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382 S. PEJOSKI, Z. HADZI-VELKOV, T. SHUMINOSKI, LYAPUNOV DRIFT-PLUS-PENALTY BASED RESOURCE ALLOCATION IN .. .
Lyapunov Drift-Plus-Penalty Based Resource Allocation
in IRS-Assisted Wireless Networks
with RF Energy Harvesting
Slavche PEJOSKI, Zoran HADZI-VELKOV, Tomislav SHUMINOSKI
Ss. Cyril and Methodious University in Skopje, 1000 Skopje, N. Macedonia
{slavchep, zoranhv, tomish}@feit.ukim.edu.mk
Submitted October 6, 2021 / Accepted June 20, 2022 / Online first August 3, 2022
Abstract. We propose a resource allocation policy for
intelligent reflective surface (IRS)-assisted wireless powered
communication network (WPCN) where the energy harvest-
ing (EH) users (EHUs) have finite energy storage and data
buffers, for storing the harvested energy and the input (sen-
sory) data, respectively. The IRS reflecting coefficients for
uplink and downlink are chosen to focus the beam towards
a specific EHU, but have additional constant phase offsets
(different for uplink and downlink) in order to account for
the direct link between the base station and the IRS targeted
EHU, and the influence to the EH process of other EHUs in
downlink. The EHUs acquire data from their sensors, re-
ceive energy in downlink and send information in uplink. We
maximize the overall average amount of sensor information
in the WPCN by optimizing the IRS reflecting coefficients for
the downlink transmissions, the amount of acquired sensor
information and the duration of the information transmission
period for each EHU in each epoch using the Lyapunov drift-
plus-penalty optimization technique. The simulation results
demonstrate the effectiveness of the proposed solution.
Keywords
Intelligent reflecting surfaces, Lyapunov drift-plus-
penalty optimization, wireless powered networks
1. Introduction
The intelligent reflective surfaces (IRS) shape the multi-
path propagation in the radio channel in such a way to aid the
communication between wireless transmitters and receivers.
Thus, the IRS promises more efficient utilization of the avail-
able spectrum and energy budget [1–3]. The IRS consists of
many sub-wavelength-sized elements that act as diffuse scat-
ters combined together into a large metasurface [4]. In one
of its many promising applications, the IRS can be merged
together with wireless powered communications networks
(WPCNs), which rely on radio frequency (RF) energy har-
vesting (EH) for charging the EH users (EHUs) [5], [6].
Namely, the EHUs, due to the physical limitations in their
design and the rapid decrease of the available energy with
distance, can harvest tiny amounts of energy. IRS can relay
and focus the signal in both directions and thus increase the
amount of harvested energy, together with the capacity of
the wireless channel. The existing literature on IRS-assisted
WPCNs typically optimize the phase shifts of each of the
IRS antenna elements (meta elements) together with other re-
source allocation parameters [7–9]. The resource allocation
schemes proposed so far typically rely on suboptimal iterative
algorithms. These algorithms have high computational com-
plexities that increase with the number of antenna elements,
which makes them unsuitable for practical implementation.
Namely, those algorithms are already computationally pro-
hibitive for a few dozen IRS antenna elements, while current
IRS implementations consist of few thousand in the exper-
imental studies presented in [4], [10], and up to 104IRS
antenna elements in the experimental study in [11]. Recently
a paper, [12], proposed a computationally simple beamform-
ing solution, where the IRS beam is sequentially focused
towards each EHU. The solution exploits the channel reci-
procity and sets the same reflection matrix for the uplink
and downlink transmissions. The paper has two model lim-
itations: it assumes that during the EH process, the EHUs
receive energy only when the IRS is aimed at them, and, it
assumes that an obstacle exists between the base station (BS)
and the EHUs that is not generally the case.
Most importantly, the previous works do not consider
the sensory data acquisition process, the data storage avail-
able to the tiny wireless devices, and additionally do not
control the energy levels of the EHU’s battery. In this paper
we are interesting in extending the system model from [12] to
be able to adjust to a scenario where direct path between the
BS and the EHUs exist, to correctly account for the receive
energy in the slots where the IRS is not pointed to the des-
ignated EHU, and, based on the objective function optimize
the IRS assisted EH process. This is achieved by offsetting
the phases of the IRS elements with respect to simple focus-
ing towards an EHU in the EH and information transmission
(IT) phases of each user. Namely, the IRS phases in the IT
DOI: 10.13164/re.2022.0382
RADIOENGINEERING, VOL. 31, NO. 3, SEPTEMBER 2022 383
and EH parts will have constant phase offset for all IRS ele-
ments. Because of the mutual coupling between EHUs (due
to the nature of the EH process), finding the constant phase
offset for the EH phase is a non-convex problem and the ap-
proach presented in [12] is unable to find the solution. Here
we aim to apply Lyapunov drift-plus-penalty method [13]
which further accounts for buffer levels in both the data ac-
quisition and the EH processes at the EHUs, and thus, brings
the system model closer to reality.
This Lyapunov drift-plus-penalty optimization tech-
nique is a powerful technique for developing scheduling
strategies for optimal utility performance of energy harvest-
ing networks with finite capacity energy storage [14]. The
use of Lyapunov drift-plus-penalty optimization technique
is also appropriate for optimizing the operation of various
communication networks where the queuing processes affect
the system performance, for example, for improving average
throughput and stabilizing the queuing in 5G nodes with ver-
tical multi-homing capabilities [15] or optimizing the flow
control and minimizing the energy consumption in stochas-
tic networks [16] and [17]. The use of Lyapunov drift-plus-
penalty techniques in WPCN have been studied in several
papers [18–20]. The paper [18] considers the control of
an energy self-sufficient receiver in a multi-access network
with simultaneous wireless information and energy transfer.
