Load-Aware Heterogeneous Cellular Networks:
Modeling and SIR Distribution
Harpreet S. Dhillon, Radha Krishna Ganti and Jeffrey G. Andrews
Abstract—Heterogeneous cellular networks (HCNs) are char-
acterized by cells whose coverage areas may vary by orders of
magnitude. It is natural therefore that their user populations
(and hence traffic loads) will vary similarly. Yet, to date, random
spatial models developed for HCNs generally assume that all base
stations (BSs) are always transmitting and hence implicitly have
the same load. This paper incorporates a flexible notion of BS
load by conditionally thinning the interference field, conditional
on the connection of a typical mobile to its serving BS. We derive
the coverage probability – i.e. the Signal-to-Interference-Ratio
(SIR) distribution – for a typical mobile in a K-tier HCN where
each tier has an arbitrary load characterized by a traffic factor
p
k
2 [0, 1], where p
k
=1is fully loaded. Fully-loaded models are
observed to be extremely pessimistic in terms of coverage, and
the analysis shows that adding lightly loaded access points (e.g.
pico or femtocells) to the macrocell network always increases the
coverage probability.
I. INTRODUCTION
Due to the increasing popularity of mobile data and video,
focus has shifted from voice-oriented applications towards
data-hungry applications such as live video streaming and
symmetric video calls [1]. Macrocell based conventional cellu-
lar networks were primarily designed to provide coverage and
are clearly not capable of accommodating this huge change in
the usage trends [2]. As a result, a typical 3G or 4G cellular
network already has microcells, picocells, distributed antennas,
and femtocells, along with the existing macrocell BSs.
This rapidly increasing heterogeneity requires new models,
e.g., a random spatial model in which the BS locations form
a realization of some random spatial point process [3]. Such
a model captures the inevitable uncertainty in their locations,
and tools from stochastic geometry, point process theory and
spatial statistics can be deployed to assist in analysis [4].
This model was introduced for HCNs in [5], [6] and extended
in [7]–[9], and is surprisingly tractable: under fairly benign
assumptions, the coverage probability could be derived in
closed-form, which is not possible even for 1-tier networks
in the deterministic hexagonal grid model. The model further
was shown to generally agree in several important ways with
more sophisticated industry (e.g. 3GPP) simulations [10] and
even early field deployments of HCNs [11].
Despite this progress, these (and other) HCN models neglect
network traffic and load, instead assuming that all the BSs
transmit concurrently all the time. Although this might be
H. S. Dhillon and J. G. Andrews are with the Wireless Networking and
Communications Group (WNCG), The University of Texas at Austin, TX,
USA (email: dhillon@utexas.edu and jandrews@ece.utexas.edu). R. K. Ganti
is with the department of EE at the Indian Institute of Technology Madras,
India (email: rganti@ee.iitm.ac.in).
justified for macrocells, it is clearly unrealistic for smaller cells
with far fewer users, on average. The main goal of this paper is
to overcome this shortcoming by incorporating a notion of BS
load, while retaining some of the tractability which makes ran-
dom spatial models, especially Poisson Point Process (PPP),
so attractive for analysis. The paper’s contributions are now
summarized.
Tractable Load Model for K-Tier HCNs. The first
contribution is the proposed model and framework, which
incorporates a simple notion of load. For an HCN where BSs
across tiers differ in terms of their transmit power, supported
data rate and deployment density, we assume that a typical
mobile connects to the strongest BS in terms of received power
and conditioned on this connection, the i
th
-tier interfering BSs
transmit independently with a probability p
i
. These BS activity
factors {p
i
} may vary significantly across the tiers due to
different coverage areas of each tier.
Coverage Probability. We derive exact expressions for the
coverage probability of a typical mobile user in both open
and closed access HCNs. Equivalently, this gives the outage
probability and characterizes the signal-to-interference-ratio
(SIR) distribution over the network.
