1

System-Level Modeling and Optimization of the
Energy Efficiency in Cellular Networks – A
Stochastic Geometry Framework

arXiv:1801.07513v1 [] 23 Jan 2018

Marco Di Renzo, Senior Member, IEEE, Alessio Zappone, Senior Member, IEEE,
Thanh Tu Lam, Student Member, IEEE, and Mérouane Debbah, Fellow, IEEE

Abstract—In this paper, we analyze and optimize the energy efficiency of downlink cellular networks. With the aid
of tools from stochastic geometry, we introduce a new closedform analytical expression of the potential spectral efficiency
(bit/sec/m2 ). In the interference-limited regime for data transmission, unlike currently available mathematical frameworks, the
proposed analytical formulation depends on the transmit power
and deployment density of the base stations. This is obtained
by generalizing the definition of coverage probability and by
accounting for the sensitivity of the receiver not only during
the decoding of information data, but during the cell association
phase as well. Based on the new formulation of the potential
spectral efficiency, the energy efficiency (bit/Joule) is given in
a tractable closed-form formula. An optimization problem is
formulated and is comprehensively studied. It is mathematically
proved, in particular, that the energy efficiency is a unimodal and
strictly pseudo-concave function in the transmit power, given the
density of the base stations, and in the density of the base stations,
given the transmit power. Under these assumptions, therefore, a
unique transmit power and density of the base stations exist,
which maximize the energy efficiency. Numerical results are
illustrated in order to confirm the obtained findings and to prove
the usefulness of the proposed framework for optimizing the
network planning and deployment of cellular networks from the
energy efficiency standpoint.
Index Terms—Cellular Networks, Energy Efficiency, Poisson
Point Processes, Stochastic Geometry, Optimization.

I. I NTRODUCTION
The Energy Efficiency (EE) is regarded as a key performance metric towards the optimization of operational cellular networks, and the network planning and deployment
Manuscript received August 2, 2017; revised November 19, 2017; accepted
January 9, 2018. Date of publication January XY, 2018; date of current
version January XY, 2018. This work was supported in part by the European
Commission through the H2020-MSCA ETN-5Gwireless project under Grant
Agreement 641985, the H2020-MSCA IF-BESMART project under Grant
Agreement 749336, and the H2020-ERC PoC-CacheMire project under Grant
Agreement 727682. The associate editor coordinating the review of this paper
and approving it for publication was S. Mukherjee. (Corresponding author:
Marco Di Renzo)
M. Di Renzo and T. Tu-Lam are with the Laboratoire des Signaux et
Systèmes, CNRS, CentraleSupélec, Univ Paris Sud, Université Paris-Saclay,
3 rue Joliot Curie, Plateau du Moulon, 91192 Gif-sur-Yvette, France. (e-mail:
marco.direnzo@l2s.centralesupelec.fr, lamthanh.tu@l2s.centralesupelec.fr).
A. Zappone and M. Debbah are with the LANEAS group of the Laboratoire des Signaux et Systèmes, CentraleSupélec, CNRS, Univ Paris Sud,
Université Paris-Saclay, 3 rue Joliot Curie, Plateau du Moulon, 91192
Gif-sur-Yvette, France. (e-mail: alessio.zappone@l2s.centralesupelec.fr, merouane.debbah@l2s.centralesupelec.fr). M. Debbah is also with the Mathematical and Algorithmic Sciences Laboratory, France Research Center, Huawei
Technologies, 20 Quai du Point du Jour, 92100 Boulogne-Billancourt, France.

of emerging communication systems [1]. The EE is defined
as a benefit-cost ratio where the benefit is given by the
amount of information data per unit time and area that can be
reliably transmitted in the network, i.e., the network spectral
efficiency, and the cost is represented by the amount of power
per unit area that is consumed to operate the network, i.e.,
the network power consumption. Analyzing and designing a
communication network from the EE standpoint necessitate
appropriate mathematical tools, which are usually different
from those used for optimizing the network spectral efficiency
and the network power consumption individually [2]. The
optimization problem, in addition, needs to be formulated in
a sufficiently simple but realistic manner, so that all relevant
system parameters appear explicitly and the utility function is
physically meaningful.
Optimizing the EE of a cellular network can be tackled in
different ways, which include [1]: the design of medium access
and scheduling protocols for optimally using the available
resources, e.g., the transmit power; the use of renewable
energy sources; the development of innovative hardware for
data transmission and reception; and the optimal planning and
deployment of network infrastructure. In the present paper, we
focus our attention on optimizing the average number of Base
Stations (BSs) to be deployed (or to be kept operational) per
unit area and their transmit power. Henceforth, this is referred
to as “system-level EE” optimization, i.e., the EE across the
entire (or a large portion of the) cellular network is the utility
function of interest.
System-level analysis and optimization are useful when the
network operators are interested in optimizing the average
performance across the entire cellular network. Hence, they
are relevant for optimally operating current networks, and for
deploying and planning future networks. In the first case, given
an average number of BSs per unit area already deployed, they
may provide information on the average number of BSs that
can be switched off based on the average load of the network,
and on their optimal transmit power to avoid coverage holes.
In the second case, they may guide the initial deployment
of cellular infrastructure that employs new types of BSs
(e.g., powered by renewable energy sources), new transmission
technologies (e.g., large-scale antennas), or that operate in new
frequency bands (e.g., the millimeter-wave spectrum).
In the last few years, the system-level modeling and analysis
of cellular networks have been facilitated by capitalizing on the
mathematical tool of stochastic geometry and, more precisely,

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on the theory of spatial point processes [3]-[5]. It has been
empirically validated that, from the system-level standpoint,
the locations of the BSs can be abstracted as points of a
homogeneous Poisson Point Process (PPP) whose intensity
coincides with the average number of BSs per unit area [6]. A
comprehensive survey of recent results in this field of research
is available in [7].
A relevant performance metric for the design of cellular
networks is the Potential Spectral Efficiency (PSE), which
is the network information rate per unit area (measured in
bit/sec/m2 ) that corresponds to the minimum signal quality
for reliable transmission. Under the PPP modeling assumption,
the PSE can be obtained in two steps: i) first by computing
the PSE of a randomly chosen Mobile Terminal (MT) and
by assuming a given spatial realization for the locations of
the BSs and ii) then by averaging the obtained conditional
PSE with respect to all possible realizations for the locations
of the BSs and MTs. In the interference-limited regime, this
approach allows one to obtain a closed-form expression of
the PSE under the (henceforth called) standard modeling
assumptions, i.e., single-antenna transmission, singular pathloss model, Rayleigh fading, fully-loaded BSs, cell association
based on the highest average received power [3]. Motivated by
these results, the PPP modeling approach for the locations of
the BSs has been widely used to analyze the trade-off between
the network spectral efficiency and the network power consumption, e.g., [8], as well as to minimize the network power
consumption given some constraints on the network spectral
efficiency or to maximize the network spectral efficiency given
some constraints on the network power consumption [9]. The
PPP modeling approach has been applied to optimize the
EE of cellular networks as well. Notable examples for this
field of research are [10]-[26]. A general study of the energy
and spectral efficiencies of multi-tier cellular networks can
be found in [27]. In the authors’ opinion, however, currently
available approaches for modeling and optimizing the systemlevel EE of cellular networks are insufficient and/or unsuitable
for mathematical analysis. This is further elaborated in the next
section.
A. Fundamental Limitations of Current Approaches for
System-Level EE Optimization
We begin with an example that shows the limitations of
the available analytical frameworks. In the interference-limited
regime, under the standard modeling assumptions, the PSE is:
PSE = λBS BW log2 (1 + γD ) Pcov (γD )
(a)
λBS BW log2 (1 + γD )
=
2 F1 (1, −2/β, 1 − 2/β, −γD )

(1)

where λBS is the density of BSs, BW is the transmission bandwidth, γD is the threshold for reliable decoding, β > 2 is the
path-loss exponent, 2 F1 (·, ·, ·, ·) is the Gauss hypergeometric
function, Pcov (·) is the coverage probability defined in [3, Eq.
(1)], and (a) follows from [3, Eq. (8)].
The main strength of (1) is its simple closed-form formulation. This is, however, its main limitation as well, especially
as far as formulating meaningful system-level EE optimization

problems is concerned. Under the standard modeling assumptions, in fact, the network power consumption (Watt/m2 ) is1
Pgrid = λBS (Ptx + Pcirc ), where Ptx is the transmit power
of the BSs and Pcirc is the static power consumption of the
BSs, which accounts for the power consumed in all hardware
blocks, e.g., analog-to-digital and digital-to-analog converters,
analog filters, cooling components, and digital signal processing [1]. The system-level EE (bit/Joule) is defined as
the ratio between (1) and the network power consumption,
i.e., EE = PSE/Pgrid . Since the PSE in (1) is independent
of the transmit power of the BSs, Ptx , and the network
power consumption, Pgrid , linearly increases with Ptx , we
conclude that any EE optimization problems formulated based
on (1) would result in the trivial optimal solution consisting
of turning all the BSs off (the optimal transmit power is
zero). In the context of multi-tier cellular networks, a similar
conclusion has been obtained in some early papers on systemlevel EE optimization, e.g., [8], where it is shown that the
EE is maximized if all macro BSs operate in sleeping mode.
A system-level EE optimization problem formulated based
on (1) would result, in addition, in a physically meaningless
utility function, which provides a non-zero benefit-cost ratio,
i.e., a strictly positive EE while transmitting zero power
(EE (Ptx = 0) = PSE/(λBS Pcirc ) > 0). In addition, the EE
computed from (1) is independent of the density of BSs. We
briefly mention here, but will detail it in Section III, that
the load model, i.e., the fully-loaded assumption, determines
the conclusion that the EE does not depend on λBS . This
assumption, however, does not affect the conclusion that the
optimal Ptx is zero. This statement is made more formal in
the sequel (see Proposition 1 and Corollary 1). It is worth
nothing that the conclusion that the PSE is independent of
Ptx is valid regardless of the specific path-loss model being
used2 . It depends, on the other hand, on the assumptions of
interference-limited operating regime and of having BSs that
emit the same Ptx .
Based on these observations, we conclude that a new
analytical formulation of the PSE that explicitly depends on
the transmit power and density of the BSs, and that is tractable
enough for system-level EE optimization is needed. From an
optimization point of view, in particular, it is desirable that
the PSE is formulated in a closed-form expression and that
the resulting EE function is unimodal and strictly pseudoconcave in the transmit power (given the density) and in the
density (given the transmit power) of the BSs. This would
imply, e.g., that the first-order derivative of the EE with respect
to the transmit power of the BSs (assuming the density given)
would have a unique zero, which would be the unique optimal
transmit power that maximizes the EE [2]. Similar conclusions
would apply to the optimal density of the BSs for a given
transmit power. Further details are provided in Section IV.
In this regard, a straightforward approach to overcome the
limitations of (1) would be to abandon the interference-limited
assumption and to take the receiver noise into account. In this
1 In the present paper, this holds true for Load Model 1 that is introduced
in Section II-D.
2 The reader may verify this statement by direct inspection of (4), where
Ptx cancels out for any path-loss models.

