#  Age of Information Under Frame Slotted ALOHA-Based Status Updating Protocol

Zhiling Yue, , Howard H. Yang, Meng Zhang, and Nikolaos Pappas The work of Z.
Yue, H. H. Yang, and M. Zhang has been supported by the National Natural
Science Foundation of China under Grants 62201504 and 62202427. The work of N.
Pappas has been supported in part by the Swedish Research Council (VR),
ELLIIT, Zenith, and the European Union (ETHER, 101096526). (_Corresponding
Author: Howard H. Yang._) Z. Yue, H. H. Yang and M. Zhang are with the
Zhejiang University/University of Illinois at Urbana-Champaign Institute,
Zhejiang University, Haining 314400, China (email<EMAIL_ADDRESS><EMAIL_ADDRESS>mengzhang@intl.zju.edu.cn). N. Pappas is with the
Department of Computer and Information Science, Linköping University,
Linköping 58183, Sweden (e-mail: nikolaos.pappas@liu.se).

###### Abstract

We propose a frame slotted ALOHA (FSA)-based protocol for a random access
network where sources transmit status updates to their intended destinations.
We evaluate the effect of such a protocol on the network’s timeliness
performance using the Age of Information (AoI) metric. Specifically, we
leverage tools from stochastic geometry to model the spatial positions of the
source-destination pairs and capture the entanglement amongst the nodes’
spatial-temporal attributes through the interference they caused to each
other. We derive analytical expressions for the average and variance of AoI
over a typical transmission link in Poisson bipolar and cellular networks,
respectively. Our analysis shows that in densely deployed networks, the FSA-
based status updating protocol can significantly decrease the average AoI and
in addition, stabilizes the age performance by substantially reducing the
variance of AoI. Furthermore, under the same updating frequency, converting a
slotted ALOHA protocol into an FSA-based one always leads to a reduction in
the average AoI. Moreover, implementing FSA in conjunction with power control
can further benefit the AoI performance, although the particular values of
framesize and power control factor must be adequately tuned to achieve the
optimal gain.

###### Index Terms:

Age of information, wireless network, interference, frame slotted ALOHA,
stochastic geometry.

##  I Introduction

Ultra-reliable and low-latency communication (URLLC) [2018Cross, 2021URLLC] is
an important technology originated from the intersection of the Internet of
Things (IoT) and the tactile Internet, promoting a broad range of real-time
applications, such as healthcare [2022health], autonomous vehicles [autoAoI],
remote sensing and control [2020Control]. These applications have stringent
requirements on the timeliness of information delivery because outdated
information can result in wrong decisions and lead to severe consequences
[2017Radio]. In supporting delay-sensitive URLLC services as above, Age of
Information (AoI) is proposed as an effective and tractable metric to quantify
the timeliness [sunmodiano2019age]. Unlike conventional metrics such as delay
and throughput, AoI is assessed from the receiver’s perspective, measuring the
time elapsed since the latest received information update was generated. Since
AoI can capture the timeliness of information deliveries where traditional
metrics cannot, the AoI-oriented network designs often generate unconventional
(and sometimes counter-intuitive) insights as well as solutions. Consequently,
AoI has attracted considerable attention to the research of the information
update and transmission timeliness in the next generation URLLC system.

Early studies of AoI primarily focused on point-to-point scenarios [2012Real,
M2016On, 2020The] and found that the corresponding AoI-oriented optimization
design is different from that for conventional metrics. Specifically,
[2012Real] provided a general method for calculating the average AoI, and
found that the conventionally update policy, namely the zero-wait policy that
maximizes throughput or minimizes latency, does not always lead to an optimal
AoI. And it observed that in fact, contrary to intuition, it is more desirable
in many cases to wait for a certain amount of time at the transmitter side
from an AoI optimum perspective. [M2016On] developed and compared the AoI
performance under three different packet management schemes. It is discovered
that the one with the replacement protocol is the best among them. Considering
that real-world applications have different sensitivity toward information
staleness, the performance under nonlinear evolution of AoI was studied in
[2020The].

In practice, information systems often consist of a large number of entities
communicating via a shared spectrum. Due to the broadcast nature of the
wireless medium, simultaneous transmissions from different nodes can interfere
with each other. Interference often incurs transmission collisions and failure
deliveries, which can significantly impede the communication quality. In
response, a line of works introduced protocol models (a.k.a. conflict graphs)
to characterize the phenomenon of transmission collisions caused by
interference. These models assume that within a certain geographic region, as
long as there are source nodes transmitting simultaneously it will lead to
failure. Based on this model, several strategies are proposed to select active
links for channel access, aiming to control the network interference and
reduce the information age. [2020Opti] obtained the optimal, as well as
suboptimal (but low-complexity) scheduling strategies for minimizing both
average AoI and peak age under different source models. In [KadSinUys:18],
authors proposed four different scheduling strategies, including Greedy,
Randomized, Max-Weight and Whittle’s Index policies, to optimize the
performance of AoI, and they derived performance guarantees for these
strategies. However, these centralized scheduling methods generally require
unified coordination and decisions, which are not applicable to scenarios
where there exist massive end-user devices or applications in which traffic is
highly bursty [2021AgeofInformation]. Therefore, distributed random access
protocols, such as slotted ALOHA (SA) and carrier-sense multiple access
(CSMA), attract considerable attention. [2020CSMA] provided closed-form
expressions for the average AoI and average peak AoI under two different
packet management schemes with and without preemption. As the low-cost devices
have low overhead budget and may not have carrier sensing capability, in which
case SA-type protocol is preferable. In [2017Statusupdates], the optimization
of AoI using SA channel access strategy was studied, showing that its
performance is inferior to that of round robin policy. In view of this, some
studies have modified SA, the work in [2020Improving] proposed a deformation
called Threshold ALOHA, by which only when a source node’s AoI exceeds a
certain threshold will it be active and generate new information with a given
probability. This channel access protocol can reduce the competition among
sources to a certain extent, so as to improve the stale sources’ probability
of successful transmission and reduce the waste of transmission power. In
[2020Closed], an index was introduced to reflect the urgency of update, and
the nodes were selected to access the wireless channel according to their
index.

However, the conflict graph model oversimplifies the physical layer features
by resorting to a binary judgment of the existence/non-existence of
interference. In addition, it does not capture the critical attributes of a
wireless system such as fading, path loss, power control, and co-channel
interference. As such, adopting the signal-to-interference-plus-noise ratio
(SINR) or signal-to-interference ratio (SIR) model to characterize
interference in the network becomes a more appealing method [2017Birth]. Based
on this model, another line of work employ stochastic geometry as a tool to
model the node position distribution as a point process, in order to account
for those intricate effects from interference incurred by simultaneous
transmissions. In [HuZhoZha:18], the authors derived the upper and lower
bounds of the cumulative distribution function (CDF) for the average AoI in
the network by taking nodes’ positions and their mutual interference into
consideration. [2021SpatialDis] improved the bound on the spatial distribution
of peak AoI and offered an exact bound on the successful transmission
probability. The work in [MankarTWC21] presented a stochastic geometric
analysis of throughput and AoI in a cellular IoT network considering the
spatial disparity in the AoI performance. The authors in [2021spatiotemporal]
provided expressions for AoI by using meta distribution, and their simulation
results showed that the AoI performance is highly dependent on the spatial
location, traffic load, and decoding threshold. [2021Understand] established a
theoretical framework to analyze the impact of spatially interacting queues on
AoI, which facilitates understanding network parameters’ effect on the age
performance. And [2022Spatio] extended the AoI analysis to networks where
source nodes have unit-size buffers and operate under the last-come first-
serve with replacement protocol, which is generally AoI-optimal. On the basis
of such spatiotemporal models, a series of studies have been carried out to
design schemes for performance optimization. In [YanArafaQue:19], a
decentralized channel access strategy was developed. Under this strategy, each
node makes the AoI-optimal transmission decision according to the observation
of its ambient communication environment. An equation related to the
surrounding parameters of each point was given in [2022Locally]. By solving
this equation, one can obtain each source’s optimal status updating rate,
adapting to its local transmission condition. Additionally, utilizing the
source nodes’ local observation, [2021powctl] designed a decentralized power
control strategy by which nodes can adjust their transmit power individually
to optimize the AoI performance.

Although these existing works have analyzed, as well as optimized, the AoI
from various aspects, their designs are mainly pertaining to time slot-based
system dynamics. In contrast, the effect of frame slotted ALOHA (FSA), an
emerging technology that regularizes transmissions in the temporal domain and
is now prevailing to IoT applications [2021Stabi], such as coordinating
massive access in machine to machine (M2M) data collection networks [2018M2M]
and communications between readers and tags in radio frequency identification
(RFID) systems [2020RFID], on information freshness remains largely
unexplored. Capitalized on the philosophy of FSA, this paper presents a new
status updating protocol that improves age performance in a wireless network.
Specifically, we organize a fixed number of consecutive time slots into a
frame, and the source nodes determine to activate in each frame (or not)
independently with a certain probability. If a source node decides to activate
in a frame, it selects one time slot in the frame uniformly at random; upon
the selected time slot, the source node generates a new update of status and
immediately sends this information to its destination. We develop a
theoretical framework that characterizes the performance of the proposed
protocol. In particular, we model the spatial positions of the source and
destination nodes as an ergodic and stationary point process. The sources
update status information to their destinations using the FSA-based protocol.
We consider an interference-limited scenario in which transmissions over a
wireless link succeeds only if the SIR received at the destination exceeds a
decoding threshold. We derive expressions for the average and quadratic AoI of
a typical node by conditioning on the network topology. Since the interference
is affected by the spatial distribution of transmission links, we employ
stochastic geometry to average the potential geographical patterns of the
nodes, and derive closed-form expressions for the average and variance of AoI
under two commonly used network models, i.e., the Poisson bipolar network and
Poisson cellular network. Leaning on the analytical results, we find that FSA
provides a way to distribute sources into different time slots for update
transmissions, thereby it can reduce the competition among sources in the
network. We show that FSA can always achieve a smaller average AoI than SA. In
addition, in the time slot of each frame after the update, since the source
has no chance to transmit again, it will sleep to reduce power consumption.
During the period of preparing this paper, we found a very recent work that
also studied the average AoI under FSA-based protocol [2022FSA]. Nonetheless,
there are marked distinctions between our work and [2022FSA]. Specifically,
[2022FSA] considered a single cell case in which multiple users transmit to a
common access point. And it adopted a protocol model to characterize the
competition among users. The authors of [2022FSA] only analyzed the average
AoI and gave a few elementary insights into the FSA status updating protocol.
In contrast, we consider a multi-cell setting, which is more practical, and
use two specific network models that have been verified to well suit real-
world scenarios [2012Pair, 2016Int]. Moreover, we apply the SINR model,
capturing more intricate features from a wireless system. We derive not only
the average, but also variance of AoI, and the analysis can be extended to
higher order moments or the Cost of Update Delay (CoUD) [2022nonlinear]. In
addition, we have conducted a more comprehensive analysis by exploring
network’s age performance in various special-case scenarios, facilitating a
thorough understanding to the impact of FSA updating protocol on AoI. Our main
contributions are summarized below.

  * •

We propose an FSA-based protocol for a set of source nodes to update their
status information toward the intended destinations in a random access
network. We develop a mathematical framework to evaluate the age performance
of transmitters under such a protocol. Our model encompasses several key
features of a wireless system, including channel fading, path loss, and
interference. By fixing the network topology, we derive the (conditional)
first and second moments of time-averaged AoI of a typical node, and verify
the accuracy via simulations.

  * •

When the nodes form a Poisson bipolar network, we obtain closed-form
expressions for the average and variance of AoI of a typical node under the
FSA-based status updating protocol. We compare the nodes’ age performance
under FSA to those under the SA protocol and identify conditions under which
FSA is instrumental in reducing the AoI. We also find that under the same
updating frequency, converting an SA protocol into an FSA one always benefits
the average AoI. Besides, the FSA-based protocol also avails the transmitters
in reducing the transmission power consumption.

  * •

In the setting of a Poisson cellular network, we derive analytical expressions
for the average and variance of AoI over a typical transmission link by
accounting for effects of transmission protocol and power control strategy.
The analysis allows us to quantify the control factors from signal power
domain and interference domain, and their joint influence on the age
performance. We also carry out several special case studies to garner useful
insights.

  * •

Numerical results reveal that: i) the FSA-based status updating protocol is
beneficial when the spatial contention among transmitters is intense, i.e.,
the wireless links are frequently activated and densely deployed in space. In
this situation, an optimal framesize exists that minimizes the average (or
variance of) AoI; ii) the gain of FSA is particularly pronounced when the
network is densified. Specifically, when the deployment density increase by
five folds, an FSA-based protocol can reduce the average and variance of AoI
by two orders of magnitude compared to those under conventional SA-based
protocol; and iii) implementing FSA in conjunction with power control strategy
can further benefit the network’s age performance, although the values of
framesize and power control factor must be adequately tuned to achieve the
optimal gain.

The remainder of this paper is organized as follows. Section II lays down the
general system model and status updating protocol for this work. By
conditioning on the spatial layout, Section III shows the preliminary analysis
framework. Simulations are also provided to validate the accuracy of our
results. In Section IV, we analyze the impact of FSA-based status updating
protocol on the AoI performance in Poisson bipolar networks. We derive closed-
form expressions for the average and variance of AoI, and provide a series of
discussions for insights. Similarly, Section V explores the nodes’ age
performance in a Poisson cellular network. We also discuss the interplay
between status updating protocol and power control policy in this section.
Finally, we conclude the paper in Section VI.

