# Age of Information in Energy Harvesting Aided Massive Multiple Access
Networks

Zhengru Fang, Jingjing Wang,
Yong Ren, Zhu Han,
H. Vincent Poor, and Lajos Hanzo

###### Abstract

Given the proliferation of the massive machine type communication devices
(MTCDs) in beyond 5G (B5G) wireless networks, energy harvesting (EH) aided
next generation multiple access (NGMA) systems have drawn substantial
attention in the context of energy-efficient data sensing and transmission.
However, without adaptive time slot (TS) and power allocation schemes, NGMA
systems relying on stochastic sampling instants might lead to tardy actions
associated both with high age of information (AoI) as well as high power
consumption. For mitigating the energy consumption, we exploit a pair of
sleep-scheduling policies, namely the multiple vacation (MV) policy and start-
up threshold (ST) policy, which are characterized in the context of three
typical multiple access protocols, including time-division multiple access
(TDMA), frequency-division multiple access (FDMA) and non-orthogonal multiple
access (NOMA). Furthermore, we derive closed-form expressions for the MTCD
system’s peak AoI, which are formulated as the optimization objective under
the constraints of EH power, status update rate and stability conditions. An
exact linear search based algorithm is proposed for finding the optimal
solution by fixing the status update rate. As a design alternative, a low
complexity concave-convex procedure (CCP) is also formulated for finding a
near-optimal solution relying on the original problem’s transformation into a
form represented by the difference of two convex problems. Our simulation
results show that the proposed algorithms are beneficial in terms of yielding
a lower peak AoI at a low power consumption in the context of the multiple
access protocols considered.

###### Index Terms:

Multiple access protocols, age of information (AoI), machine type
communication devices (MTCDs), energy harvesting (EH), concave-convex
procedure (CCP).

††This work was supported in part by National Key R&D Program of China (No.
2020YFD0901000), in part by the National Natural Science Foundation of China
(No. 62071268) and in part by the Young Elite Scientist Sponsorship Program by
CAST (No. 2020QNRC001). The work of Z. Han was partially supported by NSF
CNS-2128368, CNS-2107216, Toyota and Amazon. Moreover, L. Hanzo would like to
acknowledge the financial support of the Engineering and Physical Sciences
Research Council projects EP/P034284/1 and EP/P003990/1 (COALESCE) as well as
of the European Research Council’s Advanced Fellow Grant QuantCom (Grant No.
789028). (Corresponding author: Jingjing Wang.)††Z. Fang and Y. Ren are with
the Department of Electronic Engineering, Tsinghua University, Beijing,
100084, China. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Wang is with the School of Cyber Science and
Technology, Beihang University, Beijing 100191, China. Email:
<EMAIL_ADDRESS>Han is with the Department of Electrical and
Computer Engineering in the University of Houston, Houston, TX 77004 USA, and
also with the Department of Computer Science and Engineering, Kyung Hee
University, Seoul, South Korea, 446-701. E-mail<EMAIL_ADDRESS>V.
Poor is with the Department of Electrical and Computer Engineering, Princeton
University, Princeton, NJ 08544 USA. E-mail<EMAIL_ADDRESS>Hanzo is
with the School of Electronics and Computer Science, the University of
Southampton, Southampton SO17 1BJ, U.K. E-mail<EMAIL_ADDRESS>
##  I Introduction

###  I-A Background

Machine type communication devices (MTCDs) are set to proliferate to 70+
billion in wireless networks by 2025[1]. As a further trend, compelling new
applications, such as Industry 4.0, telemedicine and automatic driving are
finding their way into the technical requirements of wireless systems[2, 3].
Many of the above-mentioned applications are time-sensitive and/or power-
constrained, especially the family of sophisticated Internet of Things (IoT)
applications[4]. In this context next generation multiple access (NGMA)
techniques play a critical role in terms of supporting large-scale user access
as well as guaranteeing a high quality of experience.

For the sake of quantifying the ‘freshness’ of the received data, the age of
information (AoI) metric has been proposed and employed in a wide variety of
wireless networks[5], which quantifies the time elapsed since the generation
instant of the latest received status update at a destination[6]. In the IoT
era, ubiquitous MTCDs are often deployed for monitoring dynamic environmental
information[7]. To elaborate, the data collected by MTCDs can be transmitted
to so-called information fusion nodes in real-time for further analysis,
control signal extraction as well as for remote data reconstruction, where the
‘freshness’ of data directly influences the performance of real-time MTCD-
aided wireless applications[8].

These MTCDs are often characterized by comparatively small amount data as well
as by AoI-sensitive nature in typical data-collection oriented uplink (UL)
communications. In order to maintain the freshness of the data, MTCDs have to
periodically transmit their UL packets at a given frequency and data rate,
which may result in the premature depletion of their limited batteries as well
as in buffer-overflow[9]. From an economical and feasibility perspective,
disposable batteries are inadequate for scalable long-term monitoring[10]. As
a remedy, energy harvesting (EH) techniques constitute a low-cost renewable
energy source for MTCDs, which gleans energy from the ambient radio frequency
(RF) signals.

However, due to the inevitable signal attenuation and the stochastic nature of
status updates, it remains a challenge to meet the AoI requirements in the
context of time- and/or energy-sensitive applications. Specifically, in the
open literature of massive access networks, most researchers primarily
considered the network’s throughput and power consumption. However, there is a
paucity of literature on the AoI, which is a complex function of both
throughput and power consumption. Hence it is imperative to design efficient
multiple access protocols for AoI-sensitive EH-aided MTCD networks[11]. For
instance, EH-aided networks relying on time-division multiple access
(TDMA)[12], frequency-division multiple access (FDMA)[13] and non-orthogonal
multiple access (NOMA)[14] have been widely investigated both in terms of
their power consumption and throughput, which are closely linked to their
power control and resource allocation. In EH based multiple access networks,
the customizable parameters typically include the transmission slot length,
the status update rate and the EH parameters. In order to minimize the average
of the system’s peak AoI, we have to determine the optimal time-slot duration
and power allocation under the specific power consumption constraint of the
EH-aided MTCD networks considered.

###  I-B State-of-the-art

In this subsection, we review the related literature of EH-based massive
multiple access networks, with an emphasis on their energy-efficient resource
allocation, massive multiple access protocols and AoI-oriented optimization
schemes.

(1) Energy-efficient resource allocation: Numerous energy-efficient resource
allocation schemes have been designed for EH[15, 16, 17, 18, 19]. To minimize
the average outage probability at a finite battery capacity, Hu et al. [15]
harnessed the classic Lyapunov optimization for transforming the long-term
optimization problem into subproblems dedicated to each single TS. However,
the drift-plus-penalty algorithm relying on Lyapunov optimization failed to
obtain a desirable solution for devices equipped with small storage space. In
[16], Xia et al. formulated an EH-aided mobile edge computing based offloading
system and solved an online distributed battery energy management optimization
problem. However, their work did not consider the sleep-scheduling issues of
small IoT devices. Considering an amplify-and-forward network relying on EH
and relay nodes, Singh et al. [17] creatively harnessed the relaying networks
relying on EH source nodes and jointly optimized the power allocation and
throughput by solving a non-convex problem. However, since the scenario only
considered a single source node and a single relay node, the proposed methods
may not apply to multiple access aided networks.

(2) EH aided multiple access: Qiao et al. [20] optimized the throughput of an
EH-based TDMA system with the aid of superposition coding in conjunction with
fixed/variable decoding order. Vranas et al. [21] proposed a resource
allocation scheme relying on a non-linear EH model for the uplink and they
revealed that NOMA had a lower circuit power dissipation than TDMA, in the
context of no more than 3 users. As a further development, Bouzinis et al.
[22] jointly optimized decentralized power transfer and radio access based on
both NOMA and TDMA schemes. However, since the optimization of energy cost was
ignored, the proposed system was unable to extend the lifetime of nodes.
Moreover, Li et al. [23] focused their attention on EH-aided mobile edge
computing and solved the near-far problem of a NOMA-aided cooperative scheme,
which may not be realistic, since it disregarded the switching cost of the
time-division structure.

(3) AoI-oriented optimization: The AoI-oriented optimization of EH based
networks has been widely investigated in the literature[24, 25, 26, 27, 28,
29, 30, 31, 32]. Specifically, Wu et al. [24] designed an optimal online
status update scheme for minimizing the average AoI for various battery sizes
in a continuous-time setting. To overcome the ‘selfishness’ of the sensors and
to formulate an attractive AoI upgrade strategy, Zheng et al. [25] constructed
a Stackelberg game based optimization problem and solved it via the classic
Lagrangian method, where the worst case of the AoI was not considered. In
[26], Krikidis et al. studied a low-complexity EH regime, where a closed-form
AoI expression was derived as a function of the storage capacitor’s size.
Additionally, Maatouk et al.[31] investigated the AoI performance relying on
NOMA and on orthogonal multiple access (OMA). Similarly, Chiariotti et al.
[32] analyzed the AoI performance of TDMA and ALOHA-based schemes under both
OMA and NOMA frameworks. However, the aforementioned contributions have not
considered the fairness requirements of users.

###  I-C Motivation and New Contributions

TABLE I: Contrasting Our Contribution to the Litterature | [9] | [11] | [20] | [22] | [23] | [25] | [26] | [29] | [27] | [28] | [30] | Proposed work
---|---|---|---|---|---|---|---|---|---|---|---|---
Power allocation | ✓ | ✓ | ✓ |  | ✓ | ✓ | ✓ | ✓ | ✓ |  |  | ✓
TS allocation | ✓ | ✓ |  | ✓ | ✓ | ✓ |  |  | ✓ | ✓ |  | ✓
TDMA protocol |  | ✓ | ✓ | ✓ |  |  |  |  | ✓ |  | ✓ | ✓
FDMA protocol |  |  |  |  |  |  |  |  |  |  | ✓ | ✓
NOMA protocol | ✓ | ✓ |  | ✓ | ✓ |  |  |  | ✓ |  |  | ✓
AoI optimization |  |  |  |  |  | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Sleep-scheduling |  |  |  |  |  |  |  | ✓ |  | ✓ |  | ✓

Most of the aforementioned teatises assume that the transmitters of the EH
aided networks are able to operate sustainably without any sleep-scheduling
policy. By contrast, in many realistic scenarios, MTCDs have to rely on sleep-
scheduling to avoid excessive power dissipation. However, the additional
queuing delay caused by the sleep-scheduling policy might degrade the AoI
performance of our system. Furthermore, we assume that the status updates are
random. For instance, the randomness of data generation may arise from the
constraints of energy harvesting systems relying on a limited battery, or due
to an application-specific feature that needs random data sampling, hence
affecting both the peak AoI and the overall power consumption. In contrast to
the previous studies only focusing on throughput optimization, in this
contribution, we propose an AoI-optimal and energy-efficient resource
allocation scheme for EH-aided massive multiple access networks, beneficially
optimizing the TS association of data transmission, energy harvesting and the
system’s idle mode. Against this above backdrop, our main contributions are as
follows:

  * •

To the best of our knowledge, this is the first contribution optimizing both
the peak AoI and the power consumption of EH aided large-scale multiple access
networks, where the MTCDs adaptively switch between their active and idle mode
in different time slots (TS). Under the assumption of stochastic status
updates, we utilize a pair of different sleep-scheduling policies, namely the
multiple vacation (MV) policy and the start-up threshold (ST) policy defined
in[33] for the sake of mitigating the power consumption, while relying on a
queueing system. Then, we derive a closed-form expression for the average of
the users’ peak AoI in three different access schemes.

  * •

We design a beneficial TS allocation method for striking a trade-off between
the average of the users’ peak AoI and the power consumption. To solve the
original non-convex optimization problem, we decompose it into a set of convex
subproblems. By relying on an exact linear search method, the optimal energy
and TS allocation scheme is obtained. Moreover, by converting the original
problem into a typical difference of convex (DC) programming problem, we find
its near-optimal solution by the convex-concave procedure (CCP) at a low
complexity.

  * •

Numerical results show that both the MV policy and the ST policy are capable
of reducing the power consumption compared to the benchmark schemes. As for
different access schemes, we may conclude that the MV-based NOMA policy
outperforms all the others in terms of both the users’ average peak AoI and
power consumption.

Our new contributions are boldly and explicitly contrasted to the literature
at a glance in Table I.

###  I-D Organization

The remainder of this article is structured as follows. The system model is
detailed in Section II. In Section III, closed-form expressions for the
average peak AoI based on different multiple access protocols are derived. The
associated non-convex problems and their solutions are investigated in Section
IV. In Section V, we provided the performance analysis of the proposed schemes
relying on different sleep-scheduling policies, followed by our conclusions in
Section VI.

