# Efficient Sensor Scheduling Strategy Based on Spatio-temporal Scope
Information Model

Yang Liu, Chen Dong*, Xiaoqi Qin, and Xiaodong Xu  This work was supported in
part by Beijing Natural Science Foundation under Grant No. 19L2033.
(Corresponding author: Chen Dong.) Yang Liu, Chen Dong, Xiaoqi Qin, and
Xiaodong Xu are with the State Key Laboratory of Networking and Switching
Technology, Beijing University of Posts and Telecommunications, Beijing,
100876, China. (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>xuxiaodong@bupt.edu.cn;).

###### Abstract

In this paper, based on the spatio-temporal correlation of sensor nodes in the
Internet of Things (IoT), a Spatio-temporal Scope information model (SSIM) is
proposed to quantify the scope valuable information of sensor data, which
decays with space and time, to guide the system for efficient decision making
in the sensed region. A simple sensor monitoring system containing three
sensor nodes is considered, and two optimal scheduling decision mechanisms,
single-step optimal and long-term optimal decision mechanisms, are proposed
for the optimization problem. For the single-step mechanism, the scheduling
results are analyzed theoretically, and approximate numerical bounds on the
node layout between some of the scheduling results are obtained, consistent
with the simulation results. For the long-term mechanism, the scheduling
results with different node layouts are obtained using the Q-learning
algorithm. The performance of the two mechanisms is verified by conducting
experiments using the relative humidity dataset, and the differences in
performance of the two mechanisms are discussed; in addition, the limitations
of the model are summarized.

###### Index Terms:

Internet of Things, Spatio-temporal Scope Information Model, spatio-temporal
correlation, sensor scheduling.

##  I Introduction

The Internet of Things connects the physical world and transforms physical
objects from being traditional to smart by exploiting its underlying
technologies such as ubiquitous and pervasive computing, embedded devices,
communication technologies, sensor networks, Internet protocols and
applications[1]. There are a large number of sensing nodes in the IoT, whose
role is to collect data and provide information to the upper layers to make
relevant adjustments and decisions for the whole system. Typical application
scenarios include temperature and air quality monitoring in urban cities [2],
smart traffic [3] and smart agriculture [4], etc. However, due to the cost
constraints, most of the sensing nodes at the end of the network are usually
low-cost sensors powered by batteries, which are very sensitive to power
consumption, including micropower nodes and passive nodes. It has been one of
the leading research problems to improve nodes sensing efficiency and prolong
the nodes working life in these applications.

For some time-sensitive applications, the requirement for data immediacy is
stronger, i.e., outdated data is not of much value to the system. Based on
this idea, the concept of Age of Information (AoI) [5] is proposed, which is
defined as the time difference between the moment of sensing data generation
and the moment of acceptance by the receiver. On the one hand, the system
expects nodes to send frequent updates so as to be able to ensure the
immediacy of data information, but on the other hand, frequent updates can
cause congestion in the system and consume too much energy of nodes, resulting
in poor performance. Therefore, numerous scholars have studied the trade-off
strategy of the system by combining AoI with various queuing and communication
models. Specifically, the optimal service rates to minimize the average AoI in
different queuing models are derived in [5]. The problem of minimizing the
expected weighted sum AoI while simultaneously satisfying timely-throughput
constraints is addressed in [6]. Reference [7] investigates the problem of
minimizing average and peak age of information (PAoI) under general
interference constraints. A discrete-time queueing model to derive the exact
distributions of the AoI and PAoI sequences in a multisource status update
system is proposed in [8]. In [9], it is found that the optimal arrival-
independent renewal (AIR) scheduling policy to minimize the time-average AoI
is RR-ONE, i.e., scheduling terminals in a round-robin fashion, and each
terminal only retains the most up-to-date packet. Reference [10] studies the
optimal device scheduling process that jointly minimizes the average AoI and
the energy cost with two types of correlated devices.

The freshness of information is linearly decreasing in the definition of AoI,
but there is some correlation in the time domain due to the physical nature of
monitoring sources, which may cause the performance degradation due to
information aging is not a linear function of time. Therefore, some scholars
have proposed nonlinear functions about AoI and conducted related researches
Reference [11] introduces a general age penalty function to characterize the
level of dissatisfaction on data staleness. In [12], a general expression of
the generating function of AoI and the peak age of information (PAoI) metric
is provided, which provides a methodology for analyzing general non-linear age
functions. Exploiting the temporal correlation between consecutive samples of
a Markov source, reference [13] considers a generalized incremental update
scheme by sending differential updates.

In addition, the dense deployment of sensing nodes on the space inevitably
causes the observations of the nodes to be highly correlated in the spatial
domain. Therefore, some scholars have studied the layout setting of nodes on
the spatial domain and the spatial critical sampling rate, etc. The
correlation between nodes is used to reconstruct the observed physical
phenomena based on a fraction of all available sensor nodes in [14], where a
framework for the analysis of sensor density based on asymptotic analysis is
proposed. In [15], a theoretical analysis of spatio-temporal correlation
characteristics of point and field sources in WSN is performed, and the
spatio-temporal characteristics are analytically derived along with the
distortion functions. Based on the confident information coverage (CIC) model,
[16] and [17] study the critical sensor density and find the optimal placement
pattern to achieve complete coverage in randomly deployed sensor networks. In
addition, a node deployment scheme to maximize the network lifetime and ensure
CIC in a field with obstacles is proposed in [18].

Meanwhile, some scholars have simultaneously investigated the cooperative
scheduling and spatio-temporal sampling rate among nodes in conjunction with
the correlation of nodes in time and space domain. In [19], a theoretical
framework to capture the spatial and temporal correlations in wireless sensor
networks is first introduced, which constitutes a basis for the development of
such energy-efficient communication protocols for Wireless Sensor Networks
(WSN). More work related to this paper in this regard is described in detail
in subsection B.