##  II System Model

TABLE I: Notation Summary  Notation |  Definition
---|---
$\Phi_{\mathrm{s}}$; $\lambda_{\mathrm{s}}$ |  Stationary and ergodic point process modeling the locations of sources; source spatial deployment density
$\Phi_{\mathrm{d}}$; $\lambda_{\mathrm{d}}$ |  Stationary and ergodic point process modeling the locations of destinations; destination spatial deployment density
$\Phi$; $\lambda$ |  Superposition of stationary and ergodic point processes $\Phi_{\mathrm{s}}$ and $\Phi_{\mathrm{d}}$, i.e., $\Phi\triangleq{\Phi_{\mathrm{s}}}\cup{\Phi_{\mathrm{d}}}$; superposition of spatial deployment density $\lambda_{\mathrm{s}}$ and $\lambda_{\mathrm{d}}$ in Poisson bipolar network, i.e., $\lambda=\lambda_{\mathrm{s}}=\lambda_{\mathrm{d}}$
$P_{i}$; $\epsilon$ |  Transmit power of source node $i$; power control factor
$\eta$; $F$ |  Packet update rate; framesize, i.e., the number of time slots contained in each frame
$x_{i}$ |  Position of source $i$
$R_{i}$; $D_{i}$ |  Distance between source $i$ and its associated receiver; distance between source $i$ and the typical receiver
$r$; $\alpha$ |  Distance of the typical transmitter-receiver pair; path loss exponent
$\nu_{i}$; $h_{i}$ |  State indicator of source $i$; channel fading from transmitter $i$ to the typical receiver
$\theta$ |  SINR decoding threshold
$\mu_{0}^{\Phi}$ |  Transmission success probability of link $0$, conditioned on the point process $\Phi$
$\bar{\Delta}_{0}$; $\sigma^{2}_{\Delta_{0}}$ |  Average AoI over the typical link; variance of AoI over the typical link

In this section, we detail the setup of our network and the status updating
protocol. We also define the average and variance of AoI over a typical link.
The main notations used throughout this paper are listed in the Table I.

###  II-A Network Structure

####  II-A1 Spatial Configuration

We consider a wireless network deployed on the Euclidean plane, consisting of
source and destination nodes. The spatial positions of the sources and their
destinations are modeled by stationary and ergodic point processes, denoted by
$\Phi_{\mathrm{s}}$ and $\Phi_{\mathrm{d}}$, respectively. Each source
monitors an external process and transmits status updates to its intended
destination. The status update of each source is encapsulated into information
packets and transmitted over a shared spectrum. Source node $i$ transmits an
information packet with power $P_{i}$. We assume that the communication
channel between any pair of nodes is narrowband, and it is affected by two
attenuation components, i.e., the small-scale fading and large-scale path-
loss, where the channel fading varies independently across time and space
[2012Stogeo].

####  II-A2 Temporal Attribute

We consider a discrete-time system where the time is slotted into equal
durations. We assume the network is synchronized. We consider the updates of
every source node are generated at the beginning of a time slot, whereas the
transmission of an information packet takes up one time slot to finish.
Furthermore, we package a fixed number of consecutive time slots into a frame,
and the size of the frame is denoted by $F$. Since the time scale of fading
and packet transmission is much smaller than that of the spatial dynamics, we
assume the network topology is static, i.e., an arbitrary but fixed point
pattern is realized at the beginning and remains unchanged in the subsequent
time slots.

####  II-A3 Transmission Protocol

Sources adopt an FSA-based transmission strategy synchronically. Specifically,
at the beginning of each frame, every source independently decides whether it
will update in this frame or not, with probability $\eta$; if a source decides
to update in this frame, it randomly selects a time slot according to a
uniform distribution. Then, upon the transmission time slot, the source node
generates a new update and immediately transmits the information packet to the
destination. If the SIR at the destination exceeds a decoding threshold, the
packet is successfully received; otherwise, the transmission fails. By virtue
of low-cost implementation, we do not employ a MAC protocol for the re-
transmission and/or acknowledgment of a reception. As such, every source will
only send out an updated sample once.

Figure 1: An example of AoI evolution over the typical link under the FSA
status updating protocol. The framesize is set as $F=3$.

###  II-B Performance Metrics

Without loss of generality, we randomly select one transmission link in the
network to be our typical link, and denote the receiver’s location as the
origin. Note that the performance of this link is statistically identical to
the other links in the network, hence, it can serve as a representative one.

We consider AoI which captures the “freshness” of information received at
destinations. AoI measures the time elapsed since the latest received data at
a destination that is generated at the corresponding source. A formal
definition of this metric is expressed as follows:

$\displaystyle\Delta_{0}(t)=t-G_{0}(t),$ (1)

where $G_{0}(t)$ indicates the time-stamp of the most recently update received
by the typical destination at time $t$. An example of AoI evolution under the
proposed transmission protocol is illustrated in Fig. 1.

We employ the most representative metric, i.e., the average AoI over the
typical link, to quantitatively evaluate the timeliness of information
delivered in this network, which is defined as follows:

$\displaystyle\bar{\Delta}_{0}=\lim_{T\to\infty}{\frac{1}{T}\sum_{t=1}^{T}\Delta_{0}(t)}.$
(2)

In addition, we assess the variance of AoI 111It is noteworthy that the
framework developed in this paper can be extended to study the network’s age
performance under more general cost functions by using similar approaches in
[2022nonlinear].  to investigate the reliability of network age performance.
This metric is formally defined as follows:

$\displaystyle\sigma^{2}_{\Delta_{0}}=\lim_{T\to\infty}{\frac{1}{T}\sum_{t=1}^{T}\left(\Delta_{0}(t)\right)^{2}}-\left(\lim_{T\to\infty}{\frac{1}{T}\sum_{t=1}^{T}\Delta_{0}(t)}\right)^{2}.$
(3)

##  III Preliminary Results

In this section, we derive analytical expressions for the first and second
moments of AoI by fixing the network topology. We verify these results by
simulations.

###  III-A Conditional AoI Statistics

####  III-A1 SIR and Conditional Transmission Success Probability

We consider an interference-limited scenario and adopt SIR to evaluate the
transmission quality of wireless links. If the typical transmitter, situating
at $x_{0}$, transmits a packet to its intended receiver at a time slot $t$,
the received SIR can be written as:

$\displaystyle\textrm{SIR}_{0,t}=\frac{P_{0}h_{0,t}L(r)^{-1}}{\sum_{i\neq
0}P_{i}h_{i,t}\nu_{i,t}L(\|x_{i}\|)^{-1}},$ (4)

where $x_{i}$ represents the position of transmitter $i$, $r=\|x_{0}\|$ is the
distance between the typical source-destination pair, $h_{i,t}$ denotes the
channel fading from transmitter $i$ to the typical receiver,
$\nu_{i,t}\in\\{0,1\\}$ represents the state of transmitter $i$, where
$\nu_{i,t}=1$ if transmitter $i$ is active and $\nu_{i,t}=0$ otherwise, and
$L:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is a monotonically non-decreasing
function that characterizes the large-scale path loss.

Owing to the uncertainty in node positions, channel fading, and the activity
of source nodes, the SIR is a random variable. A commonly used notion to
characterize the statistical behavior of such a quantity is the conditional
transmission success probability [haenggi2016meta]. Formally, given a point
process $\Phi\triangleq{\Phi_{\mathrm{s}}}\cup{\Phi_{\mathrm{d}}}$, the
conditional transmission success probability over the typical link is defined
as:222Since the point process is stationary and the sources activate
independently from each other under the FSA-based protocol, the interference
nodes also form a stationary point process. As such, we drop the time index
$t$ from the subscript in the sequel.

$\displaystyle\mu_{0}^{\Phi}=\mathbb{P}(\mathrm{SIR_{0}}>\theta\mid\Phi),$ (5)

where $\theta$ is the decoding threshold.

####  III-A2 Conditional AoI

By fixing the spatial topology $\Phi$, dynamics of packet delivery over each
wireless link can be regarded as a Bernoulli process that varies on a frame
basis, where the active probability and transmission success probability are
$\eta$ and $\mu_{0}^{\Phi}$, respectively. Based on this abstraction, we
derive the conditional average AoI in the following.

###### Theorem 1

Conditioned on the spatial topology $\Phi$, the average AoI over the typical
link is given as:

$\displaystyle\mathbb{E}\left[\Delta_{0}|\Phi\right]=\frac{F^{2}-1}{12F}\times\eta\mu_{0}^{\Phi}+\frac{F}{\eta\mu_{0}^{\Phi}}+\frac{1-F}{2}.$
(6)

###### Proof:

Please see Appendix -A. ∎

Similarly, we can calculate the conditional quadratic AoI and it is provided
by the next theorem.

###### Theorem 2

Conditioned on the spatial topology $\Phi$, the average quadratic AoI over the
typical link is given as:

$\displaystyle\mathbb{E}\left[\Delta_{0}^{2}|\Phi\right]=\frac{2F^{2}}{(\eta\mu_{0}^{\Phi})^{2}}-\frac{F(2F\\!\\!-\\!\\!1)}{\eta\mu_{0}^{\Phi}}+\frac{F^{2}\\!\\!-\\!\\!1}{12F}\eta\mu_{0}^{\Phi}+\frac{F(F\\!\\!-\\!\\!1)}{2}.$
(7)

###### Proof:

Please see Appendix -B. ∎

Notably, $F$ not only affects AoI in an explicit manner, as we can see
directly from expressions (6) and (7), but also implicitly affects the change
of AoI through influencing $\mu_{0}^{\Phi}$. As such, the average and variance
of AoI can be obtained by deconditioning (6) and (7) with respect to point
process $\Phi$, respectively, where different distributions of $\Phi$ lead to
different results. In the sequel, we will present two types of wireless
network models, i.e., the Poisson bipolar network and Poisson cellular
network, to examine the effect of FSA on the AoI performance. Before
proceeding with a more detailed analysis, we will validate these theoretical
results via simulations.

(a) (b)

Figure 2: Simulations versus analysis: ($a$) conditional average AoI and ($b$)
conditional quadratic AoI, in which we vary the conditional transmission
probability as $\mu_{0}^{\Phi}=0.4,0.6$, and $0.8$.

###  III-B Analysis Validation

The simulations conducted in this part are dedicated to verifying the above
analysis. Specifically, we consider a source-destination pair and fix the
conditional transmission probability. We set $F=3$ and run the simulation over
$4\times 10^{6}$ time slots. We collect the AoI statistics and average them to
obtain the final results.

In Fig. 2, we plot the average AoI and quadratic AoI as a function of update
rate333In this work, the update rate is equivalent to the update probability
or the sampling probability., by fixing the values of conditional transmission
success probabilities. This figure shows that the simulations and analytical
results are almost indistinguishable, which verifies our theoretical
derivations. We also observe that an increase in the (conditional)
transmission success probability enhances the performance of the average AoI
and average quadratic AoI.

##  IV Poisson Bipolar Networks

In this section, we analyze the effect of FSA on the AoI performance in a
Poisson bipolar network. Such a model is motivated by the emerging interest in
applications like Device-to-Device (D2D) networking, mobile crowd sourcing,
and the Internet-of-Things (IoT), which do not require a centralized
infrastructure, e.g., base stations or access points.

###  IV-A Setting

Figure 3: A snapshot of the employed Poisson bipolar network, where the
squares represent the sources, continuously sampling a sequence of status
information and sending it to the destinations denoted by the circles. The
solid black line is the typical link, the solid red lines represent other
active links, and the dashed blue lines are inactive links.

In a Poisson bipolar network, transmitters form a homogeneous Poisson point
process (PPP) $\Phi_{\mathrm{s}}$ of intensity $\lambda$. Each transmitter has
a dedicated receiver in a random orientation of a constant distance $r$.
According to the displacement theorem [BacBla:09], the spatial layout of
receivers $\Phi_{\mathrm{d}}$ is also a homogeneous PPP of intensity
$\lambda$.

We assume that every source node transmits at a constant power
$P_{\mathrm{tx}}$. We also assume that the signal propagation is subjected to
small-scale Rayleigh fading and standard path loss. As such, SIR at the
typical receiver can be written as:

$\displaystyle\mathrm{SIR}_{0}^{\mathrm{B}}=\frac{P_{\mathrm{tx}}h_{0}r^{-\alpha}}{\sum_{i\neq
0}P_{\mathrm{tx}}h_{i}\nu_{i}\|x_{i}\|^{-\alpha}},$ (8)

where $\alpha$ is the path-loss exponent.

Based on (8), we analyze the AoI performance under FSA and explore the
interplay among the AoI metric and different network parameters in the
following.

###  IV-B Analysis

Similarly to [2015The], we commence the AoI analysis by averaging out the
effect of channel fading, which brings us to the following expression for the
conditional transmission success probability.

###### Lemma 1

Given point process $\Phi$, the conditional transmission success probability
over the typical link is:

$\displaystyle\mu_{0}^{\Phi}=\prod_{i\neq
0}\left(1-\frac{\eta/F}{1+\|x_{i}\|^{\alpha}/\theta r^{\alpha}}\right).$ (9)

###### Proof:

Please see Appendix -C. ∎

Using this result, we can decondition $\mu^{\Phi}_{0}$ in (6) and obtain the
analytical expression for the average AoI.

###### Theorem 3

In a Poisson bipolar network, the average AoI of the typical link under FSA
updating protocol is given by

$\displaystyle\bar{\Delta}_{0}^{\mathrm{B}}\\!=\\!\frac{F}{\eta}\exp\\!{\Big{(}\frac{C\eta}{F}\big{(}1\\!\\!-\\!\\!\frac{\eta}{F}\big{)}^{\delta-1}}\Big{)}\\!+\\!\frac{(F^{2}\\!\\!-\\!\\!1)\eta}{12F}\exp\\!{\Big{(}\\!\\!-\\!\\!\frac{C\eta}{F}\Big{)}}\\!+\\!\frac{1\\!\\!-\\!\\!F}{2},$
(10)

where $\delta=2/\alpha$ and $C=\lambda\pi
r^{2}\theta^{\delta}\Gamma(1-\delta)\Gamma(1+\delta)$, with $\Gamma(\cdot)$
being the Gamma function [2015The].

###### Proof:

Please see Appendix -D. ∎

This theorem provides a closed-form expression that accounts for several key
factors of a wireless system, i.e., the network topology, transmission
protocol, and interference, on the average AoI. The following observations can
be readily made from (10).

####  IV-B1 When $F=1$

The result in (10) reduces to the classical average AoI under the SA protocol,
given by

$\displaystyle\bar{\Delta}_{0}^{\textit{SA}}=\mathbb{E}\left[\frac{1}{\eta\mu_{0}^{\Phi}}\right]=\frac{1}{\eta}\exp{\left(C\eta(1-\eta)^{\delta-1}\right)}.$
(11)

Note that the path loss exponent $\alpha$ generally satisfies $\alpha>2$ in
practice, and consequently we have $\delta<1$. Then, following (11), we can
see that the average AoI is unbounded in regimes where the sources are
generating updates in either extremely lazy (i.e., $\eta\rightarrow 0$) or
excessively aggressive (i.e., $\eta\rightarrow 1$) manner. The former mainly
ascribes to the absence of new updates at the sources; the latter due to
potential interferers located in geographical proximity to the typical
receiver, which impedes the transmissions and hinders any possible packet
delivery. In contrast, (10) indicates that incorporating FSA to the status
updating protocol can effectively abbreviate such severe interference issue
(note that for $F>1$, the average AoI is always bounded when $\eta\rightarrow
1$).