##  II System Model

TABLE II: Summary of Key Notations  Notation |  Definition
---|---
$N$, $M$ |  Total number of MTCDs, start-up threshold for the ST policy
$\tau_{p}$, $\tau_{t}$ |  Energy harvesting slot, transmission slot
$\tau_{s}$, $\tau_{sc}$ |  Duration of the idle mode and of the switching cost
$h_{p,n}$, $h_{t,n}$ |  Channel coefficient between the power station and the MTCD, channel coefficient between the BS and the MTCD
$\varPhi_{t,n}$, $\varPhi_{th}$ |  Transmit power of the MTCD, upper bound of the power station’s charging power
$\overline{\varPhi}$, $\varPhi_{w}$ |  Average power consumption of the MTCD, power consumption in the active mode
$\varPhi_{s}$, $\varPhi_{sc}$ |  Power consumption in the idle mode, power consumption of the switching cost
$B$, $N_{0}$ |  Available bandwidth, noise spectral density
$R_{ub,n}$, $C_{n}$ |  Upper bound of channel capacity, actual average throughput
${d}_{n}^{M-B}$, $L$ |  Distance of a pair of the BS-MTCD link, length of each packet
$G_{n}$, $D_{n}$ |  Interval of the data generation, system delay of each packet
$\lambda_{n}$, $\varUpsilon_{n}$ |  Packet arrival rate, energy efficiency
$A_{p}^{MV}(\cdot)$, $A_{p}^{ST}(\cdot)$ |  Average peak AoI function for the MV policy and for the ST policy
$\overline{A}$, $A_{p}$ |  Average AoI, peak AoI
TABLE III: Summary of Key Acronyms Acronym |  Definition
---|---
MV, ST |  Multiple vacation policy, start-up threshold policy
TM, TS |  Time-division multiple access (TDMA) protocol with the MV policy, TDMA protocol with the ST policy
FM, FS |  Frequency-division multiple access (FDMA) protocol with the MV policy, FDMA protocol with the ST policy
$NM$, $NS$ |  Power-domain non-orthogonal multiple access (PD-NOMA) protocol with the MV policy, PD-NOMA protocol with the ST policy

###  II-A Network Architecture

The EH-aided massive multiple access network considered is portrayed in Fig.
1, which consists of a wireless power station as an EH source, a base station
(BS) for data collection and $N$ MTCDs denoted by
$n\in\mathcal{N}\stackrel{{\scriptstyle\triangle}}{{=}}\\{1,2,\cdots,N\\}$.
Equipped with a single antenna, each MTCD senses the surrounding environment
for collecting information and transmits it to the BS. Moreover, the power
station recharges the MTCDs, which are also equipped with RF-based EH modules.
Bearing in mind the independence of each data packet, we assume that all the
status updates of each MTCD obeys a Poissonian process. For the sake of
reducing the power consumption, each MTCDs relies on a specific sleeping
scheduling, i.e. each MTCD turns the transmit power off, if its data
transmission queue is empty111When the transmitter is active or in idle mode,
the latest data is queued instead of being transferred. When the queue is
empty, the MTCD enters into idle mode.. To reduce the co-channel interference,
the procedure of EH and data transmission of each MTCD is operated in separate
slots. Thus, each MTCD has an active mode and idle mode222For each mode, the
environment sensing is uninterrupted, relying on MTCDs’ low-power dissipation
sensors., Furthermore, the active mode is partitioned into two slots, namely
the EH slot $\tau_{p}$ and the transmission slot $\tau_{t}$. In this paper,
TDMA, FDMA and NOMA are investigated in our EH multiple access system. For
readability, the important notations and acronyms used in this paper are
listed in Table II and Table III, respectively.

Figure 1: EH-aided massive access MTCD network.

###  II-B Energy Harvesting Model

In our model, a power station charges the MTCDs relying on EH techniques. Let
$\tau_{p,n}^{\left[i\right]}$ denote the duration of EH between the power
station and the $n$th MTCD, where $\left[i\right]$ represents the index of the
status update. Moreover, let the constant $h_{p,n}\left(t\right)$ be the
channel coefficient of the wireless links between the power station and the
$n$th MTCD. At the time instant $t$, the EH power of the $n$th MTCD’s receiver
can be formulated as

$\displaystyle\varPhi_{r,n}^{\left[i\right]}\left(t\right)=\min\left\\{\eta_{k}\varPhi_{wp,n}^{\left[i\right]}h_{p,n},\varPhi_{th}\right\\},\
\forall n\in\mathcal{N},$ (1)

where $0<\eta_{p}<1$ is the EH efficiency, which substantially depends on the
specific hardware components. Furthermore,
$\varPhi_{wp,n}^{\left[i\right]}\left(t\right)$ denotes the charging power of
the power station directed towards the $n$th MTCD, while $\varPhi_{th}$ is the
upper bound of the power station’s charging power. Let $\tau_{p}^{[i]}$ be the
duration of energy harvesting for the $i$th status update. Thus, the energy
received by the MTCD from the power station for a single status update is
formulated as:

$\displaystyle
E_{h}^{[i]}=\int_{0}^{\tau_{p}^{[i]}}{\varPhi_{r,n}^{\left[i\right]}\left(t\right)dt},\
\forall n\in\mathcal{N}.$ (2)

Furthermore, the upper-bound of the battery capacity in the MTCD is set to
$C_{ub}$ which has the unit of Joule (J). While the MTCD is activated for the
$i$th status update, the conservation of energy requires that the energy
received by the MTCD must satisfy $E_{h}^{[i]}=C_{ub}$.

###  II-C Transmission Model and Channel Analysis

The most energy harvested by the MTCD from the BS is used for data
transmission. Let $\varPhi_{t,n}^{\left[i\right]}$ and
$\varPhi_{a,n}^{\left[i\right]}$ denote the power of data transmission, and
that of active state in the $i$th status update, respectively. Furthermore,
the associated battery cost values are $C_{t}$ and $C_{a}$, respectively.
Therefore, the energy constraint of the MTCD for the $i$th status update
yields
$\int_{0}^{\tau_{t}^{[i]}}{\varPhi_{t,n}^{\left[i\right]}\left(t\right)dt}=C_{t}$
and
$\int_{0}^{\tau_{t}^{[i]}}{\varPhi_{a,n}^{\left[i\right]}\left(t\right)dt}=C_{a}$.
Likewise, the energy constraint of the MTCD for the idle mode is given by
$\int_{0}^{\tau_{s}^{[i]}}{\varPhi_{s,n}^{\left[i\right]}\left(t\right)dt}=C_{s}$.
Moreover, we assume that $C_{ub}=C_{t}+C_{a}+\varepsilon C_{s}$, where
$\varepsilon>0$ denotes the coefficient of EH for the idle mode. In the
scenario considered, the MTCD only transmits a single packet having the length
of $L$ bits in a transmission time slot, and we assume that all MTCDs have the
same transmission rate. Hence, the upper bound of channel capacity should
satisfy:

$\displaystyle
R_{ub,n}=\frac{\left(\mathcal{J}+1\right)L}{\tau_{t}^{[i]}}>\mathcal{C}_{n},$
(3)

where $\mathcal{C}_{n}=\frac{L}{\tau_{b}^{[i]}}$ is the actual average
throughput of the $n$th MTCD, and $\mathcal{J}>0$ denotes the gap between the
ideal capacity and practical throughput. In a Gaussian multiple-access
channel, $N$ independent MTCDs simultaneously transmit their status updates to
the BS relying on the assumption of the additive white Gaussian noise (AWGN).

(1) TDMA: When relying on TDMA, we can allocate the complete bandwidth $B$ of
the channel to all MTCDs, but these MTCDs only transmit packets for a limited
time period. Therefore, the upper bound of the channel capacity relying on
TDMA for the $n$th MTCD is given by:

$\displaystyle
R_{ub,n}^{T,\left[i\right]}\left(t\right)=\frac{B\tau_{t}^{[i]}}{\tau_{b}^{[i]}}\log_{2}\left(1+\frac{\varPhi_{t,n}^{\left[i\right]}\left(t\right)\left|h_{t,n}\left(t\right)\right|^{2}}{BN_{0}}\right),$
(4)

where $\left|h_{t,n}\left(t\right)\right|=10^{-3}\theta_{0}d_{n}^{-a_{0}}$ is
the channel coefficient of the wireless link between the BS and the $n$th MTCD
at time $t$, while $\theta_{0}$ and $d_{n}$ denote the small-scale fading
parameter and the link length, respectively. Let $a_{0}$ denote the path loss
of the link, $B$ represents the available bandwidth and $N_{0}$ is the noise
spectral density.
(2) FDMA: As for FDMA, the available bandwidth is partitioned into $N$
subchannels, and the associated channel capacity of the $n$th MTCD is given
by:

$\displaystyle
R_{ub,n}^{F,\left[i\right]}\left(t\right)=\frac{B\tau_{t}^{[i]}}{N\tau_{b}^{[i]}}\log_{2}\left(1+\frac{\varPhi_{t,n}^{\left[i\right]}\left(t\right)\left|h_{t,n}\left(t\right)\right|^{2}}{(B/N)N_{0}}\right).$
(5)

(3) NOMA: As for power-domain NOMA (PD-NOMA), we assume that
$h_{t,1}\left(t\right)\geq h_{t,2}\left(t\right)\geq\cdots\geq
h_{t,N}\left(t\right)$. In order to ensure the successful detection of the
NOMA signal, we should control the transmit power of each MTCD to satisfy the
following conditions for successive interference cancellation (SIC) at the
BS’s receiver:

$\displaystyle{\varPhi_{t,j}^{\left[i\right]}\left(t\right)\gamma_{j}}-\sum_{k=j+1}^{N}{{\varPhi_{t,k}^{\left[i\right]}\left(t\right)\gamma_{k}\left(t\right)}}\geqslant\varPhi_{th}\
\ j=1,2,\cdots,N-1,$ (6)

where
$\gamma_{n}\left(t\right)=\frac{\left|h_{t,n}\left(t\right)\right|^{2}}{BN_{0}}$,
while $\varPhi_{th}$ is the minimum power required for successfully decoding
the NOMA signal received. Therefore, the signal to interference plus noise
ratio (SINR) for the $n$th MTCD can be formulated as follows:

$\displaystyle\varTheta_{n}^{\left[i\right]}\left(t\right)=\begin{cases}\frac{\varPhi_{t,n}^{\left[i\right]}\left(t\right)\gamma_{n}\left(t\right)}{1+\sum_{j=n+1}^{N}{\varPhi_{t,j}^{\left[i\right]}\left(t\right)\gamma_{j}\left(t\right)}}&1\leq
n\leq N-1,\\\
\varPhi_{t,n}^{\left[i\right]}\left(t\right)\gamma_{n}\left(t\right)&n=N.\\\
\end{cases}$ (7)

Because only $\left(\tau_{t}^{[i]}/\tau_{b}^{[i]}\right)\times 100\%$ TSs are
regarded as the legitimate transmission duration, the average transmission
rate of the $n$th MTCD is:

$\displaystyle
R_{ub,n}^{N,[i]}=\frac{B\tau_{t}^{[i]}}{\tau_{b}^{[i]}}\log_{2}\left(1+\varTheta_{n}^{[i]}\right).$
(8)

Note that the channel state information (CSI) is assumed to be perfectly known
both at the BS and at all the MTCDs. Then the CSI of BS can be estimated
relying on the downlink pilots at the user equipment (UE), while the CSI of
the MTCDs can be acquired through uplink channel estimation by the BS.
Furthermore, since the packet length $L$ is typically small and the time
between status updates is short, the channel can be regarded as having a near-
constant slow- and fast-fading[25].

##  III Peak AoI Model

Figure 2: Age of information of first-in-first-out (FIFO) single queue.

In this section, we carry out a closed-form worst case analysis of the
system’s AoI based on queueing theory in terms of three multiple access
protocols, namely TDMA, FDMA and NOMA. Additionally, we use different vacation
queues relying on a single server to model a pair of sleep-scheduling
policies. The AoI of the $n$th MTCD of the $i$th status update is illustrated
in Fig. 2, which can be defined by

$\displaystyle{A}_{n}^{[i]}\left(t\right)=t-\alpha^{[i]}_{n}\left(t\right),$
(9)

where $\alpha_{n}^{[i]}\left(t\right)=\alpha^{[i]}_{n}$ represents the
generation moment of the latest status information before $t$ in the BS, while
$\alpha_{n}^{\left[i\right]}$ and $\beta_{n}^{\left[i\right]}$ denote the
generation and transmission moment of the $i$th status update, respectively333
Taking into account the propagation speed of RF signal, the latency of the
transmission between the transmitter and BS can be negligible in comparison to
the system delay of the queue in each MTCD.. Thus, we assume
$\alpha_{n}^{[i]}<\beta_{n}^{[i]}$,
$\alpha_{n}^{\left[i\right]}\leq\alpha_{n}^{[i+1]}$ and
$\beta_{n}^{\left[i\right]}\leq\beta_{n}^{[i+1]}$. Moreover,
$A_{p,n}^{\left[i\right]}$ is the instantaneous peak AoI of the $n$th MTCD. In
the following section, the superscript “$[i]$” of the notations and
$\left(t\right)$ are omitted for simplicity, because the optimization
procedure for each status update is identical. Hence, $i$ can be omitted in
the following equation:

$\displaystyle\mathbb{E}\left[A_{p}\right]$
$\displaystyle=\mathbb{E}\left[\beta_{n}^{\left[i\right]}-\alpha_{n}^{\left[i-1\right]}\right]=\mathbb{E}\left[G_{n}\right]+\mathbb{E}\left[D_{n}\right].$
(10)

The average interval of the data generation can be expressed as
$\mathbb{E}\left[G_{n}\right]=\lim_{T\rightarrow\infty}T/\left(\sum_{i=1}^{\infty}\mathbbm{1}_{\\{\alpha_{n}^{\left[i\right]}<T\\}}\right)$,
where $\mathbbm{1}_{\\{\cdot\\}}$ is an indicator function, while
$\mathbb{E}\left[D_{n}\right]$ denotes the average system delay of each
packet444The system delay contains the sum of waiting time, transmission time
and idle period, while “average” indicates that we only consider their time-
averaged value under the queue’s stable condition.. Additionally, we use “per-
packet AoI” to evaluate the performing of AoI associated with per packet in EH
aided wireless networks. According to Section II-C in [25], the “per-packet
AoI” of the $i$th status update can be obtained by calculating the area of
$Q^{[i]}$ in Fig. 2 as follows:

$\displaystyle\overline{{A}}_{n}^{\left[i\right]}$
$\displaystyle=\frac{Q^{\left[i\right]}}{\beta_{n}^{\left[i\right]}-\beta_{n}^{\left[i-1\right]}}=\frac{Q^{\left[i\right]}}{2\left(G_{n}^{\left[i+1\right]}+D_{n}^{\left[i+1\right]}-D_{n}^{\left[i\right]}\right)}=\frac{D_{n}+A_{p}}{2}.$
(11)

Moreover, the expected value of the per-packet AoI is given by
$\mathbb{E}\left[\overline{A}_{n}\right]=\frac{\mathbb{E}\left[D_{n}\right]+\mathbb{E}\left[A_{p}\right]}{2}$,
which reflects the time average AoI performance. In the following section, we
further analyze the queueing system with heavily sleep-scheduling policies in
a first-in-first-out (FIFO) manner555Although last-in first-out (LIFO) queues
are well-known to minimize AoI in many situations, FIFO queues are the
standard queueing discipline in most wireless communication systems,
especially when the devices are used to achieve multiple services and not only
status updates..