The main idea of this paper is to measure the value of real-time useful
information of sensor data in the temporal and spatial domains by using the
spatio-temporal correlation of the sources. For example, the first data of
sensors collected continuously in the time domain can bring the most amount of
valuable information due to the lack of a priori knowledge, while the later
data in the time domain can bring less valuable information than the first
data due to the short time correlation of the physical properties of the
sources, i.e., the first data can provide a certain amount of priori
information for the later ones. Similarly, there is an overlap in the valuable
information provided by the data acquired simultaneously by nodes close to
each other in the spatial domain. In this paper, a model to measure real-time
valuable information of sensor data is established to represent the effective
spatio-temporal scope of data information, and the optimal node scheduling
strategy under different node layouts is investigated to improve the
efficiency of sensing information.

###  I-A Contributions and Paper Outline

In this paper, utilizing the spatio-temporal correlation of the sources in the
area to be measured, we establish a Spatio-temporal Scope Information Model,
which measure the effective information spatio-temporal domian of node data
for the whole system in real time. In simple terms, the information value of a
sensor data gradually decays in the temporal domain with the passage of time,
and it also gradually decays in the neighborhood of the node location in the
spatial domain. Moreover, this paper investigates the efficient activation
scheduling scheme for three nodes in different location layouts to ensure the
effective grasp of the system over the whole area. The main contributions of
this paper are as follows.

  1. 1.

Utilizing the spatio-temporal correlation of sensor nodes, a Spatio-temporal
Scope Information Model is proposed to quantify the valuable information of
sensor data, which decays with space and time.

  2. 2.

A single-step optimal decision-making mechanism for a three-node sensing
system is proposed. The theoretical analysis is made, and a method to solve
the boundary node distribution among various scheduling situations is
provided.

  3. 3.

A long-term optimal decision-making mechanism for a three-node sensing system
is proposed, which is modeled as a Markov decision process, and the Q-learning
algorithm is utilized to solve the optimal scheduling results.

  4. 4.

With single-step mechanism, the approximate bounds for the node layout between
partial scheduling results are obtained from theoretical analysis and
numerical calculation, which match with the simulation results. The optimal
scheduling results with long-term mechanism corresponding to different node
layout are obtained. Finally, the different performances of the two mechanisms
are experimentally verified, and the advantages and limitations of each are
summarized.

The rest of the paper is organized as follows: In Section II, the system model
and optimization problem are described. In Section III, a optimal decision-
making mechanism of single-step information acquisition is proposed for the
established model, and the related theoretical analysis is made. In Section
IV, a optimal decision-making mechanism of long-term information acquisition
is proposed and the modeling and solution methods are described. In Section V,
numerical results and simulation results are presented. In Section VI,
experimental evaluation is performed. Finally, conclusions are drawn and
discussions are made in Section VII.

###  I-B Related work

TABLE I: Comparison of References.

[b]  Reference information source sensors correlation metrics Research results
[20] Single-point Two Spatial-temporal Estimation Error optimal time shift
between two sensors [21], [22] multi-point Multiple Spatial-temporal
Estimation Error DRL -based scheduling mechanism [24], [25] Single-point
(noisy Ornstein- Uhlenbeck process) Single source and noise mutual information
closed-form VoI expressions [26], [27] Regional Multiple Spatial-temporal
error-tolerable sensing (ETS) coverage AoI violation probability, optimal
sensors transmission power [28] Regional Multiple Spatial-temporal Node
Spatial-temporal coverage radius Node activation scheduling [29], [30] time-
varying Gauss- Markov Random Field (GMRF) Multiple Spatial-temporal mean
squared estimation error Closed-form expressions for estimation error, optimal
Spatial-temporal sampling rates [31], [32] N Gaussian process Multiple
Spatial-temporal mean squared error (MSE) optimal scheduling policy [33]
Single-point Multiple Spatial-temporal Spatial-temporal mutual information
optimal update interval [34] multi-point Multiple Spatial-temporal overall
utility of information update Status update node set this work Regional Three
Spatial-temporal Spatial-temporal Scope information Node activation strategy
in any layout

Among some recent studies, the following ones are more relevant to the work of
this paper. Specifically, reference [20] considers a system consisting of two
correlated information sources ,and establishes a minimal time shift between
the two sources’ updates, for which the system estimation error is minimal. An
energy-aware scheduling mechanism based on Deep Reinforcement Learning (DRL)
is proposed in [21] and [22], which is capable of significantly prolonging the
lifetime of a network of battery-powered sensors. A measure for the freshness
of information is proposed in [23], which uses the mutual information between
the real-time source value and the delivered samples at the receiver. In [24]
and [25], a mutual-information based Value of Information (VoI) framework is
formalised to characterise how valuable the status updates are for Hidden
Markov Models. An error-tolerable sensing (ETS) coverage as the area where the
estimated information is with smaller error than the target value is defined
in [26] and [27], where the $\eta$-coverage probability is presented, and the
optimal transmission power which minimizes the average energy consumption
while guaranteeing a certain level of the $\eta$-coverage probability is
provided. In [28], a transmission scheduling approach is presented through a
geometric approach with individual node coverage model, which is a function of
the estimation accuracy in a region near the node. The performance of state
updates is studied in [29] and [30], where the status is modeled as a time-
varying Gauss-Markov Random Field (GMRF) and the estimation error of status
update at the fusion center is analyzed. In [31] and [32], an optimal
scheduling policy for transmitting observations of spatio-temporally dependent
processes over a limited number of communication channels is derived that
minimizes the time-average mean squared error (MSE), resulting in a periodic
scheduling sequence. A novel timeliness metric spatially temporally
correlative mutual information (STI) is proposed in [33], where an optimal
update interval is found by solving an integer optimization problem about slot
allocation in satellite. Assuming that each information can be commonly
observed by multiple sensors, two multi-source information update problems,
named AoI-aware Multi-Source Information Updating (AoI-MSIU) and AoI-
Reduction-aware Multi-Source Information Updating (AoIR-MSIU) problems, are
formulated in [34].