####  IV-B2 When $F>1$

In this case, let us set $\beta=\frac{\eta}{F}$ and term it as the effective
updating rate under FSA. Then, we can rewrite (10) as follows:

$\displaystyle\bar{\Delta}_{0}^{\mathrm{B}}=\bar{\Delta}_{0}^{\textit{SA}}(\beta)+\underbrace{\frac{F^{2}-1}{12}\beta\exp{\Big{(}-C\beta\Big{)}}+\frac{1-F}{2}}_{Q_{1}},$
(12)

where
$\bar{\Delta}_{0}^{\textit{SA}}(\beta)=\frac{1}{\beta}\exp{(C\beta(1-\beta)^{\delta-1}})$
denotes the average AoI of the typical link under SA with updating rate
$\beta$. From (12), we note that if $Q_{1}$ is non-negative, introducing frame
structures into SA is not indispensable since one can always obtain a smaller
average AoI without FSA by adjusting the update rate $\beta$ under SA.
Therefore, FSA-based protocol is instrumental in reducing AoI only when
$Q_{1}<0$.

An interesting observation is that regardless of the interference level,
imposing a frame structure on the SA protocol always improves the network age
performance. Below we present two approaches to bridge the SA and FSA
protocols. To facilitate exposition, we use $\eta_{\mathrm{SA}}$ to denote the
state update rate under SA protocol and $\eta_{\mathrm{FSA}}$ to denote that
under FSA. We can convert any SA at $\eta_{\mathrm{SA}}$ to:

a) FSA with $\eta_{\mathrm{FSA}}=2\eta_{\mathrm{SA}}$, $F=2$. Note that in
this scenario, the effective updating rate of the FSA protocol is
$\beta=\eta_{\mathrm{SA}}$, which is identical to that of SA. Consequently, we
can compute $Q_{1}$ as follows:

$\displaystyle
Q_{1}=\frac{\eta_{\mathrm{FSA}}}{8}\exp{\Big{(}-C\frac{\eta_{\mathrm{FSA}}}{2}\Big{)}}-\frac{1}{2}<\frac{\eta_{\mathrm{FSA}}}{8}-\frac{1}{2}<0.$
(13)

b) FSA with $\eta_{\mathrm{FSA}}=1$, $F=\frac{1}{\eta_{\mathrm{SA}}}$, if
$\frac{1}{\eta_{\mathrm{SA}}}$ is an integer. In this scenario,
$\beta=\frac{1}{F}$, which is also identical to that of SA. To demonstrate
that FSA outperforms SA, we rewrite $Q_{1}$ by the following:

$\displaystyle Q_{1}$
$\displaystyle=\frac{F^{2}\\!-\\!1}{12F}\exp{\Big{(}\\!-\\!\frac{C}{F}\Big{)}}+\frac{1\\!-\\!F}{2}$
$\displaystyle<\frac{F^{2}\\!-\\!1}{12F}+\frac{1\\!-\\!F}{2}=-\frac{(5F-1)(F-1)}{12F}<0.$
(14)

Both of the above designs can transform the commonly used SA updating protocol
into an FSA protocol and achieve better AoI performance. This is because FSA
not only decreases mutual interference amongst the transmitters, but also
equalizes the updating intervals of each source node, thereby reducing the AoI
[sunmodiano2019age].

####  IV-B3 The optimal $F$

By applying the inequality of arithmetic and geometric means to (10), we can
derive a lower bound to the average AoI over the typical link as:

$\displaystyle\bar{\Delta}_{0}$ $\displaystyle\geq
2\sqrt{\frac{(F^{2}-1)\eta\mu_{0}^{\Phi}}{12F}\times\frac{F}{\eta\mu_{0}^{\Phi}}}+\frac{1-F}{2}$
$\displaystyle=2\sqrt{\frac{F^{2}-1}{12}}+\frac{1-F}{2}.$ (15)

This inequality indicates that merely increasing $F$ does not always benefit
the AoI performance. In consequence, we shall adequately choose the framesize
in accordance with the network parameters to achieve the optimal operation
regime.

Figure 4:  A plot of function $y(F)$ verse framesize under different spatial
deployment parameters, in which we set $\alpha=3.5$, $\eta=0.8$,
$\theta=0\,\mathrm{dB}$, and vary $\lambda$ as $\lambda=2\times 10^{-2}$,
$\lambda=1\times 10^{-2}\,\mathrm{m}^{-2}$, $r$ as $r=10$, $r=15\,\mathrm{m}$.

With this understanding, we fix other parameters and explore the optimal $F$
that minimizes the average AoI. Specifically, we relax the constraint of $F$
being an integer, take the derivative of $\bar{\Delta}_{0}^{\mathrm{B}}$ with
respect to $F$, and assign
$\frac{\mathrm{d}\bar{\Delta}_{0}^{\mathrm{B}}}{\mathrm{d}F}=0$. Then, we can
obtain the theoretical optimal $F$ (denoted by $F^{*}$) by solving $y(F)=0$,
where $y(F)$ is expressed as:

$\displaystyle y(F)$
$\displaystyle=\left[\frac{1}{\eta}+\frac{C(1\\!\\!-\\!\\!\frac{\eta}{F})^{\delta-2}}{F^{2}}(\eta\delta-F)\right]\exp\\!{\Big{(}\frac{C\eta}{F}\big{(}1\\!\\!-\\!\\!\frac{\eta}{F}\big{)}^{\delta\\!-\\!1}}\Big{)}$
$\displaystyle+\frac{(F^{2}\\!\\!+\\!\\!1)F+(F^{2}\\!\\!-\\!\\!1)C\eta}{12F^{3}}\eta\exp\\!{\Big{(}\\!-\\!\frac{C\eta}{F}\Big{)}}-\frac{1}{2}.$
(16)

Note that the root of this function can be attained efficiently by popular
software such as Matlab. Based on the parameter setting as per Section IV-C,
we give a plot in Fig. 4 on what $y(F)$ looks like. This figure shows that
$y(F)$ increases monotonically, and if the solution to $y(F)=0$ exists, it
must be unique. Moreover, we shall assign the optimal framesize $F$ as the one
between $\lceil F^{*}\rceil$ and $\lfloor F^{*}\rfloor$ that makes the
$\bar{\Delta}_{0}$ smaller, where $\lceil\cdot\rceil$ and
$\lfloor\cdot\rfloor$ denote the ceil and floor functions, respectively. On
the other hand, if $y(F)=0$ does not have a solution, we set $F^{*}=1$.

####  IV-B4 Spatial throughput

Another commonly used metric in the optimization of network deployment is
spatial throughput, given by [2011Tractable]:

$\displaystyle\Theta$
$\displaystyle\triangleq\frac{\eta}{F}\mathbb{P}(\mathrm{SIR}_{0}>\theta)\log(1+\theta)$
$\displaystyle=\frac{\eta}{F}\mathbb{E}[\mu_{0}^{\Phi}]\log(1+\theta)$
$\displaystyle=\frac{\eta}{F}\exp{\left(-\frac{C\eta}{F}\right)}\log(1+\theta),$
(17)

where $\mathbb{E}[\mu_{0}^{\Phi}]=\exp(-C\frac{\eta}{F})$ follows from (-D).
This quantity measures the average rate of successful information delivery
over the typical link. It is natural to infer that maximizing spatial
throughput also optimizes AoI, because a high throughput enables packets to be
delivered swiftly, giving “fresher” information at the receiver. However,
comparing (10) and (IV-B4), we can see that jointly tuning the update rate
$\eta$ and frame size $F$ to obtain a higher throughput, i.e., making the term
$\frac{\eta}{F}\exp{\left(-\frac{C\eta}{F}\right)}$ as large as possible, does
not necessarily lead to a smaller AoI.

####  IV-B5 Transmission power consumption

According to some recent studies on energy efficiency, sleep-scheduling
strategies can effectively reduce power consumption and optimize AoI
performance [2022Energy, 2021Low]. Considering that FSA also has an implicit
“sleep scheduling” characteristic, we investigated the effectiveness of FSA in
promoting energy efficiency in addition to optimizing AoI performance. More
specifically, conditioned on the spatial topology $\Phi$, in each frame, let
us denote by $U$ (resp. $\bar{U}$) the event that the typical transmitter does
(resp. does not) generate a new update, and $S$ (resp. $\bar{S}$) that the
typical receiver does (resp. does not) receive a successful update. Moreover,
we denote $E_{0}(w)$ as the energy consumed by the typical source node since
the ($w-1$)-th successful transmission to the $w$-th one. Then, we can compute
the transmission power consumption over the typical link as:

$\displaystyle\mathbb{E}[P_{0}|\Phi]$
$\displaystyle=\lim_{W\to\infty}\frac{\sum_{w=1}^{W}E_{0}(w)}{\sum_{w=1}^{W}I_{w}}$
$\displaystyle=\frac{\mathbb{P}(U|\bar{S})(\mathbb{E}[X]-1)+1}{\mathbb{E}[I]}P_{\mathrm{tx}}=\frac{\eta
P_{\mathrm{tx}}}{F},$ (18)

where $I$ is the interval between two consecutive updates received,
$\mathbb{P}(\bar{S})=1-\mathbb{E}[\mu_{0}^{\Phi}]$, and $\mathbb{P}(U)=\eta$.
Following (IV-B5), we can clearly see that merging time slots into frames is
capable of reducing the transmission power consumption.

$\displaystyle\sigma^{2}_{\Delta^{\mathrm{B}}_{0}}$
$\displaystyle=\frac{2F^{2}}{\eta^{2}}\exp{\left(-C\sum_{k=1}^{\infty}(k+1){\delta-1\choose
k-1}\big{(}-\frac{\eta}{F}\big{)}^{k}\right)}-\frac{F^{2}}{\eta^{2}}\exp{\left(\frac{2C\eta}{F}\big{(}1-\frac{\eta}{F}\big{)}^{\delta-1}\right)}-\frac{(F^{2}-1)^{2}\eta^{2}}{144F^{2}}\exp{\left(-\frac{2C\eta}{F}\right)}$
$\displaystyle-\frac{F^{2}-1}{6}\exp{\left(\frac{C\eta}{F}\Big{(}\big{(}1-\frac{\eta}{F}\big{)}^{\delta-1}-1\Big{)}\right)}-\frac{F^{2}}{\eta}\exp{\left(\frac{C\eta}{F}\big{(}1-\frac{\eta}{F}\big{)}^{\delta-1}\right)}+\frac{(F^{2}-1)\eta}{12}\exp{\left(-\frac{C\eta}{F}\right)}+\frac{F^{2}+2F-3}{4}$
(19)

In addition to computing the average AoI, we can also derive the analytical
expression for the variance of AoI by deconditioning $\mu_{0}^{\Phi}$ in (7)
to obtain $\mathbb{E}[\Delta_{0}^{2}]$ and then following similar calculations
as the above.

###### Theorem 4

In a Poisson bipolar network, the variance of AoI over the typical link under
FSA updating protocol can be expressed as (IV-B5) at the top of this page.

###### Proof:

Please see Appendix -E. ∎

When $F=1$, this result reduces to the variance of AoI under the SA protocol.
We provide it in the following as a complement to the current studies of AoI
in random access networks.

###### Corollary 1

In a Poisson bipolar network, the variance of AoI over the typical link under
SA updating protocol is given by

$\displaystyle\sigma^{2}_{\Delta^{\textit{SA}}_{0}}=\mathbb{E}\left[\frac{2}{(\eta\mu_{0}^{\Phi})^{2}}-\frac{1}{\eta\mu_{0}^{\Phi}}\right]-\mathbb{E}\left(\left[\frac{1}{\eta\mu_{0}^{\Phi}}\right]\right)^{2}$
$\displaystyle=\frac{2}{\eta^{2}}\exp{\left(-C\sum_{k=1}^{\infty}(k+1){\delta-1\choose
k-1}(-\eta)^{k}\right)}$
$\displaystyle-\frac{1}{\eta^{2}}\exp{\left(2C\eta(1-\eta)^{\delta-1}\right)}-\frac{1}{\eta}\exp{\left(C\eta(1-\eta)^{\delta-1}\right)}.$
(20)

When $F>1$, we follow a similar analysis method to the average AoI and rewrite
(IV-B5) in terms of effective update rate, as follows:

$\displaystyle\sigma^{2}_{\Delta^{\mathrm{B}}_{0}}=\sigma^{2}_{\Delta_{0}^{\textit{SA}}}(\beta)$
$\displaystyle\underbrace{-\\!\bigg{(}\\!\frac{F^{2}\\!\\!-\\!\\!1}{12\beta}e^{\\!-\\!\frac{C}{\beta}}\\!-\\!\frac{F}{2}\bigg{)}^{\\!2}\\!\\!-\\!\\!\frac{F^{2}\\!\\!-\\!\\!1}{6}e^{\frac{C\eta}{F}\Big{(}\big{(}1\\!-\\!\frac{\eta}{F}\big{)}^{\delta\\!-\\!1}\\!-\\!1\Big{)}}\\!\\!+\\!\\!\frac{2F^{2}\\!\\!+\\!\\!2F\\!\\!-\\!\\!3}{4}}_{Q_{2}},$
(21)

where $\sigma^{2}_{\Delta_{0}^{\textit{SA}}}(\beta)$ denotes the variance of
AoI over the typical link under SA protocol with update rate $\beta$.
According to this result, we note that the FSA protocol is preferable to SA
only when $Q_{2}<0$.

###  IV-C Numerical Results

Based on the analytical results, this part shows the AoI performance under
different network operation regimes. Unless otherwise specified, we use the
following parameters: $\alpha=3.5$, $\theta=0\,\mathrm{dB}$.

(a) (b)

Figure 5: AoI performance versus the update rate under different framesizes
$F$: ($a$) average AoI and ($b$) variance of AoI, in which we set
$\lambda=1\times 10^{-2}\,\mathrm{m}^{-2}$, $r=10\,\mathrm{m}$, and vary the
framesize as $F=1$ (i.e. SA), $3$, $5$, and $7$.

