the formula
of (3) is also valid in this case with an appropriately defined
equivalent density function.

Proposition 2. The Laplace transform, LIxR ,a (s | x∗ ) ,
E(e−sIxR ,a | x∗ ), of the AP-generated interference power
IxR ,a , conditioned on x∗ , equals the right-hand side of (3)
with P = Pa , α = αa and λz (r) = λxR ,a (r), r ≥ 0, where


λa
, r > |x∗ | + |xR |
(




, |x∗ | > |xR |,
 0
, r ≤ ||x∗ | − |xR ||
λxR ,a (r) ,
λa , |x∗ | < |xR |,



λa /2
, |x∗ | = |xR |





λa 1 − π1 arccos (d) , otherwise,
(5)
r 2 −(|x∗ |2 −|xR |2 )
with d ,
, for xR 6= o, and
2r|xR |
λxR ,a (r) = λa I(r ≥ |x∗ |),

(6)

for xR = o.
Proof: See Appendix A.
The following remarks can be made:
1) The interference power experienced at an arbitrary position xR 6= o under the considered guard region
scheme is equal in distribution to the interference power
experienced at the origin without any guard region and
with interferers distributed as an inhomogeneous PPP of
an equivalent, xR -depednent density given by (5).
2) Even though derivation of the statistics of IxR ,a was
conditioned on xR and x∗ , the resulting equivalent inteferer density depends only on their norms |xR | and |x∗ |.
Although the indepedence from ∠xR might have been
expected due to the isotropic property of Φa [1], there is
no obvious reason why one would expect independence
also from ∠xR .
3) λxR ,a (r) is a decreasing function of |x∗ |, corresponding
to a (statistically) smaller AP interference power due to
an increased guard zone area.
4) For xR = o, corresponding to a standard downlink with
interference generated from other donwlink transmissions, Prop. 2 coincides with the analysis of [25], as
expected.
5) The Laplace transform of the interference in the case
of xR 6= x∗ 6= o was examined previously in [28].
However, the corresponding formulas appear as twodimensional integrals that offer limited insights compared to the simpler and more intuitive equivalent density formulation given in Prop. 2.
6) The case xR = x∗ 6= o corresponds to the nearestneighbor transmission scenario considered in [29] where
the validity of λxR ,a (r) in (5) as an equivalent density
function for computation of the Laplace transform of the
interference power was not observed.
7) In [30], the Laplace transform of the interference power
experienced at xR ∈ R2 due to a Poisson hole process,
i.e., with interferers distributed as an HPPP over R2
except in the area covered by randomly positioned disks
(holes), was considered. The holes were assumed to not
include xR and a lower bound for the Laplace transform
was obtained by considering only a single hole [30,
Lemma 5], which coincides with the result of Prop.

1

1

0.8

0.8

0.6

λx∗ ,a (r)/λa

λxR ,a (r)/λa

5

0.4
√
|xR | × 2 λa
√
|xR | × 2 λa
√
|xR | × 2 λa
√
|xR | × 2 λa

0.2

0
0

0.5

1

1.5

2
2.5
√
r × 2 λa

3

= 1/2
=1
= 3/2
=2
3.5

0.6

0.4
√
|x∗ | × 2 λa
√
∗
|x | × 2 λa
√
∗
|x | × 2 λa
√
|x∗ | × 2 λa

0.2

4

Fig. 2. Equivalent radial density of interfering
APs experienced at various
√
xR ∈ R2 , conditioned on |x∗ | = 1/(2 λa ).

2. This is not surprising as the positions of the APs,
conditioned on x∗ , are essentially distributed a singlehole Poisson process. Note that Prop. 2 generalizes [30,
Lemma 5] by allowing xR to be covered by the hole
and considers a different proof methodology.
Figure 2 shows the normalized density function λxR ,a (r)/λ
√ a
for various values of |xR | and assuming that |x∗ | = 1/(2 λa ),
the expected distance from the nearest AP. It can be seen that
the presence of the implicit circular guard region results in
reducing the equivalent interferer density in certain intervals
of the radial coordinate r, depending on the value of |xR |.
In particular, when |xR | < |x∗ |, it is guaranteed that no APs
exist within a radius |x∗ | − |xR | > 0 from xR . In contrast,
when |xR | > |x∗ | there is no protection from APs in the close
vicinity of xR .
The case |xR | = |x∗ | is particularly interesting since it
corresponds to the case when the receiver of interest is the
AP at x∗ , experiencing interference from other AP, e.g., when
operating in FD. For x∗ 6= o, corresponding to an uplink
transmission by the typical node to its nearest AP, it can be
easily shown that it holds


1
r
λx∗ ,a (r) = λa
+
+ O(r3 ), r → 0,
(7)
2 2π|x∗ |
i.e., the guard region results in the serving AP experiencing
about half of the total interfering APs density in its close vicinity, which is intuitive as for asymptotically small distances
from x∗ the boundary of the circular guard region can be
locally approximated as a line that divides the plane in two
halves, one with interferer density λa and one with interferer
density 0. Figure 3 shows the normalized λx∗ ,a (r) for various
values of |x∗ |, where its linear asymptotic behavior as well as
the advantage of a larger |x∗ | are clearly visible.
B. UE-Generated Interference
The positions of interfering UEs, conditioned on the realization of Φa , are distributed as an inhomogeneous PPP of

0

0

0.5

1

1.5

2
2.5
√
r × 2 λa

3

= 1/2
=1
= 3/2
=2
3.5

4

Fig. 3. Equivalent radial density of interfering APs experienced at x∗ .

density
λ̃u (x|Φa ) = λu I(x ∈
/ V ∗ ), x ∈ R2 .