Table I shows in detail the comparison between our work and the above
references. Throughout the above studies, although many scholars have studied
the cooperative scheduling and sampling strategies among nodes by combining
the spatio-temporal correlation of the sources, less attention has been
devoted to all possible scheduling results and change rules of the scheduling
results for different node layouts with an effective grasp of the whole
region. For detail, this paper discusses the node sequential activation
scheduling problem for different node layout cases. It is the main motivation
of this paper to study the possible optimal scheduling results for different
node layout cases, and analyze the change law of optimal scheduling results
for different node layouts.

##  II System model and Problem formulation

###  II-A System model

In this paper, we mainly consider a simple sensing system composed of three
sensing nodes, where the information sensing area is the whole two-dimensional
(2D) plane, and the importance attached to each location in the area is the
same, that is, there is no setting of sensing priority, as shown in Fig. 1.
For the convenience of preliminary analysis, it is assumed that the system
periodically decides to activate a sensing node for information acquisition,
i.e., the interval time between each decision is constant. The data
transmission conditions are assumed to be ideal, ignoring the influence of
data sending and transmission, i.e., it is assumed that the monitor can
receive sensor data immediately after the node is activated.

Figure 1: System model

###  II-B Spatio-temporal Scope Information Model

In this paper, a Gaussian random field is assumed in the two-dimensional
region to be measured, i.e., the variables to be measured between any points
in space conform to the joint Gaussian distribution. Without considering the
node hardware acquisition error, i.e., the data collected by the nodes are
considered accurate enough to completely represent the values of the random
variables to be measured. Then, for a certain node, let the random variable to
be monitored corresponding to the spatial location of the node be $X$. The
amount of valuable information that the first activation of the node can bring
to $X$ is the entropy $H(x)$. Meanwhile, for any point $p$ in the neighborhood
near the node, let its corresponding random variable be $Y_{p}$, and $Y_{p}$
and $X$ conform to the joint Gaussian distribution. Denote the correlation
coefficient as $\rho$, then the amount of information that the node data can
provide at position $p$ is

$I(p)=I(Y_{p};X)=h(Y_{p})-h(Y_{p}|X)=-\frac{1}{2}\log(1-{{\rho}^{2}})$ (1)

Then the total amount of information in the two-dimensional plane is

$\begin{split}{{I}_{D}}&=\int_{0}^{2\pi}{\int_{0}^{1}{-\frac{1}{2}\log(1-{{\rho}^{2}})\cdot\rho\cdot
d\theta d}}\rho\\\
&=-\frac{\pi}{2}\int_{0}^{1}{\log(1-{\rho}^{2})d{{\rho}^{2}}}=\frac{\pi}{2}\end{split}$
(2)

It can be seen from the above equation that although the information
calculation result in Eq. (1) tends to infinity when the correlation
coefficient $\rho$ tends to $1$, the total scope information integral in the
two-dimensional plane converges and can be calculated. In this paper, for
simplicity, we consider the spatiotemporally separable covariance function
[20]

$\rho={{e}^{-{{\lambda}_{d}}\cdot d-{{\lambda}_{t}}\cdot t}}$ (3)

where $d$ represents the spatial distance, and $t$ represents the time
difference, in addition, $\lambda_{d}$ and $\lambda_{t}$ are the scaling
parameters of information relevance with respect to space and time,
respectively. The above description is for a single node and the first
activation of the information acquisition. However, when the system has
multiple nodes and the nodes are activated in a certain order, there will be
multiple nodes with spatio-temporal association at each location in the two-
dimensional plane. In other words, at any point in space, the previous sensor
data of each node can provide a certain amount of information and eliminate
part of the uncertainty. In this paper, for the sake of simplicity, the system
keeps only the latest sensor data of each node, and only the sensor data that
can eliminate the most uncertainty and provide the most information at that
location is selected to provide a reference for that location. In other words,
it is considered that the node data closest to the location in terms of
spatio-temporal domian, i.e., the most relevant node data, provides
information for that location, and the amount of joint information provided by
multiple nodes previous data together is not considered. That is, the amount
of information available for the system at any point $p$ in the two-
dimensional plane at moment $t$ is

$I(p,t)=-\frac{1}{2}{{\log}}\left(1-{{e}^{-2{{\lambda}_{d}}\cdot|p-{{p}^{*}}(p)|-2{{\lambda}_{t}}\cdot|t-{{t}^{*}}(p)|}}\right).$
(4)

where ${p^{*}}(p)$ is the node location coordinates of the most relevant
sensor data in space-time for location $p$, and

${{p}^{*}}(p)=\mathop{\arg\min}\limits_{p_{s_{i}}}{{\lambda}_{d}}|p-p_{s_{i}}|+{{\lambda}_{t}}|t-t_{s_{i}}|,\
p_{s_{i}}\in S,$ (5)

where $S$ is the set of all node coordinates. Furthermore, $|t-{{t}^{*}}(p)|$
in Eq. (4) is the value of AoI for the most relevant data at position $p$ at
the current moment. According to the above equation, the spatio-temporal scope
information map of the system at time t for the whole 2D space can be obtained
as illustrated in Fig. 2, where the vertical height represents the amount of
valuable information, and the location of the peak is the sensing nodes
location.

