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Optimization of Wireless Powered Communication
Networks with Heterogeneous Nodes
Mohamed A. Abd-Elmagid∗, Tamer ElBatt∗†, and Karim G. Seddik‡
∗Wireless Intelligent Networks Center (WINC), Nile University, Giza, Egypt.
†Dept. of EECE, Faculty of Engineering, Cairo University, Giza, Egypt.
‡Electronics and Communications Engineering Department, American University in Cairo, AUC Avenue, New Cairo 11835, Egypt.
email: m.abdelaziz@nu.edu.eg, telbatt@ieee.org, kseddik@aucegypt.edu
Abstract—This paper studies optimal resource allocation in
a wireless powered communication network with two groups
of users; one is assumed to have radio frequency (RF) energy
harvesting capability and no other energy sources, while the
other group has legacy nodes that are assumed not to have RF
energy harvesting capability and are equipped with dedicated
energy supplies. First, the base-station (BS) with a constant
power supply broadcasts an energizing signal over the downlink.
Afterwards, all users transmit their data independently on the
uplink using time division multiple access (TDMA). We propose
two transmission schemes, namely OPIC and OPAC, subject
to different energy constraints on the system. Within each
scheme, we formulate two optimization problems with different
objective functions, namely maximizing the sum throughput and
maximizing the minimum throughput, for enhanced fairness.
We establish the convexity of all formulated problems which
opens room for efficient solution using standard techniques.
Our numerical results show the superiority of our realistic
system accommodating legacy nodes, along with RF harvesting
nodes, compared to the baseline WPCN system with RF energy
harvesting nodes only. Moreover, the results reveal new insights
and throughput-fairness trade-offs unique to our new problem
setting.
Index Terms—cellular networks, convex optimization, green
communications, RF energy harvesting.
I. INTRODUCTION
RF energy harvesting has recently emerged as a promising
solution to efficiently prolong the limited lifetime of energy-
constrained wireless networks. This is due to the fact that
RF energy harvesting allows wireless devices to continu-
ously harvest energy from the surrounding radio environment.
Since RF signals can carry energy and information at the
same time, a dynamic simultaneous wireless information and
power transfer scheme called SWIPT is proposed in [1]–[3].
The fundamental trade off between transmitting energy and
transmitting information over a point-to-point noisy link is
studied in [1]. In [2], the authors introduced dynamic power
splitting as a general SWIPT receiver operation and proposed
two practical SWIPT receiver architectures, namely, separated
and integrated information and energy receivers. In addition,
This work was made possible by grants number NPRP 5-782-2-322 and
NPRP 4-1034-2-385 from the Qatar National Research Fund (a member of
Qatar Foundation). The statements made herein are solely the responsibility
of the authors.
[3] studied and proposed SWIPT for orthogonal frequency
division multiplexing (OFDM) systems.
On the other hand, wireless powered communication net-
works (WPCNs), a newly emerging type of wireless net-
works, have recently attracted considerable attention [4]–[6].
In WPCN, users first harvest RF energy on the downlink from
wireless energy signals broadcast by a hybrid access point
(HAP). Afterwards, users transmit their information signals to
the HAP on the uplink using harvested energy in the downlink
phase, e.g., using TDMA in [4]. In addition, [5] introduced
user cooperation in WPCN as a solution for the doubly near-
far phenomenon that results in unfair rate allocation among
users as observed in [4]. Moreover, a full-duplex WPCN
scheme with energy causality has been proposed in [6], in
which all users can continuously harvest wireless power from
the HAP till its uplink information transmission slot. In this
paper, we study WPCNs with two types of nodes, with and
without RF energy harvesting capability in efficient way of
utilizing the nodes without RF energy harvesting to enhance
the network performance, even beyond the baseline WPCN
with all energy harvesting nodes [4]. This constitutes an
important step towards more realistic future wireless networks
as RF energy harvesting technology gradually penetrates the
wireless industry.