3

case, the PSE would be formulated in terms of a single-integral
that, in general, cannot be expressed in closed-form [3], [17,
Eq. (9)]. This integral formulation, in particular, results in
a system-level EE optimization problem that is not easy to
tackle. This approach, in addition, has the inconvenience of
formulating the optimization problem for an operating regime
where cellular networks are unlikely to operate in practice.
B. State-of-the-Art on System-Level EE Optimization
We briefly summarize the most relevant research contributions on energy-aware design and optimization of cellular
networks. Due to space limitations, we discuss only the
contributions that are closely related to ours. A state-of-the-art
survey on EE optimization is available in [2].
In [8], the authors study the impact of switching some macro
BSs off in order to minimize the power consumption under
some constraints on the coverage probability. Since the authors
rely on the mathematical framework in (1), they conclude
that all macro BSs need to be switched off to maximize
the EE. In [9], the author exploits geometric programming
to minimize the power consumption of cellular networks
given some constraints on the network coverage and capacity.
The EE is not studied. A similar optimization problem is
studied in [11] and [17] for two-tier cellular networks but
the EE is not studied either. As far as multi-tier cellular
networks are concerned, an important remark is necessary. In
the interference-limited regime, optimal transmit powers and
densities for the different tiers of BSs may exist if the tiers
have different thresholds for reliably decoding the data. The
PSE, otherwise, is the same as that of single-tier networks, i.e.,
it is independent of the transmit power and density of the BSs.
In [14], the authors study the EE of small cell networks with
multi-antenna BSs. For some parameter setups, it is shown
that an optimal density of the BSs exists. The EE, however,
still decreases monotonically with the transmit power of the
BSs, which implies that the EE optimization problem is not
well formulated from the transmit power standpoint. More
general scenarios are considered in [10], [12], [13], [15], [16],
[18]-[25], but similar limitations hold. In some cases, e.g.,
[20], the existence and uniqueness of an optimal transmit
power and density of the BSs are not mathematically proved
or, e.g., in [24], the problem formulation has a prohibitive
numerical complexity as it necessitates the computation of
multiple integrals and infinite series. It is apparent, therefore,
that a tractable approach for system-level EE optimization is
missing in the open technical literature. In the present paper,
we introduce a new definition of PSE that overcomes these
limitations.
C. Research Contribution and Novelty
In the depicted context, the specific novel contributions
made by this paper are as follows:
• We introduce a new closed-form analytical formulation of
the PSE for interference-limited cellular networks (during
data transmission), which depends on the transmit power
and density of the BSs. The new expression of the PSE
is obtained by taking into account the power sensitivity

of the receiver not only for data transmission but for cell
association as well.
• Based on the new expression of the PSE, a new systemlevel EE optimization problem is formulated and comprehensively studied. It is mathematically proved that the
EE is a unimodal and strictly pseudo-concave function in
the transmit power given the BSs’ density and in the BSs’
density given the transmit power. The dependency of the
optimal power as a function of the density and of the
optimal density as a function of the power is discussed.
• A first-order optimal pair of transmit power and density
of the BSs is obtained by using a simple alternating
optimization algorithm whose details are discussed in the
sequel. Numerical evidence of the global optimality of
this approach is provided as well.
• Two load models for the BSs are analyzed and compared
against each other. It is shown that they provide the same
PSE but have different network power consumptions.
Hence, the optimal transmit power and density of the
BSs that maximize their EEs are, in general, different.
Their optimal EEs and PSEs are studied and compared
against each other.
The paper is organized as follows. In Section II, the system
model is presented. In Section III, the new definition of PSE
is introduced. In Section IV, the EE optimization problem is
formulated and studied. In Section V, numerical results are
shown. Finally, Section VI concludes the paper.
Notation: The main symbols and functions used in the
present paper are reported in Table I.
II. S YSTEM M ODEL
In this section, the network model is introduced. With the
exception of the load model, we focus our attention on a
system where the standard modeling assumptions hold. One of
the main aims of the present paper is, in fact, to highlight the
differences between currently available analytical frameworks
and the new definition of PSE that is introduced. The proposed
approach can be readily generalized to more advanced system
models, such as that recently adopted in [5].
A. Cellular Network Modeling
A downlink cellular network is considered. The BSs are
modeled as points of a homogeneous PPP, denoted by ΨBS , of
density λBS . The MTs are modeled as another homogeneous
PPP, denoted by ΨMT , of density λMT . ΨBS and ΨMT are
independent of each other. The BSs and MTs are equipped
with a single omnidirectional antenna. Each BS transmits with
a constant power denoted by Ptx . The analytical frameworks
are developed for the typical MT, denoted by MT0 , that is
located at the origin (Slivnyak theorem [28, Th. 1.4.5]). The
BS serving MT0 is denoted by BS0 . The cell association
criterion is introduced in Section II-C. The subscripts 0, i and
n identify the intended link, a generic interfering link, and a
generic BS-to-MT link. The set of interfering BSs is denoted
(I)
by ΨBS . As for data transmission, the network operates in
the interference-limited regime, i.e., the noise is negligible
compared with the inter-cell interference.

4

TABLE I
S UMMARY OF MAIN SYMBOLS AND FUNCTIONS USED THROUGHOUT THE PAPER .

Symbol/Function
E{·}, Pr {·}
λBS , λMT
(I)
ΨBS , ΨMT , ΨBS
BS0 , BSi , BSn
Ptx , Pcirc , Pidle
rn , gn
l (·), Ln , L0
κ, β > 0
BW , N0
2
σN
= BW N0 , Iagg (·)
γD , γA
−α
L (x) = 1 − (1 + x/α) , α = 3.5
fX (·), FX (·)
1 (·), 2 F1 (·, ·, ·, ·), Γ(·)
max {x, y}, min {x, y}
Υ = 2 F1 (−2/β, 1, 1 − 2/β, −γD ) − 1 ≥ 0

Q (x, y, z) = 1 − exp −πx(y/η)2/β (1 + ΥL (z))
SIR, SNR
Pcov , PSE, Pgrid
.
..
z x (x, y), z x (x, y)



Definition
Expectation operator, probability measure
Density of base stations, mobile terminals
PPP of base stations, mobile terminals, interfering base stations
Serving, interfering, generic base station
Transmit, circuits, idle power consumption of base stations
Distance, fading power gain of a generic link
Path-loss, shorthand of path-loss, path-loss of intended link
Path-loss constant, slope (exponent)
Transmission bandwidth, noise power spectral density
Noise variance, aggregate other-cell interference
Reliability threshold for decoding, cell association
Probability that a base station is in transmission mode
Probability density/mass, cumulative distribution/mass function of X
Indicator function, Gauss hypergeometric function, gamma function
Maximum, minimum between x and y
Shorthand
2
Shorthand with η = κσN
γA

Signal-to-interference-ratio, average signal-to-noise-ratio
Coverage, potential spectral efficiency, network power consumption
First-order, second-order derivative with respect to x

B. Channel Modeling

D. Load Modeling

For each BS-to-MT link, path-loss and fast-fading are
considered. Shadowing is not explicitly taken into account
because its net effect lies in modifying the density of the
BSs [5]. All BS-to-MT links are assumed to be mutually
independent and identically distributed (i.i.d.).
a) Path-Loss: Consider a generic BS-to-MT link of
length rn . The path-loss is l (rn ) = κrnβ , where κ and β
are the path-loss constant and the path-loss slope (exponent).
For simplicity, only the unbounded path-loss model is studied
in the present paper. The analysis of more general path-loss
models is an interesting but challenging generalization that is
left to future research [29].
b) Fast-Fading: Consider a generic BS-to-MT link. The
power gain due to small-scale fading is assumed to follow
an exponential distribution with mean Ω. Without loss of
generality, Ω = 1 is assumed. The power gain of a generic
BS-to-MT link is denoted by gn .

Based on (2), several or no MTs can be associated to a
generic BS. In the latter case, the BS transmits zero power, i.e.,
Ptx = 0, and, thus, it does not generate inter-cell interference.
In the former case, on the other hand, two load models are
studied and compared against each other. The main objective
is to analyze the impact of the load model on the power
consumption and EE of cellular networks. Further details are
provided in the sequel. Let NMT denote the number of MTs
associated to a generic BS and BW denote the transmission
bandwidth available to each BS. If NMT = 1, for both load
models, the single MT associated to the BS is scheduled
for transmission and the entire bandwidth, BW , and transmit
power, Ptx , are assigned to it.
a) Load Model 1: Exclusive Allocation of Bandwidth
and Power to a Randomly Selected MT: If NMT > 1, the
BS randomly selects, at each transmission instance, a single
MT among the NMT associated to it. Also, the BS allocates
the entire transmission bandwidth, BW , and the total transmit
power, Ptx , to it. The random scheduling of the MTs at each
transmission instance ensures that, in the long term, all the
MTs associated to a BS are scheduled for transmission.
b) Load Model 2: Equal Allocation of Bandwidth and
Power Among All the MTs: If NMT > 1, the BS selects, at
each transmission instance, all the NMT MTs associated to it.
The BS equally splits the available transmission bandwidth,
BW , and evenly spreads the available transmit power, Ptx ,
among the NMT MTs. Thus, the bandwidth and power are
viewed as continuous resources by the BS’s scheduler: each
MT is assigned a bandwidth equal to BW /NMT and the power

C. Cell Association Criterion
A cell association criterion based on the highest average
received power is assumed. Let BSn ∈ ΨBS denote a generic
BS of the network. The serving BS, BS0 , is obtained as
follows:
BS0 = arg maxBSn ∈ΨBS {1/l (rn )}
= arg maxBSn ∈ΨBS {1/Ln }

(2)

where the shorthand Ln = l (rn ) is used. As for the intended
link, L0 = minrn ∈ΨBS {Ln } holds.