Design Insights. We observe that adding lightly loaded
small cells, such as femto or picocells, to the macrocell
network always increases the coverage probability, which is an
optimistic result for current cellular trends and counter to the
commonly held belief that adding in-band “interfering” small
cells will somehow hurt the network performance. On the other
hand, lightly loaded cells will not provide a major capacity
increase, since the load is light. If the loads on each tier are
the same, then the coverage probability neither increases or
decreases as tiers or BSs are added, and the sum throughput
increases linearly with the number of BSs and/or tiers.
II. SYSTEM MODEL
We model a downlink HCN with K classes (or tiers)
of BSs; we denote the set {1, 2,...K} by K. BSs of the
i
th
class transmit with power P
i
, have a target Signal-to-
Interference-plus-Noise-Ratio (SINR) of
i
and are assumed
to form a realization of an independent homogeneous Poisson
Point Process (PPP)
i
with density
i
. This model has been
validated vs. a real world deployment for macrocells (K =1)
in [3]. The model is likely even more sensible for K-tier
HCNs due to the increased uncertainty (“randomness”) in the
deployment of smaller cells, and as noted in the introduction
has been adopted by academia and industry alike in the past
year.
Without loss of generality, we perform analysis on a typical
mobile user located at origin and we consider the max-SINR,
equivalently max-power, connectivity model. In closed access
– considered later – the mobile connects to the strongest BS in
the allowed subset B ✓ K of tiers. Since HCNs are typically
interference-limited [12], we ignore thermal noise for nota-
tional simplicity, but this is not essential. The wireless channel
follows standard distance based path loss with exponent ↵
along with Rayleigh fading. Hence the received power at a
typical mobile from a BS located at point x 2
i
can be
expressed as P
i
h
x
kxk
↵
, where h
x
⇠ exp(1) and kxk
↵
is
the distance based path loss. Assuming Z
i
to be the set of
i
th
-tier interfering BSs (possibly thinned version of
i
), the
downlink SIR at the typical mobile user when it connects to
the BS located at point y 2
i
is:
SIR(y)=
P
i
h
y
kyk
↵
P
K
k=1
P
x2[Z
i
P
k
h
x
kxk
↵
. (1)
A. Proposed Load Model and Mathematical Preliminaries
Network “load” is modeled as the likelihood of transmission
by a given BS at a randomly chosen time instant. This can also
be visualized as the BS activity factor, formally defined as the
fraction of time for which a BS transmits. Correlation in loads
across time and space are ignored.
Thus, we assume that a typical mobile connects to the
strongest BS and conditioned on this connection, the inter-
ferer belonging to the i
th
tier transmits independently with
a probability p
i
and is idle with a probability 1 p
i
. This
conditioning makes it harder to analyze this system model
since we do not have a priori knowledge about the serving BS
and hence it is not possible to isolate the interference field.
To overcome this, we partition each tier
m
independently
into two sets of BSs
m
and
m
, where
m
and
m
are
both independent PPPs with densities p
m
m
and (1p
m
)
m
.
The set
m
represents the set of active BSs of tier m with
the possibility of one of them being a serving BS, and
m
represents the set of idle BSs of tier m with an exception that
it could also contain the serving BS since partitioning was
done independently. The advantage of this partitioning is that
the interferers are confined to the set =
S
m2K
m
.
For ease of notation, we define the maximum signal strength
from a set of nodes A as M(A)=sup
x2A
P
A
h
x
kxk
↵
and
the total received power at the origin from the set of active
BSs as I =
P
K
i=1
P
x2
i
P
i
h
x
kxk
↵
, which denotes the net
interference power if does not include the serving BS and
the interference plus signal power if it includes the serving
BS. From the definition of M(
i
) and I, it is easy to see that
1
⇣
M(
i
)
IM ( )
<
i
⌘
=1only if no active BS in the set
i
can
connect to the mobile. Similarly, 1
⇣
M(
i
)
I
<
i
⌘
=1only
if no BS in the set
i
is able to connect to the mobile. The
second event is defined to cover the possibility that a serving
BS may lie in the set
i
. Recalling that a mobile is in outage
(not in coverage) if none of the BSs in the whole network
provides SIR that is greater than the corresponding target for
that tier, the coverage probability can now be defined in terms
of these two events as:
P
c
=1 E
"
Y
i2K
1
✓
M(
i
)
I M( )
<
i
◆
1
✓
M(
i
)
I
<
i
◆
#
.