Our main contribution in this paper is three-fold. First, we
introduce a generalized problem setting for WPCN, in which
the network includes two types of nodes, with/without RF
energy harvesting capability, of which the problem setting
studied in [4] falls as a special case. Second, we develop
two optimal resource allocation schemes for the considered
WPCNs which are different in the model of how each node
without energy harvesting capability exploits its energy supply
for uplink information transmissions. Furthermore, we formu-
late two optimization problems for investigating the maximum
sum throughput and the maxmin throughput for each proposed
optimal resource allocation scheme. Third, we establish the
convexity of the formulated problems and characterize the
optimal solution in closed form for one of them and solve
the other problems efficiently using known convex problems
solvers. Our numerical results demonstrate the fundamental
trade off between achieving maximum sum throughput and
achieving fair rate allocation among different users and reveal
interesting insights about the formulated problems.
Fig. 1. WPCN with two types of nodes.
The rest of the paper is organized as follows. The system
model is presented in Section II. In Section III, the maximum
sum throughput and the maxmin throughput optimization
problems are formulated and convexity is established for OPIC
scheme. Section IV studies the maximum sum throughput
and the maxmin throughput optimization problems for OPAC
scheme. Numerical results are presented in Section V. Finally,
Section VI concludes the paper.
II. SYSTEM MODEL
We study a WPCN with two types of nodes, as shown
in Fig. 1, namely MRF energy harvesting (Type I) nodes
and Nlegacy (non-harvesting) nodes (Type II). It is assumed
that the BS and all users in both groups are equipped with a
single antenna each, operate over the same frequency channel
and the radios are half-duplex. Each user of Type I, denoted
by U1,i for i=1,··· ,M, is assumed to have no battery or
other energy supply and, hence, needs to first harvest energy
from the RF energy collected from the BS broadcast on the
downlink, primarily to energize Type I nodes. The harvested
RF energy can be stored in a rechargeable battery and then
used for data transmission on the uplink, e.g., [4]. On the other
hand, legacy Type II nodes, denoted by U2,j for j=1,··· ,N,
are not equipped with RF energy harvesting circuitry and,
hence, are assumed to have dedicated energy supplies for their
communication needs on the uplink.
The network operates in a TDMA fashion. For convenience,
we assume the block (slot) duration is normalized to 1. At
the first τ0∈[0,1] fraction of time, the BS broadcasts an
energizing signal over the downlink so that each U1,i could
harvest a certain amount of energy and charge its battery.
The remaining 1−τ0fraction of time is allocated to uplink
data transmissions where U1,i and U2,j are assigned certain
portion of time denoted by τ1,i and τ2,j , respectively, for
i=1,··· ,M and j=1,··· ,N. Hence, the slot is split
as follows:
τ0+
M
i=1
τ1,i +
N
j=1
τ2,j ≤1.(1)
The downlink channel coefficient from the BS to U1,i,
the uplink channel coefficient from U1,i to the BS and the
uplink channel coefficient from U2,j to the BS are denoted
by complex random variables h
1,i,g
1,i and g
2,j , respectively,
with channel power gains h1,i =|h
1,i|2,g1,i =|g
1,i|2and
g2,j =|g
2,j |2. In addition, it is assumed that all downlink and
uplink channels are quasi-static flat fading, i.e., all downlink
and uplink channels remain constant over a transmission block,
that is a time slot, but can change independently from one
block to another. The BS has perfect knowledge of all channel
coefficients at the beginning of each block. The transmitted
energy signal from the BS to all users, over the downlink, is
denoted by xBwith PBaverage power, i.e., E|xB|2=PB.