###  III-A TDMA Protocol

Figure 3: An example for illustrating the TDMA with sleep-scheduling. TABLE IV: The Definition of Different Sleep-Scheduling Policies. Sleep-scheduling policy | Difference | Similarity
---|---|---
MV policy |  MTCD keeps on idle mode until it find at least one packet waiting in the queue at the end of one idle period. |  If the queue becomes empty, the MTCD turns off its transmitter and switches to idle mode.
ST policy |  MTCD keeps on idle mode until it find at least $M$ packets waiting in the queue.
Figure 4: Illustrative example of sleep-scheduling slot with switching cost in
MV policy.

In TDMA, several MTCDs share the same carrier frequency channel by
partitioning the transmission frame into multiple orthogonal TSs. In terms of
the energy consumption of battery-limited MTCDs, it is unwise to maintain the
power-on state of the device. Hence, we conceive a TDMA protocol based on
sleep-scheduling, as shown in Fig. 3. In TDMA, all MTCDs share the entire
bandwidth $B$ for transmitting their packets. It can be observed that a TDMA
time slot is divided into $(N+1)$ TSs, which contain a single EH TS and $N$
transmission TSs. Specifically, all MTCDs harvest energy in the first TS of
duration $\tau_{p}$. Moreover, each MTCD occupies a single TS of duration
$\tau_{t}$. When the queue in the MTCD is empty, the transmitter is turned off
to conserve energy. As shown in Table IV, the main difference between MV
policy and ST policy is the condition of queue starting transmission. To
differentiate these two policies, we use the superscripts $TM$ and $TS$ to
denote the parameters of TDMA associated with the MV and ST policy,
respectively.

(1) TDMA Using the MV Policy: To further analyze the corresponding model, we
first introduce the concept of MV policy. Specifically, if the MTCD is woken
up from its idle mode and finds that the queue remains empty, it returns to
its idle mode for the duration of $\tau_{s}^{TM}$, which can be calculated by
the sum of the switching cost $\tau_{sc}^{TM}$ and the idle period
$\tau_{i}^{TM}$. Furthermore, the illustration of the sleep-scheduling slot
along with the switching cost of our MV policy is shown in Fig. 4 666In MV
policy, MTCD needs a switching cost $\tau_{sc}$ before MTCD starts an idle
period $\tau_{i}$. Switching cost is consumed for the monitoring of the status
updates in the MTCD’s queue.. If however there is at least one packet in the
queue during the switching cost slot, the MTCD starts to transmit its status
updates. We assume that the generation of the status update of the $n$th MTCD
follows a Poissonian process with arrival rate $\lambda_{n}$, and its
transmission time is $\tau_{t}^{TM}$, while the duration of each TDMA time
slot without idle period is $\tau_{b}^{TM}=\tau_{p}+N\tau_{t}^{TM}$.
Therefore, the procedure of status update of the $n$th MTCD using the TDMA
protocol can be modeled by a MV aided queue defined in[33] 777To streamline
the terminology, we refer to the vacation period as an idle period..

######  Lemma 1:

If the arrival interval yields an exponential distribution and its service
time is a constant, the queueing system can be regarded as a classical M/D/1
queue. Therefore, it may be readily shown that the p.g.f. of the number of
packets when the packet depart in a FIFO manner as follows:

$\displaystyle\varPi\left(z\right)=\frac{\left(1-\lambda\tau\right)\left(1-z\right)e^{-\left(\lambda-\lambda
z\right)\tau}}{e^{-\left(\lambda-\lambda z\right)\tau}-z}.$ (12)

The average queue length may be expressed as

$\displaystyle\mathbb{E}\left[\varPi\right]=\lambda\tau+\frac{\left(\lambda\tau\right)^{2}}{2\left(1-\lambda\tau\right)},$
(13)

while the average system delay may be formulated as follows:

$\displaystyle\mathbb{E}\left[D_{0}\right]=\tau+\frac{\left(\lambda\tau\right)^{2}}{2\lambda\left(1-\lambda\tau\right)},$
(14)

where $\lambda$ and $\tau$ are the arrival rate and the service time of the
queue, respectively888The entire service time of each packet is referred to as
the time interval between when a packet is generated at the MTCD and when its
last bit is successfully received at the BS. Because of the power constraint
of our EH aided MTCD, it has a relatively low transmission rate. Therefore,
the transmission duration of the packets cannot be ignored here..

######  Proof:

See Section 2.1.1 in [33]. $\blacksquare$

For the MV policy, let $\varPi_{v,n}^{0}$ denote the number of the packets
generated during a single idle period for the $n$th MTCD, while the
probability distribution of $\varPi_{v,n}^{0}$ can be expressed as

$\displaystyle{\rm{Prob}}\left\\{\varPi_{v,n}^{0}=m\right\\}=\frac{\left(\lambda_{n}\tau_{s}^{TM}\right)^{m}e^{-\lambda_{n}\tau_{s}^{TM}}}{m!}.$
(15)

Let $\varPi_{v,n}$ denote the number of packets generated at the beginning of
a time slot. Furthermore, the probability distribution of $\varPi_{v,n}$ can
be formulated as:

$\displaystyle\rm{Prob}\left\\{\varPi_{v,n}=m\right\\}$
$\displaystyle=\frac{\left(\lambda_{n}\tau_{s}^{TM}\right)^{m}e^{-\lambda_{n}\tau_{s}^{TM}}}{m!\left[1-{\rm{Prob}}\left(\varPi_{v,n}^{0}=0\right)\right]}.$
(16)

where ${\rm{Prob}}\left(\varPi_{v,n}^{0}=0\right)=e^{-\lambda_{n}\tau_{s}}$
implies the probability that there is no packet generated during one idle
period and $m!=m(m-1)\cdots(1)$. Moreover, the p.g.f. of $\varPi_{v,n}$ can be
expressed as follows:

$\displaystyle\varPi_{v,n}\left(z\right)$
$\displaystyle=\frac{e^{-\lambda_{n}\tau_{s}^{TM}}}{\left(1-e^{-\lambda_{n}\tau_{s}^{TM}}\right)}\sum_{m=1}^{+\infty}{\frac{\left(\lambda_{n}\tau_{s}^{TM}\right)^{m}}{m!}z^{m}}=\frac{e^{\lambda_{n}\tau_{s}^{TM}z}-1}{e^{\lambda_{n}\tau_{s}^{TM}}-1}.$
(17)

Then, according to the proof of Lemma 1, the probability generating function
(p.g.f.) of the average number of packets in M/D/1 queue without MV policy is
formulated as

$\displaystyle\varPi_{0}^{TM}\left(z\right)=\frac{\left(1-\lambda_{n}{\tau_{b}^{TM}}\right)\left(1-z\right)e^{-\left(\lambda_{n}-\lambda_{n}z\right){\tau_{b}^{TM}}}}{e^{-\left(\lambda_{n}-\lambda_{n}z\right){\tau_{b}^{TM}}}-z}.$
(18)

Hence, we can determine the average number of packets by using the first-order
derivative of $\varPi_{0}^{TM}\left(z\right)$. According to [1], the average
number of packets associated with the MV policy can be derived as the sum of
two independent stochastic variables, i.e., that of the number of packets with
and without idle period. Therefore, the p.g.f. of the average number of
packets associated with the MV policy is given by

$\displaystyle\varPi_{v}^{TM}\left(z\right)=\varPi_{0}^{TM}\left(z\right)\cdot\frac{1-\varPi_{v,n}\left(z\right)}{\mathbb{E}\left[\varPi_{v,n}\right]\left(1-z\right)},$
(19)

which indicates that the average system delay of the MV policy can be
decomposed into the sum of the average system delay without the MV policy and
the additional delay due to the idle mode. Under the assumption of stability,
the average system delay of the $n$th MTCD using TDMA can be derived by using
the first-order derivative:

$\displaystyle\mathbb{E}\left[\varPi_{v}^{TM}\right]=\frac{2\lambda_{n}\tau_{b}+\lambda_{n}\tau_{s}\left(1-\lambda_{n}\tau_{b}\right)-\left(\lambda_{n}\tau_{b}\right)^{2}}{2\left(1-\lambda_{n}\tau_{b}\right)}.$
(20)

Based on Little’s Law, we can express the average system delay of the MV
policy as
$\mathbb{E}\left[D_{n}^{TM}\right]=\lambda_{n}^{-1}\mathbb{E}\left[\varPi_{v}^{TM}\right]$.
Upon substituting (20) into Eq. (10), the average interval of the status
update is $\mathbb{E}\left[G_{n}^{TM}\right]=1/\lambda_{n}$, while the average
peak AoI of the $n$th MTCD based on TDMA using the MV policy can be formulated
as

$\displaystyle\mathbb{E}\left[A_{p,n}^{TM}\right]$
$\displaystyle=\mathbb{E}\left[D_{n}^{TM}\right]+\mathbb{E}\left[G_{n}^{TM}\right]$
(21)
$\displaystyle=\underset{The\,\,age\,\,from\,\,M/D/1}{\underbrace{\tau_{b}^{TM}+\frac{\lambda_{n}\left(\tau_{b}^{TM}\right)^{2}}{2\left(1-\lambda_{n}\tau_{b}^{TM}\right)}+\lambda_{n}^{-1}}}+\underset{Additional\,\,age}{\underbrace{\frac{\tau_{s}}{2}}},$

where $\mathbb{E}\left[A_{p,n}^{TM}\right]$ can be decomposed into the peak
AoI of the M/D/1 queue and the additional age of the idle period. Moreover,
for the sake of simplifying the expressions of the multiple access protocol
based on the MV policy, we define the average peak AoI function as
${A}_{p}^{MV}\left({}_{n},\tau_{b},\tau_{s}\right)=\mathbb{E}\left[A_{p,n}^{TM}\right]$.

Figure 5: The State transition of the queue for one MTCD based on TDMA with
the ST policy.

(2) TDMA Using the ST Policy: The ST policy indicates that the device in idle
mode is restarted when the total number of bits generated reaches a threshold.
With the assistance of sleep-scheduling, the energy dissipation can be
mitigated by periodically turning off the power of the MTCDs. However,
frequent powering on/off degrades the energy efficiency and hence the system’s
lifetime. Therefore, to reduce the frequency of start-up/shutdown, we improve
the TDMA protocol by introducing a start-up threshold value $M$ which
determines the number of packets in the buffer, before powering up for
transmission999In ST policy, the switching cost is ignored since the threshold
value $M$ is high enough for the extension of the idle period..

Let $\varPi^{ST}\left(t\right)$ denote the number of the packets at the time
instant $t$. As shown in Fig. 5, when the queue is empty at $t=t_{0}$, the
MTCD turns off its transmitter, switching to idle mode. Since at instant
$t_{1}$ the number of the packets does not exceed the start-up threshold $M$,
the MTCD remains in its idle mode. By contract, when the number of packets
exceeds $M$ at instant $t_{2}$, the MTCD switches to its active mode and
transmits at a rate of $\mu=1/\tau_{b}^{ST}$ until the queue becomes empty
again, where $\tau_{b}^{ST}$ denotes the duration of each TDMA time slot. Let
us denote the number of packets at the end of idle mode in the MTCD by
$\varPi_{b}^{ST}$, while its p.g.f. is given by

$\displaystyle\varPi_{b}^{ST}(z)$
$\displaystyle=\sum_{m=0}^{+\infty}{\rm{Prob}}\left(\varPi_{b}^{ST}=m\right)z^{m}=z^{M},$
(22)

where we have $\mathbb{E}\left[\varPi_{b}^{ST}\right]=M$. Let
$\varPi_{n}^{{ST}{\left[i\right]}}$ be the number of the packets in the $n$th
MTCD at the $i$th packet’s departure. Since the generation of the packets
obeys a Poissonian process of rate $\lambda_{n}$, we define
$\\{{\varPi_{n}^{ST}}^{{[i]}},i\geq 1\\}$ as an embedded Markov chain, which
can be characterized by

$\displaystyle{\varPi_{n}^{ST}}^{\left[i+1\right]}=\max\left({\varPi_{n}^{ST}}^{\left[i\right]},0\right)+M\mathbbm{1}_{\left\\{{\varPi_{n}^{ST}}^{\left[i\right]}\leqslant
0\right\\}}-1+\theta^{\left[i\right]},$ (23)

where $\theta^{\left[i\right]}$ denotes the number of packets generated
between the $i$th and the $(i+1)$st departure instant. Then, we can express
the transition probability matrix of the Markov chain as follows:

$\displaystyle\boldsymbol{P}_{n}^{ST}=\left[\begin{matrix}0&0&\cdots&\theta_{0}&\theta_{1}&\cdots\\\
\theta_{0}&\theta_{1}&\cdots&\theta_{M-1}&\theta_{M}&\cdots\\\
&\theta_{0}&\cdots&\theta_{M-2}&\theta_{M-1}&\cdots\\\
\boldsymbol{0}&&\ddots&\ddots&\ddots&\ddots\\\ \end{matrix}\right],$ (24)

where
$\theta_{m}=e^{-\lambda_{n}\tau_{b}^{ST}}\frac{\left(\lambda_{n}\tau_{b}^{ST}\right)^{m}}{m!},(\forall
m\geqslant 0)$ represents the distribution of $\theta^{\left[i\right]}$. We
omit the index $[i]$ of $\theta_{m}$ because the time of transmission and the
interval of update generation are independent of each other. According to
Lemma 2 and Eq. (10), we can derive the corresponding system delay as
$\mathbb{E}\left[D_{n}^{ST}\right]=\lambda_{n}^{-1}\mathbb{E}\left[\varPi_{n}^{ST}\right]$,
and the average interval of the status update is
$\mathbb{E}\left[G_{n}^{ST}\right]=1/\lambda_{n}$. Similar to Eq. (21), the
average peak AoI of the $n$th MTCD using TDMA having the start-up threshold
$M$ can be expressed as

$\displaystyle\mathbb{E}\left[A_{p,n}^{ST}\right]$
$\displaystyle=\mathbb{E}\left[D_{n}^{ST}\right]+\mathbb{E}\left[G_{n}^{ST}\right]$
(25)
$\displaystyle=\underset{The\,\,age\,\,from\,\,M/D/1}{\underbrace{\tau_{b}^{ST}+\frac{\lambda_{n}\left(\tau_{b}^{ST}\right)^{2}}{2\left(1-\lambda_{n}\tau_{b}^{ST}\right)}+\lambda_{n}^{-1}}}+\underset{Additional\,\,age}{\underbrace{\frac{M-1}{2\lambda_{n}}}},$

where the additional age is generated by the ST policy effect. In order to
simplify the expressions of the multiple access protocol relying on this
start-up policy, we define the average peak AoI function as
${A}_{p}^{ST}\left(\lambda_{n},\tau_{b}^{ST},M\right)=\mathbb{E}\left[A_{p,n}^{ST}\right]$.

######  Lemma 2:

When the queue is stable, i.e., $\rho=\lambda\tau_{n}<1$, the average number
of packets generated by the $n$th MTCD based on TDMA using the start-up
threshold $M$ is given by

$\displaystyle\mathbb{E}\left[\varPi_{n}^{ST}\right]=\lambda_{n}\tau_{b}+\frac{\left(\lambda_{n}\tau_{b}\right)^{2}}{2\left(1-\lambda_{n}\tau_{b}\right)}+\frac{M-1}{2},$
(26)

where $\lambda_{n}$ and $\tau_{n}^{-1}$ denote the rates of update generation
and of the packet transmission, respectively.

######  Proof:

See Appendix A. $\blacksquare$

###  III-B FDMA Protocol

Figure 6: An example that illustrates the FDMA with sleep-scheduling.

In FDMA, the bandwidth $B$ has been divided into orthogonal frequency
subchannels, and each sub-channel is allocated to a single MTCD. By sharing
the same channel, all MTCDs send their updates generated simultaneously but in
different frequency bands. As shown in Fig. 6, we assume that all MTCDs have
the same bandwidth allocation $B_{Sub}$ and the guard band101010The guard band
of the FDMA protocol represents the unused part of the radio spectrum between
each sub-channel, which avoids the adjacent channel interference. is
$\varDelta_{B}$. Furthermore, we have
$B_{Sub}=N^{-1}\left[B-\left(N-1\right)\varDelta_{B}\right]$. To differentiate
our policies, we use superscripts $FM$ and $FS$ to denote the FDMA MV policy
and ST policy, respectively.

(1) FDMA Using the MV Policy: The analysis of the average peak AoI for FDMA
using the MV policy is similar to that in Section III-A. In contrast to TDMA,
the entire transmission time $\tau_{b}^{FM}$ of a sub-channel is used for a
single MTCD, but its bandwidth is only $B_{Sub}\approx\frac{B}{N}$.
Furthermore, let $\tau_{s}^{FM}$ and ${\tau_{p}^{FM}}=\tau_{p}$ represent the
durations of the sleep-scheduling and EH within an FDMA time slot. According
to Lemma 1 and Eq. (10), we have an average peak AoI for the $n$th MTCD based
on the FDMA using the MV policy given by

$\displaystyle\mathbb{E}\left[A_{p,n}^{FM}\right]$
$\displaystyle={A}_{p}^{MV}\left(\lambda_{n},\tau_{b}^{FM},{\tau_{s}^{FM}}\right),$
(27)

where the definition of the function
${A}_{p}^{MV}\left(\lambda_{n},\tau_{b}^{FM},{\tau_{s}^{FM}}\right)$ is
obtained according to Eq. (21).

(2) FDMA Using the ST Policy: For the sake of avoiding frequently turning
on/off the MTCDs, we improve the conventional FDMA by introducing a start-up
threshold $M$. Specifically, based on the features of FDMA using multiple idle
TSs, the transmissions are only triggered, when the status update counter has
reached $M$. Then, the MTCD continues its transmissions until the MTCD queue
becomes empty. The procedure is visualized in Fig. 5. Similar to the
definitions of Section III-B, $\tau_{b}^{FS}$ and ${\tau_{s}^{FS}}$ represent
the duration of the active and idle mode, respectively. Inspired by the proof
of Lemma 2, we can express the average of the peak AoI as follows:

$\displaystyle\mathbb{E}\left[A_{p,n}^{FS}\right]$
$\displaystyle={A}_{p}^{ST}\left(\lambda_{n},\tau_{b}^{FS},M\right),$ (28)

where the definition of the function
${A}_{p}^{ST}\left(\lambda_{n},\tau_{b}^{FS},M\right)$ is obtained according
to Eq. (25).

###  III-C NOMA Protocol

Figure 7: An example that illustrates the NOMA with sleep-scheduling.

(1) NOMA Using the MV Policy: In power-domain NOMA (PD-NOMA), all MTCDs share
the same resources both in terms of time, frequency, and space. As shown in
Fig. 7, we assume that $\tau_{b,n}^{NM}=\tau_{b}$ denotes the duration of a
NOMA time slot partitioned into one EH TS and one transmission TS in the $n$th
MTCD, where $\tau_{p,n}^{NM}=\tau_{p}$ and $\tau_{t,n}^{NM}=\tau_{b}-\tau_{p}$
are the duration of EH and transmission within a time slot, respectively.
According to Lemma 1 and Eq. (10), the average peak AoI of the EH-aided
multiple access network can be formulated as:

$\displaystyle\mathbb{E}\left[A_{p}^{NM}\right]$
$\displaystyle=\frac{1}{N}\sum_{n=1}^{N}{{A}_{p}^{MV}\left(\lambda_{n},\tau_{b}^{NM},\tau_{s}^{NM}\right)}={{A}_{p}^{MV}\left(\lambda_{n},\tau_{b}^{NM},\tau_{s}^{NM}\right)}.$
(29)

(2) NOMA Using the ST Policy: In order to reduce the frequency of turning
on/off the MTCDs, we use the ST policy for improving the start-up threshold.
According to the above derivation of the different protocols based on the ST
policy of Sections III-A and III-B, we can express the average peak AoI of the
NOMA using the ST policy as

$\displaystyle\mathbb{E}\left[A_{p,n}^{NS}\right]$
$\displaystyle=\frac{1}{N}\sum_{n=1}^{N}{A}_{p}^{NS}\left(\lambda_{n},\tau_{b,n}^{NS},M\right)={A}_{p}^{NS}\left(\lambda_{n},\tau_{b}^{NS},M\right),$
(30)

where $\tau_{b,n}^{NS}$ and $M$ denote the duration of one NOMA time slot and
the start-up threshold.

##  IV Problem Formulation

In this section, based on TDMA, FDMA and PD-NOMA relying on different
policies, we will first formulate the problems of minimizing the average peak
AoI, which can be transformed into convex problems, and their optimal
solutions are obtained via an exact linear search method (Algorithm 1). The
original non-convex problems are formulated as DC programming problems, which
are solved by the CCP method (Algorithm 2) for reducing their computational
complexity.

###  IV-A TDMA-based Problem Formulation

(1) TDMA Using MV Policy: Without loss of generality, we assume that every
MTCD has the same protocol parameters, namely $\tau_{p,n}=\tau_{p}$,
$\tau_{t,n}=\tau_{t}$, $\tau_{s,n}=\tau_{s}$, $(n=1,2,\cdots,N)$. Moreover,
the energy efficiency (EE) of the status updates scheme can be defined as the
ratio of the generation rate and the sum of the transmit power consumption[9].
According to 4, the transmit power of the $n$th MTCD during the selected TS is
given by
$\varPhi_{t,n}^{TM}=\left[2^{\frac{\left(\mathcal{J}+1\right)NL}{\left(\tau_{b}^{TM}-\tau_{p}^{TM}\right){B}}}-1\right]\gamma_{n}^{-1}$.
Therefore, the EE of the considered status update scheme based on TDMA using
the MV policy can be formulated as:

$\displaystyle\varUpsilon_{n}^{TM}$
$\displaystyle=\frac{NL}{\left(\tau_{b}^{TM}-\tau_{p}^{TM}\right)\varPhi_{t,n}^{TM}}=\frac{\gamma_{n}NL}{\left(\tau_{b}^{TM}-\tau_{p}^{TM}\right)\left[2^{\frac{\left(\mathcal{J}+1\right)NL}{\left(\tau_{b}^{TM}-\tau_{p}^{TM}\right){B}}}-1\right]}.$
(31)

Furthermore, we assume that the EE of the system should not be lower than
$\varUpsilon_{min}$, namely $\varUpsilon_{n}^{TM}\geq\varUpsilon_{min}$ where
leads to the following inequality:

$\displaystyle\varUpsilon_{\min}\left(\tau_{b}^{TM}-\tau_{p}^{TM}\right)\left[2^{\frac{\left(\mathcal{J}+1\right)NL}{\left(\tau_{b}^{TM}-\tau_{p}^{TM}\right){B}}}-1\right]\leqslant
NL\gamma_{n},$ (32)

where the EH time $\tau_{p}^{TM}$ is a constant. When optimizing the average
peak AoI and the EE of the system, the associated minimization problem
$\mathbf{P}_{1}^{TM}$ is expressed as

$\displaystyle\mathbf{P}_{1}^{TM}:$
$\displaystyle\min\limits_{\boldsymbol{x}^{TM}}{A}_{p}^{MV}\left(\lambda_{n},\tau_{b}^{TM},\tau_{s}^{TM}\right)$
(33) $\displaystyle\textrm{s.t.}\quad(\ref{OP:ET-1-1}\textrm{a}):$
$\displaystyle\quad\left(\frac{\tau_{b}^{TM}-\tau_{p}^{TM}}{N}\right)\varPhi_{t,n}^{TM}\leq\tau_{p}^{TM}\varPhi_{r,n}\leq
E_{b,\max},$ $\displaystyle(\ref{OP:ET-1-1}\textrm{b}):$
$\displaystyle\quad\left[\tau_{p}^{TM},0\right]\preceq\boldsymbol{\tau}^{TM}\preceq\left[\tau_{b,\max}^{TM},\tau_{s,\max}^{TM}\right],$
$\displaystyle(\ref{OP:ET-1-1}\textrm{c}):$
$\displaystyle\quad\lambda_{n,\min}\leqslant\lambda_{n}\leqslant\lambda_{n,\max},$
$\displaystyle(\ref{OP:ET-1-1}\textrm{d}):$ $\displaystyle\quad
0<\lambda_{n}\tau_{b}^{TM}<1\ \rm{and}\ (\ref{eq:tdma-1-ee2}),$

1: Constrain $\lambda_{n}$
$\lambda_{n,\min}\leqslant\lambda_{n}\leqslant\min\left(\lambda_{n,\max},1/\tau_{b}^{TM}\right).$

2: With the fixed $\lambda_{n}$, the original problem is converted to a convex
sub-problem.

3: Initialize the starting point to $\lambda_{(0)}=\lambda_{n,\min}$, the
iteration threshold to $K$, and the iteration indicator to $k:=0$.

4: Set the accuracy (step size), to
${A}=\left[\min\left(\lambda_{n,\max},1/\tau_{b}^{TM}\right)-\lambda_{n,\min}\right]/K.$

5: repeat

6: Solve the convex sub-problem with the solution
$\boldsymbol{x}_{\left(k\right)}$, and find the optimal solution
$\boldsymbol{x}_{(*)}=\max\boldsymbol{x}_{\left(i\right)},\forall i\leqslant
k$.

7: Update $\lambda_{(k+1)}=\lambda_{(k)}+{A}$ and $k:=k+1$.

8: until $k>K$.

9: Output Optimal solution $\boldsymbol{x}_{(*)}$.

Algorithm 1 Age-optimal Exact Linear Search

where the objective function is the average peak AoI
${A}_{p}^{MV}\left(\lambda_{n},\tau_{b}^{TM},\tau_{s}^{TM}\right)$, while
$\boldsymbol{x}^{TM}=\left\\{\lambda_{n},\tau_{b}^{TM},\tau_{s}^{TM},\varPhi_{r,n}\right\\}$
denotes the vector of the associated variables and
$\boldsymbol{\tau}^{TM}=\left\\{\tau_{b}^{TM},\tau_{s}^{TM}\right\\}$ is the
vector of TS allocation. Furthermore, $\lambda_{n}$ and $\varPhi_{r,n}^{TM}$
denote the rate of status updates and the transmit power of the power station,
respectively. The inequality constraint in (33a) implies that the energy
harvested should be higher than the transmit power consumed but less than or
equal to the battery capacity $E_{b,\max}$. Furthermore, (33b) and (33c)
denote the inequality constraints of the slot allocation and the generation
rate of status updates for the $n$th MTCD, respectively. Moreover, the
optimization problem should yield the stable condition in
$(\ref{OP:ET-1-1}d)$. According to Lemma 3, we first convert the non-convex
problem in (33) to a convex one by fixing the status update rate
$\lambda_{n}$. Then, we use the exact line search based method to find the
optimal solution, while the main procedure of the search is shown in Algorithm
1.