(8)

Although the expression for λ̃u is very similar to the that of
the density λ̃a of interfering APs given in (4), the random
Voronoi guard region V ∗ appearing in (8) is significantly
more complicated than the deterministic circular guard zone
B(o, |x∗ |), which renders the analysis more challenging. To
simplify the following exposition, it will be assumed that a
rotation of the Cartesian coordinate axes is performed such
that ∠x∗ = 0. Note that this rotation has no effect in the
analysis due to the isotropic property of the HPPP [1] and
immediately renders the following results independent of the
value of ∠x∗ in the original coordinate system.
Since it is of interest to examine the UE interference
statistics conditioned only on x∗ , a natural quantity to consider,
that will be also of importance in the interference statistics
analysis, is the probabilistic cell area (PCA) function pc (x |
x∗ ), x ∈ R2 . This function gives the probability that a point
x ∈ R2 lies within V ∗ conditioned only on x∗ , i.e.,1
pc (x | x∗ ) , P(x ∈ V ∗ | x∗ ).
Lemma 3. For all x ∈ R2 , the PCA function equals
(
1
, |x| ≤ |x∗ |, ∠x = 0,
∗
pc (x | x ) =
e−λa |A| < 1 , otherwise,
where A , B(x, |x − x∗ |) \ B(o, |x∗ |). For all x 6= x∗ , it holds
(
r2 (|∠x|+θ∗ )−|x∗ |2 |∠x|+|x||x∗ | sin(|∠x|) , x∗ 6= o,
|A| = ∗ 2
π|x|
, x∗ = o,
p
with r∗ , |x − x∗ | = |x|2 + |x∗ |2 − 2|x||x∗ | cos(∠x) and



π − arcsin |x| sin(|∠x|)
, |x| cos(∠x) > |x∗ |,
r
∗


θ∗ ,
arcsin |x| sin(|∠x|)
, |x| cos(∠x) ≤ |x∗ |.
r∗
1 Recall that, under the system model, x∗ is also the AP closest to the
origin.

6

2.4
2.2
2
λa E(|V ∗ | | x∗ )

0.5

0.9

0.1

|x∗ |

1.8
1.6
1.4
1.2

Fig. 4. Contours of pc (x | x∗ ) (solid lines),
corresponding to probabilities
√
0.1 to 0.9 in steps of 0.1, for |x∗ | = 1/ λa . The points of the line segment
∗
joining o to x are the only ones with pc (x|x∗ ) = 1. The corresponding
∗ 2
contours of e−λa π|x−x | are also shown (dashed lines).

Proof: The probability that the point x ∈ R2 belongs
to V is equal to the probability that there does not exist a
point of Φa \ {x∗ } within the set A, which equals e−λa π|A|
[27]. For |x| ≤ |x∗ | and ∠x = 0, it is a simple geometrical
observation that A = Ø, therefore, |A| = 0. When x∗ = o,
A = B(x, |x|) \ B(o, 0) = B(x, |x|), and, therefore, |A| =
π|x|2 . For all other cases, |A| can be computed by the same
approach as in the proof of [31, Theorem 1]. The procedure
is straightforward but tedious and is omitted.
A simple lower bound for pc (x | x∗ ) directly follows by
noting that A ⊆ B(x, |x − x∗ |) for all x ∈ R2 .

1

analysis
simulation
0

0.5

1

1.5
√
|x∗ | × 2 λa

2

2.5

3

Fig. 5. Average area of V ∗ conditioned on x∗ as a function of the
(normalized) distance of x∗ from the origin.

∗

Corollary 4. The PCA function is lower bounded as
pc (x | x∗ ) ≥ e−λa π|x−x

with equality if and only if x∗ = o.

∗ 2

|

, x ∈ R2 ,

(9)

Remark: The right-hand side of (9) equals the probability
that x belongs to the Voronoi cell of the AP positioned at x∗
when the latter is not conditioned on being the closest AP to
the origin or any other point in R2 .
Figure
4 depicts pc (x | x∗ ) for the case where |x∗ | =
√
1/ λa (behavior is similar for other values of |x∗ | > 0). Note
that, unless x∗ = o, pc (x | x∗ ) is not circularly symmetric
w.r.t. any point in R2 , with its form suggesting that points
isotropically distributed in the vicinity of x∗ are more probable
to lie within V ∗ than points isotropically distributed in the
vicinity of o.
∗ 2
The lower bound e−λa π|x−x | , x ∈ R2 , is also shown in
Fig. 4, clearly indicating the probabilistic expansion effect of
the Voronoi cell of an AP, when the latter is conditioned on
being the closest to the origin. This cell expansion effect is
also demonstrated in Fig. 5 where the conditional average cell
area, equal to
Z