Hence, the energy harvested by an arbitrary Type I node, U1,i
in the downlink phase is given by
E1,i =ηiPBh1,iτ0,(2)
where ηi∈(0,1) is the efficiency of the RF energy har-
vesting circuitry [7], [8], at U1,i. The data signal transmit-
ted by U1,i and U2,j in the uplink phase are denoted by
x1,i ∼CN(0,P
1,i)and x2,j ∼CN (0,P
2,j ), respectively,
where CN μ, σ2stands for a circularly symmetric complex
Gaussian random variable with mean μand variance σ2.
Assuming that all the energy harvested at U1,i is used for
uplink information transmission, then the transmitted power
limits by U1,i and U2,j on the uplink are given, respectively,
by
P1,i =E1,i
τ1,i
,i=1,··· ,M, (3)
P2,j =E2,j
τ2,j
,j =1,··· ,N, (4)
where E2,j is the energy drawn by U2,j from its dedicated
energy supply (battery) within its assigned τ2,j fraction of time
for information transmission. Therefore, the received signal
at the BS in the uplink phase during τ1,i and τ2,i can be
expressed, respectively, by
y1,i =g
1,ix1,i +n1,i ,i =1,··· ,M, (5)
y2,j =g
2,j x2,j +n2,j ,j =1,··· ,N, (6)
where n1,i ∼CN(0,σ
2)and n2,j ∼CN(0,σ
2)denote the
noise at the BS within τ1,i and τ2,j , respectively. From (2)-
(6), the achievable uplink throughput of U1,i and U2,j in
bits/second/Hz is given by
R1,i (τ0,τ
1,i)=τ1,i log21+ g1,iP1,i
Γσ2
=τ1,i log21+γi
τ0
τ1,i ,
(7)
R2,j (E2,j ,τ
2,j )=τ2,j log21+ g2,j P2,j
Γσ2
=τ2,j log21+θj
E2,j
τ2,j ,
(8)
respectively, where γi=ηih1,ig1,i PB
Γσ2,θj=g2,j
Γσ2,fori=
1,··· ,M,j=1,··· ,N and Γdenotes the signal to noise
ratio gap.
In this paper, we propose two optimal resource allocation
schemes for WPCN systems with two types of nodes, namely,
RF energy harvesting and legacy (battery-powered) nodes,
which are different in the way Type II (legacy) nodes exploit
their dedicated energy supplies. Under the first formulation,
called Optimal Policy under per slot (Instantaneous) energy
Constraint (OPIC), the consumed energy per slot by each
legacy node (E2,j) is optimized subject to maximum allowable
energy consumption per slot, denoted Emax. Under the second
formulation, called Optimal Policy under Average energy
Constraint (OPAC), we relax the strong “per slot energy
requirement” of OPIC. In OPAC, the energy consumption of
Type II nodes is limited to a pre-specified fixed value per slot,
denoted by ¯
E, in an attempt to limit the overall system energy
consumption.
III. OPTIMAL POLICY UNDER INSTANTANEOUS ENERGY
CONSTRAINT (OPIC)
In this section, we formulate two optimization problems for
WPCN with two types of nodes and establish their convexity
which facilitates efficient solution using standard techniques.
First, we target maximizing sum system throughput (prob-
lem P1). Second, motivated by the fundamental fairness-
throughput trade-off, we cast the problem into a maxmin
formulation (problem P2).