5

spectral density at the detector’s (i.e., the typical MT, MT0 )
input is equal to Ptx /BW .
In the sequel, we show that the main difference between the
two load models lies in the power consumption of the BSs. In
simple terms, the more MTs are scheduled for transmission the
higher the static power consumption of the BSs is. The analysis
of general load models, e.g., based on a discrete number of
resource blocks [5], is left to future research due to space
limits.
E. Power Consumption Modeling
In the considered system model, the BSs can operate in
two different modes: i) they are in idle mode if no MTs are
associated to them and ii) they are in transmission mode if
at least one MT is associated to them. The widespread linear
power consumption model for the BSs is adopted [1], [30],
which accounts for the power consumption due to the transmit
power, Ptx , the static (circuit) power, Pcirc , and the idle power,
Pidle . If the BS is in idle mode, its power consumption is
equal to Pidle . If the BS is in transmission mode, its power
consumption is a function of Ptx , Pcirc , and depends on
the load model. Further details are provided in the sequel.
In the present paper, based on physical considerations, the
inequalities 0 ≤ Pidle ≤ Pcirc are assumed.
III. A N EW A NALYTICAL F ORMULATION

OF THE

PSE

In this section, we introduce and motivate a new definition
of coverage probability, Pcov , and PSE, which overcomes the
limitations of currently available analytical frameworks and is
suitable for system-level optimization (see Section I-A). All
symbols are defined in Table I.
Definition 1: Let γD and γA be the reliability thresholds
for the successful decoding of information data and for the
successful detection of the serving BS, BS0 , respectively. The
coverage probability, Pcov , of the typical MT, MT0 , is defined
as follows:
Pcov (γD , γA )
( 
Pr SIR ≥ γD , SNR ≥ γA
=
0

if
if

MT0 is selected
MT0 is not selected
(3)

where the Signal-to-Interference-Ratio (SIR) and the average
Signal-to-Noise-Ratio (SNR) can be formulated, for the network model under analysis, as follows:
SIR = P

(I)
BSi ∈ΨBS

Ptx /L0
.
SNR =
2
σN

Ptx g0 /L0
Ptx gi /Li 1 (Li > L0 )

(4)

Remark 1: The definition of Pcov in (3) reduces to the
conventional one if γA = 0 [3].

Remark 2: The average SNR, SNR, in (4) is averaged with
respect to the fast fading. The SIR depends, on the other hand,
on fast fading. This choice is discussed in the sequel.

Remark 3: The new definition of coverage probability, Pcov ,
in (3) is in agreement with the cell selection criterion specified

Fig. 1. Illustration of the interplay between Ptx and λBS . For simplicity,
only a cluster of seven BSs is represented by keeping the size of the region of
interest (square box) the same. The inter-site distance of the BSs (represented
as red dots), i.e., the size of the hexagonal cells, is determined by λBS . The
shape of the cells depends on the cell association in (2). The circular shaded
disk (in light yellow) represents the actual coverage region of the BSs that is
determined by Ptx : i) a MT inside the disk receives a sufficiently good signal
to detect the BS and to get associated with it, ii) a MT outside the disk cannot
detect the BS and is not in coverage. The sub-figures (a)-(c) are obtained by
assuming the same λBS but a different Ptx . The sub-figures (d) and (e) are
obtained by considering a λBS greater than that of sub-figures (a)-(c) but
keeping the same Ptx as sub-figures (a) and (b), respectively. The sub-figure
(f) is obtained by considering a λBS smaller than that of sub-figure (c) but
keeping the same Ptx as it. We observe that, for a given λBS , the transmit
power Ptx is appropriately chosen in sub-figures (a), (e) and (f). Ptx is, on
the other hand, under-provisioned in sub-figure (b) and over-provisioned in
sub-figures (c) and (d). In the first case, the MTs are not capable of detecting
the BS throughout the entire cell, i.e., a high outage probability is expected.
In the second case, the BSs emit more power than what is actually needed,
which results in a high power consumption.

by the 3rd Generation Partnership Project (3GPP) [31, Sec.
5.2.3.2].

a) Motivation for the New Definition of Pcov : The motivation for the new definition of coverage probability originates
from the inherent limitations of the conventional definition
(obtained by setting γA = 0 in (3)), which prevents one
from taking into account the strong interplay between the
transmit power and the density of the BSs for optimal cellular
networks planning. In fact, the authors of [3] have shown
that, in the interference-limited regime, Pcov is independent
of the transmit power of the BSs. If, in addition, a fullyloaded model is assumed, i.e., λMT /λBS ≫ 1, then Pcov is
independent of the density of BSs as well. This is known as
the invariance property of Pcov as a function of Ptx and λBS
[5]. The tight interplay between Ptx and λBS is, on the other
hand, illustrated in Fig. 1, where, for ease of representation,
an hexagonal cellular layout is considered. Similar conclusions
apply to the PPP-based cellular layout studied in the present
paper. In Fig. 1, it is shown that, for a given λBS , Ptx needs
to be appropriately chosen in order to guarantee that, for
any possible location of MT0 in the cell, two conditions are
fulfilled: i) the MT receives a sufficiently good signal quality,
i.e., the average SNR is above a given threshold, γA , that
ensures a successful cell association, i.e., to detect the presence
(pilot signal) of the serving BS and ii) the BSs do not overprovision Ptx , which results in an unnecessary increase of the

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power consumption. It is expected, therefore, that an optimal
value of Ptx given λBS and an optimal value of λBS given
Ptx that optimize EE exist [32].
b) Advantages of the New Definition of Pcov : The new
definition of Pcov allows one to overcome the limitations of the
conventional definition and brings about two main advantages.
The first advantage originates from direct inspection of (4). In
the conventional definition of Pcov , only the SIR is considered
and the transmit power of the BSs, Ptx , cancels out between
numerator and denominator. This is the reason why Pcov is
independent of Ptx . In the proposed new definition, on the
other hand, Ptx explicitly appears in the second constraint
and does not cancel out. The density of the BSs, λBS , appears
implicitly in the distribution of the path-loss of the intended
link, L0 . The mathematical details are provided in the sequel.
The second inequality, as a result, allows one to explicitly
account for the interplay between Ptx and λBS (shown in Fig.
1). If λBS increases (decreases), in particular, L0 decreases
(increases) in statistical terms. This implies that Ptx can be
decreased (increased) while still ensuring that the average SNR
is above γA . The second advantage is that the new definition of
Pcov is still mathematically tractable and the PSE is formulated
in a closed-form expression. This is detailed in Proposition 1.
Remark 4: The new definition of Pcov in (3) is based on the
actual value of L0 because a necessary condition for the typical
MT to be in coverage is that it can detect the pilot signal of at
least one BS during the cell association. If the BS that provides
the highest average received power cannot be detected, then
any other BSs cannot be detected either. The second constraint
on the definition of Pcov , in addition, is based on the average
SNR, i.e., the SNR averaged with respect to the fast fading,
because the cell association is performed based on long-term
statistics, i.e., based on the path-loss in the present paper, in
order to prevent too frequent handovers.

Remark 5: Compared with the conventional definition
of coverage based on the Signal-to-Interference+Noise-Ratio
(SINR) [3], the new definition in (3) is conceptually different.
Equation (3) accounts for the signal quality during both the
cell association and data transmission phases. The definition
of coverage based on the SINR, on the other hand, accounts
for the signal quality only during the data transmission phase.
In spite of this fundamental difference, Pcov in (3) may be
interpreted as an approximation for the coverage probability
based on the SINR, and, more precisely, as an alternative
method to incorporate the thermal noise into the problem
formulation. Compared with the coverage based on the SINR,
however, the new definition in (3) accounts for the impact of
thermal noise when it is the dominant factor, i.e., during the
cell association phase when the inter-cell interference can be
ignored as orthogonal pilot signals are used.

Remark 6: Figure 1 highlights that the new definition of
coverage in (3) is not only compliant with [31] but it has a
more profound motivation and wider applicability. In PPPbased cellular networks, in contrast to regular grid-based
network layouts, the size and shape of the cells are random.
This implies that it is not possible to identify a relation, based
on pure geometric arguments, between the cell size and the
transmit power of the BSs that makes the constraint on SNR

in (3) ineffective in practice. In equivalent terms, in this case,
the threshold γA may turn out to be sufficiently small to
render the constraint on SNR ineffective. This is, e.g., the
approach employed in [32, Eq. (1)], where the relation between
the transmit power and density of BSs is imposed a priori
based on the path-loss. In practice, however, cellular networks
are irregularly deployed, which makes the optimal relation
between the transmit power and density of BSs difficult to
identify because of the coexistence of cells of small and large
sizes. The constraint on SNR in (3) allows one to take into
account the interplay between the transmit power and density
of BSs in irregular (realistic) cellular network deployments. 
A. Analytical Formulation of the PSE
In this section, we provide the mathematical definitions
of the PSE for the two load models introduced in Section
II-D. They are summarized in the following two lemmas,
which constitute the departing point to obtain the closed-form
analytical frameworks derived in Section III-B.
Remark 7: The PSE is defined from the perspective of
the typical MT, MT0 rather than from the perspective of the
typical cell (or BS). This implies that the proposed approach
allows one to characterize the PSE of the so-called Crofton
cell, which is the cell that contains MT0 . This approach is
commonly used in the literature and is motivated by the lack
of results on the explicit distribution of the main geometrical
characteristics of the typical cell of a Voronoi tessellation.
Further details on the Crofton and typical cells are available
in [34] and [35].

Let N̄MT be the number of MTs that lie in the cell of
the typical MT, MT0 , with the exception of MT0 . N̄MT is
a discrete random variable whose probability mass function
in the considered system model can be formulated, in an
approximated closed-form expression, as [33, Eq. (3)]:

fN̄MT (u) = Pr N̄MT = u
(5)
3.54.5 Γ (u + 4.5) (λMT /λBS )u
≈
u+4.5 .
Γ (4.5) Γ (u + 1) (3.5 + λMT /λBS )
Remark 8: The probability mass function in (5) is an
approximation because it is based on the widely used empirical
expression of the probability density function of the area of
the Voronoi cells in [36, Eq. (1)]. A precise formula for the
latter probability density function is available in [37]. It is,
however, not used in the present paper due to its mathematical
intractability, as recently remarked in [26]. Throughout the
rest of the paper, for simplicity, we employ the sign of
equality (“=”) in all the analytical formulas that rely solely
on the approximation in (5). This is to make explicit that
our analytical frameworks are not based on any other hidden
approximations.

Based on (5), a formal mathematical formulation for the
PSE is given as follows.
Lemma 1: Let Load Model 1 be assumed. The PSE
(bit/sec/m2 ) can be formulated as shown in (6) at the top of
this page.
Proof : It follows from the definition of PSE [5], where (a)
originates from the fact that MT0 is scheduled for transmission

7



PSE (γD , γA ) = EN̄MT PSE γD , γA | N̄MT


(a)
= λMT BW log2 (1 + γD ) Pr SIR ≥ γD , SNR ≥ γA Pr N̄MT = 0
+∞
X


1
+
λMT BW log2 (1 + γD )
Pr SIR ≥ γD , SNR ≥ γA Pr N̄MT = u
u+1
u=1

+∞
X

Pr N̄MT = u
= λMT BW log2 (1 + γD ) Pr SIR ≥ γD , SNR ≥ γA
.
u+1
u=0

(6)



PSE (γD , γA ) = EN̄MT PSE γD , γA | N̄MT
+∞


(b) X
BW
=
λMT
log2 (1 + γD ) Pr SIR ≥ γD , SNR ≥ γA Pr N̄MT = u
u+1
u=0

+∞
X

Pr N̄MT = u
= λMT BW log2 (1 + γD ) Pr SIR ≥ γD , SNR ≥ γA
.
u+1
u=0

(7)

with unit probability if it is the only MT in the cell, while it is
scheduled for transmission with probability 1/(u + 1) if there
are other u MTs in the cell.