The authors in [19] consider an WPCN of body sensors that
have limited energy and data buffer sizes. Similarly, [20]
analyze a WPCN where the EHUs are equipped with finite
energy buffers and data buffers but the BS is equipped with
multiple antennas. Nonetheless, none of these papers con-
siders the design of IRS assisted networks, nor the influence
of IRS to the networks performance.
(a) System model.
(b) TDMA frame organization.
Fig. 1. System model and TDMA frame organization.
In this paper we use the Lyapunov drift-plus-penalty op-
timization technique for resource allocation and parameters
optimization in an IRS assisted WPCN consisting of EHUs
with limited energy buffers and data buffers. The solution
is of low complexity, can account for a direct communica-
tion link between BS and EHUs, takes into consideration
the complete harvested energy, optimize the EH process and
allows for maximizing the total amount of acquired sensory
data in the WPCN. To the best of our knowledge, this is the
first paper that introduces the Lyapunov drift-plus-penalty
optimization technique for parameters optimization in IRS-
assisted networks.
The paper is organized as follows: Section 2 introduces
the communication system, Section 3 presents the optimiza-
tion problem and its solution. Section 4 shows the numerical
results and Section 5 concludes the paper.
2. System Model
We consider a WPCN with a single BS and 𝐾EHUs all
equipped with a single antenna. All nodes employ TDMA
and operate in half-duplex mode over the same frequency
band. The IRS is deployed so as to maintain line of sights
(LoS) to all nodes. On the other hand the BS can directly
communicate with the different EHUs via a random fading
channel without LoS. The BS transmits at fixed power 𝑃0
and operates as an information receiver and an energy bea-
con. The EHUs transmit information to the BS. Additionally,
each EHU is equipped with a sensors from which it collects
the data that is transmitted in the IT phase. Each EHU is pro-
visioned with a finite energy storage and a finite data buffer to
store the harvested energy and to buffer the arrived data traffic
(for example, due to sensory data acquisition), respectively.
The system model is shown in Fig. 1 (a).
The communication session is divided into TDMA
frames with unit durations 𝑇=1s and lasts 𝑀TDMA frames.
Each TDMA frame is divided into two equal phases: an EH
phase and an IT phase. Each phase is subdivided into 𝐾
sub-slots. To aid simplicity of implementation, all sub-slots
of the EH phase have equal durations,𝑇/(2𝐾), and each sub-
slot is allocated to a different EHU. In the IT phase of the 𝑡th
TDMA frame, each EHU receives a slot of duration 𝜏𝑘(𝑡)𝑇
where 𝜏𝑘(𝑡)are the relative durations of the individual IT
phases such that:
0≤𝜏𝑘(𝑡) ≤ 1
2∀𝑘, (1)
𝐾
𝑘=1
𝜏𝑘(𝑡) ≤ 1
2.(2)
The system model is illustrated in Fig. 1 (b).
384 S. PEJOSKI, Z. HADZI-VELKOV, T. SHUMINOSKI, LYAPUNOV DRIFT-PLUS-PENALTY BASED RESOURCE ALLOCATION IN .. .
2.1 Channel Modeling
The IRS can be modeled as a planar antenna array
consisting of 𝑁IRS antenna elements [3]. Assuming that
each antenna element has an area of size 𝐴, such that
𝐴≤ (𝜆/4)2, the total area can be calculated as 𝑆=𝑁 𝐴.
The IRS operation is controlled by its reflection matrix
𝚯=diag(ej𝜃1,· · · ,ej𝜃𝑁). In 𝚯,𝜃𝑛∈ [0,2𝜋)is the phase
shift induced by the 𝑛th element of the IRS and it is assumed
to be configurable and programmable via an IRS controller.
During each sub-slot of the TDMA frame, the matrix 𝚯is
adjusted to the designated EHU. Namely, we assume that dif-
ferent IRS coefficients can be assigned during each sub-slot
of each phase.
Due to the LoS assumption, the channel gain and the
phase between the node and each IRS element are both
fixed and deterministic. Thus, the BS-IRS channel is mod-
eled by a deterministic vector h=[ℎ1,·· · , ℎ𝑁]T, where
ℎ𝑛=√Ω0𝑛e−j𝜙0𝑛is the channel between the BS and the
𝑛th antenna element of the IRS. Similarly, the IRS-𝑘th EHU
channel (1≤𝑘≤𝐾) is represented by a deterministic vector,
g𝑘=[𝑔𝑘1,·· · , 𝑔𝑘 𝑁 ]T, where 𝑔𝑘 𝑛 =√Ω𝑘 𝑛 e−j𝜙𝑘𝑛 ,Ω𝑘 𝑛 is the
channel gain and 𝜙𝑘𝑛 is the phase between the 𝑘th EHU and
the 𝑛th IRS element. Here, for analytical tractability, we as-
sume that the IRS is a square surface whose center is placed at
the origin of the coordinate system. In this case, the location
of the 𝑛th antenna element is [3, Eq. (22), (23)]:
𝑥𝑛=−(√𝑁−1)√𝐴
2+√𝐴mod(𝑛−1,√𝑁),(3)
𝑦𝑛=−(√𝑁−1)√𝐴
2+√𝐴𝑛−1
√𝑁.(4)
For simplicity reasons and to obtain unambiguous notation,
as shown in Fig. 1, we assume that all EHUs are located in the
first quadrant, while the BS is located in the second quadrant.