Figure 2: Spatio-temporal scope information map

At this moment, if the node $s_{i}$ is activated, the incremental information
that can be obtained for each point $p$ in the 2D plane is

$\begin{split}&inf{{o}_{gain}}(p,s_{i})=h({{Y}_{p}}|{{X}_{past}^{*}})-h({{Y}_{p}}|{{X}_{{{s}_{i}}}})\\\
&=\left[h({{Y}_{p}})-h({{Y}_{p}}|{{X}_{{{s}_{i}}}})\right]-\left[h({{Y}_{p}})-h({{Y}_{p}}|{{X}_{past}^{*}})\right]\\\
&=-\frac{1}{2}\log\frac{1-{{e}^{-2{{\lambda}_{d}}\cdot|p-{{p}_{{{s}_{i}}}}\text{
}\\!\\!|\\!\\!\text{
}}}}{1-{{e}^{-2{{\lambda}_{d}}\cdot|p-{{p}^{*}}(p)|-2{{\lambda}_{t}}\cdot
AoI^{*}(p)}}}\end{split}$ (6)

where ${Y}_{p}$ is the random variable corresponding to position p,
$X_{s_{i}}$ is the random variable corresponding to position of node $s_{i}$,
and ${X}_{past}^{*}$ is the random variable corresponding to the most relevant
node data at position $p$. In addition, $AoI^{*}(p)$ is the AoI value of the
most spatiotemporally relevant data at position $p$. However, each time a
sensor node activated by system may acquire no new valuable information in
certain regions. In other words, the previous data of other nodes provide more
information in these regions than the node activated at this moment, and the
result of the above equation takes a negative value in these regions, but
which does not meet the definition of information. Because no more information
can be provided, but temporarily no loss is caused. Thus, the information
increment that can be obtained in these areas is defined as 0, i.e., the scope
information increment in the whole two-dimensional plane of the activated node
$s_{i}$ is

$\begin{split}&{{I}_{gain}}=\iint\limits_{D}{\max\\{inf{{o}_{gain}}(p,s_{i}),0\\}}d\sigma\\\
&=\iint\limits_{D}{\max\left\\{-\frac{1}{2}\log\frac{1-{{e}^{-2{{\lambda}_{d}}\cdot|p-{{p}_{{{s}_{i}}}}\text{
}\\!\\!|\\!\\!\text{
}}}}{1-{{e}^{-2{{\lambda}_{d}}\cdot|p-{{p}^{*}}(p)|-2{{\lambda}_{t}}\cdot
AoI^{*}(p)}}},0\right\\}}d\sigma\\\ \end{split}$ (7)

###  II-C Problem formulation

In this paper, it is the research objective to find the most efficient way of
node activation scheduling given the node location layout and the spatio-
temporal correlation coefficient of the region to be measured. That is, it is
desired that the total amount of information mastered by the system for the
entire two-dimensional plane has the maximum mean value in time domian, as in
the following equation

$\max\overline{I}=\underset{T\to\infty}{\mathop{\lim}}\,\frac{1}{T}\int_{0}^{T}{\big{[}\iint\limits_{D}{I(p,t)d\sigma}\big{]}}dt$
(8)

Let the total amount of information held by the system at each activation node
moment $t_{i}$ be $I_{i}$, that is,

${{I}_{i}}=\iint\limits_{D}{I(p,{{t}_{i}})d\sigma}.$ (9)

and let the total two-dimensional information decay function after the
$i_{th}$ activation node of the system be $f_{i}(t)$ as follows

${{f}_{i}}(t)=\frac{\iint\limits_{D}{-\frac{1}{2}{{\log}_{2}}\left(1-{{e}^{-2{{\lambda}_{d}}\cdot|p-{{p}^{*}}(p)|-2{{\lambda}_{t}}\cdot[Ao{{I}^{*}}(p)+t]}}\right)}d\sigma}{\iint\limits_{D}{-\frac{1}{2}{{\log}_{2}}\left(1-{{e}^{-2{{\lambda}_{d}}\cdot|p-{{p}^{*}}(p)|-2{{\lambda}_{t}}\cdot
Ao{{I}^{*}}(p)}}\right)}d\sigma}$ (10)

Letting $T=n*\Delta t$, the expression for the information mean can be further
written as

$\begin{split}\bar{I}=\underset{n\to\infty}{\mathop{\lim}}\,\frac{\sum\limits_{i=1}^{n}{{{I}_{i}}\cdot\int_{0}^{\Delta
t}{{{f}_{i}}(t)dt}}}{n\cdot\Delta t}\end{split}$ (11)

and the following relationship holds for the total amount of two-dimensional
information and the incremental information of the activated nodes

${{I}_{i+1}}={{I}_{i}}\cdot{{f}_{i}}(\Delta t)+{{I}_{gain}}(i),\
i=1,2,3,\cdots$ (12)

Recursive induction of the above equation leads to the following result

$\begin{split}{{I}_{n}}&={{I}_{gain}}(n)+\sum\limits_{i=1}^{n-1}{\left[{{I}_{gain}}(i)\cdot\prod\limits_{j=i}^{n-1}{{{f}_{j}}(\Delta
t)}\right]},\\\ \end{split}$ (13)

then,

$\begin{split}&\sum\limits_{i=1}^{n}{{{I}_{i}}\cdot\int_{0}^{\Delta
t}{{{f}_{i}}(t)dt}}\\\
=&\sum\limits_{i=1}^{n}{\left[\left({I}_{gain}(i)+\sum\limits_{j=1}^{i-1}{{{I}_{gain}}(j)\cdot\prod\limits_{k=j}^{i-1}{{{f}_{k}}(\Delta
t)}}\right)\cdot\int_{0}^{\Delta t}{{{f}_{i}}(t)dt}\right]}\\\
=&\sum\limits_{i=1}^{n}\left\\{{{{I}_{gain}}(i)\cdot}\left[\sum\limits_{j=i}^{n}{\left(\int_{0}^{\Delta
t}{{{f}_{j}}(t)dt}\cdot\prod\limits_{k=i}^{j-1}{{{f}_{k}}(\Delta
t)}\right)}\right]\right\\}.\\\ \end{split}$ (14)