Lemma 2: Let Load Model 2 be assumed. The PSE
(bit/sec/m2 ) can be formulated as shown in (7) at the top of
this page.
Proof : It follows from the definition of PSE [5], where (b)
originates from the fact that MT0 is scheduled for transmission
with unit probability but the bandwidth is equally allocated
among the MTs in the cell, i.e., each of the u + 1 MTs is
given a bandwidth equal to BW /(u + 1).

Remark 9: By comparing (6) and (7), we note that the same
PSE is obtained for both load models. This originates from
the fact that Pcov in (3) is independent of the number of
MTs in the cell. This property follows by direct inspection
of (4) and has been used in the proof of Lemma 1 and Lemma
2. As far as the first load model is concerned, this property
originates from the fact that a single MT is scheduled at
every transmission instance. It is, however, less intuitive for
the second load model. In this latter case, as mentioned in
Section II-D, Ptx and BW are viewed as continuous resources
by the BS’s scheduler. The transmit power per unit bandwidth
of both intended and interfering links is equal to Ptx /BW .
Regardless of the number of MTs available in the interfering
cells, MT0 “integrates” this transmit power per unit bandwidth
over the bandwidth allocated to it, which depends on the total
number of MTs in its own cell. Let the number of these MTs
be u+1. Thus, the receiver bandwidth of MT0 is BW /(u + 1).
This implies that the received power (neglecting path-loss and
fast-fading) of both intended and interfering links is Prx =
(Ptx /BW ) (BW /(u + 1)) = Ptx /(u + 1). As a result, the
number of MTs, u+1, cancels out in the SIR of (4). Likewise,
the received average SNR (neglecting the path-loss) is equal to
Prx 
/(N0 BW /(u + 1)) = (Ptx /(u + 1))/(N0 BW /(u + 1)) =
2
, which is independent of the number of MTs, u + 1,
Ptx σN
and agrees with the definition of average SNR in (4). In the
next section, we show that the load models are not equivalent

in terms of network power consumption.



B. Closed-Form Expressions of PSE and Pgrid
In this section, we introduce new closed-form analytical
frameworks for computing the PSE. We provide, in addition,
closed-form expressions of the network power consumption
for the two load models under analysis. These results are
summarized in the following three propositions.
Let NMT be the number of MTs that lie in an arbitrary
(idle)
cell. The probability that the BS is in idle mode, PBS , and
(tx)
in transmission mode, PBS , can be formulated as follows [33,
Prop. 1]:
(idle)

PBS

(tx)
PBS

= Pr {NMT = 0} = 1 − L (λMT /λBS )
(idle)

= Pr {NMT ≥ 1} = 1 − PBS

= L (λMT /λBS )

(8)

where L (·) is defined in Table II. Using (8), PSE and Pgrid
are given in the following propositions.
Proposition 1: Consider either Load Model 1 or Load Model
2. Assume notation and functions given in Tables I and II.
The PSE (bit/sec/m2 ) can be formulated, in closed-form, as
follows:
λBS L (λMT /λBS )
PSE (γD , γA ) = BW log2 (1 + γD )
1 + ΥL (λMT /λBS ) (9)
× Q (λBS , Ptx , λMT /λBS ) .
Proof : See Appendix A.

Corollary 1: If γA = 0, i.e., the conventional definition of
Pcov is used, the PSE in (9) simplifies as follows:
λBS L (λMT /λBS )
.
1 + ΥL (λMT /λBS )
(10)
≫ 1, the PSE in (9) reduces to

PSE (γD , γA = 0) = BW log2 (1 + γD )

If, in addition, λMT /λBS
(1).
Proof : It follows because Q (·, ·, ·) = 1 if γA = 0 and
L (λMT /λBS ≫ 1) → 1.

Remark 10: Corollary 1 substantiates the comments made
above in this section about the need of a new definition of PSE,

8

EE (Ptx , λBS ) =

BW log2 (1 + γD ) L (λMT /λBS ) Q (λBS , Ptx , λMT /λBS )
PSE
=
. (14)
Pgrid
[1 + ΥL (λMT /λBS )] [L (λMT /λBS ) (Ptx + Pcirc − Pidle ) + Pidle + M (λMT /λBS ) Pcirc ]

as well as the advantages of the proposed analytical formulation. In particular, (10) confirms that the PSE is independent
of Ptx if γA = 0 and that the PSE is independent of Ptx and
λBS if fully-loaded conditions hold, i.e., λMT /λBS ≫ 1. 
Proposition 2: Let Load Model 1 be assumed. Pgrid
(Watt/m2 ) can be formulated as follows:

convenience of analysis, we introduce the following auxiliary
function (LM = Load Model):
M (λMT /λBS )
(
0
if
=
λMT /λBS − L (λMT /λBS ) if

(1)

Pgrid = λBS (Ptx + Pcirc ) L (λMT /λBS )
+ λBS Pidle (1 − L (λMT /λBS )) .

(11)

Proof : The network power consumption is obtained by
multiplying the average number of BSs per unit area, i.e., λBS ,
and the average power consumption of a generic BS, which
is Ptx + Pcirc if the BS operates in transmission mode, i.e.,
with probability L (λMT /λBS ), and Pidle if the BS operates
in idle mode, i.e., with probability 1 − L (λMT /λBS ).

Proposition 3: Let Load Model 2 be assumed. Pgrid
(Watt/m2 ) can be formulated as follows:
(2)

Pgrid = λBS Ptx L (λMT /λBS )
+ λMT Pcirc + λBS Pidle (1 − L (λMT /λBS )) .

(12)

Proof : It is similar to the proof of Proposition 2. The
difference is that the power dissipation of a generic BS
that operates
P in transmission mode is, in this case, equal to
Ptx +Pcirc +∞
u=1 u Pr {NMT = u} = Ptx +Pcirc (λMT /λBS ),
where NMT is the number of MTs in the cell and the last
equality follows from [33, Lemma 1].

Remark 11: The power consumption models obtained in
(11) and (12), which account for the transmit, circuits, and idle
power consumption of the BSs, have been used, under some
simplifying assumptions, in previous research works focused
on the analysis of the EE of cellular networks. Among the
many research works, an early paper that has adopted this
approach under the assumption of fully-loaded BSs and of
having a single active MT per cell is [8].

Remark 12: Since L (λMT /λBS ) ≤ λMT /λBS for every
(2)
(1)
λMT /λBS ≥ 0, we conclude that Pgrid ≥ Pgrid by assuming
the same Ptx and λBS for both load models. This originates
from the fact that, in the present paper, we assume that the
circuits power consumption increases with the number of MTs
that are served by the BSs. It is unclear, however, the best
load model to be used from the EE standpoint, especially if
Ptx and λBS are optimized to maximize their respective EEs.
In other words, the optimal Ptx and λBS that maximize the
EE of each load model may be different, which may lead to
different optimal EEs. The trade-off between the optimal PSE
and the optimal EE is analyzed numerically in Section V for
both load models.

IV. S YSTEM -L EVEL EE O PTIMIZATION : F ORMULATION
AND S OLUTION
In this section, we formulate a system-level EE optimization problem and comprehensively analyze its properties. For

LM − 1 is assumed
LM − 2 is assumed.
(13)

A unified formulation of the EE (bit/Joule) for the cellular
network under analysis is provided in (14) shown at the top of
this page, where the parameters of interest from the optimization standpoint, i.e., Ptx and λBS , are explicitly highlighted.
In the rest of the present paper, all the other parameters are
assumed to be given.
A. Preliminaries
For ease of presentation, we report some lemmas that
summarize structural properties of the main functions that
constitute (14). Some lemmas are stated without proof because
they are obtained by simply studying the sign of the first-order
and second-order derivatives of the function with respect to
the variable of interest and by keeping all the other variables
fixed. Functions of interest for this section are given in Table
II. Also, we define ∆P = Pcirc − Pidle ≥ 0.
Lemma 3: The function L (λMT /λBS ) fulfills the following properties with respect to λBS (assuming λMT
fixed): i) L (λMT /λBS ) ≥ 0 for λBS ≥ 0; ii)
L (λMT /λBS ) = 1 if λBS . → 0; iii) L (λMT /λBS ) =
0 if λBS → .. ∞; iv) LλBS (λMT /λBS ) ≤ 0 for
λBS ≥ 0; v) LλBS (λMT /λBS
.. ) ≤ 0 for λMT /λBS ≥
2α/(α − 1) = 2.8; and vi) LλBS (λMT /λBS ) ≥ 0 for
λMT /λBS ≤ 2α/(α − 1) = 2.8.
Lemma 4: As far as Load Model 2 is concerned, the function
M (λMT /λBS ) fulfills the following properties with respect to
λBS (assuming λMT fixed): i) M (λMT /λBS ) ≥ 0 for λBS ≥
0; ii) M (λMT /λBS ) →
. ∞ if λBS → 0; iii) M (λMT /λBS ) =
0 if ..λBS → ∞; iv) MλBS (λMT /λBS ) ≤ 0 for λBS ≥ 0; and
v) MλBS (λMT /λBS ) ≥ 0 for λBS ≥ 0.
Lemma 5: The function Q (λBS , Ptx , λMT /λBS )
fulfills the following properties with respect to Ptx :
i) Q (λBS , Ptx , λMT /λBS ) ≥ 0 for Ptx ≥ 0; ii)
Q (λBS , Ptx , λMT /λBS )
=
0 if Ptx
→
0; iii)
Q. (λBS , Ptx , λMT /λBS ) = 1 if Ptx → ∞; iv)
Q
..Ptx (λBS , Ptx , λMT /λBS ) ≥ 0 for Ptx ≥ 0; and v)
QPtx (λBS , Ptx , λMT /λBS ) ≤ 0 for Ptx ≥.. 0.
Proof : The result in v) follows from QPtx (·, ·, ·) in Table
II, because iv) and β > 2 hold.

Lemma 6: The function Q (λBS , Ptx , λMT /λBS ) fulfills the following properties with respect to λBS (assuming λMT fixed): i) Q (λBS , Ptx , λMT /λBS ) ≥ 0 for
λBS ≥ 0; ii) Q (λBS , Ptx , λMT /λBS ) = 0 if λBS →
0; iii) Q (λBS , Ptx , λMT /λBS ) = 1 if λBS → ∞; iv)

9

TABLE II
S UMMARY OF MAIN AUXILIARY FUNCTIONS USED THROUGHOUT THE PAPER .