Thus, the BS is located at polar coordinates (𝑑BS ,−𝛼0)with
respect to (w.r.t) the origin and the 𝑧-axis, and so its angle
w.r.t. the IRS boresight is 𝛼0. Similarly, the 𝑘th EHU is
located at polar coordinates (𝑑𝑘, 𝛼𝑘). Thus, in our system
model, the phases 𝜙0𝑛and 𝜙𝑘 𝑛 are calculated as [3, Eq. (25)]:
𝜙0𝑛=2𝜋mod ©«𝑥2
𝑛+𝑦2
𝑛+𝑑2
BS −2𝑥𝑛𝑑BS sin 𝛼0
𝜆,1ª®®¬
,(5)
𝜙𝑘𝑛 =2𝜋mod ©«𝑥2
𝑛+𝑦2
𝑛+𝑑2
BS −2𝑥𝑛𝑑𝑘sin 𝛼𝑘
𝜆,1ª®®¬
.(6)
The values of the channel power gains in the general case
when LoS is available are calculated as given with Lemma 1
from [3]. Nonetheless, for the case when both the BS and
the EHUs operate in the far-field region of the IRS (defined
by the inequality 𝑆≤9𝑑2
BS) their calculation is significantly
simplified. Namely, Ω0𝑛are approximated by (c.f. [3, Eq.
(11) and Eq. (31)])
Ω0𝑛≈𝐴cos(𝛼0)
4𝜋𝑑 2
BS ≡Ω0,∀𝑛(7)
whereas channel gains of the 𝑘th EHU, Ω𝑘𝑛 , are approxi-
mated by
Ω𝑘𝑛 ≈𝐴cos (𝛼𝑘)
4𝜋𝑑 2
𝑘≡Ω𝑘,∀𝑛. (8)
In the 𝑘th sub-slot of the EH phase of the 𝑡th TDMA
frame, the IRS aims to deliver most of the energy to
the 𝑘th EHU by setting its reflection matrix, 𝚯(𝑘 , 𝑡)=
diag(ej𝜃1(𝑘 ,𝑡 );· ·· ; ej𝜃𝑁(𝑘,𝑡 )), with the phase shifts:
𝜃𝑛(𝑘, 𝑡)=𝜙0𝑛+𝜙𝑘 𝑛 +𝜓𝑘(𝑡),∀𝑛. (9)
This phase choice allows for focusing the the IRS beam to-
wards the 𝑘th EHU as in [12]. But different form [12],
the phase offset 𝜓𝑘(𝑡)in (9) is an additional phase that can
account for the direct BS-𝑘th EHU channel and can also in-
fluence the EH process at other EHUs in the 𝑘th sub-slot of
the EH phase. Thus allowing other EHUs close to the 𝑘th
EHU to harvest larger amounts of RF energy that effectively
can lead to broadening of the EH region around the 𝑘th EHU.
The direct channel between the BS and the 𝑘th EHU
is modeled as block fading channel with random gain ℎ𝑑 𝑘
(with amplitude |ℎ𝑑 𝑘 |and a random phase 𝜙𝑑𝑘 i.e. ℎ𝑑 𝑘 =
|ℎ𝑑𝑘 |e−j𝜙𝑑 𝑘 ), where the average power gain of the channel is
Ω𝑑𝑘 =
E
|ℎ𝑑𝑘 |2.
2.2 Data Queue and Transmission Model
Let the maximum amount of inbound data, during one
frame, be denoted by 𝑖𝑘 ,max , and the instantaneous amount of
inbound data be denoted by 𝑖𝑘(𝑡). Then the data acquired by
the 𝑘th EHU (e.g., from the sensory measurements) during
frame 𝑡is given by:
0≤𝑖𝑘(𝑡) ≤ 𝑖𝑘 , max.(10)
The data transmitted by the 𝑘th EHU in the IT phase of
the 𝑡th frame is calculated as:
𝑜𝑘(𝑡)=𝜏𝑘(𝑡)𝑇𝑊 log21+𝑃𝑘|hT𝚯IT (𝑘 , 𝑡)g𝑘+ℎ𝑑𝑘 (𝑡)|2
𝑁0𝑊
(11)
where 𝑊is the bandwidth of the WPCN. From (11) it is
obvious that during the IT phase, the IRS reflection matrix
in the 𝑘th sub-slot do not influence the rate of other EHUs
(namely, the information sent by the 𝑘th EHU is of interest
only to the BS). Thus, to maximize 𝑜𝑘(𝑡), without spending
any additional resources, the phases in 𝚯IT (𝑘, 𝑡 )should be
chosen as [3]:
𝜃𝑛,IT (𝑘 , 𝑡)=𝜙0𝑛+𝜙𝑘 𝑛 −𝜙𝑑𝑘 (𝑡),∀𝑛(12)
which aligns the phases of the IRS assisted beam and the di-
rect path between the 𝑘th EHU and the BS, and thus provides
maximum SNR at the BS.