∗
∗
∗
∗
E(|V | | x ) = E
I(x ∈ V )dx x
R2
Z
=
pc (x|x∗ )dx,
(10)
R2

is plotted as a function of the distance |x∗ |, with the integral of
(10) evaluated numerically using Lemma 3. Simulation results

are also depicted serving as a verification of the validity of
Lemma 3. It can be seen that the average area of the guard
region increases with |x∗ |, which implies a corresponding
increase of the average number of UEs that lie within the
region. However, as will be discussed in the following, even
though resulting in more UEs silenced on average, an increasing |x∗ | does not necessarily imply improved protection from
UE interference, depending on the receiver position.
The interference statistics of the UE-generated interference
power are given in the following result.
Proposition 5. The Laplace transform LIxR ,u (s | x∗ ) ,
E(e−sIxR ,u | x∗ ) of the UE-generated interference power
IxR ,u , conditioned on x∗ , is lower bounded as the right-hand
side of (3) with P = Pu , α = αu and λz (r) = λxR ,u (r), r ≥
0, where


Z 2π
1
∗
λxR ,u (r) , λu 1 −
pc ((r, θ) + xR | x ) dθ ,
2π 0
(11)
with pc (x | x∗ ) as given in Lemma 3.
Proof: See Appendix B.
The following remarks can be made:
1) The statistics of the interfering UE point process when
averaged over V ∗ (equivalently, over Φa \ {x∗ }) do not
correspond to a PPP, even though this is the case when a
fixed realization of V ∗ is considered. Therefore, it cannot
be expected that Lemma 1 applies for LIxR ,u (s | x∗ ).
However, as Prop. 5 shows, when the interfering point
process is treated in the analysis as an PPP of an appropriately defined equivalent density function λxR ,u (r), a
tractable lower bound for LIxR ,u (s | x∗ ) is obtained,
which will be shown to be tight in the numerical results
section.
2) For x∗ 6= o, the non circularly-symmetric form of the
PCA function results in an integral-form expression for
the equivalent density λxR ,u (r) that depends in general
on ∠xR , in contrast to the case for λxR ,a (r) (see Sec.

7

III.A, Remark 2). A closed form expression for the
equivalent density is only available for x∗ = o discussed
below.
3) When the receiver position is specified as |xR |∠xR with
∠xR uniformly distributed over [−π, π), a straightforward extension of the proof of Prop. 5 results in the
same lower bound for the conditioned Laplace transform
of IxR ,u as in Prop. 5, with λxR ,u (r) replaced by
Z π
1
λx ,u (r)d∠xR ,
(12)
λ|xR |,u (r) ,
2π −π R
which is independent of ∠xR .
Although λxR ,u (r) is not available in closed form when x∗ 6=
o, the numerical integration over a circular contour required
in (11) is very simple. With λxR ,u (r), r ≥ 0, pre-computed,
evaluation of the lower bound for LIxR ,u (s | x∗ ) is of the same
(small) numerical complexity as the evaluation of LIxR ,a (s |
x∗ ). Moreover, a closed-form upper bound for λxR ,u (r) is
available, which can in turn be used to obtain a looser lower
bound for LIxR ,u (s | x∗ ) that is independent of ∠xR . The
bound is tight in the sense that it corresponds to the exact
equivalent density when x∗ = o.
Proposition 6. The equivalent density λxR ,u (r) of Prop. 5 is
upper bounded as
h
i
∗ 2
2
2
λxR ,u (r) ≤ λu 1 − e−λa π(|x | +|xR | +r ) I0 (λa 2πx̄r) ,
(13)
where x̄ , max(|x∗ |, |xR |) and I0 (·) denotes the zero-order
modified Bessel function of the first kind. Equality holds only
when x∗ = o.
Proof: See Appendix C.
An immediate observation of the above result is that, given
xR , the equivalent interfering UE density decreases for any
x∗ 6= o compared to the case x∗ = o. This behavior is expected
due to the cell expansion effect described before. However, as
the bound of (13) is independent on ∠xR , it does not offer
clear information on how the interferering UE density changes
when different values of x∗ 6= o are considered for a given
xR . In order to obtain futher insights on the UE-generated
interference properties, λxR ,u (r) is investigated in detail in the
following section for certain special instances of xR and/or x∗ ,
which are of particular interest in cellular networks.
IV. A NALYSIS OF S PECIAL C ASES OF UE-G ENERATED
I NTERFERENCE
A. xR = x∗
This case corresponds to an standard uplink cellular transmission with nearest AP association, no intra-cell interference,
and interference generated by UEs outside V ∗ operating in
uplink and/or D2D mode. This case was previously investigated in [16], [17], with heuristically introduced equivalent
densities considering only the average performance over x∗ .
In this section, a more detailed investigation of the equivalent
density properties is provided.
For this case, a tighter upper bound than the one in Prop. 6
is available for λx∗ ,u (r), which, interestingly, is independent
of |x∗ |, in the practical case when x∗ 6= o.