######  Lemma 3:

Consider the minimization problem in (33). For the sake of obtaining the
optimal solution, we fix the variable $\lambda_{n}$ to as a constant
$\lambda_{(k)}$, where $\lambda_{(k)}$ yields the constraint of $\lambda_{n}$.
According to the second-order condition, the original objective function is
converted to a convex function
${\widehat{{A}}}_{p}^{MV}\left(\tau_{b}^{TM},\tau_{s}^{TM}\right)\stackrel{{\scriptstyle\triangle}}{{=}}{A}_{p}^{MV}\left(\lambda_{\left(\mathrm{k}\right)},\tau_{b}^{TM},\tau_{s}^{TM}\right)$.
Similarly, the objective function
${A}_{p}^{ST}\left(\lambda_{n},\tau_{b}^{ST},M\right)$ in (36) can be
converted to a convex function
${\widehat{{A}}}_{p}^{ST}\left(\tau_{b}^{TM},M\right)$ with the aid of the
same operation.

######  Proof:

We assume that
${\widehat{{A}}}_{p}^{MV}\left(\tau_{b}^{TM},\tau_{s}^{TM}\right)$ and
${\widehat{{A}}}_{p}^{ST}\left(\tau_{b}^{TM},M\right)$ are twice
differentiable. Then we can derive their Hessian matrices as
$\nabla^{2}\widehat{{A}}_{p}^{MV}=\mathrm{diag}\left[\lambda_{n}\left(1-\lambda_{n}\tau_{b}^{TM}\right)^{-3},0\right]\succeq
0$ and
$\nabla^{2}\widehat{{A}}_{p}^{ST}=\mathrm{diag}\left[\lambda_{n}\left(1-\lambda_{n}\tau_{b}^{TM}\right)^{-3},0\right]\succeq
0$. Therefore, the optimization problems in (33) and (36) can be converted to
convex subproblems with a fixed $\lambda_{n}$.

######  Lemma 4:

The average peak AoI function of the system using the MV policy
${A}_{p}^{MV}\left(\lambda_{n},\tau_{b}^{TM},\tau_{s}^{TM}\right)$ in Eq. (21)
is the sum of the convex function $f_{1}^{TM}\left(\boldsymbol{x}^{TM}\right)$
and of the concave function $f_{2}^{TM}\left(\boldsymbol{x}^{TM}\right)$.
Thus, the original problem $\mathbf{P}_{1}^{TM}$ can be converted to a DC
programming problem:

$\displaystyle\mathbf{P}_{2}^{TM}:$
$\displaystyle\min\limits_{\boldsymbol{x}^{TM}}f_{1}^{TM}\left(\boldsymbol{x}^{TM}\right)+\boldsymbol{x}^{TM}\nabla
f_{2}^{TM}\left({\boldsymbol{x}^{TM}_{(k)}}\right)^{\mathrm{T}}$ (34) s.t.
$\displaystyle(\ref{eq:tdma-1-ee2}),\ (\ref{OP:ET-1-1}\textrm{a}),\
(\ref{OP:ET-1-1}\textrm{b}),\ (\ref{OP:ET-1-1}\textrm{c}),$
$\displaystyle\left[\lambda_{n}\left(\tau_{b}^{TM\,\,}\right)_{\left(k\right)}+\tau_{b}^{TM\,\,}\left(\lambda_{n}^{\,\,}\right)_{\left(k\right)}\right]<1+\left(\lambda_{n}\tau_{b}^{TM\,\,}\right)_{\left(k\right)},$

where ${\boldsymbol{x}^{TM}_{(k)}}$ denotes the optimal solution obtained from
the $k$th iteration, while $\nabla
f_{2}^{TM}\left({\boldsymbol{x}^{TM}_{(k)}}\right)^{\mathrm{T}}$ is the
gradient value of $f_{2}^{TM}\left({\boldsymbol{x}^{TM}_{(k)}}\right)$ at the
point ${\boldsymbol{x}^{TM}_{(k)}}$. After successively solving a series of
convex problems, we can find a locally optimal solution for the minimization
problem $\mathbf{P}_{1}^{TM}$.

######  Proof:

See Appendix B. $\blacksquare$

1: Initialize the feasible starting point $\boldsymbol{x}_{(0)}^{TM}$, the
iteration step threshold $K$ and constant $\epsilon$.

2: Initialize the convex function $f_{1}^{TM}(\boldsymbol{x})$ and the concave
function $f_{2}^{TM}(\boldsymbol{x})$ by Lemma 4.

3: Set the lower and the upper bounds of $\lambda_{n}$, $\tau_{b}$,
$\tau_{s}$, $\varPhi_{r,n}$.

4: Set the iteration indicator to $k:=0$.

5: repeat

6: Calculate the gradient value $\nabla f_{2}^{TM}$ and substitute a given
$\boldsymbol{x}_{(k)}^{TM}$.

7: Obtain the convex sub-problem in (34).

8: Calculate the locally optimal solution of (34) as
$\boldsymbol{x}_{(*)}^{TM}$.

9: Update $\boldsymbol{x}_{(k+1)}^{TM}=\boldsymbol{x}_{(*)}^{TM}$ and
$k:=k+1$.

10: until $k>K$ or
$\left|\boldsymbol{x}_{(k+1)}^{TM}-\boldsymbol{x}_{(k)}^{TM}\right|<\epsilon$.

11: Output near-optimal solution $\boldsymbol{x}_{(*)}^{TM}$.

Algorithm 2 CCP Based Peak AoI Optimization

To increase the accuracy of the solution and hence to approach the optimum,
increasing complexity of Algorithm 1 is inevitable, which is counterproductive
for EH-aided MTCDs having finite computational resources. Therefore, we
propose a low-complexity heuristic algorithm for solving the original problem.
According to Lemma 4, the objective function of the minimization problem
$\mathbf{P}_{1}^{TM}$ is non-convex, while $\mathbf{P}_{1}^{TM}$ belongs to
the family of DC programming problems, which can be solved by the CCP
algorithm. Specifically, we decompose the original objective function
${A}_{p}^{MV}\left(\boldsymbol{x}^{TM}\right)$ into the sum of a convex and a
concave function, i.e.
$\frac{\tau_{s}^{TM}}{2}+\frac{\tau_{b}^{TM}}{2}+\frac{1}{\lambda_{n}}$ and
$\frac{\tau_{b}^{TM}}{2\left(1-\lambda_{n}\tau_{b}^{TM}\right)}$. Then, we
determine an initial feasible point $\boldsymbol{x}^{TM}_{0}$ and convert
${A}_{p}^{MV}\left(\boldsymbol{x}^{TM}\right)$ to typical convex function
relying on first order Taylor approximation as follows:

$\displaystyle\widehat{{A}}_{p}^{MV}\left(\boldsymbol{x}^{TM};\boldsymbol{x}_{0}^{TM}\right)$
$\displaystyle=\widehat{{A}}_{p}^{MV}\left(\boldsymbol{x}_{0}^{TM}\right)$
(35)
$\displaystyle+\nabla\left[\widehat{{A}}_{p}^{MV}\left(\boldsymbol{x}_{0}^{TM}\right)\right]^{\mathrm{T}}\left(\boldsymbol{x}^{TM}-\boldsymbol{x}_{0}^{TM}\right).$

Then, substituting ${A}_{p}^{MV}\left(\boldsymbol{x}^{TM}\right)$ by
$\widehat{{A}}_{p}^{MV}\left(\boldsymbol{x}^{TM};\boldsymbol{x}_{0}^{TM}\right)$
in $\mathbf{P}_{1}^{TM}$ and $\mathbf{P}_{1}^{TM}$ can be converted to a
classical convex problem $\mathbf{P}_{2}^{TM}$. During each iteration,
$\mathbf{P}_{2}^{TM}$ is solved for a given $\boldsymbol{x}_{0}^{TM}$, which
can be replaced by the optimal value in the next iteration. The algorithm
stops the iteration when the stopping criterion is satisfied. The main steps
of the CCP algorithm are shown in Algorithm 2.

(2) TDMA using the ST Policy: The EE of the status update scheme based on TDMA
and the ST policy is given by
$\varUpsilon_{n}^{ST}=\frac{L}{(\tau_{b}^{ST}-\tau_{p}^{ST})\varPhi_{t,n}^{ST}}$.
Similar to Section IV-A(1), the lower bound of the EE yields
$\varUpsilon_{n}^{ST}\geq\varUpsilon_{\min}$ and the minimization problem
$\mathbf{P}_{1}^{ST}$ is expressed as

$\displaystyle\mathbf{P}_{1}^{ST}:$
$\displaystyle\min\limits_{\boldsymbol{x}^{ST}}{A}_{p}^{ST}\left(\lambda_{n},\tau_{b}^{ST},M\right)$
(36) $\displaystyle\textrm{s.t.}\quad(\ref{OP:ST-1}\textrm{a}):$
$\displaystyle\quad\left(\frac{\tau_{b}^{ST}-\tau_{p}^{ST}}{N}\right)\varPhi_{t,n}^{ST}\leq\tau_{p}^{ST}\varPhi_{r,n}\leq
E_{b,\max},$ $\displaystyle(\ref{OP:ST-1}\textrm{b}):$
$\displaystyle\quad\tau_{p}^{ST}\leq\tau_{b}^{ST}\leq\tau_{b,\max}^{ST},$
$\displaystyle(\ref{OP:ST-1}\textrm{c}):$
$\displaystyle\quad\lambda_{n,\min}\leqslant\lambda_{n}\leqslant\lambda_{n,\max},$
$\displaystyle(\ref{OP:ST-1}\textrm{d}):$ $\displaystyle\quad
0<\lambda_{n}\tau_{b}^{ST}<1,$ $\displaystyle(\ref{OP:ST-1}\textrm{e}):$
$\displaystyle\quad\varUpsilon_{n}^{ST}\geq\varUpsilon_{min},$

where
$\boldsymbol{x}^{ST}=\left\\{\lambda_{n},\tau_{b}^{ST},M,\varPhi_{r,n}\right\\}$
denotes the vector of the variables. The optimal solution of problem (36) can
be found by Algorithm 1. In order to simplify the original problem, we relax
the integrality constraint to transfer $M$ to a non-integral value, i.e.
$M\in[M_{min},M_{max}]$. Furthermore, according to Eq. (25), the objective
function of the problem $\mathbf{P}_{1}^{ST}$ is the sum of a convex function
$f_{1}^{ST}\left(\lambda_{n},\tau_{b}^{ST},M\right)=\tau_{b}^{ST}+(2\lambda_{n})^{-1}$
and of a concave function
$f_{2}^{ST}\left(\lambda_{n},\tau_{b}^{ST},M\right)=\frac{\lambda_{n}\left(\tau_{b}^{ST}\right)^{2}}{2\left(1-\lambda_{n}\tau_{b}^{ST}\right)}+\frac{M}{2\lambda_{n}}$.
Inspired by Lemma 4, $\mathbf{P}_{1}^{ST}$ can be rewritten as a DC program as
follows:

$\displaystyle\mathbf{P}_{2}^{ST}:$
$\displaystyle\min\limits_{\boldsymbol{x}^{ST}}f_{1}^{ST}\left(\boldsymbol{x}^{ST}\right)+\boldsymbol{x}^{ST}\nabla
f_{2}^{ST}\left({\boldsymbol{x}^{ST}_{(k)}}\right)^{\mathrm{T}}$ (37) s.t.
$\displaystyle(\ref{OP:ST-1}\textrm{a}),\ (\ref{OP:ST-1}\textrm{b}),\
(\ref{OP:ST-1}\textrm{c}),\ (\ref{OP:ST-1}\textrm{e}),$
$\displaystyle\left[\lambda_{n}\left(\tau_{b}^{ST\,\,}\right)_{\left(k\right)}+\tau_{b}^{ST\,\,}\left(\lambda_{n}^{\,\,}\right)_{\left(k\right)}\right]<1+\left(\lambda_{n}\tau_{b}^{ST\,\,}\right)_{\left(k\right)},$

where ${\boldsymbol{x}^{ST}_{(k)}}$ denotes the optimal solution obtained from
the $k$th iteration, and $\mathbf{P}_{2}^{ST}$ can be solved by the CCP method
of Algorithm 2. Before proceeding the following sections, we depict an example
diagram shown in Fig. 8 to summarize the analysis flow of TDMA111111Fig. 8 is
not only for TDMA protocol but also for FDMA and NOMA..

Figure 8: Flow of the mathematical analysis for TDMA protocol.