Fig. 5 illustrates the average, as well as variance, of AoI as a function of
the status updating rate, respectively, under different framesizes $F$. From
Fig. 5(a), we observe that for the considered situations, there exists an
optimal update rate that minimizes the average AoI. This mainly arises from
the tradeoff between generating fresh information at the sources and
maintaining interference at a relatively low level across the network.
Moreover, we notice that when $\eta$ is relatively low, SA attains a smaller
average AoI than FSA. Because, in this case, the interference is mild, and the
sources shall not wait for the entire frame to generate a new update. However,
when $\eta$ increases, the average AoI under SA (i.e., $F=1$) grows rapidly.
In comparison, those under FSA with $F\geq 2$ remain at relatively small
values for a wide range of $\eta$. The reason attributes to the fact that
increasing the update rate can activate more transmission links. As a result,
interference among source nodes becomes destructive, leading to the
consequence of many transmission failures. In contrast, FSA implicitly
regularizes the nodes’ transmission patterns by imposing a frame structure on
their active periods. Notably, even if two sources are situated at close
proximity in space, choosing a frame with size $F\geq 2$ can dramatically
decrease the chance that these nodes select the same time slot for sending out
the update information and result in the collision of their transmitted
packets. Consequently, we can see that when $F$ is relatively large (i.e.,
$F=7$), the average AoI declines steadily as we increase the update rate.
Because when $F$ is large, the nodes have more opportunities to pick different
time slots for generating updates (and transmitting them). As such, the
transmission benefits from low interference; hence, the more frequent the
generation of the updates, the fresher the received information. We observe
similar phenomena from Fig. 5(b), except that ($i$) the variance of AoI is
more sensitive to the change of update rates; and ($ii$) therefore, the
optimal $\eta$ that minimizes the variance of AoI is different from that of
the minimum average AoI. Nonetheless, it is noteworthy that using FSA for
status updating in a large-scale wireless network not only reduces the average
AoI but, more importantly, flats out the variations in the variance of AoI,
which is crucial for stabilizing the network operation.

(a) (b)

Figure 6:  AoI performance versus framesize under different deployment
densities $\lambda$: ($a$) average AoI and ($b$) variance of AoI, in which we
set $\eta=0.8$, $r=10\,\mathrm{m}$, and vary the deployment densities as
$\lambda=2\times 10^{-2}$, $1\times 10^{-2}$, $5\times 10^{-3}$, and $5\times
10^{-5}\,\mathrm{m}^{-2}$.

(a) (b)

Figure 7:  Comparison of AoI performance for deployment density variation
under SA and FSA strategies. First, we fixed $\eta=0.8$ in SA and
$(\eta,F)=(0.8,3)$ in FSA, then plot: ($a$) average AoI and ($b$) variance of
AoI.

Recognizing that the frame size of FSA has a remarkable influence on the AoI
performance, we plot the average and variance of AoI as functions of $F$ in
Fig. 6. The figure unveils that depending on the network configuration, there
may (or may not) exist an optimal $F$ that minimizes the average/variance of
AoI. This is because, while enlarging $F$ can mitigate the conflicts among
nodes’ transmissions, which in turn increases the transmission success
probability, it also prolongs the duration that each source generates a new
update. To this end, we can see that the framesize strikes a delicate balance
between the information freshness at the sources and the interference level
across the network. In addition, we note that although increasing the
deployment density will inevitably raise up the average and variance of AoI,
adopting FSA alleviates such deterioration by effectually dwindling the
transmission collisions. Furthermore, we notice that as $\lambda$ increases,
the optimal $F$ goes up correspondingly, entailing more stringent
regularization to curb competition among the transmitting nodes. On the other
hand, when the spatial deployment is sparse (e.g., $\lambda=2\times 10^{-2}$
in this case), interference is mild, and the FSA protocol loses its efficacy.
Actually, it may perform worse than the SA protocol under such a circumstance.

Fig. 7 depicts the AoI statistics, i.e., the average and variance, as
functions of the deployment density. In this example, we fix the framesize and
update rate as $F=3$ and $\eta=0.8$, respectively. By comparing the AoI
performance under SA and FSA, we find that networks under FSA protocol attain
a remarkable reduction in the average and variance of AoI as the
infrastructure is densified. More concretely, it can be seen from Fig. 7(a)
that under a moderate source-destination distance $r=10$, when the spatial
density increases by five folds, the average AoI under SA goes up by more than
two orders of magnitude; in comparison, the increase in the average AoI under
FSA is almost unnoticeable. Such a difference becomes more prominent when $r$
increases from 10 to 15, where the average AoI under SA surges (almost)
without limit, while that under FSA only rises fractionally. On the other
hand, Fig. 7(b) demonstrates that FSA is also effective in decreasing the
variance of AoI–with a more pronounced gain than the average AoI. These
phenomena are in line with our previous analysis, that when the source nodes
update fast and are densely distributed, FSA protocol performs better than SA
(it corresponds to that $Q_{1}$ from (13) and $Q_{2}$ from (22) are both
negative). Under this circumstance, increasing $\lambda$ and $r$ enlarges $C$,
leading to increases in the absolute values of $Q_{1}$ and $Q_{2}$, which
widen the gap between SA and FSA.

##  V Poisson Cellular Networks

This section explores the age performance in the setting of cellular networks.
Under this model, multiple source nodes transmit status update information to
a common destination. We derive analytical expressions for the considered AoI
metrics, and investigate the interplay between FSA protocol and the intra-cell
spectral competition, inter-cell interference, and the power control on the
AoI performance.

###  V-A Setting

We consider the uplink of a Poisson cellular network, as depicted in Fig. 8,
where spatially distributed sensors need to update status information to their
targeted data fusing centers. The data fusing centers are deployed according
to a homogeneous PPP $\Phi_{\mathrm{d}}$ with spatial density
$\lambda_{\mathrm{d}}$. The sensors are scattered as an independent
homogeneous PPP $\Phi_{\mathrm{s}}$ of intensity $\lambda_{\mathrm{s}}$. Every
sensor associates with the closest data fusing center in geographical space.

In the context of cellular networks, distances between different transmitter-
receiver pairs vary significantly, leading to a crucial impact from path loss
on the signal attenuation. In view of this challenge, we consider each sensor
adopts a power control strategy for its transmission. Specifically, we denote
by $\epsilon\in[0,1]$ and $R_{i}$ the power control factor and the distance
between the $i$-th sensor located at $x_{i}$ to its associated data fusing
center, respectively; then, the transmit power of sensor $i$ is
$P_{\mathrm{tx}}R_{i}^{\alpha\epsilon}$. Accordingly, the SIR of the typical
receiver can be written as:

$\displaystyle\mathrm{SIR}_{0}^{\mathrm{C}}=\frac{P_{\mathrm{tx}}h_{0}r^{\alpha(\epsilon-1)}}{\sum_{i\neq
0}P_{\mathrm{tx}}h_{i}\nu_{i}R_{i}^{\alpha\epsilon}\|x_{i}\|^{-\alpha}}.$ (22)

Using this expression, we will analyze the average and variance of AoI under
FSA in the following section and investigate the AoI performance under
different power control strategies.

Figure 8: An example of the Poisson cellular network, where the sources and
destinations are denoted by the squares and circles, respectively. The solid
black line is the typical link, the solid red lines represent other active
links, and the solid blue circles are inactive sources with no connection.

###  V-B Analysis

Similar to the previous section, we commence our analysis with deriving an
initial expression for the conditional transmission success probability.
Specifically, by substituting (22) into (5) and averaging out the randomness
from channel fading and interferers’ active states, we have:

$\displaystyle\mu_{0}^{\Phi}$
$\displaystyle=\mathbb{P}\left(\mathrm{SIR}_{0}^{\mathrm{C}}>\theta|\Phi\right)=\prod_{i\neq
0}\left(1-\frac{\eta/F}{1+\frac{\|x_{i}\|^{\alpha}}{\theta
R_{i}^{\alpha\epsilon}r^{\alpha(1-\epsilon)}}}\right).$ (23)

Based on this result, we can decondition $\mu_{0}^{\Phi}$ in (6) and obtain
the expression of the average AoI over the typical link. However, owing to
effects of power control, the spatial locations of interference nodes is a
non-stationary point process where the intensity depends on the distance
between an interferor to the typical data fusing center, hindering the
derivation of an exact analysis. In light of this, we adopt the approximations
developed in [2016User] for the probability density function (pdf). According
to [2016User], the pdf of the distance, $R_{i}$, between a generic sensor $i$
and its associated data fusing center can be approximated as

$\displaystyle f_{r}(u)$ $\displaystyle=\frac{5}{2}\lambda_{\mathrm{d}}\pi
u\exp{\left(-\frac{5}{4}\lambda_{\mathrm{d}}\pi u^{2}\right)}.$ (24)

The distance between sensor $i$ and the typical data fusing center, which
located at the origin, is given by

$\displaystyle f_{x}(u)$
$\displaystyle=2\pi\lambda_{\mathrm{d}}u\exp{(-\lambda_{\mathrm{d}}\pi
u^{2})}.$ (25)

Due to the association policy, we know that $R_{i}$ and $D_{i}$ are not
independent. In fact, $R_{i}$ cannot be larger than $D_{i}$, since otherwise
sensor $i$ will associate to another data fusing center. We formalize this
correlation by using the conditional distribution function given by:

$\displaystyle f_{r}(u|D_{i})=\frac{\frac{5}{2}\lambda_{\mathrm{d}}\pi
u\exp{(-\frac{5}{4}\lambda_{\mathrm{d}}\pi
u^{2})}}{1-\exp{(-\frac{5}{4}\lambda_{\mathrm{d}}\pi D_{i}^{2})}}.$ (26)

Armed with the above preparation, we are now ready to present the analytical
expression for the average AoI.

###### Theorem 5

In a Poisson cellular network, the average AoI over the typical link under FSA
updating protocol is given by

$\displaystyle\bar{\Delta}_{0}^{\mathrm{C}}$
$\displaystyle=\\!\\!\int_{0}^{\infty}\\!\\!e^{-z}\Bigg{(}\frac{F^{2}\\!\\!-\\!\\!1}{12F}\eta\exp{\Big{(}\\!\\!-\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\frac{\eta}{F}g_{\theta}(1,z;\alpha,\epsilon)\Big{)}}$
$\displaystyle+\frac{F}{\eta}\exp{\Big{(}\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\frac{\eta}{F}g_{\theta}(1-\frac{\eta}{F},z;\alpha,\epsilon)\Big{)}}\Bigg{)}\mathrm{d}z+\frac{1\\!-\\!F}{2},$
(27)

where $g_{\theta}(m,z;\alpha,\epsilon)$ is given as:

$\displaystyle
g_{\theta}(m,z;\alpha,\epsilon)\\!=\frac{4}{5}z^{2}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\int_{0}^{1}\\!\frac{q\exp{(-zqs)}\mathrm{d}s\mathrm{d}q}{(m\\!+\\!\frac{q^{\alpha(1-\epsilon)/2}}{\theta
s^{\alpha\epsilon/2}})(1-e^{-zq})}.$ (28)

###### Proof:

Please see Appendix -F. ∎

The theorem above captures the interplay amongst the update rate, power
control, and interferes’ spatial dependencies, as well as their composite
effects on the average AoI. The resultant expression is, nevertheless, a bit
involved. By assuming the positions of interference sensors being i.i.d. and
constitutes a Poisson marked process [2014On], we can obtain a tight
approximation to it.

###### Corollary 2

Under the setting of a cellular network with power control, the average AoI
over the typical link under FSA updating protocol can be approximated by the
following:

$\displaystyle\bar{\Delta}_{0}^{\mathrm{C}}$
$\displaystyle\approx\frac{F^{2}\\!\\!-\\!\\!1}{12F}\eta\int_{0}^{\infty}\\!\\!e^{-z-C_{1}z^{1-\epsilon}}\\!\mathrm{d}z\\!+\\!\frac{F}{\eta}\int^{\infty}\\!\\!e^{-z+C_{2}z^{1-\epsilon}}\\!\mathrm{d}z\\!+\\!\frac{1\\!-\\!F}{2}$
$\displaystyle=\frac{1\\!-\\!F}{2}\\!+\\!\sum_{n=0}^{\infty}\frac{\Gamma\big{(}1\\!+\\!(1\\!-\\!\epsilon)n\big{)}}{n!}\bigg{(}\frac{F^{2}\\!\\!-\\!\\!1}{12F}\eta(-C_{1})^{n}\\!+\\!\frac{F}{\eta}{C_{2}}^{n}\bigg{)},$
(29)

where
$C_{1}=\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\frac{\eta}{F}\theta^{\delta}\Gamma(1-\delta)\Gamma(1+\delta)\Gamma(1+\epsilon)$
and
$C_{2}=\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\frac{\eta}{F}\big{(}1-\frac{\eta}{F}\big{)}\theta^{\delta}\Gamma(1-\delta)\Gamma(1+\delta)\Gamma(1+\epsilon)$.

###### Proof:

Please see Appendix G. ∎

In order to garner more insights from Theorem 5, we resort to the following
special cases.

####  V-B1 When $F=1$

The average AoI given in (5) reduces to that under SA protocol, which can be
expressed as:

$\displaystyle\bar{\Delta}_{0}^{\textit{SA}}$
$\displaystyle=\mathbb{E}\left[\frac{1}{\eta\mu_{0}^{\Phi}}\right]$
$\displaystyle=\frac{1}{\eta}\int_{0}^{\infty}\\!\\!\exp{\left(\\!-z+\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\eta
g_{\theta}(1\\!-\\!\eta,z;\alpha,\epsilon)\right)}\mathrm{d}z.$ (30)

This result accounts for the effect of power control on the AoI performance in
a Poisson cellular network, which complements the current development of AoI
analysis in wireless networks.

####  V-B2 When $F>1$

Similar to the scenario under the Poisson bipolar network, we can rewrite (5)
as follows:

$\displaystyle\bar{\Delta}_{0}^{\mathrm{C}}=\bar{\Delta}_{0}^{\textit{SA}}(\beta)$
$\displaystyle+\underbrace{\frac{F^{2}-1}{12}\beta\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\\!\exp{\Big{(}-z-\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\beta
g_{\theta}(1,z;\alpha,\epsilon)\Big{)}}\mathrm{d}z\\!+\\!\frac{1\\!-\\!F}{2}}_{Q_{2}},$
(31)

where $\bar{\Delta}_{0}^{\textit{SA}}(\beta)$ denotes the average AoI under
the SA update protocol when the update rate is $\beta$. From (V-B2) we can see
that FSA is not effectual in reducing average AoI unless $Q_{2}$ is negative.
Such a condition can be formally rewritten as the following:

$\displaystyle\int_{0}^{\infty}\exp{\Big{(}-z-\frac{\lambda_{\mathrm{s}}}{2\lambda_{\mathrm{d}}}\eta
g_{\theta}(1,z;\alpha,\epsilon)\Big{)}}\mathrm{d}z<\frac{4}{\eta}.$ (32)

As power control plays a critical role in the AoI performance, we investigate
the average AoI under three specific cases.