Denoting 𝐷𝑘(𝑡)to be the length of the data queue of
the 𝑘th EHU at the beginning of the frame 𝑡, the dynamics of
the data queue is described by:
𝐷𝑘(𝑡+1)=max [𝐷𝑘(𝑡) − 𝑜𝑘(𝑡),0]+𝑖𝑘(𝑡).(13)
RADIOENGINEERING, VOL. 31, NO. 3, SEPTEMBER 2022 385
2.3 Energy Harvesting and Consumption
Model
Each EHU is equipped with an rechargeable battery
with fixed energy storage capacity denoted by Λ𝑘. In this pa-
per we assume that EHUs transmit at a very low power levels
and that the distance between the EHUs are such that the har-
vested energy during the IT phase is negligible. During the
𝑢th sub-slot of the EH phase, the 𝑘th EHU harvests energy
of the amount 𝜂𝑘𝑇
2𝐾𝑃0|hT𝚯EH (𝑢, 𝑡)g𝑘+ℎ𝑑 𝑘 (𝑡)|2, where 𝜂𝑘
is the EH efficiency of the 𝑘th EHU. Thus, the amount of
energy harvested by the 𝑘th EHU during the EH phase is:
𝑒in
𝑘(𝑡)=𝜂𝑘𝑃0𝑇
2𝐾
𝐾
𝑢=1|hT𝚯EH (𝑢, 𝑡)g𝑘+ℎ𝑑 𝑘 (𝑡)|2.(14)
Please note that the choice of 𝚯EH (𝑢, 𝑡)for 𝑢=1, ..., 𝐾 im-
pacts 𝑒in
𝑘(𝑡), unlike 𝚯IT (𝑢, 𝑡)that does not affect 𝑜𝑘(𝑡)for
𝑢≠𝑘. This justifies the need for different IRS matrices for
the sub-slots of the same EHU during the IT and EH phases.
The energy spent by the 𝑘th EHU, 𝑒out
𝑘(𝑡), is determined
by the power consumption of the sensory data acquisition and
the power consumption of the RF information transmission,
as:
𝑒out
𝑘(𝑡)=𝜖𝑘𝑖𝑘(𝑡) + 𝑃𝑘𝜏𝑘(𝑡)𝑇(15)
where 𝜖𝑘is the energy consumption per one bit of data ac-
quired by the 𝑘th EHU sensor and 𝑃𝑘is the transmit RF
power of the 𝑘th EHU. We assume that 𝑃𝑘is predetermined
constant parameter associated with the hardware settings of
the 𝑘th EHU.
Based on the EH and the energy consumption model,
the length of the energy queue at the 𝑘th EHU in the 𝑡th frame
is 𝐸𝑘(𝑡)and its dynamics, can be described as:
𝐸𝑘(𝑡+1)=min[𝐸𝑘(𝑡) + 𝑒in
𝑘(𝑡) − 𝑒out
𝑘(𝑡),Λ𝑘].(16)
3. Resource Allocation
Similar to [19] and [20], our proposed scheme is de-
signed by maximizing the average amount of acquired sen-
sory data by all the EHUs in the WPCN:
Maximize
𝑖𝑘(𝑡),𝜓𝑘(𝑡),𝜏𝑘(𝑡)lim
𝑀−>∞
1
𝑀
𝑀−1
𝑡=0
𝐾
𝑘=1
𝑖𝑘(𝑡).(17)
s.t. (1),(2),(9)−(16)
The optimization problem (17) is non-convex problem
due to the the constraint associated with (14) which is non-
convex with respect to 𝜓𝑘(𝑡),∀𝑘. In order to solve (17) we re-
sort to the Lyapunov-drift-plus penalty (LDPP) method [13].
The LDPP method guarantees stability of the energy and data
queues, 𝐸𝑘(𝑡)and 𝐷𝑘(𝑡)respectively. Namely, using (13),
(16) and the assumption that both buffers are limited in capac-
ity it can be easily shown that they are stable [13]. The method
consists of defining the Lyapunov function based on the queu-
ing model, finding the upper bound on the LDPP function
and sub-optimally minimizing it under the constraints of the
original problem, independently in each TDMA epoch.
Let us denote the state of the energy queue by E(𝑡)=
[𝐸1(𝑡), ..., 𝐸𝐾(𝑡)]Tand the state of the energy queue by
D(𝑡)=[𝐷1(𝑡), ..., 𝐷𝐾(𝑡)]T. Then, the queue vector is
defined as S(𝑡)=[D(𝑡),E(𝑡)]T. The Lyapunov function is
defined to evaluate the length of the data queue 𝐷𝑘(𝑡)and
the free space of battery Λ𝑘−𝐸𝑘(𝑡)[19] as:
𝐿(S(𝑡)) =1
2"𝐾
𝑘=1
𝜇1𝑘𝐷2
𝑘(𝑡) + 𝜇2𝑘(Λ𝑘−𝐸𝑘(𝑡))2#(18)
where 𝜇1𝑘and 𝜇2𝑘are nonnegative constants used to bal-
ance among the different nature of the buffers [20]. Please
note that by minimizing 𝐿(S(𝑡)) the data buffer can approach
to an empty state level, and the energy queue can approach
a level of Λ𝑘.
The one-step conditional Lyapunov drift is defined as:
Δ(S(𝑡)) =
E
{𝐿(S(𝑡+1)) − 𝐿(S(𝑡)) | S(𝑡)}.(19)
The Lyapunov drift-plus-penalty expression is defined as:
Δ(S(𝑡)) − 𝑉·
E
(𝐾
𝑘=1
𝑖𝑘(𝑡) | S(𝑡))(20)
where 𝑉is a nonnegative control parameter.
The LDPP method seeks to minimize the upper bound
of (20), in each 𝑡, for all possible values of the state S(𝑡),
and for all control parameters 𝑉 > 0. By expanding (19), we
obtain
Δ(S(𝑡)) =1
2
E
(𝐾
𝑘=1𝜇1𝑘(𝐷2
𝑘(𝑡+1) − 𝐷2
𝑘(𝑡))
+𝜇2𝑘((Λ𝑘−𝐸𝑘(𝑡+1))2
−(Λ𝑘−𝐸𝑘(𝑡))2)|S(𝑡)).