(2)
For tractability, we assume that the target-SIRs
i
are
greater than 0 dB, i.e.,
i
> 1, 8 i. This is in fact the case for a
large fraction of mobile users and only a few edge users might
violate this assumption. Moreover, in [6], we have shown that
the results derived under this weaker assumption hold down
until around 4dB which covers a large fraction of cell edge
users as well. The reason why this assumption is helpful is
because it ensures that at most one BS in the active set
meets the target SIR requirements for a typical mobile user.
Refer to [6] for a detailed discussion on this assumption and
its application in coverage analysis of a fully-loaded K-tier
HCN.
B. Coverage Regions
To understand the effect of proposed model on the coverage
footprints of various BSs, consider a realization of a three tier
HCN shown in Fig. 1. In the left figure, we plot the coverage
regions assuming a fully loaded network by tessellating the
space according to max-SIR connectivity model. Clearly, this
plot does not resemble a classical Voronoi tessellation due
to the differences in the transmit powers of BSs across tiers.
Moreover, it should be noted that the “cell edges” are not
as sharp in reality due to fading and shadowing, which are
averaged out for these illustrative plots. The effect of incor-
porating the proposed load model can be understood in two
equivalent ways: i) thinning of the interference field (middle
figure), ii) biasing of a typical mobile towards its serving BS
(right figure). While the former is a direct result of conditional
thinning, the latter is an indirect consequence of the expansion
of coverage regions in the thinned interference field.
III. COVERAGE PRO B A BIL ITY
This is the main technical section of this paper where we
derive the probability that a typical mobile is in coverage. We
first derive coverage probability for an open access network,
where the mobile is allowed to connect to any BS in the
network. The result for the closed access network or the closed
subscriber group, where the mobile is restricted to connect
only to the B ✓ K tiers, immediately follow.
A. Exact Expression for Coverage Probability
We start by stating the Laplace transform of I, i.e., L
I
(s)=
E [exp(sI)], in Lemma 1, which will be useful in the
derivation of coverage probability. The proof is given in [6].
Lemma 1. The Laplace transform of I can be expressed as:
L
I
(s)=exp
s
2/↵
C(↵)
K
X
l=1
p
l
l
P
2/↵
l
!
, (3)
where C(↵) is given by C(↵)=
2⇡
2
csc
(
2⇡
↵
)
↵
.
Fig. 1. Coverage regions in a realization of a three-tier network with
2
=2
1
,
3
=4
1
, P
1
=100P
2
and P
1
=1000P
3
. The big circles represent
macrocells, squares represent picocells, small diamonds represent femtocells and big triangle represents typical mobile. The left figure depicts fully loaded
system and the other two have the interference field thinned by p
1
= .6 and p
2
= p
3
= .4. To highlight the removal of certain interferers, their original
coverage regions are removed as well in the middle figure. The coverage regions in the right figure are redrawn based only on the active set of interferers. It
highlights that the typical mobile is now “biased” towards its serving BS and the new coverage regions are enlarged due to thinning of the interference field.
The following Lemma deals with fractional moments of
interference and is the main technical result required for
evaluating the coverage probability for this model.
Lemma 2. Let
i
denote the set of active transmitters of tier
i and
i
=
i
/(1 +
i
). Let I denote the total received power
from the BSs in the set and for notational simplicity define
T = 1
✓
max
i2K
M(
i
)
i
<I
◆
I
2/↵
. Then
E [T
m
]=
m!g(m)
(A)
m
,
where
g(m)=
✓
A
⌘
◆
m
(
1
(1 +
2m
↵
)
B
⌘
⇡(1 +
2
↵
)
(1 +
(m+1)2
↵
)
)
, (4)
and
A = ⇡(1 + 2/↵)
X
l2K
(1 p
l
)
l
P
2/↵
l
2/↵
l
(5)
B =
X
i2K
i
p
i
P
2
↵
i
2
↵
i
2
F
1
(1,
2m
↵
, 1+
(m+1)2
↵
,
1
1+
i
)
(1 +
i
)
2m/↵
(6)
⌘ = C(↵)
K
X
l=1
p
l
l
P
2/↵
l
(7)
The hypergeometric function is denoted by
2
F
1
(a, b, c, z)=
(c)
(b)(cb)
R
1
0
t
b1
(1t)
cb1
(1tz)
a
dt.