Lemma 7. For xR = x∗ , the equivalent density λxR ,u (r) =
λx∗ ,u (r) of Prop. 5 is upper bounded as


2
λx∗ ,u (r) ≤ λu 1 − e−λa πr ,
(14)
with equality if and only if x∗ = o.
Proof:
Follows
by
replacing
the
term
pc ((r, θ) + xR | x∗ ) = pc ((r, θ) + x∗ | x∗ ) appearing in
(11) with its bound given in Cor. 4, which evaluates to
2
e−λa πr .
The bound of (14) indicates that λx∗ ,u (r) tends to zero
at least as fast as O(r2 ) for r → 0, irrespective of x∗ .
The following exact statement on the asymptotic behavior of
λx∗ ,u (r) shows that λx∗ ,u (r) ∼ cr2 , r → 0, with the value of
c independent of |x∗ | when x∗ 6= o.
Proposition 8. For xR = x∗ , the equivalent density
λxR ,u (r) = λx∗ ,u (r) of Prop. 5 equals
λx∗ ,u (r) = λu λa bπr2 + O(r3 ), r → 0,

(15)

with b = 1 for x∗ = o and b = 1/2 for x∗ 6= o.

Proof: For x∗ = o, the result follows directly from
Lemma 7. For x∗ 6= o, it can be shown by algebraic manipulation based on Lemma 3, that the term pc ((r, θ) + xR | x∗ ) =
pc ((r, θ) + x∗ | x∗ ) appearing in (11) equals
λu λa (qπ + (−1)q arccos(sin(θ)) + cos(θ) sin(θ)) r2 +O(r3 ),

for r → 0, where q = 0 for θ ∈ [−π, −π/2] ∪ [π/2, π) and
q = 1 for θ ∈ (−π/2, π/2). Substituting this expression in
(11) and performing the integration leads to the result.
Equation (15) indicates that imposing a guard region V ∗
when x∗ 6= o reduces the interfering UE density by half in
the close vicinity of x∗ , compared to the case x∗ = o. This
effect is similar to the behavior of λx∗ ,a (r) discussed in Sec.
III. A. However, λx∗ ,a (0) = λa /2 , whereas λx∗ ,u (0) = 0,
clearly demonstrating the effectiveness of the guard region for
reducing UE-generated interference in the vicinity of x∗ .
For non-asymptotic values of r, the behavior of λx∗ ,u (r)
can only be examined by numerical evaluation of (11). Figure 6 depicts the normalized equivalent density λx∗ ,u (r)/λu
for various values
of |x∗ | in the range of practical
in√
√
∗
terest [0, 3/(2
λ
)]
(note
that
P(|x
|
>
3/(2
λ
))
=
a
a
√
2
e−λa π(3/(2 λa )) ≈ 8.5 × 10−4 ). It can be seen that for nonasymptotic values of r, λx∗ ,u (r) is a decreasing function of
|x∗ | for all r > 0, corresponding to a reduced UE-generated
interference. This is expected since, as evident from Fig. 4, the
cell expansion effect with increasing |x∗ | can only improve the
interference protection at x∗ .
Interestingly, the dependence of λx∗ ,u (r) on |x∗ |, although
existing, can be seen to be rather small. This observation along
with Prop. 8 strongly motivates the consideration of a single
curve as an approximation to the family of curves in Fig. 6.
A natural approach is to consider the upper bound of (14),
which has the benefit of being available in closed form. This
is essentially the approach that was heuristically employed in
[17] without observing that it actually corresponds to a (tight)
bound for the equivalent interferer density.

1

1

0.8

0.8

λxR ,u (r)/λu

λx∗ ,u (r)/λu

8

0.6

0.4

√
exact (|x∗ | × 2 λa = 0, 0.5, 1, 1.5, 2, 2.5, 3)
piecewise-linear upper bound
piecewise-linear approx. in [14]

0.2

0

0

0.5

1

1.5

2
2.5
√
r × 2 λa

3

0.6

0.4

0.2

3.5

0

4

Fig. 6. Equivalent density function λx∗ ,u (r) for various |x∗ | (solid curves),
with the direction of the arrow corresponding to increasing |x∗ |. A piecewiselinear approximation and bound are also shown.

Fig. 7.
|xR |.

√
|xR | × 2 λa = 0, 0.5, 1, 1.5, 2, 2.5, 3
0

0.5

1

1.5

2
2.5
√
r × 2 λa

3

3.5

4

Equivalent density function λxR ,u (r) when x∗ = o for various

B. x∗ = o
Another approach is to consider a piecewise-linear curve,
which, in addition to having a simple closed form, results
in a closed form approximate expression for the bound of
LIx∗ ,u (s | x∗ ) as given below (details are straightforward and
omitted).
Lemma 9. With the piecewise-linear approximation
λx∗ ,u (r) ≈ λu min (δr, 1) , r ≥ 0,

(16)

for some δ > 0, the lower bound for LIx∗ ,u (s | x∗ ) equals




1 2
exp −λu π C(Pu s)2/α + 2
F̃ (3) − F̃ (2)
δ
3
where
,
(2π/α)/
sin(2π/α) and F̃ (x)
 C

x
x
−1
with 2 F1 (·) denoting
2 F1 1, α , 1 + α , sPu δ α
hypergeometric function.

,
the

A piecewise-linear approximation as in (16) was heuristically proposed in [16] for a slightly more general system
model than the one considered here. A closed form formula
for δ was provided, only justified as obtained by means of
curve fitting without providing any details on the procedure.
For the system model √
considered in this paper, this formula
results in δ ≈ 0.82687 λa and the corresponding piecewiselinear approximation of the equivalent density is also depicted
in Fig. 6. It can be seen that this approximation is essentially
an attempt to capture the average behavior of λx∗ ,u (r) over
|x∗ |, thus explaining its good performance reported in [16].
However, it is not clear why this particular value of δ is
more preferable than any other (slightly) different value. A
more rigorous approach for the selection of δ is to consider
the tightest piecewise-linear upper bound for λx∗ ,u (r), which
leads to another (looser) lower bound for LIxR√
(s | x∗ ). This
,u
bound corresponds to a value of δ ≈ 1.13118 λa , found by
numerical optimization, and is also shown in Fig. 6.