The above equation is difficult to continue to derive, mainly because
$f_{i}(t)$ is challenging to get a clear closed form, so here it is proposed
to unify the total information decay of all moments approximately expressed as
$f(t)$, that is,

${{f}_{i}}(t)\approx f(t),i=1,2,3\cdots.$ (15)

In other words, it is assumed that the total information decay trend is
uniform at all activation node moments. The reason for such an approximate
assumption here is mainly motivated by the consideration that the system tends
to acquire more and smoother information, which leads to similar total
information at each activation node moment, thus causing the decay trend may
be approximately the same. Then Eq. (14) can be further written as

$\begin{split}&\int_{0}^{\Delta
t}{f(t)dt}\cdot\sum\limits_{i=1}^{n}{\left({{I}_{gain}}(i)\cdot\sum\limits_{j=0}^{n-i}{f{{(\Delta
t)}^{j}}}\right)}\\\ =&\int_{0}^{\Delta
t}{f(t)dt}\cdot\sum\limits_{i=1}^{n}{\left({{I}_{gain}}(i)\cdot\frac{1-f{{(\Delta
t)}^{n-i+1}}}{1-f(\Delta t)}\right)}\\\ =&\frac{\int_{0}^{\Delta
t}{f(t)dt}}{1-f(\Delta
t)}\cdot\sum\limits_{i=1}^{n}{\left[{{I}_{gain}}(i)\cdot\left(1-f{{(\Delta
t)}^{n-i+1}}\right)\right]}\end{split}$ (16)

Since $f(\Delta t)$ ¡ 1, for any positive number $\varepsilon$ that is greater
than 0 and small enough,

$\begin{split}&1-f{{(\Delta t)}^{n-i+1}}<1-\varepsilon\\\
&i>n+1-{{\log}_{f(\Delta t)}}\varepsilon,\\\ \end{split}$ (17)

so when $n$ tends to infinity, the percentage of terms with information
increment $I_{gain}(i)$ coefficients less than $1-\varepsilon$ is

$\underset{n\to\infty}{\mathop{\lim}}\,\frac{{{\log}_{f(\Delta
t)}}\varepsilon}{n}=0.$ (18)

Therefore the coefficients can be approximated by 1, i.e., the optimization
problem becomes the following form

$\begin{split}\max\ &\bar{I}=\frac{\int_{0}^{\Delta t}{f(t)dt}}{\Delta
t\left(1-f(\Delta
t)\right)}\cdot\underset{n\to\infty}{\mathop{\lim}}\,\cdot\frac{1}{n}\sum\limits_{i=1}^{n}{{{I}_{gain}}(i)}\\\
\text{s}\text{.t}\text{.
}&Se({{t}_{i}})={{s}_{i}}\in\left\\{{{s}_{1}},{{s}_{2}},{{s}_{3}}\right\\}\\\
&{{t}_{i}}=0,\Delta t,2\Delta t,\cdots,n\Delta t,\\\ \end{split}$ (19)

where $Se(t_{i})$ is the node activated at the moment $t_{i}$. From the above
expression, the original problem is converted to the problem of maximizing the
mean value of the incremental information obtained by each activated node.

##  III Single-step optimal mechanism

Regarding the above-mentioned problems, the first optimization mechanism
readily comes to mind is the single-step decision method, which means that the
system computationally finds out the sensing node that can obtain the most
information increment at the current moment based on the known spatio-temporal
information residual map, then makes it active and updates the spatio-temporal
information map. The system takes the same actions at each discrete decision
moment after that. The mathematical expression as a rule for each node
activation $s_{i}$ can be written as follows

$Se(t_{i})={\underset{s_{i}\in\left\\{{{s}_{1}},{{s}_{2}},{{s}_{3}}\right\\}}{\arg\max}\,\iint\limits_{D}{\max\\{inf{{o}_{gain}}(p,s_{i}),0\\}}d\sigma}.$
(20)

where ${{p}_{{{s}_{i}}}}$ is the position coordinate of the activated sensing
node. According to whether the value of the latter term in the integral
equation is greater than 0, the integration region can be divided into two
parts, $D^{\prime}$ and $D^{\prime\prime}$, thus the integral in Eq. (20) is
equivalent to

$\iint\limits_{D^{\prime}}{-\frac{1}{2}\log\frac{1-{{e}^{-2{{\lambda}_{d}}\cdot|p-{{p}_{{{s}_{i}}}}\text{
}\\!\\!|\\!\\!\text{
}}}}{1-{{e}^{-2{{\lambda}_{d}}\cdot|p-{{p}^{*}}(p)|-2{{\lambda}_{t}}\cdot
AoI^{*}(p)}}}}d\sigma,$ (21)

where $D^{\prime}$ and $D^{\prime\prime}$ are the set consisting of points
that respectively satisfy the following conditions:

$\left\\{\begin{split}&p\in D^{\prime},\
|p-{{p}_{{{s}_{i}}}}|-|p-{{p}^{*}}(p)|<\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot|t-{{t}^{*}}(p)|\\\
&p\in D^{\prime\prime},\
|p-{{p}_{{{s}_{i}}}}|-|p-{{p}^{*}}(p)|\geq\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot|t-{{t}^{*}}(p)|.\end{split}\right.$
(22)

In addition, the most spatially and temporally correlated node $p^{*}(p)$
varies for different locations $p$ with the following equations:

$\begin{split}{{p}^{\text{*}}}(p)\text{=}&\arg\min{{\lambda}_{d}}|p-{{p}^{*}}(p)|+{{\lambda}_{t}}|t-{{t}^{*}}(p)|\\\
=&\begin{cases}{{p}_{{{s}_{1}}}},&p\in{{S}_{1}}\\\
{{p}_{{{s}_{2}}}},&p\in{{S}_{2}}\\\ {{p}_{{{s}_{3}}}},&p\in{{S}_{3}},\\\
\end{cases}\end{split}$ (23)

where $S_{1}$, $S_{2}$ and $S_{3}$ are the sets of points that respectively
meet the conditions as Eq. (24),