Proof: See Appendix A.
Using these Lemmas, we now derive the main result which
characterizes the coverage probability of a typical mobile in
the network.
Theorem 1 (Open Access). The downlink coverage probabil-
ity for a typical mobile user in a K-tier open access network
assuming
i
> 1, 8 i, is
P
c
=
⇡
C(↵)
P
i2K
p
i
i
P
2/↵
i
2/↵
i
P
K
i=1
p
i
i
P
2/↵
i
1
X
m=1
g(m), (8)
Proof: The coverage probability is given by (2). Since
the point processes
i
and the corresponding fading random
variables are independent, conditioning on the common in-
terference, we can move the expectation inside the product.
Hence
1P
c
= E
"
K
Y
i=1
1
✓
M(
i
)
I M( )
<
i
◆
E [1 (M(
i
) <
i
I)]
#
,
(9)
where the inner expectation is with respect to the inactive
transmitter sets. The inner expectation can be simplified to:
E
"
Y
x2
i
1
P
i
hkxk
↵
<
i
I
#
(10)
(a)
= E
"
Y
x2
i
1 exp
i
P
1
i
Ikxk
↵
#
(11)
(b)
=exp
✓
(1 p
i
)
i
Z
x2R
2
e
i
P
1
i
Ikxk
↵
dx
◆
(12)
(c)
=exp
✓
(1 p
i
)
i
2
↵
i
I
2
↵
P
2
↵
i
⇡
✓
1+
2
↵
◆◆
, (13)
where (a) follows form the fact that fading is Rayleigh
distributed, i.e., h ⇠ exp(1), (b) follows from the probability
generating functional (PGFL) of PPP [13] and (c) follows
from some algebraic manipulations to reduce the integral to a
Gamma function. Now recalling the expression of A given by
(5), we can write:
1 P
c
= E
1
✓
max
i2K
M(
i
)
i
<I
◆
exp(AI
2/↵
)
. (14)
Using the Taylor series expansion of exp( x ) , exchanging the
infinite summation and expectation,
1 P
c
=
1
X
m=0
(A)
m
m!
E
1
✓
max
i2K
M(
i
)
i
<I
◆
I
2m/↵
.
The summation can be split as
1P
c
= P
✓
max
i2K
M(
i
)
i
<I
◆
+
1
X
m=1
(A)
m
m!
E [T
m
] . (15)
The term 1 P
⇣
max
i2K
M(
i
)
i
<I
⌘
is the coverage proba-
bility in a fully loaded heterogeneous network where the m-th
tier density is p
m
m
. This is derived in [6] and is given by:
1 P
✓
max
i
M(
i
)
i
<I
◆
=
⇡
C(↵)
K
P
i=1
p
i
i
P
2
↵
i
2
↵
i
P
K
i=1
p
i
i
P
2
↵
i
. (16)
Using Lemma 2 to evaluate E [T
m
], we obtain the result.
We note that the expression of coverage probability involves
infinite summation over the sequence g(m). Therefore, we first
show that the infinite summation converges by showing that
|g(m)| ! 0 as m !1. Observe that:
|g(m)|
✓
A
⌘
◆
m
1
(1 +
2m
↵
)
(A/⌘)
m
b1+
2m
↵
c!
=
(A/⌘)
m
d
2m
↵
e!
=
(A/⌘)
m
d
2m
↵
e
d
2m
↵
e
d
2m
↵
e!