Function Definition
−α

L (λMT /λBS ) = 1 − (1 + (1/α) λMT /λBS )
M (λMT /λBS ) = λMT /λBS − L (λ
MT /λBS )


Q (λBS , Ptx , λMT /λBS ) = 1 − exp −πλBS (Ptx /η)2/β (1 + ΥL (λMT /λBS ))

.

2/β

QPtx (λBS , Ptx , λMT /λBS ) = πλBS
(1/η)

..

2/β−1

(2/β) (1 + ΥL (λMT /λBS )) Ptx

× exp −πλBS (Ptx /η)2/β (1 + ΥL (λMT /λBS ))
2/β

QPtx (λBS , Ptx , λMT /λBS ) = hπλBS (1/η)

2/β−1

(2/β) (1 + ΥL i(λMT /λBS )) Ptx

× −Q̇Ptx (λBS , Ptx , λMT /λBS )
2/β

+πλBS(1/η)

2/β−2

(2/β) (2/β − 1) (1 + ΥL (λMT /λ
 BS ))Ptx
2/β

× exp −πλBS (Ptx /η)
(1 + ΥL (λMT /λBS ))
 2 
.
−(α+1)
LλBS (λMT /λBS ) = − λMT λBS (1h + (1/α) λMT /λBS )
i
 2 
.
−(α+1)
MλBS (λMT /λBS ) = − λMT λBS 1 − (1 + (1/α) λMT /λBS )
i
h
.
.
2/β
1 + ΥL (λMT /λBS ) + ΥλBS LλBS (λMT /λBS )
QλBS (λBS , Ptx , λMT /λBS ) = π(Ptx /η)


2/β
× exp −πλBS (Ptx /η)
(1 + ΥL (λMT /λBS ))


..
LλBS (λMT /λBS ) = h λMT λ3BS (1 + (1/α) λMT /λBS )−(α+1)
i
−1
× 2 − (1 + α) (1/α) λMT /λBS (1 + (1/α)λMT /λBS )
i

h
..
−(α+1)
MλBS (λMT /λBS ) = 2 λMT λ3BS 1 − (1 + (1/α) λMT /λBS )


+ (1
+ α) (1/α) λ2MT λ4BS (1 + (1/α) λMT /λBS )−(α+2)



.Q(λBS ,Ptx ,λMT /λBS ) − (Ptx + ∆P) − Pcirc M (λMT /λBS )
SP (Ptx ) = L λλMT
BS
QPtx (λBS ,Ptx ,λMT /λBS )
 .


 
.  λMT   λMT 
λMT
Pcirc
−
L
M
L λλMT
SD (λBS ) = .
λBS
λ
BS
λ
λBS M λBS
BS
BS
LλBS (λMT /λBS )



 .
M
(λ
. λBS MT /λBS )
+ΥL2 λλMT
(Ptx + ∆P) + ΥPcirc L2 λλMT
BS
BS
LλBS (λMT /λBS )
.



L(λ
/λ )Q
(λ ,P ,λ
/λ )
− . MT BS λBS BS tx MT BS 1 + ΥL λλMT
BS
LλBS (λMT /λBS )Q(λBS ,Ptx ,λMT /λBS )

× [L (λMT /λBS ) (Ptx + ∆P) + Pidle + Pcirc M (λMT /λBS )]

.

Q
..λBS (λBS , Ptx , λMT /λBS ) ≥ 0 for λBS ≥ 0; and v)
QλBS (λBS , Ptx , λMT /λBS ) ≤ 0 for λBS ≥ 0.
.
Proof : The result in iv) follows from QλBS (·, ·, ·)
in Table
II because, for λBS ≥ 0, L (λMT /λBS ) +
.
λBS LλBS (λMT /λBS ) ≥ 0. This latter inequality holds true
because 1 + x (1 + 1/α) ≤ (1 + x/α)(α+1) for x.. ≥ 0. The
result in v)
. follows without explicitly computing QλBS (·, ·, ·)
because QλBS (·, ·, ·) in Table II is the composition of two
increasing and concave functions in λBS , i.e., the function in
the square brackets in the first row and the exponential function
in the second row.

Lemma 7: The EE in (14) fulfills the following properties
with respect to Ptx and λBS : i) EE (Ptx , λBS ) = 0 if Ptx → 0
or λBS → 0; and ii) EE (Ptx , λBS ) = 0 if Ptx → ∞ or
λBS =→ ∞.
Proof : This immediately follows from Lemmas 3-6.

B. Optimal Transmit Power Given the Density of the BSs
In this section, we analyze whether there exists an optimal
(opt)
and unique transmit power, Ptx , that maximizes the EE formulated in (14), while all the other parameters, including λBS ,

are fixed and given. In mathematical terms, the optimization
problem can be formulated as follows:
EE (Ptx , λBS )
h
i
(min)
(max)
subject to Ptx ∈ Ptx , Ptx
.

maxPtx

(min)

(15)

(max)

where Ptx
≥ 0 and Ptx
≥ 0 are the minimum and
maximum power budget of the BSs, respectively. One may
(max)
(min)
assume, without loss of generality, Ptx
→ 0 and Ptx
→
∞.
The following theorem completely characterizes the solution
of (15).
Theorem 1: Let SP (·) be the function defined in Table II.
The EE in (14) is a unimodal and strictly pseudo-concave function in Ptx . The optimization problem
in (15)nhas a uniqueoo
son
(opt)
(min)
(max)
lution given by Ptx = max Ptx , min P∗tx , Ptx
,
where P∗tx is the only stationary point of the EE in (14) that
is obtained as the unique solution of the following equation:

.

EEPtx (P∗tx , λBS ) = Pidle − SP (P∗tx ) = 0
⇔ SP (P∗tx ) = Pidle .

(16)

10

Proof : See Appendix B.



C. Optimal Density Given the Transmit Power of the BSs
In this section, we analyze whether there exists an optimal
(opt)
and unique density of BSs, λBS , that maximizes the EE formulated in (14), while all the other parameters, including Ptx ,
are fixed and given. In mathematical terms, the optimization
problem can be formulated as follows:
EE (Ptx , λBS )
i
h
(max)
(min)
subject to λBS ∈ λBS , λBS
maxλBS

(min)

(17)

(max)

where λBS
≥ 0 and λBS
≥ 0 are the minimum and
maximum allowed density of the BSs, respectively. One may
(min)
(min)
assume, without loss of generality, λBS → 0 and λBS →
∞.
The following theorem completely characterizes the solution
of (17).
Theorem 2: Let SD (·) be the function defined in Table II.
The EE in (14) is a unimodal and strictly pseudo-concave function in λBS . The optimization problem
in (17)nhas a uniqueoo
son
(opt)
(min)
(max)
lution given by λBS = max λBS , min λ∗BS , λBS
,
∗
where λBS is the only stationary point of the EE in (14) that
is obtained as the unique solution of the following equation:

.

EEλBS (Ptx , λ∗BS ) = SD (λ∗BS ) − Pidle = 0
⇔ SD (λ∗BS ) = Pidle .
Proof : See Appendix C.

(18)


D. On the Dependency of Optimal Transmit Power and Density of the BSs
The optimal transmit power and BSs’ density that maximize
the EE are obtained from the unique solutions of (16) and
(18), respectively. These equations, however, cannot be further
simplified and, therefore, explicit analytical expressions for
(opt)
(opt)
Ptx
and λBS cannot, in general, be obtained. This is
an inevitable situation when dealing with EE optimization
problems, and, indeed, a closed-form expression of the optimal
transmit power for simpler EE optimization problems does not
exist either [1]. In some special cases, the transmit power can
be implicitly expressed in terms of the Lambert-W function,
which, however, is the solution of a transcendental equation
[2]. Notable examples of these case studies include even basic
point-to-point communication systems without interference
[40]. Based on these considerations, it seems hopeless to
attempt finding explicit analytical expressions from (16) and
(18), respectively. However, thanks to the properties of the
EE function, i.e., unimodality and strict pseudo-concavity,
(opt)
(opt)
proved in Theorem 1 and Theorem 2, Ptx and λBS can be
efficiently computed with the aid of numerical methods that
are routinely employed to obtain the roots of non-linear scalar
equations, e.g., the Newton’s method [42]. For example, the
unique solutions of (16) and (18) may be obtained by using the
functions FSolve in Matlab and NSolve in Mathematica.
Theorem 1 and Theorem 2 are, however, of paramount importance, since they state that an optimum maximizer exists and
is unique.

(opt)

Even though explicit analytical formulas for Ptx
and
(opt)
λBS cannot be obtained, it is important to understand how
these optimal values change if any other system parameter
changes. For instance, two worthwhile questions to answer are:
(opt)
“How does Ptx change as a function of λBS ?” and “How
(opt)
does λBS change as a function of Ptx ?”. These questions
are relevant to optimize the deployment of cellular networks
from the EE standpoint, since they unveil the inherent interplay
between transmit power and density of BSs discussed in
Section III and illustrated in Fig. 1. A general answer to these
two questions is provided in the following two propositions.
∗
Proposition 4: Let Ptx be the unique solution of (16) if
λBS = λBS . Let nthe optimal P
to Theorem 1
ntx ∗ accordingoo
(opt)
(min)
(max)
be Ptx
= max Ptx , min Ptx , Ptx
. Let λBS ≶

.

λBS be another BSs’ density. Let EEPtx (·, ·) be the first-order
derivative in (16). The following holds:

 (opt)
(opt)
.
(opt)
(19)
Ptx ⋚ Ptx ⇔ EEPtx Ptx , λBS ⋚ 0.

Proof : Theorem 1 states that the EE function has a single
stationary point that is its unique
global maximizer. In mathe.
∗
(P
matical
terms,
this
implies
EE
tx , λBS ) > 0 if Ptx < Ptx
P
tx
.
∗
and EEPtx (Ptx , λBS ) < 0 if Ptx > Ptx for every λBS ≥ 0.
Therefore, the optimal transmit power needs to be increased
(decreased) if the first-order derivative of the EE is positive
(negative). Based on this, (19) follows because min {·, ·} and
max {·, ·} are increasing functions.

∗
Proposition 5: Let λBS be the unique solution of (18) if
Ptx = Ptx . Letn the optimalnλBS according
ooto Theorem 2 be
(opt)
∗
(min)
(max)
λBS = max λBS , min λBS , λBS
. Let Ptx ≶ Ptx

.

be another transmit power. Let EEλBS (·, ·) be the first-order
derivative in (18). The following holds:


(opt)
.
(opt)
(opt)
⋚ 0.
(20)
λBS ⋚ λBS ⇔ EEλBS Ptx , λBS

Proof : It follows from Theorem 2, similar to the proof of
Proposition 4.