This case corresponds to the typical node being an AP, i.e.,
a typical AP. Note that, in contrast to the cases when x∗ 6= o,
the Voronoi cell of the typical AP is not conditioned to cover
any point in R2 , therefore, no cell expansion effect is observed
and, for xR 6= o, it is possible that the receiver lies outside
the guard region. This is a different operational scenario from
the common downlink with nearest AP association [25] where
the intended receiver by default lies within the Voronoi cell
of the serving AP. It may arise, for example, in case of D2D
communications, where both communicating UEs receive data
(e.g., control information) from the same AP.
The equivalent interfering UE density is given in Prop. 6 and
is depicted in Fig. 7 for various values of |xR |. Note that the
case xR = o corresponds to the typical AP in receive mode,
whereas for xR 6= o the typical AP transmits. It can be seen
that increasing |xR | effectively results in reduced protection
in the close vicinity of the receiver since the latter is more
probable to lie near the edge or even outside the Voronoi guard
region.
C. xR = o
This case corresponds to the typical node being at receive
mode. One interesting scenario which corresponds to this case
is downlink cellular transmission with nearest AP association,
where interference is due to transmitting UEs (e.g., in uplink
or D2D mode) that lie outside the serving cell.
When x∗ = o, the setup matches the case xR = x∗
∗
with
above, and it holds λo,u (r|x∗ ) =
 x = o, discussed

−λa πr 2
∗
λu 1 − e
. For x 6= o, the asymptotic behavior of
∗
λo,u (r|x ) can be obtained by the same approach as in the
proof of Prop. 8.
Proposition 10. For x∗ 6= o the equivalent density function
λxR ,u (r) = λo,u (r|x∗ ) of Prop. 5 equals
8
λo,u (r) = λu λa |x∗ |r + O(r2 ), r → 0.
π

9

1

λo,u (r)/λu

0.8

0.6

0.4

0.2

0

√
|x∗ | × 2 λa = 0, 0.5, 1, 1.5, 2, 2.5, 3
0

0.5

1

1.5

2
2.5
√
r × 2 λa

3

3.5

4

Fig. 8. Equivalent density function λo,u (r) for various |x∗ |, with the
direction of the arrow corresponding to increasing |x∗ |.

In stark contrast to the behavior of λx∗ ,u (r), λo,u (r) tends
to 0 only linearly (instead of quadradically) as r → 0 when
x∗ 6= o, and with a rate that is also proportional to (instead
of independent of) |x∗ |. This implies that the typical node
is not as well protected from interfering UEs as its nearest
AP, experiencing an increased equivalent interferer density in
its close vicinity as |x∗ | increases. This can be understood by
noting that with increasing |x∗ |, the origin, although contained
in a guard region of (statistically) increasing area, is more
probable to lie near the cell edge (Fig. 1 demonstrates such a
case.)
The strong dependence of λo,u (r) on |x∗ | observed for r →
0 also holds for all values of r. This is shown in Fig. 8 where
the numerically evaluated λo,u (r) is depicted for the same
values of |x∗ | as those considered in Fig. 6. It can be seen
that λo,u (r) increases with |x∗ | for all r up to approximately
√
1 − 1.5 times the average value of |x∗ | (equal to 1/(2 λa )),
whereas it decreases with |x∗ | for greater r .
This behavior of λo,u (r), especially for r → 0, does not
permit the use of single function to be used as a reasonable
approximation or bound for λo,u (r), irrespective of |x∗ |, as
was done for the case for λx∗ ,u (r). Noting that the upper
bound of (13) has the same value whenever |x∗ | or |xR | are
zero, it follows that the curves shown in Fig. 7 with |xR |
replaced by |x∗ | are upper bounds for the corresponding curves
of Fig. 8. Unfortunately, it can be seen that the bound is very
loose for small r and/or large |x∗ |.
D. |xR | ∈ (0, |x∗ |), ∠xR = 0

The previous cases demonstrate how different the properties of the UE-generated interference become when different
receiver positions are considered. This observation motivates
the question of which receiver position is best protected from
UE interfernce given the positions o and x∗ 6= o of the typical
node and nearest AP, respectively. This question is relevant in,
e.g., relay-aided cellular networks where the communication
between a UE and its nearest AP is aided by another node (e.g.,

an inactive UE) [32]. Although the performance of a relayaided communication depends on multiple factors including
the distances among the nodes involved and considered transmission scheme [33], a reasonable choice for the realy position
is the one experiencing less interference.
By symmetry of the system geometry, this position should
lie on the line segment joining the origin with x∗ (inclusive).
However, as can be seen from Figs. 6 and 8, the shape of
the equivalent interferer density does not allow for a natural
ordering of the different values of xR . For example, λx∗ ,u (r) is
smaller than λo,u (r) for small r, whereas the converse holds
for large r. Noting that the receiver performance is mostly
affected by nearby interference [1], a natural criterion to order
the interfering UE densities for varying xR is their asymptotic
behavior as r → 0. For |xR | = |x∗ | and |xR | = o, this is
given in Props. 8 and 10, respectively. For all other positions
the asymptotic density is given by the following result.
Proposition 11. For |xR | ∈ (0, |x∗ |), ∠xR = 0, and x∗ 6= o,
the equivalent density function λxR ,u (r) of Prop. 5 equals
λxR ,u (r) = λu λa