(21)
Using (16) we easily obtain (Λ𝑘−𝐸𝑘(𝑡+1))2=(min[𝐸𝑘(𝑡)+
𝑒in
𝑘(𝑡) − 𝑒out
𝑘(𝑡) − Λ𝑘,0])2. Applying the inequality
(min[𝑎, 0])2≤𝑎2we obtain:
(Λ𝑘−𝐸𝑘(𝑡+1))2≤ (𝐸𝑘(𝑡) + 𝑒in
𝑘(𝑡) − 𝑒out
𝑘(𝑡) − Λ𝑘)2
≤ (𝐸𝑘(𝑡) − Λ𝑘)2+𝑒in
𝑘(𝑡)2+𝑒out
𝑘(𝑡)2
−2(𝐸𝑘(𝑡) − Λ𝑘)(𝑒out
𝑘(𝑡) − 𝑒in
𝑘(𝑡)).
(22)
Similarly, using (13) we obtain 𝐷2
𝑘(𝑡+1)=
(max [𝐷𝑘(𝑡) − 𝑜𝑘(𝑡),0]+𝑖𝑘(𝑡))2. Additionally expanding
𝐷2
𝑘(𝑡+1), using (max[𝑎, 0])2≤𝑎2and max[𝑐−𝑑, 0]< 𝑐
for nonnegative 𝑎, 𝑐 and 𝑑, we obtain:
𝐷2
𝑘(𝑡+1)≤(𝐷𝑘(𝑡) − 𝑜𝑘(𝑡))2+𝑖𝑘(𝑡)2
+2 max [𝐷𝑘(𝑡) − 𝑜𝑘(𝑡),0]𝑖𝑘(𝑡)
≤ (𝐷𝑘(𝑡) − 𝑜𝑘(𝑡))2+𝑖𝑘(𝑡)2+2𝐷𝑘(𝑡)𝑖𝑘(𝑡).
(23)
Introducing (22) and (23) in (21) and (20), and notic-
ing that all parameters in (20) are non-negative, the one-step
conditional LDPP expression from (20) is upper bounded by:
386 S. PEJOSKI, Z. HADZI-VELKOV, T. SHUMINOSKI, LYAPUNOV DRIFT-PLUS-PENALTY BASED RESOURCE ALLOCATION IN .. .
Δ(S(𝑡)) − 𝑉·
E
(𝐾
𝑘=1
𝑖𝑘(𝑡) | S(𝑡))
≤𝑈+
E
(𝐾
𝑘=1
𝜇2𝑘(𝐸𝑘(𝑡) − Λ𝑘)𝑒in
𝑘(𝑡) | S(𝑡))
+
E
(𝐾
𝑘=1(𝜇1𝑘𝐷𝑘(𝑡) − 𝜇2𝑘𝜖𝑘(𝐸𝑘(𝑡) − Λ𝑘))𝑖𝑘(𝑡) | S(𝑡))
−𝑉·
E
(𝐾
𝑘=1
𝑖𝑘(𝑡) | S(𝑡))
−
E
(𝐾
𝑘=1
𝜇1𝑘𝐷𝑘(𝑡)𝑜𝑘(𝑡) + 𝜇2𝑘(𝐸𝑘(𝑡) − Λ𝑘)𝑃𝑘𝜏𝑘(𝑡)𝑇|S(𝑡))
(24)
where 𝑈is a finite constant related to the worst-case second
moments of 𝑜𝑘(𝑡),𝑖𝑘(𝑡),𝑒in
𝑘(𝑡), and 𝑒out
𝑘(𝑡)processes, i.e.:
𝑈=1
2
E
(𝐾
𝑘=1
𝜇1𝑘(𝑜𝑘(𝑡)2+𝑖𝑘(𝑡)2)
+𝜇2𝑘(𝑒in
𝑘(𝑡)2+𝑒out
𝑘(𝑡)2) | S(𝑡)(25)
and is of no importance to the optimization process [13].
The LDPP method minimizes the right-hand-side of the
inequality (24) (after removing 𝑈), by choosing the most ap-
propriate control policy action in each epoch. This leads
to removing the averaging from the right hand side of (24),
dropping 𝑈and minimizing it under the constraints from
(17). Please note that in (24) there are three different parts:
•the part Í𝐾
𝑘=1𝜇2𝑘(𝐸𝑘(𝑡) − Λ𝑘)𝑒in
𝑘, affected by the EH
process and the IRS configuration (in our optimization
problem influenced by 𝜓𝑘(𝑡)), is called IRS configura-
tion sub-problem;
•the second part is Í𝐾
𝑘=1(𝜇1𝑘𝐷𝑘(𝑡) − 𝜇2𝑘𝜖𝑘(𝐸𝑘(𝑡) −
Λ𝑘))𝑖𝑘(𝑡) −𝑉Í𝐾
𝑘=1𝑖𝑘(𝑡), which depends on 𝑖𝑘(𝑡), and,
is termed data acquisition sub-problem;
•the third part −Í𝐾
𝑘=1𝜇1𝑘𝐷𝑘(𝑡)𝑜𝑘(𝑡) + 𝜇2𝑘(𝐸𝑘(𝑡) −
Λ𝑘)𝑃𝑘𝜏𝑘(𝑡)𝑇, which depends on 𝜏𝑘(𝑡), is the infor-
mation transmission sub-problem.