###  IV-B FDMA-based Problem Formulation

(1) FDMA Using the MV Policy: In FDMA, the bandwidth of the $n$th MTCD
subchannel is $B_{Sub}=N^{-1}\left[B-\left(N-1\right){A}_{B}\right]$. Thus,
the EE of the FDMA using the MV policy can be formulated as
$\varUpsilon_{n}^{FM}=\frac{L}{\left(\tau_{b}^{FM}-\tau_{p}^{FM}\right)\varPhi_{t,n}^{FM}}$,
where the transmit power is given by
$\varPhi_{t,n}^{FM}=\gamma_{n}^{-1}\left[2^{\frac{\left(\mathcal{J}+1\right)L}{\left(\tau_{b}^{FM}-\tau_{p}^{FM}\right)B_{Sub}}}-1\right]$.
Therefore, the lower bound of the EE yields

$\displaystyle\varUpsilon_{\min}\left(\tau_{b}^{FM}-\tau_{p}^{FM}\right)\left[2^{\frac{\left(\mathcal{J}+1\right)L}{\left(\tau_{b}^{FM}-\tau_{p}^{FM}\right)B_{Sub}}}-1\right]\leqslant\gamma_{n}L.$
(38)

The problem of optimizing the average peak AoI and the EE of the system may be
formulated as the minimization problem $\mathbf{P}^{FM}$:

$\displaystyle\mathbf{P}^{FM}:$
$\displaystyle\min\limits_{\boldsymbol{x}^{FM}}{A}_{p}^{MV}\left(\lambda_{n},\tau_{b}^{FM},\tau_{s}^{FM}\right)$
(39) $\displaystyle\textrm{s.t.}\quad(\ref{OP:EF-1}\textrm{a}):$
$\displaystyle\quad\left(\tau_{b}^{FM}-\tau_{p}^{FM}\right)\varPhi_{t,n}^{FM}\leq\tau_{p}^{FM}\varPhi_{r,n}\leq
E_{b,\max},$ $\displaystyle(\ref{OP:EF-1}\textrm{b}):$
$\displaystyle\quad\left[\tau_{p}^{FM},0\right]\preceq\boldsymbol{\tau}^{FM}\preceq\left[\tau_{b,\max}^{FM},\tau_{s,\max}^{FM}\right],$
$\displaystyle(\ref{OP:EF-1}\textrm{c}):$
$\displaystyle\quad\lambda_{n,\min}\leqslant\lambda_{n}\leqslant\lambda_{n,\max},$
$\displaystyle(\ref{OP:EF-1}\textrm{d}):$ $\displaystyle\quad
0<\lambda_{n}\tau_{b}^{FM}<1\ \rm{and}\ (\ref{eq:fdma-1-ee2}),$

where the objective function is the average peak AoI, while
$\boldsymbol{x}^{FM}=\left\\{\lambda_{n},\tau_{b}^{FM},\tau_{s}^{FM},\varPhi_{r,n}\right\\}$
denotes the vector of the variables and $\boldsymbol{\tau}^{FM}$ is the vector
of TS allocation.

(2) FDMA Using the ST Policy: Under the ST policy, the MTCD only wakes up from
its idle mode, if the number of status updates in the queue is more than $M$.
Moreover, the EE of the system yields
$\varUpsilon_{n}^{FS}=\frac{NL}{(\tau_{b}^{FS}-\tau_{p}^{FS})\varPhi_{t,n}^{FS}}\geq\varUpsilon_{\min}$.
Similarly to the minimization problem above, let
$\boldsymbol{x}^{FS}=\left\\{\lambda_{n},\tau_{b}^{FS},M,\varPhi_{r,n}\right\\}$
denote the vector of variables, and the optimization problem is

$\displaystyle\mathbf{P}^{FS}:$
$\displaystyle\min\limits_{\boldsymbol{x}^{FS}}{A}_{p}^{ST}\left(\lambda_{n},\tau_{b}^{FS},M\right)$
(40) $\displaystyle\textrm{s.t.}\quad(\ref{OP:FST-1}\textrm{a}):$
$\displaystyle\quad\left(\tau_{b}^{FS}-\tau_{p}^{FS}\right)\varPhi_{t,n}^{FS}\leq\tau_{p}^{FS}\varPhi_{r,n}\leq
E_{b,\max},$ $\displaystyle(\ref{OP:FST-1}\textrm{b}):$
$\displaystyle\quad\tau_{p,\min}^{FS}\leq\tau_{b}^{FS}\leq\tau_{b,\max}^{FS},$
$\displaystyle(\ref{OP:FST-1}\textrm{c}):$
$\displaystyle\quad\lambda_{n,\min}\leqslant\lambda_{n}\leqslant\lambda_{n,\max},$
$\displaystyle(\ref{OP:FST-1}\textrm{d}):$ $\displaystyle\quad
0<\lambda_{n}\tau_{b}^{FS}<1,$ $\displaystyle(\ref{OP:FST-1}\textrm{e}):$
$\displaystyle\quad\varUpsilon_{n}^{FS}\geq\varUpsilon_{min}.$

According to Lemmas 3 and 4, the optimal results of $\mathbf{P}^{FM}$ and
$\mathbf{P}^{FS}$ can be obtained by Algorithm 1. To reduce the computational
complexity, the previous problems may be rewritten as DC programming problems.
Algorithm 2 provides a tractable iterative procedure based on a CCP method to
solve it.

###  IV-C NOMA-based Problem Formulation

(1) NOMA Using the MV Policy: For the case of NOMA, we assume that all the $N$
MTCDs share the same spectrum along with the same status update rate and TS
allocation. Since SIC is utilized by the BS receiver, we should accurately
control the transmit power and the EH of the different MTCDs. The EE of the
NOMA scheme using the MV policy is the ratio between the status update rate
and the average power consumption, expressed as:

$\displaystyle\varUpsilon^{NM}=\frac{\gamma
LN}{\left(\tau_{b}^{NM}-\tau_{p}^{NM}\right)\left[2^{\frac{\left(\mathcal{J}+1\right)NL}{\left(\tau_{b}^{NM}-\tau_{p}^{NM}\right)B}}-1\right]}\geq\varUpsilon_{\min}.$
(41)

In contrast to the TDMA and FDMA protocols, the transmit power of NOMA should
satisfy the power constraint of Eq. (6). Therefore, the corresponding
minimization problem is

$\displaystyle\mathbf{P}^{NM}:$
$\displaystyle\min\limits_{\boldsymbol{x}^{NM}}{A}_{p}^{MV}\left(\lambda,\tau_{b}^{NM},\tau_{s}^{NM}\right)$
(42) $\displaystyle\textrm{s.t.}\quad(\ref{OP:EN-1}\textrm{a}):$
$\displaystyle\quad\left(\tau_{b}^{NM}-\tau_{p}^{NM}\right)P_{\Sigma}\leq\tau_{p}^{NM}\varPhi_{\Sigma}\leq
E_{b,\max},$ $\displaystyle(\ref{OP:EN-1}\textrm{b}):$
$\displaystyle\quad\left[\tau_{p}^{NM},0\right]\preceq\boldsymbol{\tau}^{NM}\preceq\left[\tau_{b,\max}^{NM},\tau_{s,\max}^{NM}\right],$
$\displaystyle(\ref{OP:EN-1}\textrm{c}):$
$\displaystyle\quad\lambda_{\min}\leqslant\lambda\leqslant\lambda_{\max},$
$\displaystyle(\ref{OP:EN-1}\textrm{d}):$ $\displaystyle\quad
0<\lambda\tau_{b}^{NM}<1,$ $\displaystyle(\ref{OP:EN-1}\textrm{f}):$
$\displaystyle\quad\varUpsilon^{NM}\geq\varUpsilon_{\min},\ {\rm{and}}\
(\ref{OP:EN-1-ee}),$

where $(\ref{OP:EN-1}\textrm{a})$ is the transmit power constraint of NOMA,
and we have
$P_{\Sigma}=\sum_{n=1}^{N}{\varPhi_{t,n}^{NM}}=\left[2^{\frac{\left(\mathcal{J}+1\right)NL}{\left(\tau_{b}^{NM}-\tau_{p}^{NM}\right)B}}-1\right]/\gamma$,
where $\varPhi_{t,n}^{NM}\geq 0,\forall{n}\in\mathcal{N}$. Let
$\boldsymbol{x}^{NM}=\left\\{\lambda,\tau_{b}^{NM},\tau_{s}^{NM},\varPhi_{\Sigma}\right\\}$
denote the vector of the variables. Moreover, we omit the subscript $n$ since
all MTCDs have the same average peak AoI along with the same TS allocation and
status update rate.

(2) NOMA Using the ST Policy: In this policy, the optimization problem
requires the minimization of the average peak AoI
${A}_{p}^{ST}\left(\lambda_{n},\tau_{b}^{NS},M\right)$ under the constraints
of transmit power, TS allocation etc. Thus, the problem for the case of the
NOMA using the ST policy can be formulated as follows, while we drop the
subscript $n$ for simplicity

$\displaystyle\mathbf{P}^{NS}:$
$\displaystyle\min\limits_{\boldsymbol{x}^{NS}}{A}_{p}^{ST}\left(\lambda_{n},\tau_{b}^{NS},M\right)$
(43) $\displaystyle\textrm{s.t.}\quad(\ref{OP:NST-1}\textrm{a}):$
$\displaystyle\quad\left(\tau_{b}^{NS}-\tau_{p}^{NS}\right)P_{\Sigma}\leq\tau_{p}^{NS}\varPhi_{\Sigma}\leq
E_{b,\max},$ $\displaystyle(\ref{OP:NST-1}\textrm{b}):$
$\displaystyle\quad\tau_{p}^{NS}\leq\tau_{b}^{NS}\leq\tau_{b,\max}^{NS},$
$\displaystyle(\ref{OP:NST-1}\textrm{c}):$
$\displaystyle\quad\lambda_{n,\min}\leqslant\lambda_{n}\leqslant\lambda_{n,\max},$
$\displaystyle(\ref{OP:NST-1}\textrm{d}):$ $\displaystyle\quad
0<\lambda_{n}\tau_{b}^{NS}<1,$ $\displaystyle(\ref{OP:NST-1}\textrm{e}):$
$\displaystyle\quad\varUpsilon_{n}^{NS}\geq\varUpsilon_{min},$

where the transmit power $\varPhi_{t,n}^{NS}$ and the EE of the system
$\varUpsilon_{n}^{NS}$ are obtained by (41) and (43).

According to Lemma 3, we obtain the optimal solutions of $\mathbf{P}^{NM}$ and
$\mathbf{P}^{NS}$ via Algorithm 1. Furthermore, in order to mitigate the
computational complexity of devices, Algorithm 2 is used for solving
$\mathbf{P}^{NM}$ and $\mathbf{P}^{NS}$, respectively.

##  V Simulation Results And Discussions

This section presents numerical results for verifying the performance of both
sleep-scheduling policies. We also use the three multiple access protocols
without sleep-scheduling as our benchmark designs.

###  V-A Parameters Settings

As for the $n$th MTCD, in our simulation scenario, we assume that the distance
between each MTCD and the BS or power station is the same121212We assume that
all MTCDs are on the perpendicular bisector of the line segment between the
power station and the BS., i.e., $d_{n}^{P-M}=d_{n}^{M-B}$. The distance set
is defined as
$\boldsymbol{d}^{M-B}=\left(d_{1}^{M-B},\cdots,d_{N}^{M-B}\right)$, which
represents a linearly spaced vector in the range of $[3,5]$m and satisfies
$d_{1}^{M-B}<\cdots<d_{N}^{M-B}$. The available bandwidth is 5MHz with a
carrier center frequency of 470MHz, while the TGn path-loss factor of 2 is
used[22]. The noise power spectral density is $N_{0}=-60$dBm/Hz. The
efficiency of EH is $\eta_{p}=0.9$. The maximum transmission time is
$\tau_{b,max}=40$ms, while the EH time is a constant 10ms. Furthermore,
$\varPhi_{r,max}$ and $\varPhi_{t,max}$ are 4W and 400mW, respectively. The
probability of an MTCD becoming active in a EH or transmission mode is
$\rho=\lambda_{n}\tau_{b}$[33]. Therefore, we define the average power
consumption for the protocols based on sleep-scheduling as follows:

$\displaystyle\overline{\varPhi}=\rho\left(\frac{\tau_{b}-\tau_{p}}{\tau_{b}}\varPhi_{t}+\varPhi_{w}\right)+\left(\frac{1-\rho}{\tau_{s}}\right)\left(\varPhi_{s}\tau_{i}+\varPhi_{sc}\tau_{sc}\right),$
(44)

where $\varPhi_{w}=100$mW, $\varPhi_{s}=10$mW and $\varPhi_{sc}=100$mW
represent the power consumption in the active mode, idle mode and the
switching cost, respectively. Moreover, we suppose
$\tau_{s}=\tau_{i}+\tau_{sc}$, where $\tau_{i}=9\tau_{sc}$. Substituting
$\varPhi_{s}=\varPhi_{w}$ into (44), we obtain the average power consumption
of our benchmark designs.

(a) MV policy

(b) ST policy

Figure 9: The average peak AoI versus $L$ under a pair of sleep-scheduling
policies. Figure 10: The average per-packet AoI versus $L$ under a pair of
sleep-scheduling policies.

(a) MV policy

(b) ST policy

Figure 11: The average power consumption versus $L$ under a pair of sleep-
scheduling policies.

In Fig. 9(a), we consider the scenario of 10 MTCDs relying on the MV policy,
where the maximum status update rate is $\lambda_{max}=15$ packets per second.
It is observed that for the different protocols relying on the increment of
$L$, the average peak AoI increases under the MV policy. A large $L$ results
in an increased throughput and tell-traffic load, hence leading to a higher
peak AoI. Moreover, NOMA has a lower peak AoI than FDMA and TDMA, but the peak
AoI of NOMA increases marginally when the gap between the channel capacity and
the current throughput is substantial. Although CCP is a heuristic algorithm,
it can be seen that the results based on CCP are very close to the optimal
solutions. Fig. 9(b) characterizes the average peak AoI using the ST policy.
Similar to the protocols based on the MV policy, it can be observed that the
freshness of data is directly related to the packet length and that NOMA under
the ST policy is superior to both TDMA and FDMA in terms of its peak AoI.
Observe that harnessing the CCP method in the NOMA protocol strikes a trade-
off between the average peak AoI attained and the computational complexity
imposed. Additionally, Fig. 10 illustrates the results of average per-packet
AoI. According to Eq. (11), the relationship between the average per-packet
AoI and the average peak is linear in terms of the same status update rate
$\lambda$, and thus Fig. 10 follows the same trend as Fig. 9. Furthermore, it
can be observed that when $L$ is less than $160$ bits, TDMA-MV and FDMA-MV
have a better per-packet AoI performance than NOMA-ST, because the delay of
NOMA is degraded by the ST policy.