8 (|x∗ |/|xR |)2 3
r + O(r4 ), r → 0. (17)
9π |x∗ | − |xR |

It follows that the optimal receiver position must have
|xR | ∈ (0, |x∗ |) since, in this case, the density scales as O(r3 )
instead of O(r) and O(r2 ) when |xR | = 0 and |xR | = |x∗ |,
respectively. Its value can be easily obtained by minimization
of the expression in (17) w.r.t. |xR |.

Corollary 12. The receiver position that experiences the
smallest equivalent UE interferer density in its close vicinity
∗
lies on the line
√ segment joining the origin to x and is at dis∗
tance |x |/ 2 from the origin. The corresponding equivalent
32 3
r + O(r4 ), r → 0.
density equals λxR ,u (r) = λu λa 9π
V. N UMERICAL E XAMPLES
In this section, numerical examples will be given, demonstrating the performance of various operational scenarios in
cellular networks that can be modeled as special cases of the
analysis presented in the previous sections. The system performance metric that will be considered is LIxR ,k (ραk θ|x∗ ),
with k ∈ {a, u} depending on whether AP- or UE-generated
interference is considered, with ρ > 0, θ > 0 given. Note that
this metric corresponds to the coverage probability that the
signal-to-interference ratio (SIR) at xR is greater than θ, when
the distance between transmitter and receiver is ρ, the direct
link experiences a path loss exponent ak and Rayleigh fading,
and the transmit power is equal to Pk [1]. For simplicity and
w.l.o.g., all the following examples correspond to λa = 1 and
αa = αu = 4.
The analytical results of the previous sections will be
compared against simulated system performance where, in
a single experiment, the distribution of interfering APs and
UEs is generated as follows. Given the AP position closest
to the origin, x∗ , the interfering AP positions are obtained as
a realization of a PPP of density as in (4). Afterwards, an
independent realization of a HPPP of density λu is generated,
which, after removing the points lying within V ∗ , represents
the interferering UE positions.

10

1

1

0.8
coverage probability

0.8
coverage probability

simulation
analysis

1
4

1
2

0.6
3
4

0.4

1
|xR |
|x∗ |

0.2

0
−20

=

5
4

−10

0.6

=

1
4

1
2
3
4

0.4
1
0.2

sim. (PPP)
sim. (VPLP)
analysis
−15

|xR |
|x∗ |

5
4

−5

0

5

10

15

20

θ (dB)

(a)

0
−20

−15

−10

−5

0

5

10

15

20

θ (dB)

(b)

Fig. 9. Coverage probability of the link between the typical node and an isotropically positioned receiver under UE interference for (a) λu = λa , (b)
λu = 10λa .

In addition, for cases where the UE-generated interference
is of interest, a VPLP process of density λu = λa is also
simulated, corresponding to scenarios where one UE from each
Voronoi cell in the system transmits (as in the standard uplink
scenario with nearest AP association).
A. D2D Link Not Necessarily Contained in a Single Cell
In this example, UE-generated inteference is considered at
a receiver position xR 6= o with ∠xR uniformly distributed
in [−π, π) and ρ = |xR |. This case may model a D2D link
where the typical node is a UE that directly transmits to a
receiver isotropically distributed at a distance |xR | > 0 . A
guard region V ∗ is established by the closest AP to the typical
node, however, it is not guaranteed to include xR , i.e., the D2D
nodes may not be contained within the same cell, which is a
common scenario that arises in practice, referred to as crosscell D2D communication [21].
With interference generated from other UEs in D2D and/or
uplink mode, a lower bound for LIxR ,u (ραu θ|x∗ ) can be
obtained using Prop. 5 with an equivalent density function
λ|xR |,u (r) as given in (12). Figure 9 shows this lower bound
for λu = λa (Fig. 9a) and λu = 10λa (Fig. 9b). The
position x∗ √is assumed to be isotropically distributed with
|x∗ | = 1/(2 λa ) 2 and various values of the ratio |xR |/|x∗ |
are considered. It can be seen that, compared to the simulation
under the PPP interference assumption, the quality of the
analytical lower bound depends on λu . For λu = λa , it
is very close to the exact coverage probability, whereas for
λu = 10λa , it is reasonably tight and able to capture the
behavior of the exact coverage probability over varying |xR |.
Also, for λu = λa and a VPLP interference, the analytical
expression provides a reasonably accurate approximation for
the performance of this analytically intractable case. As expected, performance degradation is observed with increasing
2 Recall that this value is the average distance of the closest AP from the
typical node.