$\left\\{\begin{split}{{S}_{1}}=\left\\{p\in
D\Big{|}|p-{{p}_{{{s}_{1}}}}|-|p-{{p}_{{{s}_{2}}}}|<\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot({{AoI}_{{{s}_{2}}}}-{{AoI}_{{{s}_{1}}}}),\
|p-{{p}_{{{s}_{1}}}}|-|p-{{p}_{{{s}_{3}}}}|<\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot({{AoI}_{{{s}_{3}}}}-{{AoI}_{{{s}_{1}}}})\right\\}&\\\
{{S}_{2}}=\left\\{p\in
D\Big{|}|p-{{p}_{{{s}_{2}}}}|-|p-{{p}_{{{s}_{1}}}}|<\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot({{AoI}_{{{s}_{1}}}}-{{AoI}_{{{s}_{2}}}}),\
|p-{{p}_{{{s}_{2}}}}|-|p-{{p}_{{{s}_{3}}}}|<\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot({{AoI}_{{{s}_{3}}}}-{{AoI}_{{{s}_{2}}}})\right\\}&\\\
{{S}_{3}}=\left\\{p\in
D\Big{|}|p-{{p}_{{{s}_{3}}}}|-|p-{{p}_{{{s}_{1}}}}|<\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot({{AoI}_{{{s}_{1}}}}-{{AoI}_{{{s}_{3}}}}),\
|p-{{p}_{{{s}_{3}}}}|-|p-{{p}_{{{s}_{2}}}}|<\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot({{AoI}_{{{s}_{2}}}}-{{AoI}_{{{s}_{3}}}})\right\\}&.\end{split}\right.$
(24)

in which the ${{AoI}_{{{s}_{1}}}}$, ${{AoI}_{{{s}_{2}}}}$ and
${{AoI}_{{{s}_{3}}}}$ are the AoI of the latest sensing data of each node. The
increment of information that can be obtained by the activated node $s_{i}$
from the above expression can be written as

$\begin{split}I_{gain}=&\sum\limits_{j=1}^{3}{\iint\limits_{{{S}_{j}}}{\max\\{inf{{o}_{gain}}(p,s_{i}),0\\}}d\sigma}\\\
=&\sum\limits_{j=1}^{3}{\iint\limits_{{{S}_{j}}^{\prime}}{{info_{gain}}(p,s_{i})}d\sigma},\end{split}$
(25)

where $S_{j}^{\prime}$ is the set of points that meet the condition as

$|p-{{p}_{{{s}_{i}}}}|-|p-{{p}_{{{s}_{j}}}}|<\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot{AoI}_{{{s}_{j}}},\
p\in{{S}_{j}},\ j\in\left\\{1,2,3\right\\}.$ (26)

From the fact that the trajectory of a point in plane geometry where the
difference between the distances to two points is fixed is a hyperbola, we can
preliminarily determine that the boundary of the above set $D^{\prime}$ may
contain several curves. For example, when the system only has two nodes, and
${{p}_{{{s}_{i}}}}$, ${{p}^{*}}(p)$ are taken as ${{p}_{{{s}_{1}}}}$,
${{p}_{{{s}_{2}}}}$, respectively, let the coordinates of node $s_{1}$ and
node $s_{2}$ in 2D space be $(-c,0)$, $(c,0)$ as shown in Fig. 3(a). From the
property that the difference between two sides of a triangle is less than the
third side, it is easy to obtain the inequality as follows:

$|p-{{p}_{{{s}_{1}}}}|-|p-{{p}_{{{s}_{2}}}}|<d_{s_{1},s_{2}}=2c.$ (27)

Abbreviating $\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot\Delta t$ to $2a$,
the inequality corresponding to the set $D^{\prime\prime}$ can be written as

$\begin{matrix}&2c>|p-{{p}_{{{s}_{1}}}}|-|p-{{p}_{{{s}_{2}}}}|\geq\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot\Delta
t=2a>0\\\
\Leftrightarrow&\frac{{x}^{2}}{{a}^{2}}-\frac{{{y}^{2}}}{{{c}^{2}}-{{a}^{2}}}\geq
1,\ x>0,\ c>a>0,\end{matrix}$ (28)

and $D^{\prime\prime}$ is an empty set when $c<a$, i.e.,
$d_{s_{1},s_{2}}<\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot\Delta t$, in
which case, a certain increment of information can be obtained in the whole 2D
plane. However, it should be noted that activating node $s_{1}$ or node
$s_{2}$ can obtain the same incremental information at this point, and there
is no optimal activation scheme from the perspective of information
acquisition only. Therefore, in the subsequent analytical modelling, the
distance between the nodes must be at least greater than
$\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot\Delta t$.

The area distribution of $D^{\prime\prime}$ that can be obtained from Eq. (28)
is shown in the shaded part of Fig. 3(a), and no new information increment can
be obtained by activating node $s_{1}$ in this area. The remaining blank areas
are where new information increments can be obtained, that is, the set
$D^{\prime}$. The boundary curve formula of the $D^{\prime\prime}$ is required
for the expression of the information increment’s integral limit to calculate
the information increment’s value.

(a)

(b)

Figure 3: Schematic diagram of the distribution of set elements and boundary
curves. (a) Schematic of the set distribution and boundary curve of the second
activation node of the two-point system. (b) Schematic of the curve after
rotating $\theta$ counterclockwise around the origin, where the dashed line
represents the asymptote of the curve.