In Fig. 11(a), we evaluate the average energy consumption $\overline{P}$
versus the packet length. Observe that a shorter packet is beneficial in terms
of saving energy. Moreover, relying on our proposed MV policy, NOMA
outperforms the other multiple access protocols. However, upon using the CCP
method, FDMA using the MV policy exhibits a reduced power for $60<L<160$ bits.
Fig. 11(b) shows the impact of the packet length on the average power
consumption $\overline{P}$. With the increment of $L$, more status updates
have to be transmitted by the MTCDs, which requires more energy from the power
station to reduce the peak AoI. For reducing the computational complexity, the
following experiments are conducted relying on CCP method rather than the
exact linear search.

###  V-B Performance Versus Maximum Status Update Rate

(a) TDMA protocol (b) FDMA protocol (c) NOMA protocol

Figure 12: The average peak AoI versus different status update rate
constraints.

(a) TDMA protocol (b) FDMA protocol (c) NOMA protocol

Figure 13: The average per-packet AoI versus different status update rate
constraints.

(a) TDMA protocol

(b) FDMA protocol

(c) NOMA protocol

Figure 14: The average power consumption versus different status update rate
constraints.

This subsection studies the impact of the maximum status update rate of both
policies on the AoI and on the energy consumption, where $N=10$ and
$L=100$bits, for all three multiple access protocols. Fig. 12(a) and Fig.
12(b) indicate that the networks based on TDMA-MV and FDMA-MV have a lower
peak AoI, when the maximum status update rate increases. This is because the
status updates become infrequent when the maximum status update rate
$\lambda_{max}$ is low, which in turn increases the average peak AoI. Since
NOMA has a better energy efficiency than the other protocols, its average peak
AoI is monotonically reduced in Fig. 12(c). Moreover, Fig. 12 also
demonstrates that the sleep-scheduling policies degrade the AoI for the sake
of reducing the energy consumption. At $\lambda_{max}=1.4$ kbps, the ST policy
imposes 37.5%, 36.6% and 40.4% average peak AoI degradation in TDMA, FDMA and
NOMA, respectively. By contrast, the average peak AoI values of the MV policy
only increase by 5.8%, 5.6% and 5.7% relying on TDMA, FDMA and NOMA,
respectively. Therefore, the MV policy is more suitable for a low-delay
scenarios than the ST policy. Additionally, Fig. 13 illustrates the average
per-packet AoI vs. the status update rate $\lambda_{max}$, which has a similar
AoI trend as Fig. 12. Specifically, at $\lambda_{max}=1.4$ kbps, the average
per-packet AoI increases by 58.6%, 58.7% and 67.7% in TDMA, FDMA and NOMA,
respectively. As for the MV policy, the AoI degradations are 8.2%, 8.1% and
9.4% for TDMA, FDMA and NOMA, respectively. Observe that the sleep-scheduling
policies have a more grave impact on the per-packet AoI than on the peak AoI.
Furthermore, the AoI degradation of NOMA is higher than that of the other
protocols.

Fig. 14 demonstrates the performance of the MTCD’s average power consumption
for the different policies. As shown in the figure, compared to the benchmark
protocols, the power consumption is reduced by both the ST and MV policies.
Moreover, the average power consumption increases with $\lambda_{max}$, since
the packets within the queue are transmitted more frequently to maintain the
queue’s stability, which leads to an increased energy consumption.
Specifically, when $\lambda_{max}=1.4$ kbps and the MV policy is adopted, the
average energy consumption of MTCDs can be reduced by 11.9.5%, 17.2% and 20.7%
for TDMA, FDMA and NOMA, respectively. As for the ST policy, the average
energy consumption can be reduced by 13.8%, 20.4% and 24.69% for TDMA, FDMA
and NOMA, respectively. Therefore, it can be concluded that the sleep-
scheduling policies reduce the energy consumption at the cost of an increased
AoI. Although the ST policy is superior to the MV policy in terms of its
energy dissipation, the MV policy may be better suited for near real-time
status update scenario.

###  V-C Performance Versus the Number of MTCDs

(a) Average peak AoI

(b) Average power consumption

In Fig. LABEL:fig:AoI-EE-MV, we evaluate the average peak AoI and the sum of
the average power consumption (i.e. $P_{\Sigma}=N\cdot\overline{P}$) for
different number of MTCDs $N$, where we have $L=100$bits and
$\lambda_{max}=15$. Observe that upon increasing $N$, the optimal average peak
AoI increases. The reason for this trend is that the total throughput of the
network rises sharply when the number of MTCDs is increased, hence the packets
are more likely to be queued for a long time and require more energy from the
power station. Furthermore, for the MV policy, the average peak AoI of FDMA is
better than that of TDMA, and NOMA always has the edge. That is because the
spectral resources of FDMA and NOMA are capable of supporting an increased
number of MTCDs, but the TDMA has a limited number of time slots.

##  VI Conclusions

The family of EH-aided NGMA systems relying on the multiple vacation policy
and start-up threshold policy has been investigated and closed-form
expressions for the peak AoI of both policies have been derived. Since the
latency caused by sleep-scheduling leads to the degradation of AoI, we have
formulated the associated time slot allocation problems for minimizing the
average peak AoI and obtained the optimal solutions of the resultant non-
convex problems by an exact linear search method. For the sake of reducing the
computational complexity, we have formulated the original problems as DC
problems, which can be solved by CCP based heuristic algorithms. Our
simulation results have shown that the proposed sleep-scheduling policies are
more energy-efficiency than the benchmark protocols. Additionally, under the
premise of fair rate allocation among MTCDs, NOMA outperforms both TDMA and
FDMA in terms of its peak AoI and power consumption.

##  Appendix A Proof of Lemma 2

######  Proof:

Let $\boldsymbol{\pi_{\varPi}}=\left(\pi_{1},\pi_{2},\cdots\right)$ be the
stationary distribution of $\varPi_{n}^{ST}$, and
$\boldsymbol{e}=(1,1,\cdots)^{\mathrm{T}}$. According to the equilibrium
equation of
$\boldsymbol{\pi_{\varPi}}\boldsymbol{P}_{0}=\boldsymbol{\pi_{\varPi}}$ and
Eq. (24), we can formulate the stationary distribution as follows:

$\displaystyle\pi_{m}=\begin{cases}\sum_{i=1}^{m+1}{\pi_{i}\theta_{m+1-i}}&0\leqslant
k\leqslant M-2,\\\
\pi_{0}\theta_{m-M+1}+\sum_{i=1}^{m+1}{\pi_{i}\theta_{m+1-i}}&k\geqslant
M-1.\\\ \end{cases}$ (45)

Therefore, the p.g.f. of the stationary distribution
$\boldsymbol{\pi_{\varPi}}(z)$ can be expressed as follows:

$\displaystyle\boldsymbol{\pi}_{\varPi}\left(z\right)$
$\displaystyle=\sum_{i=M-1}^{+\infty}{z^{i}\left[\pi_{0}\theta_{i-M+1}+\sum_{j=1}^{i+1}{\pi_{j}\theta_{i+1-j}}\right]}+\sum_{i=0}^{M-2}{z^{i}\sum_{j=1}^{i+1}{\pi_{j}\theta_{i+1-j}}}$
(46)
$\displaystyle=\frac{\left[\boldsymbol{\pi}_{\varPi}\left(z\right)-\pi_{0}\right]+\pi_{0}z^{M}}{z}e^{-\lambda_{n}\tau_{b}\left(1-z\right)}.$

Upon substituting $\boldsymbol{\pi_{\varPi}}(1)=1$ into the above equation, we
may arise at $\pi_{0}=\left(1-\lambda_{n}\tau_{b}\right)M^{-1}$. Therefore,
Eq. (46) can be simplified as

$\displaystyle\boldsymbol{\pi}_{\varPi}\left(z\right)=\frac{\left(1-\lambda_{n}\tau_{b}\right)\left(1-z\right)\left(1-z^{M}\right)e^{-\lambda_{n}\tau_{b}\left(1-z\right)}}{M\left[e^{-\lambda_{n}\tau_{b}\left(1-z\right)}-z\right]\left(1-z\right)}.$
(47)

Furthermore, the average number of the packets generated in the $n$th MTCD can
be expressed as

$\displaystyle\mathbb{E}\left[\varPi_{n}^{ST}\right]$
$\displaystyle=\left.\dfrac{d\boldsymbol{\pi}_{\varPi}\left(z\right)}{dz}\right|_{z=1}=\lambda_{n}\tau_{b}+\frac{\left(\lambda_{n}\tau_{b}\right)^{2}}{2\left(1-\lambda_{n}\tau_{b}\right)}+\frac{M-1}{2}.$
(48)

Based on Little’s Law, we can express the average system delay of the packets
in the $n$th MTCD as
$\mathbb{E}\left[D_{n}^{ST}\right]=\lambda_{n}^{-1}\mathbb{E}\left[\varPi_{n}^{ST}\right]$.

##  Appendix B Proof of Lemma 4

######  Proof:

The variables $\lambda_{n},\tau_{b}^{TM},\tau_{s}^{TM}$ and
${A}_{p}^{MV}\left(\lambda_{n},\tau_{b}^{TM},\tau_{s}^{TM}\right)$ are defined
in Section III-A. Moreover, we decompose the original objective function into
the pair of functions $f_{1}^{TM}$ and $f_{2}^{TM}$, namely
${A}_{p}^{MV}=f_{1}^{TM}+f_{2}^{TM}$, which are given by

$\displaystyle\begin{cases}f_{1}^{TM}\left(\boldsymbol{x}^{TM}\right)=\frac{\tau_{s}^{TM}}{2}+\frac{\tau_{b}^{TM}}{2}+\frac{1}{\lambda_{n}},\\\
f_{2}^{TM}\left(\boldsymbol{x}^{TM}\right)=\frac{\tau_{b}^{TM}}{2\left(1-\lambda_{n}\tau_{b}^{TM}\right)}.\end{cases}$
(49)

While $\frac{1}{\lambda_{n}}$ denotes the power function of $\lambda_{n}$ when
$\lambda_{n}>0$. Therefore, $f_{1}^{TM}$ is convex on $\mathbf{R}_{+}^{2}$,
because the nonnegative, nonzero sum of convex functions is convex. To prove
that $f_{2}^{TM}$ is a concave function, we derive the second-order condition,
namely the Hessian matrix, as follows:

$\displaystyle\nabla^{2}f_{2}^{TM}=\frac{1}{\left(1-\lambda_{n}\tau_{b}^{TM}\right)^{3}}\left[\begin{matrix}\frac{2\left(1-\lambda_{n}\tau_{b}^{TM}\right)^{3}}{\lambda_{n}^{3}}+\left(\tau_{b}^{TM}\right)^{3}&\tau_{b}^{TM}&0\\\
\tau_{b}^{TM}&\lambda_{n}&0\\\ 0&0&0\\\ \end{matrix}\right].$ (50)

It is clear that $\nabla^{2}f_{2}^{TM}\preceq 0$ since the determinants obey
$D_{i}\geq 0$ for odd $i$ and $D_{i}\leq 0$ for even $i$, and we have
$D_{1}=\frac{\partial^{2}f_{2}^{TM}}{\partial\left(\tau_{b}^{TM}\right)^{2}}>0$,
$D_{2}\geq 0$ and $D_{3}=\left|\nabla^{2}f_{2}^{TM}\right|\leq 0$. Therefore,
it can be concluded that $f_{2}^{TM}$ is a concave function. By exploiting the
second-order condition, we can prove that the inequality constraints (32),
(33a)-(33c) are convex based on the second-order condition, while (33d) has to
be rendered convex as
$\left[\lambda_{n}\left(\tau_{b}^{TM\,\,}\right)_{\left(k\right)}+\tau_{b}^{ST\,\,}\left(\lambda_{n}^{\,\,}\right)_{\left(k\right)}\right]<1+\left(\lambda_{n}\tau_{b}^{TM\,\,}\right)_{\left(k\right)}$,
where the subscript $(k)$ denotes the value obtained from the $k$th iteration.