! 0,
where the limiting argument follows from the fact that the
sequence of the form x
n
/n! ! 0. In addition to proving that
the series converges, this upper bound on |g(m)| also sheds
light on the behavior of the sequence g(m). If A/⌘ < 1, the
bound decreases monotonously with m and hence it is enough
to consider only a few significant terms to closely approximate
the infinite sum. However, if A/⌘ > 1, especially if A/⌘ 1,
the upper bound first increases until d
2m
↵
e(A/⌘)
m
d
2m
↵
e
and
decreases thereafter. Therefore, the number of significant terms
of g(m) required to approximate the infinite sum would be
relatively higher in this case.
We now provide the exact expression for the coverage prob-
ability in a closed access network in the following Theorem.
We recall that that coverage probability in closed-access is
given by (2) with the only change that the product is over
B instead of K. By definition, coverage probability in closed
access is less than that of open access. Using this definition,
the proof proceeds exactly same as that of Theorem 1, and
hence is not provided.
Theorem 2 (Closed Access). The downlink coverage proba-
bility of a typical mobile in a K-tier closed access network
where a mobile is allowed to connect to B ✓ K tiers assuming
i
> 1, 8 i, is
P
c
=
⇡
C(↵)
P
i2B
p
i
i
P
2/↵
i
2/↵
i
P
K
i=1
p
i
i
P
2/↵
i
1
X
m=1
g
c
(m), (17)
where g
c
(m) and corresponding A and B are given by (4),
(5) and (6), respectively, with the only difference that the
summations defined over set K are now over set B.
B. Special Cases of Interest
We now use the results derived in this section to study
some special cases and compare the system performance with
already known results for fully-loaded system. First, we note
that for a fully-loaded system, the value of A =0and hence
g(m)=g
c
(m)=0, 8 m. Therefore, the coverage probability
in this case can be expressed as:
P
c
=
⇡
C(↵)
P
i2K
i
P
2/↵
i
2/↵
i
P
K
i=1
i
P
2/↵
i
, (18)
which is the same as the Corollary 1 in [6]. The coverage
probability in closed access is also given by the same expres-
sion with the only difference that the summation over the set
K is now over the set B.
For a single-tier open-access network, the coverage proba-
bility derived in Theorem 1 can be simplified and is expressed
as the following Corollary.
Corollary 1 (Single-Tier). The coverage probability for the
single tier open access network with BS activity factor p is
P
c
=
⇡
2/↵
C(↵)
1
X
m=1
g(m), (19)
where the terms A/⌘ and B/⌘ appearing in g(m) are
A
⌘
=
⇡(1 +
2
↵
)(1 p)
C(↵)p
2/↵
(20)
B
⌘
=
2
F
1
(1,
2m
↵
, 1+
(m+1)2
↵
,
1
1+
)
C(↵)
2/↵
(1 + )
2m/↵
. (21)
Remark 1 (Scale-invariance of a single-tier network). From
Corollary 1, we note that for any BS activity factor p, the
coverage probability in a single-tier open-access network is
independent of the BS density and transmit power P . This
is henceforth referred to as “scale-invariance” of cellular
networks to changes in the BS density and their transmit
powers.
Remark 1 is a generalization of a similar result derived for
fully-loaded networks in [6], which can easily be seen from
(18). In addition to single-tier networks, (18) also shows that
the general fully-loaded open-access multi-tier networks also
exhibit scale-invariance if the target SIRs for all the tiers are
the same. Motivated by this observation, we study the coverage
probability for our proposed load model in open-access multi-
tier networks under the assumption that the target SIR is the
same for all tiers in the next Corollary.
Corollary 2 (Coverage Probability: K-Tier with same ).
The coverage probability for a K-tier open-access network
assuming target SIRs to be the same (= ) for all the tiers is
given by (19), with the difference that A/⌘ appearing in g(m)
is now defined as:
A
⌘
=
⇡(1 +
2
↵
)
C(↵)
2/↵
P
K
l=1
(1 p
l
)
l
P
2/↵
l
P
K
l=1
p
l
l
P
2/↵
l
, (22)
and B/⌘ is given by (21).