Please note that even for more complex optimization prob-
lems with more variables, the same sub-problem structure
remains [20].
3.1 IRS Configuration Sub-Problem
The IRS configuration sub-problem is formulated as:
min
𝜓𝑘(𝑡)
𝐾
𝑘=1
𝜇2𝑘(𝐸𝑘(𝑡) − Λ𝑘)𝑒in
𝑘.(26)
Due to (16) we have (𝐸𝑘(𝑡) − Λ𝑘) ≤ 0∀𝑘. The problem
can be solved by setting the first derivative of the objec-
tive function of (26) with respect to variable 𝜓𝑘(𝑡)to zero,
and finding the root of the resulting transcendental equation,
which yields:
𝜓𝑘(𝑡)=atan −Í𝐾
𝑗=1𝜇2𝑗𝜂𝑗Δ𝐸𝑗(𝑡)|𝐵𝑗 ,𝑘 ℎ𝑑 𝑗 (𝑡) | sin (𝜙𝑑 𝑗 (𝑡)+Δ𝜙𝑗, 𝑘 )
Í𝐾
𝑗=1𝜇2𝑗𝜂𝑗Δ𝐸𝑗(𝑡)|𝐵𝑗,𝑘 ℎ𝑑 𝑗 (𝑡) | cos (𝜙𝑑 𝑗 (𝑡)+Δ𝜙𝑗, 𝑘 )
(27)
where Δ𝐸𝑗(𝑡)= Λ𝑗−𝐸𝑗(𝑡),𝐵𝑗 , 𝑘 =|𝐵𝑗 ,𝑘 |ejΔ𝜙𝑗, 𝑘 =
hT𝚯0(𝑗, 𝑡)g𝑘and 𝚯0(𝑗, 𝑡)is obtained from 𝚯(𝑗 , 𝑡)when
𝜓𝑘(𝑡)=0in (9).
3.2 Data Acquisition Sub-Problem
The Data acquisition sub-problem is given by:
min
𝑖𝑘(𝑡)
𝐾
𝑘=1(𝜇1𝑘𝐷𝑘(𝑡) + 𝜇2𝑘𝜖𝑘(Λ𝑘−𝐸𝑘(𝑡)) − 𝑉)𝑖𝑘(𝑡).(28)
s.t. (10)
Problem (28) is constrained linear problem and its so-
lution is:
𝑖𝑘(𝑡)=(𝑖𝑘, max, 𝑉 ≥𝜇1𝑘𝐷𝑘(𝑡) + 𝜇2𝑘𝜖𝑘(Λ𝑘−𝐸𝑘(𝑡))
0, 𝑉 < 𝜇1𝑘𝐷𝑘(𝑡) + 𝜇2𝑘𝜖𝑘(Λ𝑘−𝐸𝑘(𝑡))
(29)
3.3 Information Transmission Sub-Problem
The information transmission sub-problem is given as:
min
𝜏𝑘(𝑡)−
𝐾
𝑘=1
𝜇1𝑘𝐷𝑘(𝑡)𝑜𝑘(𝑡) + 𝜇2𝑘(𝐸𝑘(𝑡) − Λ𝑘)𝑃𝑘𝜏𝑘(𝑡)𝑇 .
(30)
s.t. (1)and (2)
Problem (30) is a convex problem with closed form
solution given by:
𝜏𝑘(𝑡)=(1/2, 𝑘 =argmax𝑢𝑄(𝑢)and 𝑓 𝑓 (𝑘)>0
0,otherwise (31)
where 𝑄(𝑢)=𝜇1𝑢𝐷𝑢(𝑡)𝑊log21+𝑃𝑢|hT𝚯IT (𝑢,𝑡 )g𝑢+ℎ𝑑𝑢 (𝑡) |2
𝑁0𝑊−
𝜇2𝑢(Λ𝑢−𝐸𝑢(𝑡))𝑃𝑢.
3.4 LDPP Method Resource Allocation
The LDPP method for our paper is named IRS aware
LDPP algorithm (ILA). Using (27), (29), (31) and the equa-
tions (13) and (16) describing the dynamics of the buffers,
the complete ILA algorithm is shown in Algorithm 1.
Algorithm 1. ILA Algorithm.
Initialize: 𝐷𝑘(0),𝐸𝑘(0) ∀𝑘. Set 𝑡=1;
repeat
Step 1: Choose 𝜓𝑘(𝑡)based on (27)
Step 2: Choose 𝑖𝑘(𝑡)based on (29)
Step 3: Choose 𝜏𝑘(𝑡)based on (31)
Step 4: Update 𝐷𝑘(𝑡)based on (13)
Step 5: Update 𝐸𝑘(𝑡)based on (16)
Step 6: Set 𝑡=𝑡+1
until 𝑡≤𝑀
RADIOENGINEERING, VOL. 31, NO. 3, SEPTEMBER 2022 387
4. Numerical Results
In this section, we illustrate the performance of the ILA
algorithm presented in Algorithm 1, termed "Proposed ILA
algorithm".
The area of each IRS antenna element is set to
𝐴=(𝜆/4)2with 𝜆=0.1m (corresponding to the car-
rier frequency of 3 GHz). The IRS consists of 𝑁=2500
IRS antenna elements. The BS transmit power is set to
𝑃0=4W, and the BS thermal noise power density is set to
𝑁0=10−19 W/Hz. The BS is placed at an angle of 𝛼0=𝜋/4.