####  V-B3 No power control

In this case, $\epsilon=0$, each sensor transmits information with the same
power. Correspondingly, we can derive the average AoI over the typical link as
follows:

$\displaystyle\bar{\Delta}_{0}^{\mathrm{C,N}}\\!\\!=\\!\begin{cases}\infty,&\mbox{if
}\mbox{$\frac{\lambda_{s}}{\lambda_{d}}\Omega_{\delta}\frac{\eta}{F}(1\\!\\!-\\!\\!\frac{\eta}{F})^{\delta-1}\geq
1$},\\\
\frac{\frac{F}{\eta}}{1\\!-\\!\frac{\lambda_{s}}{\lambda_{d}}\Omega_{\delta}\frac{\eta}{F}(1\\!-\\!\frac{\eta}{F})^{\delta\\!-\\!1}}\\!\\!\\!\\!\\!\\!&+\frac{\frac{F^{2}\\!-\\!1}{12F}\\!\times\\!\eta}{1+\frac{\lambda_{s}}{\lambda_{d}}\frac{\eta}{F}\Omega_{\delta}}\\!+\\!\frac{1\\!-\\!F}{2},\mbox{
otherwise},\end{cases}$ (33)

where $\Omega_{\delta}=\theta^{\delta}\Gamma(1-\delta)\Gamma(1+\delta)$. The
result in (33) provides a closed-form expression that explicitly illustrates
the influence of spatial contention on the average AoI, from which we can see
that increasing the density of sensors degrades the average AoI reciprocally,
as they need to vie for the ratio resources to transmit information packets.

####  V-B4 Full path inversion

In this case, $\epsilon=1$, and the impact of path loss on the useful signal
power can be alleviated by power control. Nonetheless, interference may also
go up as other transmitters are also raising their transmit power.
Consequently, we can derive the average AoI over the typical link under FSA
protocol as follows:

$\displaystyle\bar{\Delta}_{0}^{\mathrm{C,F}}\\!\\!=\\!\frac{F^{2}\\!\\!-\\!\\!1}{12F}\eta\cdot\exp\\!{\Bigg{(}\frac{4}{5}\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\frac{\eta}{F}\int_{0}^{\infty}\\!\\!\\!\int_{0}^{1}\\!\frac{ze^{-zs}\mathrm{d}s\mathrm{d}z}{(1+\frac{s^{\delta}}{\theta})(1\\!-\\!e^{-z})}\Bigg{)}}$
$\displaystyle+\\!\frac{F}{\eta}\exp\\!{\Bigg{(}\frac{4}{5}\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\frac{\eta}{F}\\!\int_{0}^{\infty}\\!\\!\\!\int_{0}^{1}\\!\frac{ze^{-zs}\mathrm{d}s\mathrm{d}z}{(1\\!\\!-\\!\\!\frac{\eta}{F}\\!+\\!\frac{s^{\delta}}{\theta})(1\\!-\\!e^{-z})}\Bigg{)}}\\!+\frac{1\\!-\\!F}{2}.$
(34)

Additionally, in view of the accurate expression being complicated, we adopt
Corollary 2 to get an approximated result as follows:

$\displaystyle\bar{\Delta}_{0}^{\mathrm{C,F}}$
$\displaystyle\approx\frac{F^{2}\\!-\\!1}{12F}\eta\exp\left(-\frac{2\eta\lambda_{\mathrm{s}}}{F\lambda_{\mathrm{d}}}\Omega_{\delta}\right)$
$\displaystyle+\frac{F}{\eta}\exp\left(\frac{2\eta\lambda_{\mathrm{s}}}{F\lambda_{\mathrm{d}}}\Omega_{\delta}\Big{(}1-\frac{\eta}{F}\Big{)}^{\delta-1}\right)+\frac{1\\!-\\!F}{2}.$
(35)

####  V-B5 Maximum power constraint

In practice, transmission power is often limited to a maximum value.
Therefore, in this part, we consider a maximum power constraint model, which
is described as follows:

$\displaystyle
P_{i}=\begin{cases}P_{\mathrm{tx}}R_{i}^{\alpha\epsilon},&R_{i}<p^{\frac{1}{\alpha\epsilon}},\\\
P_{\mathrm{max}},&R_{i}\geq p^{\frac{1}{\alpha\epsilon}},\end{cases}$ (36)

where $P_{\mathrm{max}}$ indicates the maximum transmission power, and
$p=\frac{P_{\mathrm{max}}}{P_{\mathrm{tx}}}$. In this case, we can derive the
average AoI over the typical link as follows:

$\displaystyle\bar{\Delta}_{0}^{\mathrm{C,M}}=\frac{1\\!-\\!F}{2}+$
$\displaystyle\frac{F^{2}\\!\\!-\\!\\!1}{12}\eta\\!\Bigg{[}\\!\\!\int_{0}^{C_{\epsilon}}\\!\\!\\!\\!\\!\exp\\!\Big{(}\\!\\!-\\!\\!z\\!+\\!g_{\theta}^{C_{\epsilon}}\\!((1,\\!0),\\!z;\\!\alpha,\\!\epsilon)\\!+\\!g_{\theta,1}^{C_{\epsilon}}\\!((1,\\!\frac{\epsilon}{\delta},\\!1),\\!z;\\!\alpha,\\!\epsilon)\\!\Big{)}\\!\mathrm{d}\\!z$
$\displaystyle+\\!\\!\int_{C_{\epsilon}}^{\infty}\\!\\!\\!\exp\\!\Big{(}\\!\\!-\\!\\!z\\!+\\!g_{\theta}^{C_{\epsilon}}\\!((1,\\!\frac{\epsilon}{\delta}),\\!z;\\!\alpha,\\!\epsilon)\\!+\\!g_{\theta,\\!-\\!1}^{C_{\epsilon}}\\!((1,\\!0,\\!1),\\!z;\\!\alpha,\\!\epsilon)\\!\Big{)}\\!\mathrm{d}\\!z\Bigg{]}$
$\displaystyle+\frac{F}{\eta}\\!\Bigg{[}\\!\\!\int_{0}^{C_{\epsilon}}\\!\\!\\!\\!\\!\exp\\!\Big{(}\\!\\!-\\!\\!z\\!+\\!g_{\theta}^{C_{\epsilon}}\\!((0,\\!0),\\!z;\\!\alpha,\\!\epsilon)\\!+\\!g_{\theta,1}^{C_{\epsilon}}\\!((0,\\!\frac{\epsilon}{\delta},\\!-\\!1),\\!z;\\!\alpha,\\!\epsilon)\\!\Big{)}\\!\mathrm{d}\\!z$
$\displaystyle+\\!\\!\int_{C_{\epsilon}}^{\infty}\\!\\!\\!\exp\\!\Big{(}\\!\\!-\\!\\!z\\!+\\!g_{\theta}^{C_{\epsilon}}\\!((0,\\!\frac{\epsilon}{\delta}),\\!z;\\!\alpha,\\!\epsilon)\\!+\\!g_{\theta,\\!-\\!1}^{C_{\epsilon}}\\!((1,\\!0,\\!-\\!1),\\!z;\\!\alpha,\\!\epsilon)\\!\Big{)}\\!\mathrm{d}\\!z\\!\Bigg{]}\\!,$
(37)

where $C_{\epsilon}=\frac{5}{4}\lambda_{\mathrm{d}}\pi
p^{\frac{\delta}{\epsilon}}$, while
$g_{\theta}^{C_{\epsilon}}((m,\\!a),\\!z;\\!\alpha,\\!\epsilon)$ and
$g_{\theta,n}^{C_{\epsilon}}((m,\\!a,\\!b),\\!z;\\!\alpha,\\!\epsilon)$ are
respectively given as

$\displaystyle
g_{\theta}^{C_{\epsilon}}((m,\\!a),\\!z;\\!\alpha,\\!\epsilon)\\!=\\!\frac{4}{5}z^{2}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{0}^{1}\\!\frac{qe^{-zqs}(1\\!-\\!e^{-\frac{48}{25}zq})\mathrm{d}s\mathrm{d}q}{(m\\!+\\!\frac{q^{\frac{\alpha(1-\epsilon)}{2}}}{\theta
s^{\frac{\alpha\epsilon}{2}}}(\frac{C_{\epsilon}}{z})^{a})(1\\!-\\!e^{-zq})},$
(38) $\displaystyle
g_{\theta,n}^{C_{\epsilon}}((m,\\!a,\\!b),\\!z;\\!\alpha,\\!\epsilon)\\!=\\!\frac{4}{5}z^{2}\\!\\!\int_{C_{\epsilon}z^{n}}^{\infty}\\!\\!\\!\int_{C_{\epsilon}z^{n}}^{1}\\!\frac{qe^{-zqs}(1\\!-\\!e^{-\frac{48}{25}zq})}{1-e^{-zq}}$
$\displaystyle\bigg{[}\frac{1}{\big{(}m\\!+\\!\theta
q^{-\frac{\alpha}{2}}(\frac{C_{\epsilon}}{z})^{a}\big{)}^{b}}-\frac{1}{\big{(}m\\!+\\!\theta
s^{\frac{\alpha\epsilon}{2}}q^{\frac{\alpha(\epsilon-1)}{2}}(\frac{C_{\epsilon}}{z})^{a-\frac{\epsilon}{\delta}}\big{)}^{b}}\bigg{]}\mathrm{d}s\mathrm{d}q.$
(39)

$\displaystyle\sigma^{2}_{\Delta^{\mathrm{C}}_{0}}$
$\displaystyle=\int_{0}^{\infty}e^{-z}\Bigg{[}\frac{2F^{2}}{\eta^{2}}\exp{\bigg{(}\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\frac{2\eta}{F}G_{\theta}((1\\!\\!-\\!\\!\frac{\eta}{F},\\!2,\\!2,\\!1\\!\\!-\\!\\!\frac{\eta}{F},\\!1,\\!\frac{\eta}{F}),z;\alpha,\epsilon)\bigg{)}}-\frac{F^{2}}{\eta^{2}}\exp{\bigg{(}\\!-\\!z\\!+\\!\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\frac{2\eta}{F}G_{\theta}((1\\!\\!-\\!\\!\frac{\eta}{F},\\!1,\\!0,\\!0,\\!0,\\!0),z;\alpha,\epsilon)\bigg{)}}$
$\displaystyle-\frac{(F^{2}\\!\\!-\\!\\!1)^{2}\eta^{2}}{144F^{2}}\exp{\bigg{(}\\!-\\!z\\!-\\!\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\frac{2\eta}{F}G_{\theta}((1,\\!1,\\!0,\\!0,\\!0,\\!0),z;\alpha,\epsilon)\bigg{)}}-\frac{F^{2}\\!\\!-\\!\\!1}{6}\exp{\bigg{(}\\!-\\!z\\!+\\!\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}(\frac{\eta}{F})^{2}[G_{\theta}((1\\!\\!-\\!\\!\frac{\eta}{F},\\!1,\\!1,\\!1,\\!-1,\\!0),z;\alpha,\epsilon)\bigg{)}}$
$\displaystyle-\frac{F^{2}}{\eta}\exp{\bigg{(}\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\frac{\eta}{F}G_{\theta}((1\\!\\!-\\!\\!\frac{\eta}{F},\\!1,\\!0,\\!0,\\!0,\\!0),z;\alpha,\epsilon)\bigg{)}}+\frac{(F^{2}-1)\eta}{12}\exp{\bigg{(}\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\frac{\eta}{F}G_{\theta}((1,\\!1,\\!0,\\!0,\\!0,\\!0),z;\alpha,\epsilon)\bigg{)}}\Bigg{]}\mathrm{d}z+\frac{F^{2}+2F-3}{4}$
(40)

Similarly, we can also obtain the analytical expression for the variance of
AoI by deconditioning $\mu_{0}^{\Phi}$ in (7) and taking the second moment of
$\mu_{0}^{\Phi}$ minus the square of the average.

###### Theorem 6

In a Poisson cellular network, the variance of AoI over the typical link under
the FSA updating protocol can be expressed as (V-B5) at the top of this page,
where

$\displaystyle
G_{\theta}((a,\\!b,\\!c,\\!d,\\!l,\\!\varrho),\\!z;\\!\alpha,\\!\epsilon)$
$\displaystyle=\frac{4}{5}z^{2}\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{0}^{1}\\!\\!\frac{q\exp{(-zqs)}\mathrm{d}s\mathrm{d}q}{\frac{(a+\frac{q^{\alpha(1-\epsilon)/2}}{\theta
s^{\alpha\epsilon/2}})^{b}}{c(d+\frac{q^{\alpha(1\\!-\\!\epsilon)/2}}{\theta
s^{\alpha\epsilon/2}})^{l}+\varrho}(1-e^{-zq})}.$ (41)

###### Proof:

Similar to the derivation in (-F), we can calculate the second moment of
$\frac{1}{\mu_{0}^{\Phi}}$ as follows:

$\displaystyle\mathbb{E}\left[\frac{1}{(\mu_{0}^{\Phi})^{2}}\right]=\mathbb{E}\left[\prod_{i\neq
0}\Bigg{(}1+\frac{\eta/F}{1-\frac{\eta}{F}+\frac{D_{i}^{\alpha}}{\theta
R_{i}^{\alpha\epsilon}r^{\alpha(1\\!-\\!\epsilon)}}}\Bigg{)}^{2}\right]$
$\displaystyle=\\!\\!\mathbb{E}\Bigg{[}\\!\prod_{i\neq
0}\\!\int_{0}^{D_{i}}\\!\\!\bigg{(}\\!1\\!+\\!\frac{\eta/F}{1\\!\\!-\\!\\!\frac{\eta}{F}\\!\\!+\\!\\!\frac{D_{i}^{\alpha}u^{\\!-\\!\alpha\epsilon}}{\theta
r^{\alpha(1\\!-\\!\epsilon)}}}\\!\bigg{)}^{\\!2}\frac{\frac{5}{2}\lambda_{\mathrm{d}}\pi
u\exp{(\\!-\\!\frac{5}{4}\lambda_{\mathrm{d}}\pi
u^{2})}\mathrm{d}u}{1\\!-\\!\exp{(\\!-\\!\frac{5}{4}\lambda_{\mathrm{d}}\pi
D_{i}^{2})}}\Bigg{]}$
$\displaystyle=\\!\\!\mathbb{E}\Bigg{[}\exp\\!\bigg{(}2\pi\lambda_{\mathrm{s}}\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{0}^{\kappa}\\!\\!\Big{(}2\\!+\\!\frac{\eta/F}{1\\!-\\!\frac{\eta}{F}\\!+\\!\frac{D_{i}^{\alpha}u^{-\alpha\epsilon}}{\theta
r^{\alpha(1-\epsilon)}}}\Big{)}$
$\displaystyle\qquad\qquad\qquad\times\frac{\frac{\eta}{F}\\!\\!\cdot\\!\\!\frac{5}{2}\lambda_{\mathrm{d}}\pi
u\exp{(-\frac{5}{4}\lambda_{\mathrm{d}}\pi
u^{2})}\mathrm{d}u\kappa\mathrm{d}\kappa}{(1\\!-\\!\frac{\eta}{F}\\!+\\!\frac{\kappa^{\alpha}u^{-\alpha\epsilon}}{\theta
r^{\alpha(1-\epsilon)}})(1\\!-\\!\exp{(-\frac{5}{4}\lambda_{\mathrm{d}}\pi\kappa^{2})})}\bigg{)}\\!\Bigg{]}$
$\displaystyle=\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp{\\!\Bigg{(}\\!\\!\\!-\\!\\!z\\!\\!+\\!\\!\frac{4}{5}\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\\!z^{2}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\\!\int_{0}^{1}\\!\\!\frac{\frac{\eta}{F}qe^{\\!-\\!zsq}\big{(}2\\!\\!-\\!\\!\frac{\eta}{F}\\!\\!+\\!\\!\frac{2q^{\frac{\alpha(1\\!-\\!\epsilon)}{2}}}{\theta
s^{\frac{\alpha\epsilon}{2}}}\big{)}\mathrm{d}s\mathrm{d}q}{\big{(}1\\!\\!-\\!\\!\frac{\eta}{F}\\!+\\!\frac{q^{\frac{\alpha(1\\!-\\!\epsilon)}{2}}}{\theta
s^{\frac{\alpha\epsilon}{2}}}\big{)}^{2}\big{(}1\\!-\\!e^{-zq}\big{)}}\\!\Bigg{)}}\\!\mathrm{d}z.$
(42)

Substituting the expressions for the first moment of $\mu_{0}^{\Phi}$ as per
(-F), while the first and the second moment of $\frac{1}{\mu_{0}^{\Phi}}$
given in (-F) and (V-B5) respectively, into the expression for the variance of
AoI after deconditioning $\mu_{0}^{\Phi}$ (-E), we can obtain the result in
this theorem. ∎

When $F=1$, this result also reduces to the variance of AoI under SA protocol,
given as the following.

###### Corollary 3

In a Poisson cellular network, the variance of AoI over the typical link SA
protocol is given by

$\displaystyle\sigma^{2}_{\Delta^{\textit{SA}}_{0}}$
$\displaystyle=\\!\\!\int_{0}^{\infty}\\!\\!e^{-z}\Bigg{[}\frac{2}{\eta^{2}}\exp{\bigg{(}2\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\eta
g_{\theta}((1\\!\\!-\\!\\!\eta,\\!2,\\!2,\\!1\\!\\!-\\!\\!\eta,\\!1,\\!\eta),z;\alpha,\epsilon)\bigg{)}}$
$\displaystyle-\frac{1}{\eta^{2}}\exp{\bigg{(}\\!-\\!z\\!+\\!2\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\eta
g_{\theta}((1\\!\\!-\\!\\!\eta,\\!1,\\!0,\\!0,\\!0,\\!0),z;\alpha,\epsilon)\bigg{)}}$
$\displaystyle-\frac{1}{\eta}\exp{\bigg{(}\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\eta
g_{\theta}((1\\!\\!-\\!\\!\eta,\\!1,\\!0,\\!0,\\!0,\\!0),z;\alpha,\epsilon)\bigg{)}}\Bigg{]}\mathrm{d}z.$
(43)

###### Remark 1

Given the specific values of the parameters, integrals given above can be
numerically evaluated via using two integral functions, namely integral and
integral2, in Matlab. The infinite upper limits can be expressed by inf or
replaced by a large and appropriate finite value according to the required
accuracy.

###  V-C Numerical Results

In this section, we utilize the developed theoretical expressions to
numerically evaluate and investigate the effect of FSA on the average AoI in a
cellular network. Unless otherwise specified, we use the same set of
parameters as in Section IV-C.

(a) (b)

Figure 9: Average AoI versus update rate under different framesizes $F$ and
different power control strategies: ($a$) no power control ($\epsilon=0$) and
($b$) full path inversion ($\epsilon=1$), in which we set
$\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}=5$, and vary the framesize
as $F=1$ (i.e. slotted ALOHA), $3$, $5$, and $7$.

Fig. 9 illustrates the relationship between the average AoI and update rate
under different framesizes. In this figure, we compare the AoI performance
under two different power control policies, namely, the unified transmit power
and full path inversion. The figure reveals a similar phenomenon as that in
the bipolar network, i.e., there exists an optimal $\eta$ that minimizes the
average AoI, and as $\eta$ increases, the FSA updating protocol outperforms
the traditional SA protocol. Such an observation validates the importance of
adopting frame structure in the status updating protocol. Besides, by
comparing Fig. 9(a) and Fig. 9(b), we see that using power control strategy
can substantially reduce the average AoI, especially when there is no frame
and/or $\eta$ is large. Specifically, in Fig. 9(a), the average AoI under SA
protocol increases sharply when $\eta$ is still small. As $F$ increases, AoI
can maintain a lower value in a wider range of $\eta$. In Fig. 9(b), after the
implementation of power control, the change of the average AoI becomes slower,
especially the performance of SA is obviously better.

(a) (b)

Figure 10: AoI performance verse the power control factor $\epsilon$: ($a$)
average AoI and ($b$) variance of AoI, in which we set $\eta=0.4$,
$\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}=5$, and vary the framesize
as $F=1$ (i.e. slotted ALOHA), $3$, $5$, and $7$.

In Fig. 10, we plot the average and variance of AoI as a function of power
control factor $\epsilon$ under different framesizes. From this figure, we
immediately notice that power control has a direct influence on the
performance of AoI, whereas for a varying value of $F$, there exists an
optimal $\epsilon$ that minimizes the average or variance of AoI. Moreover,
the optimal $\epsilon$ is closely related to the particular value of $F$. To
be more precise, when $F=1$, i.e., the updating protocol is SA, there are no
frame to regularize the potential collisions amongst the transmitters. Since
the typical source node could locate in a long distance to the data fussing
center, the power attenuation caused by path loss, together with the
deterioration from interference, can lead to a significant degradation to the
average and variance of AoI. Therefore, one shall increase the power control
factor to compensate the path loss. Nonetheless, increasing the source nodes’
transmit power not only strengthens their signal power, especially for those
located remotely to the receivers, but also enlarges the accumulated
interference. Hence, the power control factor needs to be adequately tuned so
as to optimize the age performance. In contrast, under the FSA protocol (with
$F\geq 2$), the adoption of frame reduces the competition between nodes and
reducing the requirement of power control. As such, the optimal $\epsilon$ in
FSA will be smaller than that under SA. Additionally, we note that although a
larger framesize provides more possibility of small interference transmission
environment for remote sources, improves the probability of successful
transmission, so that the network still maintains well performance, merely
increasing $F$ does not always benefit the AoI.

(a) (b)

Figure 11:  AoI performance versus framesize with different deployment
densities $\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}$ and different
power control strategies: ($a$) no power control ($\epsilon=0$) and ($b$) full
path inversion ($\epsilon=1$), in which we set $\eta=0.4$, and vary the
average number of sources that each data fusing center can contact as
$\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}=2$, $4$, $6$, and $8$.

In Fig. 11, we depict the network average AoI as a function of $F$, under
different deployment densities. Unlike the bipolar network, cellular network
deployment affects AoI only in terms of the density ratio of sources and
destination nodes $\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}$, that
is, the average number of sensors connected to each data fusing center. First,
we observe a phenomenon similar to Fig. 6, that is, depending on the network
spacial deployment, there may be an optimal $F$, and increasing
$\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}$ will definitely
deteriorate AoI performance, but employing FSA protocol can alleviate this
deterioration. Second, we observe that the evolution trend of the average AoI
with $F$ is different under different power control strategies. Specifically,
Fig. 11(a) shows that without power control, the average AoI under SA protocol
has reached a very large value even when
$\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}$ is still very small (e.g.,
$\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}=2$). However, the
implementation of FSA strategy can obtain a relatively good performance, and
with the increase of the ratio, we need a larger frame to maintain the average
AoI performance in a better state. In Fig. 11(b), the power control strategy
can increase the competitiveness of effective signals and then improve the
performance of AoI. In this case, when
$\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}$ is relatively small, FSA
does not play a significant role, and even worse than the performance of AoI
under SA protocol. As the ratio increases, namely the competition between
sources increases, FSA will show its advantage.

##  VI Conclusion

In this paper, we undertook an analytical study toward understanding the
effect of FSA-like status updating protocol on the AoI performance in wireless
networks. We adopted a general model that accounts for channel fading, path
loss, and interference. We derived closed-form expressions for the average and
its variance in Poisson bipolar and cellular networks, respectively. Based on
the analysis, we identified the operating regime under which FSA is
instrumental in reducing AoI. We also provided the optimal framesize that
minimizes the average AoI for a given configuration of network parameters. The
numerical results confirmed that when interference is severe, i.e., if the
network is densely deployed and/or the source nodes are updating status
information aggressively, employing FSA in the transmission protocol can
substantially reduce the average and variance of AoI. In contrast, such a
scheme is ineffective for AoI reduction in a sparsely deployed network, as
interference is mild in this scenario, and the nodes should update more
frequently to attain a small AoI. Our numerical results also showed that
implementing FSA jointly with power control can further benefit the age
performance in a wireless system, while the particular values of framesize and
power control factor need to be adequately tuned to achieve the optimal gain.

The mathematical framework presented in this paper lays the foundation for
analyzing large-scale wireless networks with frame-based traffic patterns. The
model can be extended to study the effects of FSA-based status updating
protocols on the age performance under non-linear cost functions.
Additionally, studying the optimal design of frame structure with a variable
size adapted to each transmitter’s local geometry is another promising future
research that can be extended from the fixed frame size setting and results
shown in this paper.

###  -A Proof of Theorem 1

We adopt a graphical method to calculate the average AoI. Specifically, we
denote by $X$ the number of frames between two successfully received updates,
$I$ the interval between two consecutive updates received, $A$ the area
between the two successful updates as illustrated in Figure 1, and use the
subscript $w$ to represent the value of the $w$-th corresponding variable
(e.g. define $I_{w}$ as the time elapsed since the $(w-1)$-th acceptance to
the update before a new update is received again). Thus, the time average AoI
can be expressed as follows:

$\displaystyle\bar{\Delta}_{0}=\lim_{W\to\infty}\frac{\sum_{w=1}^{W}A_{w}}{\sum_{w=1}^{W}I_{w}}=\frac{\mathbb{E}[\sum_{i=1}^{I}i]}{\mathbb{E}[I]}=\frac{1}{2}+\frac{\mathbb{E}[I^{2}]}{\mathbb{E}[I]},$
(44)

in which $I_{w}$ is given by:

$\displaystyle I_{w}=FX_{w}+(N_{w}-N_{w-1}),$ (45)

numbering time slots in the frame, $N_{w}\in\\{1,2,...,F\\}$ represents the
index of the time slot that is an update received for the $w$ time (e.g., in
Figure 1, $N_{w-1}=3$ and $N_{w}=2$). Note that $X$ obeys a geometric
distribution with parameter $\eta\mu_{0}^{\Phi}$, therefore,
$\mathbb{E}[X]=\frac{1}{\eta\mu_{0}^{\Phi}}$; $N_{w-1}$ and $N_{w}$ have the
same distribution, being a discrete uniform distribution independent of $X$ in
the range of $\\{1,...F\\}$. By denoting $\mathbb{E}[N]=\mathbb{E}[N_{w}]$, we
have:

$\displaystyle\mathbb{E}[I]$ $\displaystyle=F\mathbb{E}[X],$ (46)
$\displaystyle\mathbb{E}[I^{2}]$
$\displaystyle=F^{2}\mathbb{E}[X]+2(\mathbb{E}[N^{2}]-\mathbb{E}[N]^{2}).$
(47)

On the other hand, $\mathbb{E}[N]$ and $\mathbb{E}[N^{2}]$ can be respectively
computed as follows:

$\displaystyle\mathbb{E}[N]$
$\displaystyle=\sum_{n=1}^{F}n\mathbb{P}[N=n]=\frac{1}{F}\sum_{d=1}^{F}d=\frac{F+1}{2},$
(48) $\displaystyle\mathbb{E}[N^{2}]$
$\displaystyle=\sum_{n=1}^{F}n^{2}\mathbb{P}[N=n]=\frac{1}{F}\sum_{d=1}^{F}d^{2}=\frac{(F+1)(2F+1)}{6}.$
(49)

Combining the above fragments, we can obtain the conditional average AoI.

###  -B Proof of Theorem 2

The similar graphical method is applied to Theorem 2, we can express the
average quadratic AoI as follows:

$\displaystyle\bar{\Delta}_{0}^{2}=\frac{\mathbb{E}[\sum_{i=1}^{I}i^{2}]}{\mathbb{E}[I]}=\frac{2\mathbb{E}[I^{3}]+3\mathbb{E}[I^{2}]+\mathbb{E}[I]}{6\mathbb{E}[I]}.$
(50)

According to the expression for $I_{w}$ in (45), we can obtain:

$\displaystyle\mathbb{E}[I^{3}]=F^{3}\mathbb{E}[X^{3}]+6F(\mathbb{E}[N^{2}]-\mathbb{E}[N]^{2})\mathbb{E}[X].$
(51)

Since $X$ obeys a geometric distribution with parameter $\eta\mu_{0}^{\Phi}$,
we can carry out the following calculation:

$\displaystyle\mathbb{E}[X^{3}]$
$\displaystyle=\sum_{k=1}^{\infty}k^{3}\mathbb{P}(X=k)=\sum_{k=1}^{\infty}k^{3}\eta\mu_{0}^{\Phi}(1-\eta\mu_{0}^{\Phi})^{k-1}$
$\displaystyle=-\eta\mu_{0}^{\Phi}\frac{\mathrm{d}\sum_{k=1}^{\infty}k^{2}(1-\eta\mu_{0}^{\Phi})^{k}}{\mathrm{d}(\eta\mu_{0}^{\Phi})}$
$\displaystyle=-\eta\mu_{0}^{\Phi}\frac{\mathrm{d}(1-\eta\mu_{0}^{\Phi})\mathbb{E}[X^{2}]/(\eta\mu_{0}^{\Phi})}{\mathrm{d}(\eta\mu_{0}^{\Phi})}$
$\displaystyle=\frac{(\eta\mu_{0}^{\Phi})^{2}-6\eta\mu_{0}^{\Phi}+6}{(\eta\mu_{0}^{\Phi})^{3}}.$
(52)

Then, combining the above results, we get the result.