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Transmission Probability (p)
Coverage Probability (P
c
)
Proposed Model
Fully−Loaded
Fig. 2. Coverage probability as a function of transmission probability is a
single tier network ( =1and ↵ =3.8).
Remark 2 (Scale-Invariance of K-tier HCNs with same ).
From Corollary 2, we note that the coverage probability for K-
tier HCNs is not scale-invariant in general, even when target
SIRs of all the tiers are the same. However, the invariance
property does hold when the BS activity factors of all the tiers
are the same. Interestingly, the coverage probability in this
case is the same as that of a single-tier network given by
Corollary 1.
To understand this remark, we consider the following simple
example.
Example 1 (Scale-invariance in a 2-tier HCN). Consider a
two-tier network with BS activity factors p
1
and p
2
. If p
1
<p
2
,
increasing the density of the first tier leads to a higher increase
in the intended power due to the higher likelihood of having
a closer tier-1 BS as the serving BS but a relatively smaller
increase in the interference power. The coverage probability
in this case is expected to increase. On the other hand, if
p
1
>p
2
, increasing the density of tier-1 BSs leads to higher
increase in the interference power as compared to the intended
power, leading to a decrease in the coverage probability. The
two effects cancel each other when the activity factors of the
two tiers are the same.
IV. NUMERICAL RESULTS
Since most of the analytical results derived in this paper are
fairly self-explanatory and do not require separate numerical
treatment, we will provide only those results which help better
visualize certain important trends.
First, we compare the coverage results with those of a fully-
loaded system in Fig. 2. Although a huge difference in the
coverage guarantees was expected for very low BS activity
factors, it is indeed interesting that the coverage estimates
assuming full load are quite pessimistic even for reasonably
high load scenarios, such as p = .7 .8.
To study scale invariance and the effect of adding small
cells, we now consider a two-tier system and plot the coverage
0 10 20 30 40 50 60 70 80 90 100
0.72
0.74
0.76
0.78
0.8
0.82
0.84
0.86
0.88
0.9
Density of second tier BSs (λ
2
)
Coverage Probability (P
c
)
p
2
= .8
p
2
= .6
p
2
= .4
Fig. 3. Coverage probability in a two-tier network as a function of
2
( =[1, 1], P =[1,.01],
1
=1, p
1
= .6 and ↵ =4).
probability as a function of the density of second tier for
various BS activity factors in Fig. 3. The target SIR is
considered to be the same for both the tiers. We first note
that the network is invariant to the changes in density when
p
1
= p
2
as discussed in the last section. More importantly,
we note that the coverage probability increases with
2
when
the second tier BSs are less active than the first tier. This is a
possibly important result from the perspective of small-cells,
which are generally less active than macrocell BSs. Therefore,
the coverage probability of the network should increase with
the addition of small-cells in this regime. On the other hand, if
a tier of BSs is added which is more active than the macrocells,
the coverage would decrease, although this case seems pretty
unlikely given the high load handled by the macrocells.
V. C ONCLUSIONS
In this paper, we have developed a tractable load model
for K-tier HCNs by defining a notion of conditional-thinning
of interference, conditional on the connection of a typical
mobile to its serving BS. Using tools from stochastic geometry
and point process theory, we derived a simple expression
for the average coverage probability of a typical mobile.
Apart from other design insights, our analysis shows that the
addition of small cells to macrocellular networks will increase
the overall coverage probability of the network and hence
provides a strong rebuttal to the viewpoint that unplanned
infrastructure might bring down a cellular network due to
increased interference.
APPENDIX A
PROO F OF LEMMA 2:
Being consistent with the definition of T , we note that:
T
m
= 1
✓
max
i2K
M(
i
)
i
<I
◆
I
2m/↵
. (23)
![Fig. 3. Coverage probability in a two-tier network as a function of 2 ( = [1, 1], P = [1, .01], 1 = 1, p1 = .6 and ↵ = 4).](/figures/fig-3-coverage-probability-in-a-two-tier-network-as-a-30oy6b3o.webp)