Remark 13: It is worth mentioning that the approach utilized
to prove Proposition 4 and Proposition 5 is applicable to study
(opt)
(opt)
the dependency of Ptx and λBS , respectively, with respect
to any other system parameters. The findings in Proposition 4
and Proposition 5 are especially relevant for cellular network
planning. Let us consider, e.g., (19).
. By simply studying the
sign of the first-order derivative EEPtx (·, ·), one can identify,
with respect to an optimally deployed cellular network, the set
of BSs’ densities that would require to increase or decrease the
transmit power while still operating at the optimum. In Section
(opt)
V, numerical examples are shown to highlight that Ptx may
either decrease or increase as λBS increases or decreases. 

E. Joint Optimization of Transmit Power and Density of the
BSs
In Sections IV-B and IV-C, either λBS or Ptx are assumed
to be given, respectively. In practical applications,
however,


(opt)
(opt)
it is important to identify the optimal pair Ptx , λBS

11

TABLE III
A LTERNATING OPTIMIZATION OF THE EE.

Algorithm h
i
h
i
(min)
(max)
(min)
(max)
; λBS ∈ λBS , λBS
;
Let Ptx ∈ Ptx , Ptx
i
h
(opt)
(max)
(min)
(initial guess); V = 0; ǫ > 0;
Set λBS = λBS ∈ λBS , λBS
Repeat
V0 = V ;


n
n

oo
.
∗
(opt)
(opt)
∗
(min)
(max)
= 0; Ptx = max Ptx , min Ptx , Ptx
Ptx ← EEPtx Ptx , λBS
;
 (opt)
n
n ∗

oo
.
∗
(opt)
(min)
(max)
λBS ← EEλBS Ptx , λBS = 0; λBS = max λBS , min λBS , λBS
;
 (opt) (opt) 
V = EE Ptx , λBS ;
Until |V − V0 | /V ≤ ǫ;
(opt)

Return Ptx

(opt)

= Ptx

(opt)

; λBS

(16)
(18)
(14)

(opt)

= λBS .

that jointly maximizes the EE in (14). This joint optimization
problem can be formulated as follows:

TABLE IV
S ETUP OF PARAMETERS ( UNLESS OTHERWISE STATED ). I T IS WORTH
NOTHING THAT THE SETUP γD = γA CONSTITUTES JUST A CASE STUDY

EE (Ptx , λBS )
i
h
i
h
(max)
(min)
(max)
(min)
Ptx ∈ Ptx , Ptx
, λBS ∈ λBS , λBS
(21)

AND THAT THE MAIN FINDINGS OF THE PRESENT PAPER HOLD TRUE FOR
EVERY γA > 0.

maxPtx ,λBS
subject to

where a notation similar to that used in (15) and (17) is
adopted.
In Theorem 1 and Theorem 2, we have solved the optimization problem formulated in (21) with respect to Ptx for a given
λBS and with respect to λBS for a given Ptx , respectively. By
leveraging these results, a convenient approach for tackling
(21) with respect to Ptx and λBS is to utilize the alternating
optimization method, which iteratively optimizes Ptx for a
given λBS and λBS for a given Ptx until convergence of the
EE in (14) within a desired level of accuracy [41, Proposition
2.7.1]. The algorithm that solves (21) based on the alternating
optimization method is reported in Table III. Its convergence
and optimality properties are summarized as follows.
(opt)

(opt)

Proposition 6: Let Ptx (m), λBS (m), and EE(m) be
Ptx , λBS and EE obtained from the algorithm in Table III at the
mth iteration, respectively. The sequence EE(m) is monotonically increasing and converges. In addition,
 every limit point
(opt)
(opt)
of the sequence Ptx (m) , λBS (m) fulfills the KarushKuhn-Tucker (KKT) first-order optimality conditions of the
problem in (21).
Proof : At the end of each iteration of the algorithm in Table
III, the value of EE does not decrease. The sequence EE(m),
hence, converges, because the EE in (14) is a continuous
function over the compact feasible set of the problem in
(21) and, thus, it admits a finite maximum by virtue of the
Weierstrass extreme value theorem [41]. From [41, Proposition
2.7.1], the alternating optimization method fulfills the KKT
optimality conditions, provided that i) the objective and constraint functions are differentiable, ii) each constraint function
depends on a single variable, and iii) each subproblem has a
unique solution. The first and second requirements follow by
direct inspection of (21). The third requirement is ensured by
Theorem 1 and Theorem 2.


Parameter
β
2
κ = 4πfc /3 · 108
N0
BW
Pcirc
Pidle
Ptx

λBS = 1/ πR2cell BSs/m2
λMT = 1/ πR2MT = 121 MTs/km2
γD = γA

Value
3.5
fc = 2.1 GHz
-174 dBm/Hz
20 MHz
51.14 dBm [8]
48.75 dBm [8]
43 dBm [8]
Rcell = 250 m
RMT = 51.29
5 dB

Remark 14: The optimization problems in Theorem 1 and
Theorem 2 can be efficiently solved by using the Newton’s
method, which allows one to find the root of real-valued objective functions via multiple iterations of increasing accuracy
and at a super-linear (i.e., quadratic if the initial guess is
sufficiently close to the actual root) convergence rate [42].
The properties of convergence of the alternating maximization
algorithm in Table III to a stationary point of the objective
function in (21) are discussed in [41, Proposition 2.7.1]. Under
mild assumptions that hold for the specific problem at hand,
the algorithm in Table III is locally q-linearly convergent to
a local maximizer of the objective function provided that the
initial guess is sufficiently close to the actual root [43, Section
2]. Further details can be found in [43].

In Section V, numerical evidence of the global optimality
of the algorithm in Table III is given as well. In addition,
numerical results on the average (with respect to the initial
guess) number of iterations as a function of the tolerance of
convergence, ǫ > 0, are illustrated.
V. N UMERICAL R ESULTS
In this section, we show numerical results to validate the
proposed analytical framework for computing the PSE and

12

(a)

56

(b)

35

(a)

100
LM-1
LM-2

LM-1
LM-2

(b)

100

P tx = 35 [dBm]

90

90

80

80

70

70

P tx = 45 [dBm]

50

EE [bit/mJoule]

EE(P(opt)
) [bit/mJoule]
tx

P(opt)
[dBm]
tx

52

25

20

15

EE [bit/mJoule]

30
54

60
50
40
30

10

P tx = 45 [dBm]

10
00

0
100

200

300

0

100

Rcell [m]

200

300

0
00

(a)

100

Rcell [m]

70

EE [bit/mJoule]

80

70
60
50
40
30

85

R cell = 75 [m]

30

R cell = 100 [m]

75
25

60
50
40

20

15
30

20

R cell = 75 [m]

10

10

10

20

30

Ptx [dBm]

40

50

5

0

0
0

55
45
35

15
LM-1
LM-2

10

R cell = 100 [m]

0

65

25

20

R cell = 50 [m]

00

(b)

95

R cell = 50 [m]

90

80

(a)

35

P(opt)
[dBm]
tx

90

EE [bit/mJoule]

0 10 20 30 40 50 60 70 80

Rcell [m]

Fig. 4. Energy efficiency versus Rcell for Load Model 1 (a) and Load Model
2 (b). Solid lines: Mathematical Framework from (14). Markers: Monte Carlo
simulations.

(b)

100

00

0
0 10 20 30 40 50 60 70 80

Rcell [m]

Fig. 2. Optimal transmit power (a) and energy efficiency (b) versus Rcell .
Solid lines: Optimum from Theorem 1. Markers: Optimum from a brute-force
search of (15). Special case with β = 6.5 and λMT = 21 MTs/km2 .

40

10

P tx = 55 [dBm]

EE(P(opt)
) [bit/mJoule]
tx

0

50

20

P tx = 35 [dBm]

5

46

60

30

20

48

P tx = 55 [dBm]

0

10

20

30

40

50

0

Ptx [dBm]

0

50

100

150

Rcell [m]

200

250

LM-1
LM-2

5
0

00
0

50

100

150

200

250

00

Rcell [m]

Fig. 3. Energy efficiency versus the transmit power for Load Model 1 (a)
and Load Model 2 (b). Solid lines: Framework from (14). Markers: Monte
Carlo simulations.

Fig. 5. Optimal transmit power (a) and energy efficiency (b) versus Rcell .
Solid lines: Optimum from Theorem 1. Markers: Optimum from a brute-force
search of (15). LM-1: Load Model 1 and LM-2: Load Model 2.

EE, as well as to substantiate the findings originating from
the analysis of the system-level EE optimization problems as
a function of the transmit power and density of the BSs. Unless
otherwise stated, the simulation setup is summarized in Table
IV. For ease of understanding, the BSs’ density is represented
via the inter-site distance (Rcell ) defined in Table IV. A similar
comment applies to the density of the MTs that is expressed
in terms of their average distance (RMT ). As far as the choice
of the setup of parameters is concerned, it is worth mentioning
that the power consumption model is in agreement with [8]
and [30]. The density of the MTs coincides with the average
density of inhabitants in France.
a) Validation Against Monte Carlo Simulations: In Figs.
3 and 4, we validate the correctness of (14) against Monte
Carlo simulations. Monte Carlo results are obtained by simulating several realizations, according to the PPP model, of
the cellular network and by empirically computing the PSE

according to its definition in (6) and (7), as well as the power
consumption based on the operating principle described in
the proofs of Proposition 2 and Proposition 3. It is worth
mentioning that, to estimate the PSE, only the definitions in
the first line of (6) and (7) are used. The results depicted
in Figs. 3 and 4 confirm the good accuracy of the proposed
mathematical approach. They highlight, in addition, the unimodal and pseudo-concave shape of the EE as a function
of the transmit power, given the BSs’ density, and of the
BSs’ density, given the transmit power. If the same transmit
power and BSs’ density are assumed for both load models,
we observe, as expected, that the first load model provides a
better EE than the second load model.
b) Validation of Theorem 1 and Theorem 2: In Figs.
5 and 6, we compare the optimal transmit power and BSs’
density obtained from Theorem 1 and Theorem 2, i.e., by
computing the unique zero of (16) and (18), respectively,

13

20

90
85

120
20

60

P(opt)
[dBm]
tx

EE(R (opt)
) [bit/mJoule]
cell

30

LM-1
LM-2

140

50
40
30

100
18

16

80

60

14

20

0

20

40

10

60

Ptx [dBm]

0

20

40

60

80

00
00

(a)

(b)

250

LM-1
LM-2

50

100

150

RMT [m]

200

0
50

10

15

20

45
0

5

10

15

20

0

γD = γA [dB]

5

10

15

20

γD = γA [dB]

LM-1
LM-2

85
80

[bit/Joule]

85

80

EE

75
0
1
0
0
-1
-1
-1

70

100

150

RMT [m]

200

75
70

(opt)

EE(P(opt)
,R(opt)
) [bit/mJoule]
tx
cell

R(opt)
[m]
cell

P(opt)
[dBm]
tx

10
50

5

LM-1
LM-2

90

100

15

60

90

LM-1
LM-2

150

20

65

Fig. 8. Optimal transmit power (a), density of BSs (Rcell ) (b), and energy
efficiency (c) versus the reliability thresholds (γD = γA ). Solid lines:
Optimum from the algorithm in Table III. Markers: Optimum from a bruteforce search of (21). LM-1: Load Model 1 and LM-2: Load Model 2.