The EHUs are uniformly distributed along an arc of radius
𝑑𝑘within a range of polar angles from 𝜋/6to 𝜋/3. For the
direct path, we assume that the average power gain depends
on the distance as Ω𝑑𝑘 =10−3𝑑−3
𝑑𝑘 , where 𝑑𝑑 𝑘 is the dis-
tance of the direct path between the BS and the 𝑘th EHU
(𝑑𝑑𝑘 is calculated based on 𝑑BS,𝑑𝑘,𝛼0and 𝛼𝑘using the
cosine theorem).
The capacity of the data buffers of each EHU is set to
109bits, whereas the capacity of the energy buffer of each
EHU is set to Λ𝑘=1J. The initial values for the energy and
data buffers are set to 10% of their maximum value. The
number of TDMA frames is set to 𝑀=106and 𝜂𝑘=0.9∀𝑘.
Firstly, we compare the performance of the proposed
algorithm with a corresponding scheme based on conven-
tional convex optimization developed in [12] (denoted by the
"Benchmark IRS"). The BS and the EHUs are placed along
an arc centered at the IRS with radius 𝑑BS =𝑑𝑘=15 m. It
is assumed that the radiation pattern of the IRS in the bench-
mark is independent of the angle between the IRS and its
LoS to the corresponding node. Note, since all EHUs are at
same distance from the IRS, the max-min resource alloca-
tion considered in [12] reduces to the max-sum-rate resource
allocation that corresponds to the objective function of the
resource allocation problem (17). Note, the scheme consid-
ered in [12] assumes that the IRS can attain a very narrow
beam aimed exactly at the targeted EHU, such that other
EHUs cannot harvest RF energy outside their dedicated EH
sub-slots.
Additionally, to facilitate a fair comparison, we set
Ω𝑘𝑑 =0, as the benchmark scheme ignores the direct LoS
between BS and EHUs. Since the benchmark scheme also
ignores the information acquisition process, such that the
sensory data is always available to the EHUs, we also set
𝜖𝑘=0,∀𝑘. Additionally, to ensure that the acquired data
is equal to the transmitted data we choose 𝑉,𝜇1and 𝜇2to
keep the buffers in "Proposed ILA algorithm" at levels very
close to their initial levels. The transmit power of the EHUs
was set to 4μW and the communication channel bandwidth
was set to 1MHz. As a performance metric, we adopt the
system’s sum achievable rate which for the "Proposed ILA
algorithm" is defined by:
𝑅sum =1
𝑊
1
𝑀
𝐾
𝑘=1
𝑀−1
𝑡=0
𝑖𝑘(𝑡).
The performance comparison is shown in Fig. 2. Fig-
ure 2 shows that our proposed scheme significantly outper-
forms the benchmark for 𝐾≥10. The performance gap
between the proposed and the benchmark scheme increases
with increasing the number of EHUs, 𝐾. In the proposed
scheme, as 𝐾increases, the distance among EHUs decreases,
the amount of harvested energy by the neighboring EHUs in
a given EH sub-slot increases, and, thus, the sum achievable
rate increases. On the other hand, the performance of the
benchmark scheme is independent of 𝐾, because its problem
formulation assumes that during a given EH sub-slot, only
one specific EHU collects the broadcasted RF energy through
the IRS, while the other EHUs do not collect RF energy no
matter how close they are to the EHU to which the beam
is directed.
Next, we compare "Proposed ILA algorithm" for three
different system settings: 1. "Symmetric IRS assisted
WPCN", 2. "IRS assisted WPCN" and 3. "WPCN without
IRS", respectively. Specifically, "Symmetric IRS assisted
WPCN" is obtained when 𝚯EH (𝑘, 𝑡)is the same as 𝚯IT (𝑘 , 𝑡)
i.e. 𝜓𝑘(𝑡)=−𝜙𝑑𝑘 (𝑡). The "IRS assisted WPCN" is obtained
when in Step 1 of Algorithm 1, the value of 𝜓𝑘(𝑡)is set to
𝜓𝑘(𝑡)=0. This gives a simple focusing of the IRS beam
towards EHU 𝑘but doesn’t consider for the influence of the
BS-EHU 𝑘direct path. The setting "WPCN without IRS" is
obtained when no IRS is present. We specifically study the
average sum of the acquired data:
𝐼sum =1
𝑀
𝐾
𝑘=1
𝑀−1
𝑡=0
𝑖𝑘(𝑡).
We set 𝜇1𝑘=1.6×10−10 and 𝜇2𝑘=5×1012. If
not stated otherwise we set 𝑉=10000. Also, the max-
imum amount of collected data at the sensors is set to
𝑖𝑘, max =10 kbit ∀𝑘, and, the monitoring power consump-
tion is set to 𝜖𝑘=0.01 mJ/kbit ∀𝑘as in [19] and [21], which
corresponds to acquisition rates for temperature, glucose and
accelerometer and the typical acquisition power consump-
tion. Additionally, the transmission power of each EHU is
set to 𝑃𝑘=20 nW ∀𝑘. The bandwidth of the communication
session is set to 100 kHz and 𝑑𝑘=5 m ∀𝑘.
5 10 15 20 25 30
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Number of EHUs
Sum rate (bits/symbol)
Optimal IRS assisted WPCN P0=4W
Optimal IRS assisted WPCN P0=2W
Benchmark P0=4W
Benchmark P0=2W
Fig. 2. The sum rate 𝑅sum v.s. the number of EHUs.