## References

  * [1] X. Chen, D. W. K. Ng, W. Yu, E. G. Larsson, N. Al-Dhahir, and R. Schober, “Massive access for 5G and beyond,” _IEEE Journal on Selected Areas in Communications_ , vol. 39, no. 3, pp. 615–637, Sep. 2020.
  * [2] J. Wang, C. Jiang, H. Zhang, Y. Ren, K.-C. Chen, and L. Hanzo, “Thirty years of machine learning: The road to Pareto-optimal wireless networks,” _IEEE Communications Surveys & Tutorials_, vol. 22, no. 3, pp. 1472–1514, Jan. 2020.
  * [3] Y. Yuan, Z. Yuan, and L. Tian, “5G non-orthogonal multiple access study in 3GPP,” _IEEE Communications Magazine_ , vol. 58, no. 7, pp. 90–96, Jul. 2020.
  * [4] X. Lu, P. Wang, D. Niyato, and Z. Han, “Resource allocation in wireless networks with RF energy harvesting and transfer,” _IEEE Network_ , vol. 29, no. 6, pp. 68–75, Dec. 2015.
  * [5] R. D. Yates, Y. Sun, D. R. Brown, S. K. Kaul, E. Modiano, and S. Ulukus, “Age of information: An introduction and survey,” _IEEE Journal on Selected Areas in Communications_ , vol. 39, no. 5, pp. 1183–1210, Mar. 2021.
  * [6] M. K. Abdel-Aziz, S. Samarakoon, C.-F. Liu, M. Bennis, and W. Saad, “Optimized age of information tail for ultra-reliable low-latency communications in vehicular networks,” _IEEE Transactions on Communications_ , vol. 68, no. 3, pp. 1911–1924, Mar. 2019.
  * [7] L. Liang, L. Xu, B. Cao, and Y. Jia, “A cluster-based congestion-mitigating access scheme for massive M2M communications in Internet of things,” _IEEE Internet of Things Journal_ , vol. 5, no. 3, pp. 2200–2211, Jun. 2018\.
  * [8] M. Zhang, A. Arafa, J. Huang, and H. V. Poor, “Pricing fresh data,” _IEEE Journal on Selected Areas in Communications_ , vol. 39, no. 5, pp. 1211–1225, May 2021.
  * [9] J. Tang, J. Luo, M. Liu, D. K. So, E. Alsusa, G. Chen, K.-K. Wong, and J. A. Chambers, “Energy efficiency optimization for NOMA with SWIPT,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 13, no. 3, pp. 452–466, Feb. 2019.
  * [10] X. Pei, W. Duan, M. Wen, Y.-C. Wu, H. Yu, and V. Monteiro, “Socially aware joint resource allocation and computation offloading in NOMA-aided energy-harvesting massive IoT,” _IEEE Internet of Things Journal_ , vol. 8, no. 7, pp. 5240–5249, Apr. 2020.
  * [11] Q. Wu, W. Chen, D. W. K. Ng, and R. Schober, “Spectral and energy-efficient wireless powered IoT networks: NOMA or TDMA?” _IEEE Transactions on Vehicular Technology_ , vol. 67, no. 7, pp. 6663–6667, Jan. 2018.
  * [12] K. Chi, Z. Chen, K. Zheng, Y.-h. Zhu, and J. Liu, “Energy provision minimization in wireless powered communication networks with network throughput demand: TDMA or NOMA?” _IEEE Transactions on Communications_ , vol. 67, no. 9, pp. 6401–6414, Sep. 2019.
  * [13] S. Mao, S. Leng, S. Maharjan, and Y. Zhang, “Energy efficiency and delay tradeoff for wireless powered mobile-edge computing systems with multi-access schemes,” _IEEE Transactions on Wireless Communications_ , vol. 19, no. 3, pp. 1855–1867, Feb. 2019.
  * [14] F. Zhou and R. Q. Hu, “Computation efficiency maximization in wireless-powered mobile edge computing networks,” _IEEE Transactions on Wireless Communications_ , vol. 19, no. 5, pp. 3170–3184, May 2020.
  * [15] Y. Hu, C. Qiu, and Y. Chen, “Lyapunov-optimized two-way relay networks with stochastic energy harvesting,” _IEEE Transactions on Wireless Communications_ , vol. 17, no. 9, pp. 6280–6292, Sep. 2018.
  * [16] S. Xia, Z. Yao, Y. Li, and S. Mao, “Online distributed offloading and computing resource management with energy harvesting for heterogeneous MEC-enabled IoT,” _IEEE Transactions on Wireless Communications, (DOI: 10.1109/TWC.2021.3076201)_ , 2021.
  * [17] K. Singh, M.-L. Ku, J.-C. Lin, and T. Ratnarajah, “Toward optimal power control and transfer for energy harvesting amplify-and-forward relay networks,” _IEEE Transactions on Wireless Communications_ , vol. 17, no. 8, pp. 4971–4986, May 2018.
  * [18] J. Hu, K. Yang, G. Wen, and L. Hanzo, “Integrated data and energy communication network: A comprehensive survey,” _IEEE Communications Surveys & Tutorials_, vol. 20, no. 4, pp. 3169–3219, Jul. 2018.
  * [19] J. Wang, C. Jiang, K. Zhang, X. Hou, Y. Ren, and Y. Qian, “Distributed Q-learning aided heterogeneous network association for energy-efficient IIoT,” _IEEE Transactions on Industrial Informatics_ , vol. 16, no. 4, pp. 2756–2764, Nov. 2019.
  * [20] D. Qiao and J. Han, “Achievable throughput of energy harvesting fading multiple-access channels under statistical QoS constraints,” _IEEE Transactions on Wireless Communications_ , vol. 17, no. 5, pp. 2906–2917, Feb. 2018.
  * [21] C. K. Vranas, P. S. Bouzinis, V. K. Papanikolaou, P. D. Diamantoulakis, and G. K. Karagiannidis, “On the gain of NOMA in wireless powered networks with circuit power consumption,” _IEEE Communications Letters_ , vol. 23, no. 9, pp. 1657–1660, Jun. 2019.
  * [22] P. S. Bouzinis, P. D. Diamantoulakis, L. Fan, and G. K. Karagiannidis, “Pareto-optimal resource allocation in decentralized wireless powered networks,” _IEEE Transactions on Communications_ , vol. 69, no. 2, pp. 1007–1020, Oct. 2020.
  * [23] B. Li, F. Si, W. Zhao, and H. Zhang, “Wireless powered mobile edge computing with NOMA and user cooperation,” _IEEE Transactions on Vehicular Technology_ , vol. 70, no. 2, pp. 1957–1961, Feb. 2021.
  * [24] X. Wu, J. Yang, and J. Wu, “Optimal status update for age of information minimization with an energy harvesting source,” _IEEE Transactions on Green Communications and Networking_ , vol. 2, no. 1, pp. 193–204, Nov. 2017.
  * [25] H. Zheng, K. Xiong, P. Fan, Z. Zhong, and K. B. Letaief, “Age of information-based wireless powered communication networks with selfish charging nodes,” _IEEE Journal on Selected Areas in Communications_ , vol. 39, no. 5, pp. 1393–1411, Mar. 2021.
  * [26] I. Krikidis, “Average age of information in wireless powered sensor networks,” _IEEE Wireless Communications Letters_ , vol. 8, no. 2, pp. 628–631, Jan. 2019.
  * [27] H. Azarhava, M. P. Abdollahi, and J. M. Niya, “Age of information in wireless powered IoT networks: NOMA vs. TDMA,” _Ad Hoc Networks_ , vol. 104, p. 102179, Jul. 2020.
  * [28] A. M. Bedewy, Y. Sun, R. Singh, and N. B. Shroff, “Optimizing information freshness using low-power status updates via sleep-wake scheduling,” in _Proceedings of the Twenty-First International Symposium on Theory, Algorithmic Foundations, and Protocol Design for Mobile Networks and Mobile Computing (Mobihoc)_ , New York, NY, Oct. 2020, pp. 51–60.
  * [29] A. K. Aydin and N. Akar, “Energy management for age of information control in solar-powered IoT end devices,” _Wireless Networks_ , pp. 1–14, May 2021\.
  * [30] H. Pan and S. C. Liew, “Information update: TDMA or FDMA?” _IEEE Wireless Communications Letters_ , vol. 9, no. 6, pp. 856–860, Feb. 2020.
  * [31] A. Maatouk, M. Assaad, and A. Ephremides, “Minimizing the age of information: NOMA or OMA?” in _IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS)_ , Paris, France, May, 2019, pp. 102–108.
  * [32] F. Chiariotti, I. Leyva-Mayorga, Č. Stefanović, A. E. Kalør, and P. Popovski, “Spectrum slicing for multiple access channels with heterogeneous services,” _Entropy_ , vol. 23, no. 6, p. 686, May 2021.
  * [33] N. Tian and Z. G. Zhang, _Vacation Queueing Models: Theory and Applications_. Springer Science & Business Media, 2006, vol. 93.

|  Zhengru Fang (S’20) received his B.S. degree in electronics and information
engineering (with honors) from the Huazhong University of Science and
Technology (HUST), Wuhan, China in 2019. He is currently pursuing the M.S.
degree in electronics and communication engineering from Tsinghua University,
Beijing, China. He has been serving as a Reviewer for IEEE JSAC, IEEE IoTJ,
IEEE Systems Journal, IEEE Access, IEEE GLOBECOM and IEEE ICCC. His research
interests lie in the area of Internet of Underwater Things, age of
information, queueing theory and mobile edge computing.
---|---
|  Jingjing Wang (S’14-M’19-SM’21) received his B.S. degree in Electronic
Information Engineering from Dalian University of Technology, Liaoning, China
in 2014 and the Ph.D. degree in Information and Communication Engineering from
Tsinghua University, Beijing, China in 2019, both with the highest honors.
From 2017 to 2018, he visited the Next Generation Wireless Group chaired by
Prof. Lajos Hanzo, University of Southampton, UK. Dr. Wang is currently an
associate professor at School of Cyber Science and Technology, Beihang
University. His research interests include AI enhanced next-generation
wireless networks, swarm intelligence and confrontation. He has published over
100 IEEE Journal/Conference papers. Dr. Wang was a recipient of the Best
Journal Paper Award of IEEE ComSoc Technical Committee on Green Communications
& Computing in 2018, the Best Paper Award of IEEE ICC and IWCMC in 2019.
---|---
|  Yong Ren (M’11-SM’16) received his B.S, M.S and Ph.D. degrees in electronic
engineering from Harbin Institute of Technology, China, in 1984, 1987, and
1994, respectively. He worked as a post doctor at Department of Electrical
Engineering, Tsinghua University, China from 1995 to 1997. Now he is a full
professor of Department of Electronic Engineering and serves as the director
of the Complexity Engineered Systems Lab in Tsinghua University. He has
authored or co-authored more than 400 technical papers in the area of computer
network and mobile telecommunication networks. He has served as a reviewer of
more than 40 international journals or conferences. His current research
interests include complex system theory and its applications to the
optimization of the Internet, Internet of Things and ubiquitous network,
cognitive networks and cyber-physical systems.
---|---
|  Zhu Han (S’01–M’04-SM’09-F’14) received the B.S. degree in electronic
engineering from Tsinghua University, in 1997, and the M.S. and Ph.D. degrees
in electrical and computer engineering from the University of Maryland,
College Park, in 1999 and 2003, respectively. From 2000 to 2002, he was an R&D
Engineer of JDSU, Germantown, Maryland. From 2003 to 2006, he was a Research
Associate at the University of Maryland. From 2006 to 2008, he was an
assistant professor at Boise State University, Idaho. Currently, he is a John
and Rebecca Moores Professor in the Electrical and Computer Engineering
Department as well as in the Computer Science Department at the University of
Houston, Texas. His research interests include wireless resource allocation
and management, wireless communications and networking, game theory, big data
analysis, security, and smart grid. Dr. Han received an NSF Career Award in
2010, the Fred W. Ellersick Prize of the IEEE Communication Society in 2011,
the EURASIP Best Paper Award for the Journal on Advances in Signal Processing
in 2015, IEEE Leonard G. Abraham Prize in the field of Communications Systems
(best paper award in IEEE JSAC) in 2016, and several best paper awards in IEEE
conferences. Dr. Han was an IEEE Communications Society Distinguished Lecturer
from 2015-2018, AAAS fellow since 2019 and ACM distinguished Member since
2019. Dr. Han is 1% highly cited researcher since 2017 according to Web of
Science. Dr. Han is also the winner of 2021 IEEE Kiyo Tomiyasu Award, for
outstanding early to mid-career contributions to technologies holding the
promise of innovative applications, with the following citation: “for
contributions to game theory and distributed management of autonomous
communication networks.”
---|---
|  H. Vincent Poor (S’72, M’77, SM’82, F’87) received the Ph.D. degree in EECS
from Princeton University in 1977. From 1977 until 1990, he was on the faculty
of the University of Illinois at Urbana-Champaign. Since 1990 he has been on
the faculty at Princeton, where he is currently the Michael Henry Strater
University Professor. During 2006 to 2016, he served as the dean of
Princeton’s School of Engineering and Applied Science. He has also held
visiting appointments at several other universities, including most recently
at Berkeley and Cambridge. His research interests are in the areas of
information theory, machine learning and network science, and their
applications in wireless networks, energy systems and related fields. Among
his publications in these areas is the forthcoming book Machine Learning and
Wireless Communications. (Cambridge University Press). Dr. Poor is a member of
the National Academy of Engineering and the National Academy of Sciences and
is a foreign member of the Chinese Academy of Sciences, the Royal Society, and
other national and international academies. He received the IEEE Alexander
Graham Bell Medal in 2017.
---|---
|  Lajos Hanzo (http://www-mobile.ecs.soton.ac.uk,
https://en.wikipedia.org/wiki/Lajos_Hanzo) (FIEEE’04) received his Master
degree and Doctorate in 1976 and 1983, respectively from the Technical
University (TU) of Budapest. He was also awarded the Doctor of Sciences (DSc)
degree by the University of Southampton (2004) and Honorary Doctorates by the
TU of Budapest (2009) and by the University of Edinburgh (2015). He is a
Foreign Member of the Hungarian Academy of Sciences and a former Editor-in-
Chief of the IEEE Press. He has served several terms as Governor of both IEEE
ComSoc and of VTS. He has published 2000+ contributions at IEEE Xplore, 19
Wiley-IEEE Press books and has helped the fast-track career of 123 PhD
students. Over 40 of them are Professors at various stages of their careers in
academia and many of them are leading scientists in the wireless industry. He
is also a Fellow of the Royal Academy of Engineering (FREng), of the IET and
of EURASIP. He is the recipient of the 2022 Eric Sumner Field Award.
---|---