(c)

95

200

25

20
0

γD = γA [dB]

LM-1
LM-2

30

10

Ptx [dBm]

Fig. 6. Optimal density of BSs (Rcell ) (a) and energy efficiency (b) versus
the transmit power. Solid lines: Optimum from Theorem 2. Markers: Optimum
from a brute-force search of (17). LM-1: Load Model 1, LM-2: Load Model
2.

35

70

50

LM-1
LM-2

0
-40 -20

80

75

40

12

0
-40 -20

80

55

10
LM-1
LM-2

1
00
00
-1 0

(c)

95

22

70

40

(b)

150
LM-1
LM-2

80

50

(a)

24

EE(P(opt)
,R(opt)
) [bit/mJoule]
tx
cell

60

R(opt)
[m]
cell

(b)

95
90

R(opt)
[m]
cell

(a)

70

65
50

100

150

200

RMT [m]

Fig. 7. Optimal transmit power (a), density of BSs (Rcell ) (b), and energy
efficiency (c) versus the density of MTs (RMT ). Solid lines: Optimum from
the algorithm in Table III. Markers: Optimum from a brute-force search of
(21). LM-1: Load Model 1 and LM-2: Load Model 2.

against a brute-force search of the optimum of (15) and (17),
respectively. We observe the correctness of Theorem 1 and
Theorem 2 for the load models analyzed in the present paper.
Figures 5 and 6, in addition, confirm two important remarks
that we have made throughout this paper. The first is that a
joint pair of transmit power and BSs’ density exists. This is
highlighted by the fact that the EE evaluated at the optimal
transmit power, given the BSs’ density, and at the optimal
BSs’ density, given the transmit power, is still a unimodal
and pseudo-concave function. This motivates one to use the
alternating optimization algorithm proposed in Section IV-E.
The second is related to the difficulty of obtaining an explicit
closed-form expression of the optimal transmit power as a
function of the BSs’ density and of the BSs’ density as a
function of the transmit power. Figure 6(a), for example,

65
60
55
50
0

0.5

1

1.5

2

PSE (opt) [kbit/sec/m2]

2.5

3

0
0

Fig. 9. Analysis of the EE vs. PSE trade-off. Solid lines: Optimum from the
algorithm in Table III. Markers: Optimum from a brute-force search of (21).
LM-1: Load Model 1 and LM-2: Load Model 2.

clearly shows that the behavior of the optimal transmit power
is not monotonic as a function of the BSs’ density. This
is in contrast with heuristic optimization criteria based on
the coverage probability metric [32]. Figure 5(a), on the
other hand, provides more intuitive trends according to which
the optimal transmit power increases as the density of the
BSs decreases. This is, however, just a special case that is
parameter-dependent. A counter-example is, in fact, illustrated
in Fig. 2, where, for a different set of parameters, it is shown
that the optimal transmit power may increase, decrease and
then increase again as a function of the average inter-site
distance of the BSs (Rcell). In this case, the density of the MTs
coincides with the average density of inhabitants in Sweden
and a large path-loss exponent is assumed to highlight the
peculiar performance trend. These numerical examples clearly
substantiate the importance of Theorem 1 and Theorem 2, and

14

16

(a)

(b)

4

γD = γA = 0 dB

γD = γA = 0 dB
γD = γA = 20 dB

Average Number of Iterations

Average Number of Iterations

3.8

γD = γA = 10 dB

14

12

10

8

6

γD = γA = 10 dB
γD = γA = 20 dB

3.6
3.4
3.2
3
2.8
2.6
2.4

4
2.2
2
10-10

10-5

100

2
10-10

ǫ

00
10-5

100

00

ǫ

Fig. 10. Number of iterations of the algorithm in Table III as a function of
ǫ > 0. The number of iterations
is averaged
(15000 trials) over the initial
h
i
(opt)
(min)
(max)
guess λBS = λBS ∈ λBS , λBS
. (a) Load Model 1 and (b) Load
(min)

Model 2. Setup: Rcell
(max)
Ptx
= 60 dBm.

(max)

= 10 m, Rcell

(min)

= 2000 m, Ptx

= −20 dBm,

highlight the complexity of the optimization problem that is
analyzed and successfully solved in the present paper.
c) Validation of the Alternating Optimization Algorithm
in Table III: In Figs. 7 and 8, we provide numerical evidence
of the convergence of the alternating optimization algorithm
introduced in Section IV-E towards the global optimum of
the optimization problem formulated in (21). The study is
performed by computing the joint optimal transmit power and
BSs’ density as a function of the density of the MTs (Fig. 7)
and of the reliability thresholds (Fig. 8). We observe a very
good agreement between the algorithm in Table III and a bruteforce search of the optimum of (21). Similar studies have been
conducted as a function of other system parameters, but they
are not reported in the present paper due to space limitations.
d) Comparison Between Load Model 1 and 2: With the
exception of Figs. 3 and 4, all the figures reported in this
section illustrate the achievable EE of the two load models
analyzed in the present manuscript when they operate at
their respective optima. Based on the obtained results, we
conclude that, for the considered system setup, the first load
model outperforms the second one in terms of EE. Figures
7 and 8 show, for example, that this may be obtained by
transmitting a higher power but, at the same time, by reducing
the deployment density of the BSs. It is worth mentioning
that, even though both load models provide the same PSE and
serve, in the long time-horizon, all the MTs of the network,
they have one main difference: the MTs under the first load
model experience a higher latency (i.e., the MTs experience
a longer delay before being served, since they are randomly
chosen among all the available MTs in the cell), since a single
MT is served at any time instance. We evince, as a result, that
the higher EE provided by the first load model is obtained
at the price of increasing the MTs’ latency. The analysis and
optimization of energy-efficient cellular networks with latency
constraints is, therefore, an important generalization of the

study conducted in the present paper.
e) Analysis of the EE vs. PSE Trade-Off: In Fig. 9,
we illustrate the trade-off between EE and PSE, which is
obtained by setting the transmit power and density of the
BSs at the optimal values that are obtained by solving the
optimization problem in (21) with the aid of the algorithm in
Table III. Figure 9 provides a different view of the comparison
between Load Model 1 and 2 introduced in Section II-D.
The Load Model 1 is a suitable choice to obtain a high
EE at low-medium PSEs, while the Load Model 2 is a
more convenient option torequired to converge within obtain
a good EE at medium-high PSEs. Based on these results,
the optimization of the EE vs. PSE trade-off constitutes an
interesting generalization of the study carried out in the present
paper.
f) Convergence Analysis of the Maximization Algorithm
in Table III: Motivated by Remark 14, Fig. 10 shows the
average number of iterations of the alternating optimization
algorithm in Table III as a function of the convergence
accuracy ǫ. We observe that the algorithm necessitates more
iterations for Load Model 1. In general, however, we observe
that the number of iterations that are required to converge
within the defined convergence accuracy is relatively small.
VI. C ONCLUSION
In the present paper, we have introduced a new closedform analytical expression of the potential spectral efficiency
of cellular networks. Unlike currently available analytical
frameworks, we have shown that the proposed approach allows
us to account for the tight interplay between transmit power
and density of the base stations in cellular networks. Therefore,
the proposed approach is conveniently formulated for the
optimization of the network planning of cellular networks,
by taking into account important system parameters. We have
applied the new approach to the analysis and optimization of
the energy efficiency of cellular networks. We have mathematically proved that the proposed closed-form expression of the
energy efficiency is a unimodal and strictly pseudo-concave
function in the transmit power, given the density, and in the
density, given the transmit power of the base stations. Under
these assumptions, as a result, a unique transmit power and
density of the base stations exist, which can be obtained by
finding the unique zero of a simple non-linear function that
is provided in a closed-form expression. All mathematical
derivations and findings have been substantiated with the
aid of numerical simulations. We argue that the applications
of the proposed approach to the system-level modeling and
optimization of cellular networks are countless and go beyond
the formulation of energy efficiency problems.
Extensions and generalizations of the analytical and optimization frameworks proposed in the present paper, include,
but are not limited to, the system-level analysis and optimization of i) the energy efficiency versus spectral efficiency
trade-off, ii) uplink cellular networks, iii) three-dimensional
network topologies with elevated base stations and spatial
blockages, iv) cache-enabled cellular networks, v) cellular
networks with network slicing, vi) cellular networks with

15

renewable energy sources and energy harvesting, and vii)
multi-tier (heterogeneous) cellular networks.

A PPENDIX A
P ROOF OF P ROPOSITION 1
Under the assumption that MT0 is selected, from (3) and
(4), we have:
)
(


2
g0 /L0 1 L0 ≤ Ptx γA σN
≥ γD
Pcov (γD , γA ) = Pr P
(I) gi /Li 1 (Li > L0 )
BSi ∈Ψ
BS

2
Ptx /(γA σN
)

=

Z

Pr

0

(

|

)
g0 /x
P
≥ γD fL0 (x) dx
(I) g /Li 1 (Li > x)
BSi ∈ΨBS i
{z
}
G(γD ;x)

(22)

−1 2/β−1 −πλ (x/κ)2/β
BS
x
e
is
where fL0 (x) = 2πλBS κ2/β β
the probability density function of L0 that is obtained by
applying the displacement theorem of PPPs [5, Eq. (21)]. It
is worth mentioning that (22) is exact if the Crofton cell is
considered, while it is an approximation if the typical cell is
considered (see Remark 7 for further details).
The probability term, G (·; ·), in the integrand function of
(22) can be computed as follows:
!
−1
Z +∞ 
2/β−1
(a)
y
(tx) y
2πλBS 2/β dy
1+
G (γD ; x) = exp −
xγD
κ β
x


(b)
2/β
= exp −πλ(tx)
Υ
BS (x/κ)
(23)
where (a) follows from the probability generating functional
theorem of PPPs [3] by taking into account that, based on
(8), the interfering BSs constitute a PPP of intensity equal
(tx)
(tx)
to λBS = λBS PBS = λBS L (λMT /λBS ), and (b) follows by
(tx)
solving the integral. The intensity of the interfering PPP, λBS ,
is obtained by taking into account that only that BSs that are
in transmission mode contribute to the inter-cell interference.
(tx)
The analytical expression of λBS is, in particular, obtained
with the aid of the independent thinning theorem of PPPs,
similar to [5] and [38]. The impact of the spatial correlation
that exists among the BSs that operate in transmission mode
[39], is, on the other hand, postponed to future research.
By inserting (23) in (22) and by applying some changes of
variable, we obtain:
Pcov (γD , γA ) = πλBS κ−2/β
2/β

2
γA σN
))
(Ptx /(Z

×

0



exp −πλBS κ−2/β (1 + ΥL (λMT /λBS )) z dz.