388 S. PEJOSKI, Z. HADZI-VELKOV, T. SHUMINOSKI, LYAPUNOV DRIFT-PLUS-PENALTY BASED RESOURCE ALLOCATION IN .. .
3 4 5 6 7 8 9
0
0.5
1
1.5
2
2.5
3
3.5
4x 104
dBS[m]
Average sum of the acquired data in bits
Proposed ILA algorithm
Symmetric IRS assisted WPCN
IRS assisted WPCN
WPCN without IRS
Fig. 3. The average sum of the acquired data 𝐼sum v.s. 𝑑BS for
𝐾=15.
5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
3.5 x 104
Number of EHUs
Average sum of the acquired data in bits
Proposed ILA algorithm
Symmetric IRS assisted WPCN
IRS assisted WPCN
WPCN without IRS
Fig. 4. The average sum of the acquired data 𝐼sum v.s. 𝐾for
𝑑BS =5m.
0 2 4 6 8 10
x 105
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Session duration (in TDMA frames)
Energy buffer level in J
V = 1000
V = 10000
V = 25000
V = 40000
(a) Energy buffer.
0 2 4 6 8 10
x 105
0
1
2
3
4
5
6
7
8x 108
Session duration (in TDMA frames)
Data buffer level in bits
V = 1000
V = 10000
V = 25000
V = 40000
(b) Data buffer.
Fig. 5. The energy and data buffers for the EHU with 𝑘=1for the session duration parametrized with different values of 𝑉for 𝐾=15.
Figure 3 shows the 𝐼sum as a function of 𝑑BS . With
the increase in 𝑑BS the average sum of the acquired data is
reduced. For the considered scenario, the introduction of
IRS increases the 𝐼sum approximately twice, and the opti-
mal choice of 𝜓𝑘(𝑡)increases 𝐼sum for approximately another
50% compared to the "WPCN without IRS". The optimal
choice of 𝜓𝑘(𝑡)leads to performance improvement compared
to the choice 𝜓𝑘(𝑡)=−𝜙𝑑𝑘 (𝑡)which justifies the different
IRS reflection matrices for uplink and downlink.
In Fig. 4, the average sum of the acquired data for differ-
ent number of EHUs is shown. The increase in the number
of EHUs increases 𝐼sum . "Proposed ILA algorithm" due to
its awareness for the influence of the EH phase of one EHU
to the EH process of other EHUs allows for highest value of
𝐼sum over the whole considered range of 𝐾.
In Fig. 5, the energy and data buffer levels of the EHU
with 𝑘=1for the session duration are shown. The results
in the figure are parametrized for different values of 𝑉. It
should be noted that similar figures with small variations
can be found for the EHUs with different values of 𝑘. We
have found that for the analyzed scenario, the values for 𝑉
can range from 1000 to 45000 for the buffers to be away
from overloading or emptying. Nonetheless, those ranges
for 𝑉should be observed in correlation with 𝜇1𝑘,𝜇2𝑘and
the analyzed scenario. In Fig. 5 three regions in the figures
can be observed. In the first region the EHU energy buffer
approaches its steady level, in the second region, the data
buffer approaches its steady level, and in the third region
both buffers are at their steady levels. The figures show that
the increase in 𝑉leads to increase in the data buffer level,
and decrease in the energy buffer level. This comes from the
chosen Lyapunov function in (18). Namely, for small values
of 𝑉, the influence of the Lyapunov drift in (24) is stronger
than the penalty influence. Thus, the buffers are closer to the
values obtained by just minimizing the Lyapunov function.
As 𝑉increases the penalty becomes more important and the
buffer levels move in opposite direction.
5. Conclusion and Future Work
The paper uses the Lyapunov drift-plus-penalty opti-
mization technique for resource allocation and IRS parame-
ters optimization in an IRS assisted wireless powered com-
munication networks, composed of EHUs with limited en-
ergy buffers and data buffers. The provided solution is of low
complexity, can be used in presence of a direct communica-
tion link between BS and EHUs, optimizes the IRS assisted
EH process and maximizes the average acquired sensory data.
RADIOENGINEERING, VOL. 31, NO. 3, SEPTEMBER 2022 389
By presenting the average sum of the acquired data for differ-
ent numbers of EHUs and for different positions of the base
station the paper shows the superiority of the optimal solution
compared to the benchmarks. Additionally, by presenting the
energy and data buffers for the EHUs, it is shown that the in-
crease in the control parameter V leads to increasing of the
data buffer level, and to decrease in the energy buffer level.
In our future work we will develop enhanced resource allo-
cation schemes for more practical communication systems,
including multi-antenna BS, non-orthogonal multiple access
(NOMA), and active IRS vs. passive IRS comparison.
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About the Authors . . .
Slavche PEJOSKI received his bachelor, master and Ph.D.
degrees from the Ss. Cyril and Methodius University in
Skopje, N. Macedonia, in 2007, 2010, and 2015, respec-
tively. He is currently an Associate Professor at Ss. Cyril
and Methodius University in Skopje, N. Macedonia.
Zoran HADZI-VELKOV is a Professor of Telecommunica-
tions Engineering at the Ss. Cyril and Methodius University
in Skopje, Macedonia. Between 2012 and 2014, he was
visiting professor at the University of Erlangen-Nuremberg,
Germany. Between 2012 and 2016, he served on the editorial
board of the journal IEEE Communications Letters.
Tomislav SHUMINOSKI was born in Struga, Macedonia.
He received the B.Sc. (2008), M.Sc. (2010) and Ph.D. (2016)
degrees from the Faculty of Electrical Engineering and Infor-
mation Technologies, University Ss. Cyril and Methodius in
Skopje, N. Macedonia, where he is an Associate Professor.