A. Sum Throughput Maximization
The motivation behind introducing this scheme is to char-
acterize the maximum sum throughput for a WPCN with
two types of nodes compared to the performance of the
baseline WPCN with only energy harvesting nodes introduced
in [4]. Towards this objective, we find the optimal transmission
durations (τ0for harvesting, as well as Type I and Type II
nodes) and the optimal consumed energy by each Type II
node per slot (E2,j ) that maximize the system sum throughput
subject to a constraint on the total energy consumption per slot
(Instantaneous), denoted by Emax, and the total transmission
block time constraint. For fair comparison with [4], we take
Emax to be the total per slot energy consumption of [4] under
similar conditions, i.e. number of nodes of each type, channel
gains, etc. Having the total per slot energy consumption
available for each slot is a strong assumption that we relax in
the next section. Nevertheless, we adopt it in this formulation
in order to assess the best performance achievable by WPCN
with two types of nodes under the same amount of resources
available to the reference system in [4]. Therefore, from (7)
and (8), the problem of sum throughput maximization can be
formulated as
P1 :
max
τ,E2
M
i=1
τ1,i log21+γi
τ0
τ1,i +
N
j=1
τ2,j log21+θj
E2,j
τ2,j
s.t. τ0+
M
i=1
τ1,i +
N
j=1
τ2,j ≤1,
M
i=1
ηiPBh1,iτ0+
N
j=1
E2,j ≤Emax,
τ≥0,
E2≥0,
(9)
where τ=[τ0,τ
1,1,··· ,τ
1,M ,τ
2,1,··· ,τ
2,N ]and
E2=[E2,1,E
2,2,··· ,E
2,N ].
Theorem 1. P1 is a convex optimization problem.
Proof: τ1,i log21+γi
τ0
τ1,i is the perspective func-
tion of the concave function log2(1 + γiτ0)which pre-
serves the concavity of R1,i with respect to (τ0,τ
1,i).Also,
τ2,j log21+θj
E2,j
τ2,j is the perspective function of the
concave function log2(1 + θjE2,j)which preserves the con-
cavity of R2,j with respect to (τ2,j,E
2,j ). Since the non-
negative weighted sum of concave functions is also concave
[9], then the objective function of P1, which is the non-
negative weighted summation of concave functions, i.e., R1,i
and R2,j for i=1,··· ,M and j=1,··· ,N, is a concave
function in (τ,E2). In addition, all constraints of P1 are affine
in (τ,E2). This establishes the proof.
Given the sum throughput maximization objective in P1,
OPIC allocates more energy to the nodes with better channel
power gains and, hence, more uplink transmission time. This,
in turn, leads to unfair rate allocation among different users
as will be shown in Fig. 3 in the simulation section.
B. Maxmin Fairness Formulation
Motivated by the fairness limitations of P1, we formulate a
second optimization problem targeting the fairness in the well-
known maxmin sense [10] subject to the same constraints of
P1 as follows.
P2 :max
τ,E2
min
i,j (R1,i (τ0,τ
1,i),R
2,j (E2,j ,τ
2,j ))
s.t. τ0+
M
i=1
τ1,i +
N
j=1
τ2,j ≤1,
M
i=1
ηiPBh1,iτ0+
N
j=1
E2,j ≤Emax,
τ≥0,
E2≥0.
(10)
Along the lines of the proof of Theorem 1, problem P2
convexity can be established. Details are omitted due to space
limitations. An equivalent optimization problem to P2 can be
formulated as
P2∗:max
t,τ,E2
t
s.t. τ0+
M
i=1
τ1,i +
N
j=1
τ2,j ≤1,
M
i=1
ηiPBh1,iτ0+
N
j=1
E2,j ≤Emax,
τ1,i log21+γi
τ0
τ1,i ≥t, i =1,··· ,M,
τ2,j log21+θj
E2,j
τ2,j ≥t, j =1,··· ,N,
τ≥0,
E2≥0,
(11)
where tis an auxiliary variable that denotes the minimum
throughput achieved by each user.
IV. OPTIMAL POLICY UNDER AVERAGE ENERGY
CONSTRAINT (OPAC)
Motivated by the strong requirement of knowing the per
slot total energy consumption of the system with only RF
energy harvesting nodes in Section III and with the purpose of
having a more practically viable problem formulation, we relax
the aforementioned requirement. We introduce an alternative
formulation that, instead, relies on a fixed energy budget for
type II nodes allowed to consume per transmission block (i.e.,
slot), denoted by ¯
E.