However, due to different node positions and different angles of the
coordinate system, when the horizontal axis of the curve is not parallel to
the x-axis, a 2D rotation matrix can be used to derive the equation of the
curve $\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1$ after
rotating $\theta$ counterclockwise around the origin as

$(\frac{{{\cos}^{2}}\theta}{{{a}^{2}}}-\frac{{{\sin}^{2}}\theta}{{{b}^{2}}}){{x}^{2}}+(\frac{{{\sin}^{2}}\theta}{{{a}^{2}}}-\frac{{{\cos}^{2}}\theta}{{{b}^{2}}}){{y}^{2}}+\sin
2\theta(\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}})xy=1.$ (29)

The above equation corresponds to a hyperbola, while the dividing curve of
different regions in the information residual map is only one side of the
hyperbola. Moreover, the expression equation of a single curve may be
conveniently written as only one of two forms $y=f(x)$ or $x=g(y)$ with
different rotation angles. For example, when the rotation angle is 0, the
expression equation of the curve located on the right side of the y-axis is
only conveniently written as $x=g(y)$. Because there are two corresponding $y$
values for each value of $x$, the curve cannot be expressed as a unique
expression of $y=f(x)$. And the primary method of discrimination is based on
the maximum number of intersections of the two asymptotic lines of the curve
with the horizontal and vertical lines. For example, as shown in Fig. 3(b),
the curve can be expressed as $x=g(y)$ when there is at most one intersection
point between the two asymptotes of the curve and the line parallel to the
x-axis. Meanwhile, the curve can be expressed as $y=f(x)$ when there is at
most one intersection point between the two asymptotes of the curve and the
line parallel to the y-axis. The applicability of the two expression forms can
then be obtained as follows:

$\left\\{\begin{split}&x=g(y)\Leftrightarrow\theta\in[0,\arctan\frac{b}{a}]\cup[\pi-\arctan\frac{b}{a},\pi]\\\
&y=f(x)\Leftrightarrow\theta\in[\frac{\pi}{2}-\arctan\frac{b}{a},\frac{\pi}{2}+\arctan\frac{b}{a}],\\\
\end{split}\right.$ (30)

where $\frac{b}{a}$ is the absolute value of the slope of the hyperbola
asymptote. It is easy to find that when $\arctan\frac{b}{a}<\frac{\pi}{4}$,
i.e., $c<\sqrt{2}a$, the range of values of $\theta$ in Eq. (30) does not
completely cover $[0,\pi]$. In other words, there are certain ranges where a
single curve cannot be expressed in the above two functional forms, and one
solution in this case is to adjust the angle of the coordinate system for
subsequent expression of the curve equation.

When the three nodes present an equilateral triangular layout, it is easy to
analyze that the optimal scheduling strategy under the single-step decision
mechanism is alternate activation of the three nodes in turn according to the
decay of information over time and the equal spatial correlation between
individual nodes. However, it is essential to note that the above-mentioned
alternate activation of nodes presupposes that the distance between nodes is
greater than $\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot 2\Delta t$. The
reason is that if the node distance is smaller than this value, the valuable
information contained in the sensed data of each node can be covered by the
data of other nodes after two moments so that there exists the same increment
of information available to two nodes at each decision moment; therefore there
is no optimal decision at this time from the perspective of information
acquisition. The following is a preliminary consideration to analyze the
possible scheduling scenarios by changing the distance between only two nodes
and keeping other conditions unchanged, i.e., keeping the nodes in an
isosceles triangular layout. The location distribution of the three nodes is
assumed as shown in Fig. 4(a).

(a) isosceles triangle

(b) general triangular

Figure 4: The location distribution of three nodes. (a) The node location
distribution with three nodes presenting an isosceles triangle layout. (b) The
node location distribution with three nodes presenting a general triangle
layout, where the shaded area is the optional set of node $s_{1}$.

###  III-A Three-node isosceles triangle layout

This subsection mainly analyzes the possible scheduling scenarios when only
the distance of node $s_{2}$ and $s_{3}$ is changed, and other conditions
remain unchanged. From the previous analysis, it is known that when the three
nodes are laid out in an equilateral triangle, the scheduling strategy is to
activate them alternately, and the node activation sequence can be set to
$123123\cdots$. Thus from a long-term perspective, when node $s_{3}$ is
activated, the AoI values of the latest sensing data of nodes $s_{1}$,
$s_{2}$, $s_{3}$ are $2\Delta t$, $\Delta t$, $3\Delta t$, respectively, which
are abbreviated as $[2,\ 1,\ 3]$. At this time, the amount of incremental
information obtained by activating node $s_{3}$ is recorded as
$info_{[2,1,3]}({{s}_{3}})$. Putting the AoI vector $[2,\ 1,\ 3]$ into Eq.
(24), the following results are obtained:

$\begin{split}&\left\\{\begin{split}&{{S}_{1}}:\left\\{\begin{split}&|p-{{p}_{{{s}_{2}}}}|-|p-{{p}_{{{s}_{1}}}}|>\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot\Delta
t\\\
&|p-{{p}_{{{s}_{1}}}}|-|p-{{p}_{{{s}_{3}}}}|<\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot\Delta
t\\\ \end{split}\right.\\\
&{{S}_{2}}:\left\\{\begin{split}&|p-{{p}_{{{s}_{2}}}}|-|p-{{p}_{{{s}_{1}}}}|<\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot\Delta
t\\\
&|p-{{p}_{{{s}_{2}}}}|-|p-{{p}_{{{s}_{3}}}}|<\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot
2\Delta t\\\ \end{split}\right.\\\
&{{S}_{3}}:\left\\{\begin{split}&|p-{{p}_{{{s}_{1}}}}|-|p-{{p}_{{{s}_{3}}}}|>\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot\Delta
t\\\
&|p-{{p}_{{{s}_{2}}}}|-|p-{{p}_{{{s}_{3}}}}|>\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot
2\Delta t\\\ \end{split}.\right.\\\ \end{split}\right.\\\ \end{split}$ (31)