(24)

The proof follows from (6) and (7)
the aid of some
 and by
P+∞ simplifications
−1
the
identity
Pr N̄MT = u
u=0 (u + 1)
−1
(λMT /λBS ) L (λMT /λBS ) [33, Proposition 2].

with
using
=

A PPENDIX B
P ROOF OF T HEOREM 1

In this section, we are interested in the functions that
depend on Ptx . For ease of writing, we adopt the simplified
notation:. Ptx → P, L (·) →. L, M (·) → M, Q (·, Ptx , ·) →
Q (P), QPtx (·, Ptx , ·) → Q (P),
. = Pi ,
. Pcirc = Pc , Pidle
EE (Ptx , ·) → EE (P), and EEPtx (Ptx , ·) → EE (P). A
similar notation is adopted for higher-order derivatives with
respect to P.
The stationary points of (14) are the zeros of the first-order
derivative
of EE (·) with respect to P. From (14), we obtain
.
EE (P) = 0 ⇔ Pi − SP (P) = 0, which can be re-written as
follows:
..
(25)
Q (P) Q (P) − P = ∆P + Pi /L + Pc M/L .
{z
}
{z
} |
|
Wleft (P)

Wright

With the aid of some algebraic manipulations and by
exploiting Lemmas 3-6, the following holds: i) Wright ≥ 0
is a non-negative function that is independent of P, ii)
Wleft (P). ≥ 0 is a non-negative and increasing
.. function of
P, i.e., W left (P) ≥ 0, since Q (P) ≥ 0 and Q (P) ≤ 0 from
Lemma 5, iii) Wleft (P → 0) = 0 and Wleft (P → ∞) = ∞.
This implies that Wleft (·) and Wright intersect each other
∗
in just one point.
. Therefore, a unique∗ stationary
. point, P ,
exists. Also, EE (P) > 0 for P < P and EE (P) < 0 for
P > P∗ . Finally, by taking into account the constraints on the
transmit power, it follows 
that the unique
 optimal maximizer
of the EE is P(opt) = max P(min) , min P∗ , P(max) , since
P ∈ P(min) , P(max) . This concludes the proof.

A PPENDIX C
P ROOF OF T HEOREM 2

In this section, we are interested in the functions that
depend on λBS . For ease of writing, we adopt the simplified
notation: λBS → λ, L (·/λBS ) → .L (λ), M (·/λBS ) →
M
. (λ), Q (λBS , ·, ·/λBS ) → Q (λ), QλBS (λBS , ·, ·/λBS ) →
Q. (λ), Pcirc = Pc., Pidle = Pi , EE (·, λBS ) → EE (λ),
EEλBS (·, λBS ) → EE (λ), Ptx → P. Similar notation applies
to higher-order derivatives.
The proof is split in two parts: i) λMT /λ ≥ 2.8 and ii)
λMT /λ ≤ 2.8. This is necessary because, from Lemma 3,
L (·) is concave in λ if λMT /λ ≥ 2.8 and convex in λ if
λMT /λ ≤ 2.8.
a) Case Study λMT /λ ≥ 2.8: The stationary points of
(14) are the zeros of the first-order derivative
of EE (·) with
.
respect to λ. From (14), we obtain EE (λ) = 0 ⇔ SD (λ) −
Pi = 0. This stationary equation can be re-written as follows

16

P5
(Wright (λ) = ℓ=1 Wℓ (λ)):

..
 .
.

Pi = − L (λ) L (λ) Q (λ) Q (λ)
|{z} |
{z
}
W
left

W1 (λ)

× [1 + ΥL (λ)] [L (λ) (P + ∆P) + Pi + Pc M (λ)]
{z
}
|

+ Pcirc
|



.

W2 (λ)

..

M (λ) L (λ) L (λ) − M (λ)
{z
}
W3 (λ)

..
.
+ ΥL (λ) (P + ∆P) + ΥPc L2 (λ) M (λ) L (λ) .
{z
} |
|
{z
}
2

W4 (λ)

W5 (λ)

(26)

With the aid of some algebraic manipulations and by
exploiting Lemmas 3-6, the following holds: i) Wleft ≥ 0 is a
non-negative function that is independent of λ, ii) Wright (λ) ≥
0 is a non-negative function of λ, since Wℓ (λ) ≥ 0 for
ℓ = 1, . . . , 5 if λMT /λ ≥ 2.8. In particular, W3 (λ) ≥ 0 if
λMT /λ ≥ 1.4 and Wℓ (λ) ≥ 0 for λ ≥ 0 if ℓ = 1, 2, 4, 5, iii)
Wright (λ → 0) = ∞ and Wright (λ → ∞) = 0. This implies
that Wleft and Wright (·) would intersect each other in just
a. single point if Wright is a decreasing function in λ, i.e.,
W right (λ) ≤ 0 for λMT /λ ≥ 2.8. A sufficient condition for
this to hold is that W.ℓ (·) for ℓ = 1, . . . , 5 are decreasing
functions in λ, i.e., Wℓ (λ) ≤ 0 for λMT /λ ≥. 2.8. This
holds to be true and. can be proved as follows. W2 (λ) ≤ 0
for λ ≥ 0 and W4 (λ) ≤ 0 for λ ≥ 0 because L (·)
and M (·) are. decreasing functions in λ (see. Lemma 3 and
Lemma 4). W3 (λ) ≤ 0 for λ ≥ 0 and W5 (λ) ≤ 0 for
λ ≥ 0 immediately follow by inserting into them the firstorder derivatives of L (·) and M (·) with respect to λ and with
the aid of simple algebraic manipulations. Less evident is the
behavior of W1 (·) as a function of λ. Using some algebra,
the first-order derivative satisfies the following:

.
. 2
.
..
W 1 (λ) Q (λ) L (λ) = −L (λ) L (λ) Q (λ) Q (λ)
{z
}
|
A1 (λ)

 .2
. 
+ −L (λ) Q (λ) Q (λ)
|
{z
}

.

A2 (λ)
2

.

+ L (λ) L (λ) Q (λ)
|
{z
}
A (λ)

.. 3

(27)

.

+ L (λ) L (λ) Q (λ) Q (λ) .
{z
}
|
A4 (λ)

A sufficient condition for W1 (·) to be a decreasing function
in λ is that Aℓ (λ) ≤ 0 for ℓ = 1, . . . , 4. From Lemmas 3-6,
this can be readily proved. In particular, Aℓ (λ) ≤ 0 for λ ≥ 0
if ℓ = 1, 2, 3 and A4 (λ) ≤ 0 for λMT /λ ≥ .2.8. Therefore,
∗
a unique stationary
EE (λ) > 0 for
. point, λ , exists. Also,
∗
λ < λ and EE (λ) < 0 for λ > λ∗ . Finally, by taking
into account the constraints on the density of BSs, it follows
that 
the unique optimal
maximizer of the EE is λ(opt) =
 ∗ (max)
(min)
max λ
, min λ , λ
, since λ ∈ λ(min) , λ(max) .

b) Case Study λMT /λ ≤ 2.8: As for this case study,
we leverage a notable result in fractional optimization [2]:
the ratio between a i) non-negative, differentiable and concave
function, and a ii) positive, differentiable and convex function
is a pseudo-concave function. It is, in addition, a unimodal
function with a finite maximizer if the ratio vanishes when
the variable of interest (i.e., the BSs’ density) tends to zero
and to infinity. As for the case study under analysis, the EE
in (14) can be re-written, by neglecting unnecessary constants
that are independent of λ and do not affect the properties of
the function, as follows:
EE (λ) =
Q (λ)
.
[1 + ΥL (λ)] [(P + ∆P) + Pi /L (λ) + Pc M (λ)/L (λ)]
(28)
From Lemma 6, the numerator of (28) is a non-negative,
differentiable, increasing and concave function for λ ≥ 0.
From Lemma 7, the EE in (28) tends to zero if λ → 0
and λ → ∞. Therefore, a sufficient condition to prove the
unimodality and pseudo-concavity of the EE is to show that
the denominator of (28) is a positive, differentiable and convex
function in λ for λMT /λ ≤ 2.8. From Lemma 3 and Lemma 4,
the first two properties are immediately verified. To complete
the proof, the convexity of the denominator of (28) needs to
be analyzed.
Let Den (·) be the denominator
of (28). Let us introduce the
.2 .
..
function K (λ) = 2L (λ) L (λ) − L (λ). The second-order
derivative of Den (·), as a function of λ, is as follows:

..

..

..

Den (λ) = Υ(P + ∆P)L (λ) + ΥPc M (λ)
{z
} | {z }
|
D1 (λ)

D2 (λ)


. 
+ Pc L (λ) 2λMT λ
L (λ) + λL (λ)
|
{z
}|
{z
}






2

D3 (λ)





3

D4 (λ)



+ Pc M (λ) L (λ) K (λ) + Pc L (λ) K (λ)
|
|
{z
}
{z
}


D5 (λ)

2

2

D6 (λ)

+ Pi L (λ) K (λ) .
|
{z
}
D7 (λ)

(29)

A sufficient condition for proving that Den (·) is a convex
function in λ is to show that Dℓ (λ) ≥ 0 for ℓ = 1, 2, . . . , 7
and K (λ) ≥ 0 if λMT /λ ≤ 2.8. This can be proved as
follows. D1 (λ) ≥ 0 for λMT /λ ≤ 2.8 follows from Lemma
3. Dℓ (λ) ≥ 0 for ℓ = 2, 5 if λ ≥ 0 follows from Lemma 4.
Dℓ (λ) ≥ 0 for ℓ = 3, 6, 7 if λ ≥ 0 follows from Lemma 3.
D4 (·) and K (·) require deeper analysis. Define ξ = λMT /λ.
D4 (·) and K (·) are positive functions in ξ if:
D4 (ξ) ≥ 0 ⇔
D4 (ξ) = 1 − (1 + ξ/α)−α − x(1 + ξ/α)−(α+1) ≥ 0

(30)

K (ξ) ≥ 0 ⇔
−α

K (ξ) = (1 + ξ/α)

(31)
−1

+ [2 + (1 + 1/α) x] [2 − (1 − 1/α) x]

≥ 1.

17

By direct inspection of (30) and (31), it is .not difficult to
prove the following: i) D4 (ξ → 0) =. 0 and D 4 (ξ) ≥ 0 for
ξ ≥ 0, and ii) K (ξ → 0) = 1 and K (ξ) ≥ 0 for ξ ≤ 2.8.
These two conditions imply D4 (λ) ≥ 0 for λ ≥ 0 and
K (λ) ≥ 0 for λMT /λ ≤ 2.8. This concludes the proof.
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