A. Sum Throughput Maximization
In this subsection, we formulate the sum throughput max-
imization problem for the OPAC scheme. From (7) and (8),
the problem can be formulated as
P3 :
max
τ
M
i=1
τ1,i log21+γi
τ0
τ1,i +
N
j=1
τ2,j log21+θj
¯
E
τ2,j
s.t. τ0+
M
i=1
τ1,i +
N
j=1
τ2,j ≤1,
τ≥0.
(12)
The objective function of problem P3 is optimized over
transmission durations (τ0for harvesting, as well as Type I
and Type II nodes) subject to the total transmission block time
constraint with a fixed ( ¯
E) for type II nodes specified to satisfy
an average energy consumption constraint on the network.
Theorem 2. P3 is a convex optimization problem and the
optimal time allocations are given by
τ∗
0=⎧
⎨
⎩
x∗
1−¯
EA2−1
x∗
1+A1−1,if (x∗
1−1) ≥¯
EA2,
0,otherwise,
(13)
τ∗
1,i =⎧
⎨
⎩
γi(x∗
1−¯
EA2−1)
(x∗
1−1)(x∗
1+A1−1),if (x∗
1−1) ≥¯
EA2,
0,otherwise,
(14)
τ∗
2,j =⎧
⎪
⎨
⎪
⎩
θj¯
E
x∗
1−1,if (x∗
1−1) ≥¯
EA2,
θj
A2
,otherwise,
(15)
for i=1,··· ,M and j=1,··· ,N, where A1=M
i=1 γi,
A2=N
j=1 θjand x∗
1>1is the solution of f(x1)=A1,
where
f(x)=xln(x)−x+1.(16)
Proof: Please refer to the Appendix.
Based on Theorem 2, it is clear that the optimal time
allocated to each user for uplink information transmission
depends on its distance to the BS, i.e., the near users (with
better channel power gains) to the BS are allocated more
uplink transmission time than the far users, which demon-
strates the doubly near-far phenomenon [4]. Moreover, it is
observed that τ∗
2,j is proportional to ¯
E, i.e., as the amount of
allocated energy per transmission block for each legacy node
increases, the uplink allocated time for legacy nodes increases
and that allocated for RF energy harvesting nodes decreases.
Taking into consideration the above two observations, the
sum throughput maximization results in an unfair achievable
throughput among different users.
B. Maxmin Fairness Formulation
Targeting fairness among users, we adopt, once more,
the maxmin fairness formulation within the OPAC WPCN
paradigm as follows.
P4 :max
τmin
i,j R1,i (τ0,τ
1,i),R
2,j ¯
E,τ2,j
s.t. τ0+
M
i=1
τ1,i +
N
j=1
τ2,j ≤1,
τ≥0.
(17)
Along the lines of Theorem 2, problem P4 convexity can
be established. Details are omitted due to space limitations.
Moreover, an equivalent optimization problem to P4 can be
formulated as in P2∗.
V. NUMERICAL RESULTS
In this section, we provide numerical results showing the
merits of the formulated optimization problems and the asso-
ciated trade-offs. Motivated by the convexity of the formulated
problems, we use standard optimization tools, e.g., CVX [9],
to obtain the optimal solutions. We consider the channel
power gains are modeled as h1,i =g1,i =10
−3ρ2
1,id−α
1,i for
i=1,··· ,M and g2,j =10
−3ρ2
2,j d−α
2,j for j=1,··· ,N,
where d1,i denotes the distance between U1,i and the BS, d2,j
denotes the distance between U2,j and BS and αdenotes the
pathloss exponent. ρ1,i and ρ2,j are the standard Rayleigh
short term fading; therefore ρ2
1,i and ρ2
2,j are exponentially
distributed random variables with unit mean. We consider
same parameters as in [4], we use PB=20dBm, σ2=
−160 dBm/Hz, ηi=0.5for i=1,··· ,M,Γ=9.8
dB and the bandwidth is set to be 1 MHz. Moreover, each
throughput curve shown later is obtained by averaging over
1000 randomly generated channel realizations. We compare
the performance of our system with two types of nodes in
OPIC and OPAC with the performance of the baseline WPCN
with only Type I nodes [4] subject to same amount of available
resources.