###  -C Proof of Lemma 1

According to the transmission protocol, we note that every source decides
whether it will update in a typical frame independently with probability
$\eta$, and if a source decides to update in this frame, it randomly selects a
time slot according to a uniform distribution. Consequently, at any given time
slot, a source activates with probability $\eta/F$, i.e.,
$\mathbb{P}(\nu_{i}=1)=\frac{\eta}{F}$. As such, we can substitute (8) into
(5) and get the following:

$\displaystyle\mu_{0}^{\Phi}$
$\displaystyle=\mathbb{P}\left(\frac{P_{\mathrm{tx}}h_{0}r^{-\alpha}}{\sum_{i\neq
0}P_{\mathrm{tx}}h_{i}\nu_{i}\parallel\\!\\!x_{i}\\!\\!\parallel^{-\alpha}}>\theta\Big{|}\Phi\right)$
$\displaystyle=\mathbb{P}\left(h_{0}>\theta r^{\alpha}\sum_{i\neq
0}h_{i}\nu_{i}\parallel\\!\\!x_{i}\\!\\!\parallel^{-\alpha}\Big{|}\Phi\right)$
$\displaystyle=\mathbb{E}\left[\prod_{i\neq 0}\exp\left(-\theta
r^{\alpha}h_{i}\nu_{i}\parallel\\!\\!x_{i}\\!\\!\parallel^{-\alpha}\right)\Big{|}\Phi\right]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}\left[\prod_{i\neq
0}\frac{1}{1+\theta
r^{\alpha}\nu_{i}\parallel\\!\\!x_{i}\\!\\!\parallel^{-\alpha}}\right]$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\prod_{i\neq
0}\left(\frac{\eta/F}{1+\theta
r^{\alpha}\parallel\\!\\!x_{i}\\!\\!\parallel^{-\alpha}}+1-\frac{\eta}{F}\right),$
(53)

where (a) follows since $\\{h_{i}\\}_{i=1}^{\infty}$ are i.i.d. random
variables following the exponential distribution with unit mean, and (b)
follows as $\\{\nu_{i}\\}_{i=1}^{\infty}$ are independent of each other. Then,
after a further simplification, we obtain the result shown in Lemma 1.

###  -D Proof of Theorem 3

Deconditioning $\mu_{0}^{\Phi}$ in (6), we can get the expression of average
AoI as:

$\displaystyle\bar{\Delta}_{0}=\frac{F^{2}-1}{12F}\eta\mathbb{E}[\mu_{0}^{\Phi}]+\frac{F}{\eta}\mathbb{E}\bigg{[}\frac{1}{\mu_{0}^{\Phi}}\bigg{]}+\frac{1-F}{2}.$
(54)

Using Lemma 1, we can compute the mean of $\mu_{0}^{\Phi}$ as follows:

$\displaystyle\mathbb{E}[\mu_{0}^{\Phi}]$
$\displaystyle=\mathbb{E}\Bigg{[}\prod_{i\neq
0}\left(1-\frac{\eta/F}{1+\|x_{i}\|^{\alpha}/\theta
r^{\alpha}}\right)\Bigg{]}$
$\displaystyle=\mathbb{E}\Bigg{[}\exp\bigg{(}-\sum_{i\neq
0}-\log\bigg{(}1-\frac{\eta/F}{1+\|x_{i}\|^{\alpha}/\theta
r^{\alpha}}\bigg{)}\Bigg{)}\Bigg{]}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\exp\Bigg{(}\\!-\\!\lambda\int_{x\in\mathbb{R}^{2}}\left[1-\left(1-\frac{\eta/F}{1+\|x\|^{\alpha}/\theta
r^{\alpha}}\right)\right]\mathrm{d}x\Bigg{)}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\exp\Bigg{(}-\lambda\pi\delta\int_{0}^{\infty}\frac{\eta/F}{1+u/\theta
r^{\alpha}}u^{\delta-1}\mathrm{d}u\Bigg{)}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}\exp\Bigg{(}-\lambda\pi
r^{2}\theta^{\delta}\Gamma(1-\delta)\Gamma(1+\delta)\frac{\eta}{F}\Bigg{)},$
(55)

where (a) follows by using the probability generating functional (PGFL) of PPP
[2022Spatio], (b) changes variables from rectangular to polar coordinates and
sets $u=\|x\|^{\alpha}$, and (c) is due to the result
$\int_{0}^{\infty}\frac{u^{\delta-1}\mathrm{d}u}{u+m}=m^{\delta-1}\frac{\pi}{\sin{(\pi\delta)}}$
[2015The]. Similarly, we can calculate $\mathbb{E}[\frac{1}{\mu_{0}^{\Phi}}]$
as follows:

$\displaystyle\mathbb{E}\bigg{[}\frac{1}{\mu_{0}^{\Phi}}\bigg{]}$
$\displaystyle=\mathbb{E}\Bigg{[}\prod_{i\neq
0}\left(1-\frac{\eta/F}{1+\|x_{i}\|^{\alpha}/\theta
r^{\alpha}}\right)^{-1}\Bigg{]}$
$\displaystyle=\exp\Bigg{(}\\!\\!-\lambda\\!\\!\int_{x\in\mathbb{R}^{2}}\\!\\!\left[1-\left(1-\frac{\eta/F}{1+\|x\|^{\alpha}/\theta
r^{\alpha}}\right)^{\\!\\!-1}\right]\mathrm{d}x\Bigg{)}$
$\displaystyle=\exp\Bigg{(}\\!\\!-\lambda\\!\\!\int_{x\in\mathbb{R}^{2}}\frac{\eta/F}{1-\frac{\eta}{F}+\frac{\|x\|^{\alpha}}{\theta
r^{\alpha}}}\mathrm{d}x\Bigg{)}$
$\displaystyle=\exp\Bigg{(}\lambda\pi\delta\int_{0}^{\infty}\frac{\eta/F}{1-\frac{\eta}{F}+\frac{u}{\theta
r^{\alpha}}}u^{\delta-1}\mathrm{d}u\Bigg{)}$
$\displaystyle=\exp\Bigg{(}\lambda\pi
r^{2}\theta^{\delta}\Gamma(1-\delta)\Gamma(1+\delta)\frac{\eta}{F}\Big{(}1\\!\\!-\\!\\!\frac{\eta}{F}\Big{)}^{\delta-1}\Bigg{)}.$
(56)

By substituting (-D) and (-D) into (54), we can obtain the result shown in
Theorem 1.

###  -E Proof of Theorem 4

Similar to the steps taken in Appendix -D, we can calculate the variance of
AoI as follows:

$\displaystyle\sigma^{2}_{\Delta_{0}}=\mathbb{E}[\Delta_{0}^{2}]-(\mathbb{E}[\Delta_{0}])^{2}$
$\displaystyle=\\!\frac{2F^{2}}{\eta^{2}}\mathbb{E}\\!\left[\frac{1}{(\mu_{0}^{\Phi})^{2}}\right]\\!-\\!\frac{2F^{2}\\!\\!-\\!\\!F}{\eta}\mathbb{E}\\!\left[\frac{1}{\mu_{0}^{\Phi}}\right]\\!+\\!\frac{(F^{2}\\!\\!-\\!\\!1)\eta}{12F}\mathbb{E}[\mu_{0}^{\Phi}]\\!+\\!\frac{F^{2}\\!\\!-\\!\\!F}{2}$
$\displaystyle-\Bigg{[}\frac{F^{2}-1}{12F}\eta\mathbb{E}[\mu_{0}^{\Phi}]+\frac{F}{\eta}\mathbb{E}\Big{[}\frac{1}{\mu_{0}^{\Phi}}\Big{]}+\frac{1-F}{2}\Bigg{]}^{2},$
(57)

where $\mathbb{E}[\frac{1}{(\mu_{0}^{\Phi})^{2}}]$ can be derived by the
following:

$\displaystyle\mathbb{E}\left[\frac{1}{(\mu_{0}^{\Phi})^{2}}\right]=\mathbb{E}\Bigg{[}\prod_{i\neq
0}\left(1-\frac{\eta/F}{1+\|x_{i}\|^{\alpha}/\theta
r^{\alpha}}\right)^{-2}\Bigg{]}$
$\displaystyle=\exp{\bigg{(}-\lambda\pi\delta\int_{0}^{\infty}\Big{[}1-\Big{(}1-\frac{\frac{\eta}{F}\theta
r^{\alpha}}{u+\theta
r^{\alpha}}\Big{)}^{-2}\Big{]}u^{\delta-1}\mathrm{d}u\bigg{)}}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\exp{\bigg{(}-\lambda\pi\delta\int_{0}^{\infty}\Big{[}1-\sum_{k=0}^{\infty}(k+1)\Big{(}\frac{\frac{\eta}{F}\theta
r^{\alpha}}{u+\theta
r^{\alpha}}\Big{)}^{k}\Big{]}u^{\delta-1}\mathrm{d}u\bigg{)}}$
$\displaystyle=\exp{\bigg{(}\lambda\pi\delta\sum_{k=1}^{\infty}(k+1)\Big{(}\frac{\eta}{F}\theta
r^{\alpha}\Big{)}^{k}\int_{0}^{\infty}\frac{u^{\delta-1}\mathrm{d}u}{(u+\theta
r^{\alpha})^{k}}\bigg{)}}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\exp{\bigg{(}-\lambda\pi\theta^{\delta}r^{2}\frac{\pi\delta}{\sin{(\pi\delta)}}\sum_{k=1}^{\infty}(k+1)\Big{(}-\frac{\eta}{F}\Big{)}^{k}\frac{\Gamma(\delta)}{\Gamma(k)\Gamma(\delta\\!-\\!k\\!+\\!1)}\bigg{)}}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}\exp{\bigg{(}-C\sum_{k=1}^{\infty}(k+1){\delta-1\choose
k-1}\Big{(}-\frac{\eta}{F}\Big{)}^{k}\bigg{)}},$ (58)

where (a) leverages the relationship that
$\frac{1}{(1+x)^{2}}=\sum_{k=0}^{\infty}(k+1)(-x)^{k},\ \mathrm{for}\
\left|x\right|<1$, (b) uses

$\displaystyle\int_{0}^{\infty}\frac{u^{\delta-1}\mathrm{d}u}{(u+\theta
r^{\alpha})^{k}}=(\theta
r^{\alpha})^{\delta-k}\\!\\!\times\\!\\!\frac{(-1)^{k+1}\pi}{\sin(\pi\delta)}\\!\\!\times\\!\\!\frac{\Gamma(\delta)}{\Gamma(k)\Gamma(\delta\\!-\\!k\\!+\\!1)},$

and (c) follows from the Odd Element formula
$\Gamma(\delta)\Gamma(1-\delta)=\frac{\pi}{\sin(\pi\delta)},\ \mathrm{for}\
0<\delta<1$ and the recursion of the Gamma function
$\Gamma(1+\delta)=\delta\Gamma(\delta)$. Then, by substituting the expression
for $\mathbb{E}[\mu_{0}^{\Phi}]$, $\mathbb{E}[\frac{1}{\mu_{0}^{\Phi}}]$ and
$\mathbb{E}[\frac{1}{(\mu_{0}^{\Phi})^{2}}]$ into (-E), we get the result.

###  -F Proof of Theorem 5

Since the sources only communicate with their nearest data fusing centers,
when we condition on the distance between the user and the nearest center as
$u$, then there is no data fusion center in area $A=2\pi u^{2}$. Then, we can
deduce that the distance $R$ between users and its dedicated receiver follows
a distribution with the following probability density function:

$\displaystyle f_{R}(u)=e^{-\lambda_{\mathrm{d}}\pi
u^{2}}2\pi\lambda_{\mathrm{d}}u.$ (59)

Similar to the derivation process of bipolar network, we can first obtain the
conditional transmission success probability as:

$\displaystyle\mu_{0}^{\Phi}=\prod_{i\neq
0}\left(1-\frac{\eta/F}{1+\frac{D_{i}^{\alpha}}{\theta
R_{i}^{\alpha\epsilon}r^{\alpha(1-\epsilon)}}}\right).$ (60)

Next, we can calculate the first moment of $\mu_{0}^{\Phi}$ as follows:

$\displaystyle\mathbb{E}[\mu_{0}^{\Phi}]=\mathbb{E}\Bigg{[}\\!\prod_{i\neq
0}\\!\left(\\!1\\!\\!-\\!\\!\int_{0}^{D_{i}}\\!\\!\\!\frac{\frac{\eta}{F}\frac{5}{2}\lambda_{\mathrm{d}}\pi
u\exp{(\\!-\\!\frac{5}{4}\lambda_{\mathrm{d}}\pi
u^{2})}\mathrm{d}u}{\big{(}1\\!+\\!\frac{D_{i}^{\alpha}u^{\\!-\\!\alpha\epsilon}}{\theta
r^{\alpha(1\\!-\\!\epsilon)}}\big{)}\big{(}1\\!-\\!\exp{(\\!-\\!\frac{5}{4}\lambda_{\mathrm{d}}}\pi
D_{i}^{2})\big{)}}\\!\\!\right)\\!\\!\Bigg{]}$
$\displaystyle=\mathbb{E}\Bigg{[}\\!\exp\\!\bigg{(}\\!\sum_{i\neq
0}\\!\log\Big{(}\\!1\\!\\!-\\!\\!\int_{0}^{D_{i}}\\!\\!\\!\frac{\frac{\eta}{F}\frac{5}{2}\lambda_{\mathrm{d}}\pi
u\exp{(\\!-\\!\frac{5}{4}\lambda_{\mathrm{d}}\pi
u^{2})}\mathrm{d}u}{\big{(}1\\!+\\!\frac{D_{i}^{\alpha}u^{\\!-\\!\alpha\epsilon}}{\theta
r^{\alpha(1\\!-\\!\epsilon)}}\big{)}\\!\big{(}1\\!-\\!e^{\\!-\\!\frac{5}{4}\lambda_{\mathrm{d}}\pi
D_{i}^{2}}\big{)}}\\!\Big{)}\\!\bigg{)}\Bigg{]}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}_{r}\Bigg{[}\\!\exp{\Bigg{(}\\!\\!-\\!2\pi\lambda_{\mathrm{s}}\int_{0}^{\infty}\\!\\!\\!\\!\int_{0}^{\kappa}\\!\\!\frac{\frac{\eta}{F}\frac{5}{2}\lambda_{\mathrm{d}}\pi
ue^{\\!-\\!\frac{5}{4}\lambda_{\mathrm{d}}\pi
u^{2}}\mathrm{d}u\kappa\mathrm{d}\kappa}{\big{(}1\\!+\\!\frac{\kappa^{\alpha}u^{\\!-\\!\alpha\epsilon}}{\theta
r^{\alpha(1\\!-\\!\epsilon)}}\big{)}\big{(}1\\!-\\!e^{\\!-\\!\frac{5}{4}\lambda_{\mathrm{d}}\pi\kappa^{2}}\big{)}}\\!\Bigg{)}\\!}\Bigg{]}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\\!\\!\int_{0}^{\infty}\\!\\!\exp{\\!\Bigg{(}\\!\\!-\\!z\\!-\\!\frac{4}{5}\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}z^{2}\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{0}^{1}\frac{\frac{\eta}{F}q\exp{(-zsq)}\mathrm{d}s\mathrm{d}q}{\big{(}1\\!+\\!\frac{q^{\alpha(1\\!-\\!\epsilon)/2}}{\theta
s^{\alpha\epsilon/2}}\big{)}\big{(}1\\!-\\!e^{-zq}\big{)}}\Bigg{)}}\mathrm{d}z,$
(61)

where (a) follows from adopting the PGFL of PPP, and (b) results from the
variable substitution: $s=\frac{u}{\kappa}$, $q=(\frac{\kappa}{r})^{2}$ and
$z=\frac{5}{4}\lambda_{\mathrm{d}}\pi r^{2}$. Similarly, we can calculate the
expectation of $\frac{1}{\mu_{0}^{\Phi}}$ as follows:

$\displaystyle\mathbb{E}\bigg{[}\frac{1}{\mu_{0}^{\Phi}}\bigg{]}=\mathbb{E}\left[\prod_{i\neq
0}\Bigg{(}1+\frac{\eta/F}{1-\frac{\eta}{F}+\frac{D_{i}^{\alpha}}{\theta
R_{i}^{\alpha\epsilon}r^{\alpha(1\\!-\\!\epsilon)}}}\Bigg{)}\right]$
$\displaystyle=\\!\\!\int_{0}^{\infty}\\!\\!\exp{\\!\Bigg{(}\\!\\!-\\!z\\!+\\!\frac{4}{5}\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{0}^{1}\\!\\!\frac{z^{2}\frac{\eta}{F}q\exp{(-zsq)}\mathrm{d}s\mathrm{d}q}{\big{(}1\\!-\\!\frac{\eta}{F}\\!+\\!\frac{q^{\alpha(1\\!-\\!\epsilon)/2}}{\theta
s^{\alpha\epsilon/2}}\big{)}\big{(}1\\!-\\!e^{-zq}\big{)}}\\!\Bigg{)}}\mathrm{d}z.$
(62)

Substituting (-F) and (-F) into (54), we can obtain the average AoI of the
cellular network.

###  -G Proof of Corollary

We remove the spatial dependency of interference nodes’ locations and
approximate them as a marked PPP $\\{x_{i},W_{x_{i}}\\}_{i=1}^{\infty}$, where
$W_{x_{i}}=R_{i}^{\alpha\epsilon}$ [2014On]. Similar to the derivation process
of Theorem 3, we can first condition on the distance of the typical
transmitter-receiver pair $r$ and calculate the first moment of
$\mu_{0}^{\Phi}$ as follows:

$\displaystyle\mathbb{E}[\mu_{0}^{\Phi}|r]=\mathbb{E}\Bigg{[}\prod_{i\neq
0}\Bigg{(}1-\frac{\eta/F}{1+\frac{D_{i}^{\alpha}}{\theta
R_{i}^{\alpha\epsilon}r^{\alpha(1-\epsilon)}}}\Bigg{)}\Bigg{]}$
$\displaystyle\approx\mathbb{E}\Bigg{[}\exp{\Bigg{(}\\!-\\!\lambda_{\mathrm{s}}\int_{x\in\mathbb{R}^{2}}\bigg{[}1-\bigg{(}1-\frac{\eta/F}{1+\frac{\|x\|^{\alpha}}{\theta
W_{x}r^{\alpha(1-\epsilon)}}}\bigg{)}\bigg{]}\mathrm{d}x\Bigg{)}}\Bigg{]}$
$\displaystyle=\exp{\Bigg{(}-\mathbb{E}\bigg{[}\int_{x\in\mathbb{R}^{2}}\frac{\lambda_{\mathrm{s}}\eta/F\mathrm{d}x}{1+\frac{\|x\|^{\alpha}}{\theta
W_{x}r^{\alpha(1-\epsilon)}}}\bigg{]}\Bigg{)}}$
$\displaystyle=\exp{\Bigg{(}-\lambda_{\mathrm{s}}\frac{\eta}{F}\pi\mathbb{E}\bigg{[}\int_{0}^{\infty}\theta
W_{x}r^{\alpha(1-\epsilon)}\frac{\delta u^{\delta-1}\mathrm{d}u}{u+\theta
W_{x}r^{\alpha(1-\epsilon)}}\bigg{]}\Bigg{)}}$
$\displaystyle=\exp{\Bigg{(}-\lambda_{\mathrm{s}}\frac{\eta}{F}\pi\theta^{\delta}\Gamma(1-\delta)\Gamma(1+\delta)r^{2(1-\epsilon)}\mathbb{E}\big{[}W_{x}^{\delta}\big{]}\Bigg{)}}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\exp{\Bigg{(}-\lambda_{\mathrm{s}}\frac{\eta}{F}\pi\theta^{\delta}\Gamma(1-\delta)\Gamma(1+\delta)\frac{\Gamma(\epsilon+1)}{(\lambda_{\mathrm{d}}\pi)^{\epsilon}}r^{2(1-\epsilon)}\Bigg{)}},$
(63)

where (a) results from the following

$\displaystyle\mathbb{E}\big{[}W_{x}^{\delta}\big{]}$
$\displaystyle=\mathbb{E}\big{[}R_{i}^{2\epsilon}\big{]}\approx\int_{0}^{\infty}2\pi\lambda_{\mathrm{d}}ue^{-\lambda_{\mathrm{d}}\pi
u^{2}}u^{2\epsilon}\mathrm{d}u$
$\displaystyle=\int_{0}^{\infty}\frac{t^{\epsilon}}{(\lambda_{\mathrm{d}}\pi)^{\epsilon}}e^{-t}\mathrm{d}t=\frac{\Gamma(\epsilon+1)}{(\lambda_{\mathrm{d}}\pi)^{\epsilon}}.$
(64)

Then, we decondition (-G) on $r$ and get the following:

$\displaystyle\mathbb{E}[\mu_{0}^{\Phi}]\approx\int_{0}^{\infty}2\pi\lambda_{\mathrm{d}}re^{-\lambda_{\mathrm{d}}\pi
r^{2}}\mathbb{E}[\mu_{0}^{\Phi}|r]\mathrm{d}r$
$\displaystyle=\int_{0}^{\infty}\\!\\!\exp\Big{(}\\!-\\!z\\!-\\!\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\frac{\eta}{F}\theta^{\delta}\Gamma(1\\!-\\!\delta)\Gamma(1\\!+\\!\delta)\Gamma(1\\!+\\!\epsilon)z^{1-\epsilon}\Big{)}\mathrm{d}z.$
(65)

The expression above can be equivalently written as follows:

$\displaystyle\mathbb{E}[\mu_{0}^{\Phi}]\approx\mathbb{E}\big{[}e^{-C_{1}Z^{1-\epsilon}}\big{]}$
(66)

where $Z$ is a random variable that obeys the exponential distribution with
unit mean. Denote by $Y=Z^{1-\epsilon}$, we can compute the probability
density function (pdf) of $Y$ as follows:

$\displaystyle f_{Y}(y)$ $\displaystyle=\frac{\mathrm{d}\mathbb{P}(Y\leq
y)}{\mathrm{d}y}=\frac{\mathrm{d}\mathbb{P}\big{(}Z\leq
y^{\frac{1}{1-\epsilon}}\big{)}}{\mathrm{d}y}$
$\displaystyle=\frac{\mathrm{d}\big{(}1-e^{-\frac{y}{1-\epsilon}}\big{)}}{\mathrm{d}y}=\frac{1}{1-\epsilon}y^{\frac{1}{1-\epsilon}-1}e^{-\frac{y}{1-\epsilon}}.$
(67)

This expression indicates that $Y$ follows a Weibull distribution with the
shape parameter $(1-\epsilon)$ and unit scale parameter. Therefore, we can
obtain the approximate result as
$\mathbb{E}[\mu_{0}^{\Phi}]\approx\mathbb{E}\big{[}e^{-C_{1}Z^{1-\epsilon}}\big{]}=\sum_{n=0}^{\infty}\frac{(-C_{1})^{n}}{n!}\Gamma\big{(}1+(1-\epsilon)n\big{)}$.

In the same way, we can calculate
$\mathbb{E}\Big{[}\frac{1}{\mu_{0}^{\Phi}}|r\Big{]}$ as follows:

$\displaystyle\mathbb{E}\Big{[}\frac{1}{\mu_{0}^{\Phi}}\big{|}r\Big{]}=\mathbb{E}\Bigg{[}\prod_{i\neq
0}\Bigg{(}1+\frac{\eta/F}{1-\frac{\eta}{F}+\frac{D_{i}^{\alpha}}{\theta
R_{i}^{\alpha\epsilon}r^{\alpha(1-\epsilon)}}}\Bigg{)}\Bigg{]}$
$\displaystyle\approx\exp{\Bigg{(}\mathbb{E}\bigg{[}\int_{x\in\mathbb{R}^{2}}\frac{\lambda_{\mathrm{s}}\eta/F\mathrm{d}x}{1-\frac{\eta}{F}+\frac{\|x\|^{\alpha}}{\theta
W_{x}r^{\alpha(1-\epsilon)}}}\bigg{]}\Bigg{)}}$
$\displaystyle=\exp{\\!\Bigg{(}\\!\lambda_{\mathrm{s}}\frac{\eta}{F}\Big{(}1\\!\\!-\\!\\!\frac{\eta}{F}\Big{)}^{\delta\\!-\\!1}\\!\pi\theta^{\delta}r^{2(1-\epsilon)}\Gamma(1\\!-\\!\delta)\Gamma(1\\!+\\!\delta)\frac{\Gamma(\epsilon\\!+\\!1)}{(\lambda_{\mathrm{d}}\pi)^{\epsilon}}\\!\Bigg{)}}.$
(68)

By deconditioning the above with respect to $r$, we can obtain
$\mathbb{E}\Big{[}\frac{1}{\mu_{0}^{\Phi}}\Big{]}$ as follows:

$\displaystyle\mathbb{E}\Big{[}\frac{1}{\mu_{0}^{\Phi}}\Big{]}$
$\displaystyle\approx\int_{0}^{\infty}\\!\\!\exp\Big{(}\\!-\\!z\\!$
$\displaystyle\qquad+\\!\frac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{d}}}\frac{\eta}{F}\Big{(}1\\!\\!-\\!\\!\frac{\eta}{F}\Big{)}^{\delta-1}\theta^{\delta}\Gamma(1\\!-\\!\delta)\Gamma(1\\!+\\!\delta)\Gamma(1\\!+\\!\epsilon)z^{1\\!-\\!\epsilon}\\!\Big{)}\\!\mathrm{d}z$
$\displaystyle=\sum_{n=0}^{\infty}\frac{{C_{2}}^{n}}{n!}\Gamma\big{(}1+(1-\epsilon)n\big{)}.$
(69)

Substituting expressions for $\mathbb{E}[\mu_{0}^{\Phi}]$ and
$\mathbb{E}\Big{[}\frac{1}{\mu_{0}^{\Phi}}\Big{]}$ into (50), we can get the
approximate average AoI.

|  Zhiling Yue (S’22) received the B.E. degree in Electronic Information
Engineering of Honors Program from China Agricultural University (CAU), China,
in 2021. She is currently pursuing the Ph.D. degree with the Zhejiang
University/University of Illinois at Urbana-Champaign Institute (ZJU-UIUC
Institute), Zhejiang University, China. She is also affiliated with the
College of Information Science and Electronic Engineering, Zhejiang
University, China. Her research interests include the analysis of large-scale
wireless networks leveraging tools from stochastic geometry and queuing
theory.
---|---
|  Howard H. Yang (S’13–M’17) received the B.E. degree in Communication
Engineering from Harbin Institute of Technology (HIT), China, in 2012, and the
M.Sc. degree in Electronic Engineering from Hong Kong University of Science
and Technology (HKUST), Hong Kong, in 2013. He earned the Ph.D. degree in
Electrical Engineering from Singapore University of Technology and Design
(SUTD), Singapore, in 2017. He was a Postdoctoral Research Fellow at SUTD from
2017 to 2020, a Visiting Postdoc Researcher at Princeton University from 2018
to 2019, and a Visiting Student at the University of Texas at Austin from 2015
to 2016. Currently, he is an assistant professor with the Zhejiang
University/University of Illinois at Urbana-Champaign Institute (ZJU-UIUC
Institute), Zhejiang University, Haining, China. He is also an adjunct
assistant professor with the Department of Electrical and Computer Engineering
at the University of Illinois Urbana-Champaign, IL, USA Dr. Yang’s research
interests cover various aspects of wireless communications, networking, and
signal processing, currently focusing on the modeling of modern wireless
networks, high dimensional statistics, graph signal processing, and machine
learning. He serves as an editor for the IEEE Transactions on Wireless
Communications. He received the IEEE Signal Processing Society Best Paper
Award in 2022, the IEEE WCSP 10-Year Anniversary Excellent Paper Award in
2019, and the IEEE WCSP Best Paper Award in 2014.
---|---