|xR |, due to both increasing path loss of the direct link as well
as increased probability of the receiver lying outside V ∗ .
B. AP-to-D2D-Receiver Link
In this example, AP-generated inteference is considered at
a receiver position xR 6= o with ∠xR uniformly distributed
in [−π, π) and ρ = |x∗ − xR |. This case is similar to the
previous, however, considering AP-generated interference and
the receiver at xR receiving data from the AP node at x∗ . This
case models the scenario where the typical node establishes a
D2D connection with a node at xR , and the AP at x∗ sends
data to xR via a dedicated cellular link (e.g., for control purposes or for implementing a cooperative transmission scheme).
Note that, in contrast to the previous case, the link distance
ρ is random and equal to ρ = |x∗ | + |xR | − 2|x∗ ||xR | cos ψ,
with ψ uniformly distributed in [−π, π). The exact coverage
probability of this link can be obtained using Prop. 2 followed
by a numerically computed expectation over ψ.
αa
Figure 10 shows E(LIxR ,a (ρ√
θ|x∗ )) for an isotropically
∗
∗
distributed x with |x | = 1/(2 λa ), and for various values
of the ratio |xR |/|x∗ |. Note that the case |xR | = 0 corresponds
to the standard downlink transmission model with nearest
AP association [25]. Monte Carlo evaluation of the coverage
probability (not shown here) perfectly matches the analytical
curves. It can be seen that increasing |xR | reduces the coverage
probability for small SIR but increases it for high SIR. This is
in direct proportion to the behavior of the equivalent interferer
density λxR ,a (r) with increasing |xR | shown in Fig. 2.
C. Uplink with Nearest AP Association
In this example, UE-generated interference is considered
at a receiver position xR = x∗ with ρ = |x∗ | and λu = λa
corresponding to the conventional uplink transmission scenario
with nearest AP association and one active UE per cell (no
cross-mode interference). Figure 11 shows the lower bound

11

1

1

|xR |
|x∗ |

0.6
5
4

0.4

1

1
3 2
4

0.8

=0
coverage probability

coverage probability

0.8

1
4

0.6

0.4
√
0.2 |x∗ | × 2 λa = 2

0.2

−15

−10

−5

0

5

10

15

0
−20

20

θ (dB)

Fig. 10. Coverage
probability for the link between the AP positioned at x∗
√
(|x∗ | = 1/(2 λa )) and an isotropicaly positioned receiver.

of LIxR ,u (ραu θ|x∗ ) obtained by Prop. 5 as well as the
looser, but more easily computable, lower bound obtained
using the closed
form expression given in Lemma 7 with
√
δ = 1.13118 λa (resulting in the tightest bound possible with
a piecewise-linear equivalent density function). The coverage
√
probability is computed for an AP at a distance |x∗ |=c/(2 λa )
from the typical node with c = 1/2, 1, 2, roughly corresponding to a small, average, and large uplink distance (performance
is independent of ∠x∗ ).
As was the case in Fig. 9a, Prop. 5 provides a very tight
lower bound for the actual coverage probability (under the PPP
model for the interfering UE positions). The bound of Lemma
7, although looser, is nevertheless a reasonable approximation
of the performance, especially for smaller values of |x∗ |.
Compared to the VPLP model, the PPP model provides an
optimistic performance prediction that is, however, reasonably
tight, especially for large |x∗ |. This observation motivates its
usage as a tractable approximation of the actual inteference
statistics as was also reported in [16], [17]. Interestingly,
the bound of Lemma 7 happens to provide an even better
approximation for√the VPLP performance for |x∗ | close to or
smaller than 1/(2 λa ).

−15

−10

1
2

1

−5

0

5

10

15

20

θ (dB)

Fig. 11. Coverage probability of the uplink between the typical node its
nearest AP (λu = λa ).

1
Case A
Case B
Case C

0.8
coverage probability

0
−20

sim. (PPP)
sim. (VPLP)
Prop. 5
Lemma 7

0.6

0.4

0.2

√
ρ × 2 λa = 2

0
−20

−15

−10

1
2

1

−5

0

5

10

15

20

θ (dB)

Fig. 12. Coverage probability under UE interference for the links corresponding to the cases described in Sec. V. D.

D. Effect of Guard Region on UE-Generated Interference
Protection

Case C (receiver imposes a Voronoi guard region): This
case corresponds to xR = x∗ = o modeling an nonstandard uplink transmission (see also description in
Table I)

In order to see the effectiveness of a Voronoi guard region
for enhancing link quality under UE-generated interference,
the performance under the following operational cases is
examined.
• Case A (no guard region is imposed): This results in the
standard transmission model under an HPPP of interferers
positions of density λu [1]. The exact coverage probability is well known (see, e.g., [1, Eq. 3.29]).
• Case B (transmitter imposes a Voronoi guard region):
This case corresponds to x∗ = o, xR 6= o, modeling an
non-standard downlink transmission (see also description
in Table I).

Cases B and C correspond to the analysis considered in Sec.
V. B. Assuming the same link distance ρ for all cases, Fig. 12
shows the analytically obtained coverage probability (exact for
Case A, lower bound for Cases B and C), assuming λu = λa .
It can be seen that imposing a Voronoi guard region (Cases B
and C) is always beneficial, as expected. However, for large
link distances, Case B provides only marginal gain as the
receiver is very likely to be located at the edge or even outside
the guard region. In contrast, the receiver is always guaranteed
to be protected under case C, resulting in the best performance
and significant gains for large link distances.