In Fig. 2, we compare the average maximum achievable sum
throughput for the 3 studied systems vs. the pathloss exponent.
We consider the same scenario for all schemes with N=1,
M=1,d1,1=10mand d2,1=5mwhere the baseline WPCN
system with energy harvesting nodes only is considered to
have two users with same distances d1,1and d2,1given above.
Our objective is to fairly compare the three systems, baseline
WPCN, OPIC and OPAC, with same total amount of energy
resources. Towards this objective, at each pathloss exponent
value, the total harvested energy for each channel realization
at the baseline WPCN is assumed to be Emax in P1 (i.e., a per
slot energy constraint). Also, ¯
Efor Type II nodes is computed
using the optimal derived closed form time allocations (for
P3) such that the average energy consumptions over the 1000
channel realizations are equal in both the baseline WPCN and
the OPAC system.
A number of observations are now in order. First, we note
that the average maximum sum throughput of the three studied
systems monotonically decreases as the pathloss exponent
increases. Second, as expected, the average maximum sum
throughput achieved by P1 is the highest due to the fact that
OPIC allocates more energy to the user with better channel
power gains that is, the legacy node in our scenario, to
maximize the sum throughput. Therefore, in our scenario, the
average maximum sum throughput is obtained via allocating
more energy to the legacy node than the energy harvesting
node and, hence, reducing τ0. Unlike the baseline WPCN with
energy harvesting nodes only where the amount of harvested
energy by the farther user cannot be efficiently utilized for
uplink data transmissions and cannot be reduced through
reducing the τ0as in P1 since the user closer to the BS
is also an energy harvesting node which harvests its energy
during that τ0fraction of time. Therefore, it is clear that the
average maximum sum throughput of P1 outperforms the one
of baseline WPCN with energy harvesting nodes only. Third,
the average maximum sum throughput obtained by P3 is less
than that achieved by P1 due to the OPAC that yields fixed ¯
E
that guarantees equal “average” consumed energy.
In Fig. 3, Jain’s fairness index [11] is plotted for the three
systems under consideration against the pathloss exponent
considering the same scenario in Fig. 2. It is observed that
the fairness index of both proposed schemes, namely OPIC
and OPAC, is less than that of the baseline WPCN with energy
harvesting nodes only. This, in turn, highlights one instance of
the fundamental throughput-fairness trade-off, where the su-
perior sum throughput performance in P1 and P3 compared to
baseline WPCN came at the expense of a modest degradation
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
0
0.5
1
1.5
2
2.5
3
3.5
4
Pathloss Exponent
Average Sum Throughput (Mbps)
Baseline WPCN
OPIC (P1)
OPAC (P3)
Fig. 2. Average maximum sum throughput for the 3 systems.
in the fairness.
Motivated by the inherent unfairness witnessed for the sum
throughput maximization formulation for the 3 studied sys-
tems, Fig. 4 shows the average maxmin throughput comparison
with the same set of parameters as in Fig. 2. It is observed
that the average maxmin throughput achieved by P2 and P4 is
larger than that of the baseline WPCN with energy harvesting
nodes only. It is also observed that twice the average maxmin
throughput of each system at each pathloss exponent value
(which is the average sum throughput given that we have only
two users) is less than the average maximum sum throughput
for the same system (Fig. 2). This, in turn, demonstrates
the fundamental trade off between achieving maximum sum
throughput and achieving fair throughout allocations among
different users.
Fig. 5 shows the effect of scaling the system with larger
number of nodes on the network performance via comparing
the average maximum sum throughput of P1 and P3 with the
one of the baseline WPCN with energy harvesting nodes only.
Towards this objective, we consider The network has six users
with same distance d=10
6mfor the three systems under
consideration. It is observed that as the number of Type II
nodes (N) in both P1 and P3 increases, the average maximum
sum throughput increases since increasing Nreduces the
allocated time for energy harvesting (τ0)and, hence, the
average maximum sum throughput increases via assigning that
reduction in (τ0)for uplink data transmission through Type
II nodes. Therefore, it is clear that the maximum achievable
sum throughput is obtained by the extreme case of N=6and
M=0(τ∗
0=0).
VI. CONCLUSION
This paper studies a generalized problem setting for wireless
powered communication networks. In which, the network
has two types of nodes, energy harvesting nodes assumed
to have RF energy harvesting capability and legacy nodes
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Pathloss Exponent
Jain’s Fairness Index
Baseline WPCN
OPIC (P1)
OPAC (P3)
Fig. 3. Jain’s fairness index for the 3 systems.
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Pathloss Exponent
Average Maxmin Throughput (Mbps)
Baseline WPCN
OPIC (P2)
OPAC (P4)
Fig. 4. Average maxmin throughput for the 3 systems.
assumed not to have RF energy harvesting capability and
equipped with dedicated energy supplies. A BS first broadcasts
an energizing signal over the downlink so that each energy
harvesting user could harvest a certain amount of energy
for uplink information transmission, then all users send their
information independently to the BS using TDMA. We provide
two optimal resource allocation schemes, namely OPIC and
OPAC, subject to different energy constraints on the system.
Furthermore, We formulate two optimization problems to
investigate the maximum sum throughput and the maxmin
throughput in both proposed schemes. Our numerical results
reveal that both OPIC and OPAC schemes outperform the
baseline WPCN with energy harvesting nodes only in terms
of the maximum sum throughput and the maxmin throughput.
They also demonstrate the fundamental trade off between
achieving maximum sum throughput and achieving fairness
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
8.5
9
9.5
10
10.5
11
11.5
12
12.5
13
13.5
Pathloss Exponent
Average Sum Throughput (Mbps)
Baseline WPCN
M=3,N=3 (P1)
M=2,N=4 (P1)
M=1,N=5 (P1)
M=0,N=6 (P1)
M=3,N=3 (P3)
M=2,N=4 (P3)
M=1,N=5 (P3)
M=0,N=6 (P3)
Fig. 5. Average maximum sum throughput for the 3 systems.
among different users.
APPENDIX
Since τ1,i log21+γi
τ0
τ1,i is the perspective
function of the concave function log2(1 + γiτ0)
which preserves the concavity of R1,i with respect
to (τ0,τ
1,i).Also, ∂2
∂τ2
2,j τ2,j log21+θj
¯
E
τ2,j =
−θ2
j¯
E2
ln(2)τ2,j (τ2,j +θj¯
E)2<0for τ2,j ∈[0,1], therefore
R2,j is also concave function in τ2,j. A non-negative