•

12

VI. C ONCLUSION
This paper considered the analytical characterization of
cross-mode inter-cell interference experienced in future cellular networks. By employing a stochastic geometry framework,
tractable expressions for the interference statistics were obtained that are exact in the case of AP-generated interference
and serve as a tight lower bound in the case of UE-generated
interference. These expressions are based on appropriately
defined equivalent interferer densities, which provide an intuitive quantity for obtaining insights on how the interference
properties change according to the type of interference and
receiver position. The considered system model and analysis
are general enough to capture many operational scenarios
of cellular networks, including conventional downlink/uplink
transmissions with nearest AP association as well as D2D
transmissions between UEs that do not necessarily lie in the
same cell. The analytical expressions of this paper can be
used for sophisticated design of critical system aspects such
as mode selection and resource allocation towards getting the
most out D2D- and/or FD-enabled cellular communications.
Interesting topics for future research is investigation of the
joint properties of AP- and UE-generated interference as well
as extension of the analysis to the case of heterogeneous,
multi-tier networks.
A PPENDIX A
P ROOF OF P ROPOSITION 2
Let Φ̃a , Φa \ {x∗ } denote the interfering AP point
process. Given x∗ , Φ̃a is an inhomogeneous PPP of density
λ̃a (x), x ∈ R2 , defined in (4). Density λ̃a (x) is circularly
symmetric w.r.t. xR = o, which, using Lemma 1, directly
leads to the Laplace transform expression of (3) with radial
density as in (6). Considering the case xR 6= o, it directly
follows from Lemma 1 and (2) that
 Z

LIxR ,a (s) = exp −
λ̃a (x + xR )γ(sPa |x|−αa )dx
R2

by a change of integration variable. By switching to polar
coordinates (centered at o) for the integration, the right-hand
side of (3) results with P = Pa , α = αa and λxR (r) =
λxR ,a (r), where
Z 2π
1
λ̃a ((r, θ) + xR )dθ
λxR ,a (r) =
2π 0
Z
λa 2π
=
I {(r, θ) + xR ∈
/ B(o, |x∗ |)} dθ
2π 0
Z
λa 2π
=
I {(r, θ) ∈
/ B(−xR , |x∗ |)} dθ
2π 0
Z
λa 2π
=
I {(r, θ) ∈
/ B((|xR |, 0), |x∗ |)} dθ,
2π 0
with xR representing the polar coordinates of the receiver in
the first equality, with a slight abuse of notation. The last
equality follows by noting that the integral does not depend on
the value of ∠xR . Evaluation of the final integral is essentially
the evaluation of an angle (see Fig. 13), which can be easily
obtained by elementary Euclidean geometry, resulting in the
expression of (5).

B((|xR |, 0), |x∗ |)
C(o, r)

|xR |
θ0

Fig.
the angle θ0 , which is equal to quantity
R 2π 13. Geometrical figure depicting
/ B((|xR |, 0), |x∗ |)) dθ appearing in the proof of Prop. 2.
0 I ((r, θ) ∈

A PPENDIX B
P ROOF OF P ROPOSITION 5
It holds
LIxR ,u (s | x∗ )



=E E e−sIxR ,u | Φa | x∗


 Z

(a)
= E exp −
λ̃u (x|Φa )γ(sPu |x − xR |−αu )dx x∗
R2
 Z



∗
−αu
≥ exp −
E λ̃u (x + xR |Φa ) | x γ(sPu |x|
)dx ,
R2

where (a) follows by noting that, conditioned on Φa , the interfering UE point process equals Φu \V ∗ , which is distributed
as an inhomogeneous PPP of density λ̃u (x|Φ1 ), x ∈ R2
as given in (8), and using Lemma 1. The inequality is an
application of Jensen’s inequality with a change
of integration


variable. Result follows by noting that E λ̃u (x|Φa ) | x∗ =
λu [1 − pc (x | x∗ )] and switching to polar coordinates for the
integration.
A PPENDIX C
P ROOF OF P ROPOSITION 6
Replacing the PCA function in (11) with its lower bound
as per Cor. 4 results in the equivalent density bound
λxR ,u (r)


Z 2π
1
−λa π|(r,θ)+xR −x∗ |2
e
dθ
≤λu 1 −
2π 0
!
∗ 2 Z 2π
(a)
e−λa π|x |
−λa π|(r,θ)+xR |2
≤ λu 1 −
e
dθ
2π
0
!
∗ 2 Z 2π
2
e−λa π|x |
(b)
=λu 1 −
e−λa π|(r,θ)+(|xR |,0)| dθ
2π
0
!
∗ 2
2
2 Z
e−λa π(|x | +|xR | +r ) 2π λa π2r|xR | cos θ
(c)
= λu 1 −
e
dθ ,
2π
0
where (a) follows by application of the triangle inequality,
(b) by noting that the integral is independent of ∠xR and
(c) since |(r, θ) + (|xR |, 0)|2 = r2 + |xR |2 − 2r|xR | cos(π −
θ) (cosine law). The integral in the last equation is equal to
2πI0 (λa 2π|xR |r). By exchanging the roles x∗ and xR in (a)
and following the same reasoning, another upper bound of the
form of (c) results, with the only difference that the integrand
has |x∗ | instead of |xR | and evaluates to 2πI0 (λa 2π|x∗ |r).
Considering the minimum of these two bounds leads to (13).

13

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