weighted sum of concave functions is concave [9], then the
objective function of P3 which is the non negative weighted
summation of concave functions, i.e., R1,i and R2,j for
i=1,··· ,M and j=1,··· ,N, is concave function in
τ=[τ0,τ
1,1,··· ,τ
1,M ,τ
2,1,··· ,τ
2,N ]. Furthermore, all
constraints in P3 are affine in τ, thus it is clear that P3 is
convex optimization problem and its Lagrangian is given by
L(τ,λ)=Rsum (τ)−λ⎛
⎝τ0+
M
i=1
τ1,i +
N
j=1
τ2,j −1⎞
⎠,
(18)
where Rsum (τ)=M
i=1 R1,i (τ0,τ
1,i)+N
j=1 R2,j ¯
E,τ2,j
and λis the non negative Lagrangian dual variable associated
with the constraint in (1). Hence, the dual function can be
expressed as
G(λ) = max
τ∈S L(τ,λ),(19)
where Sis the feasible set specified by τ≥0. It can be
easily shown that there exists a τthat strictly satisfies all
constraints of P3. Hence, according to Slater’s condition [9],
strong duality holds for this problem; therefore, the Karush-
Kuhn-Tucker (KKT) conditions are necessary and sufficient
for the global optimality of P3, which are given by
τ∗
0+
M
i=1
τ∗
1,i +
N
j=1
τ∗
2,j ≤1,(20)
λ∗⎛
⎝τ∗
0+
M
i=1
τ∗
1,i +
N
j=1
τ∗
2,j −1⎞
⎠=0,(21)
∂
∂τ0
Rsum (τ∗)−λ∗=0,(22)
∂
∂τ1,i
Rsum (τ∗)−λ∗=0,i=1,··· ,M, (23)
∂
∂τ2,j
Rsum (τ∗)−λ∗=0,j=1,··· ,N, (24)
where τ∗and λ∗denote, respectively, the optimal primal and
dual solutions of P3. Since Rsum (τ)is a monotonic increasing
function in τ, therefore τ∗
0+M
i=1 τ∗
1,i +N
j=1 τ∗
2,j =1must
hold. From (22), (23) and (24), we have
M
i=1
γi
1+γi
τ∗
0
τ∗
1,i
=λ∗ln(2),(25)
ln 1+γi
τ∗
0
τ∗
1,i −
γi
τ∗
0
τ∗
1,i
1+γi
τ∗
0
τ∗
1,i
=λ∗ln(2),i=1,··· ,M,
(26)
ln 1+ ¯
Eθj
τ∗
2,j −
¯
Eθj
τ∗
2,j
1+ ¯
Eθj
τ∗
2,j
=λ∗ln(2),j=1,··· ,N. (27)
Therefore, from (26) and (27) we have
γ1τ∗
0
τ∗
1,1
=γ2τ∗
0
τ∗
1,2
=···γMτ∗
0
τ∗
1,M
=¯
Eθ1
τ∗
2,1
=¯
Eθ2
τ∗
2,2
=··· ¯
EθN
τ∗
2,N
.
(28)
Taking into consideration that τ∗
0+M
i=1 τ∗
1,i +N
j=1 τ∗
2,j =1
and (28). Hence, τ∗
1,i and τ∗
2,j can be expressed, respectively,
by
τ∗
1,i =γiτ∗
0(1 −τ∗
0)
A1τ∗
0+¯
EA2
,i=1,··· ,M, (29)
τ∗
2,j =θj¯
E(1 −τ∗
0)
A1τ∗
0+¯
EA2
,j=1,··· ,N, (30)
where A1=M
i=1 γiand A2=N
j=1 θj. From (25), (26)
and (27), it follows that
x1ln(x1)−x1+1=A1,(31)
where x1=1+γi
τ∗
0
τ∗
1,i
=1+ ¯
Eθj
τ∗
2,j
. From (29) and (30), it
is clear that x1>1if A1>0,A2>0and 0<τ
∗
0<1.
According to [4, Lemma 3.2], there exists a unique solution
x∗
1>1for (31). Thus from (29)-(31), the optimal time
allocations are given by
τ∗
0=x∗
1−¯
EA2−1
x∗
1+A1−1,(32)
τ∗
1,i =γix∗
1−¯
EA2−1
(x∗
1−1) (x∗
1+A1−1),i=1,··· ,M, (33)
τ∗
2,j =θj¯
E
x∗
1−1,j=1,··· ,N. (34)
If (x∗
1−1) <¯
EA2, then the total block time will be assigned
to the Type II nodes for uplink information transmissions.
Therefore, from (20) and (27), the optimal time allocations
are given by
τ∗
0=0,(35)
τ∗
1,i =0,i=1,··· ,M, (36)
τ∗
2,j =θj
A2
,j=1,··· ,N. (37)
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