It is evident that the formulas of the boundary curves for different regions
in the information distribution map can be obtained, which satisfy the
conditions as

$\left\\{\begin{split}&C_{S_{1},S_{2}}:|p-{{p}_{{{s}_{2}}}}|-|p-{{p}_{{{s}_{1}}}}|=\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot\Delta
t\\\
&C_{S_{1},S_{3}}:|p-{{p}_{{{s}_{1}}}}|-|p-{{p}_{{{s}_{3}}}}|=\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot\Delta
t\\\
&C_{S_{2},S_{3}}:|p-{{p}_{{{s}_{2}}}}|-|p-{{p}_{{{s}_{3}}}}|=\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot
2\Delta t,\\\ \end{split}\right.$ (32)

and the sufficient and necessary conditions for the existence of the above
boundary curves are

$\left\\{\begin{split}&d_{s_{1},s_{2}}>\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot\Delta
t\\\ &d_{s_{1},s_{3}}>\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot\Delta t\\\
&d_{s_{2},s_{3}}>\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot 2\Delta
t.\end{split}\right.$ (33)

If one of the conditions is not met, the corresponding boundary curve does not
exist. Taking the midpoint of the base of the isosceles triangle as the origin
and the base as the $x$-axis direction to establish a rectangular coordinate
system, and setting the base length as $d$ and the height of the triangle as
$h$, the coordinates of the three nodes are $(0,d)$, $(-\frac{d}{2},0)$,
$(\frac{d}{2},0)$, respectively. Then the information residual distribution
map of the system can be drawn according to Eq. (31) and Eq. (32), as shown in
Fig. 5(a), where the upper triangular, star and circular marker areas
represent the sets $S_{1}^{\prime}$, $S_{2}^{\prime}$, and $S_{3}^{\prime}$,
respectively, meanwhile, the solid lines are the boundaries of the different
set regions, and the dashed lines are the extensions of the boundary curves.
The three boundary curves necessarily intersect at a point, which can be
proved by the geometric distance property of the hyperbola. When the rotation
angle $\theta$ of the curve is 0 and the centre of the curve is at the origin,
the general formula of a single curve is

$x=h(y,a,c,sign)=sign\cdot
a\cdot\sqrt{1+\frac{{{y}^{2}}}{{{c}^{2}}-{{a}^{2}}}},$ (34)

where $sign$ is a symbolic variable that takes the value $\pm 1$, and $c$ is
the hyperbolic focal length, which is half of the distance between the two
nodes. Thus the set $S_{2}$ and $S_{3}$ boundary curve expression can be
written as

$C_{S_{2},S_{3}}:x=h(y,2a,\frac{d}{2},1)=2a\cdot\sqrt{1+\frac{{{y}^{2}}}{\frac{{{d}^{2}}}{4}-4{{a}^{2}}}}.$
(35)

The general expression of $S_{1}$, $S_{2}$ boundary curve and $S_{1}$, $S_{3}$
boundary curve can be written as Eq. (36) by combining Eq. (29) with Eq. (30),

$\left\\{\begin{split}&y=f(x)={{y}_{o}}-\frac{C}{2B}\cdot(x-{{x}_{o}})\pm\frac{1}{\sqrt{|B|}}\sqrt{1+(\frac{{{C}^{2}}}{4B}-A){{(x-{{x}_{o}})}^{2}}},\
\theta\in[\frac{\pi}{2}-\arctan\frac{b}{a},\frac{\pi}{2}+\arctan\frac{b}{a}]\\\
&x=g(y)={{x}_{o}}-\frac{C}{2A}\cdot(y-{{y}_{o}})\pm\frac{1}{\sqrt{|A|}}\sqrt{1+(\frac{{{C}^{2}}}{4A}-B){{(y-{{y}_{o}})}^{2}}},\
\theta\in[0,\arctan\frac{b}{a}]\cup[\pi-\arctan\frac{b}{a},\pi].\\\
\end{split}\right.$ (36)

where $({{x}_{o}},{{y}_{o}})$ are the coordinates of the center point of the
corresponding curve, i.e., the intersection of the two asymptotes, and
$\theta$ is the rotation angle of the curve. In addition, $c$ is the
hyperbolic focal length, and

$\left\\{\begin{split}&a=\frac{{{\lambda}_{t}}\cdot\Delta
t}{2{{\lambda}_{d}}},\ b=\sqrt{c^{2}-a^{2}}\\\
&A=\frac{{{\cos}^{2}}\theta}{{{a}^{2}}}-\frac{{{\sin}^{2}}\theta}{{{b}^{2}}}\\\
&B=\frac{{{\sin}^{2}}\theta}{{{a}^{2}}}-\frac{{{\cos}^{2}}\theta}{{{b}^{2}}}\\\
&C=\sin 2\theta(\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}).\end{split}\right.$
(37)

For ease of subsequent expression, the Eq. (36) is abbreviated to the
following form:

$\left\\{\begin{split}&y=f(x,a,\text{c},{{x}_{o}},{{y}_{o}},\theta,sign),\theta\in{{\Theta}_{1}}\\\
&x=g(y,a,c,{{x}_{o}},{{y}_{o}},\theta,sign),\theta\in{{\Theta}_{2}},\\\
\end{split}\right.$ (38)

where $sign$ takes the value $\pm 1$, representing the selection of positive
or negative signs in the two expressions in Eq. (36). Then the $S_{1}$,
$S_{2}$ and $S_{1}$, $S_{3}$ dividing lines can be obtained as Eq. (39) and
Eq. (40).

$C_{S_{1},S_{2}}:\left\\{\begin{split}&y=f\left(x,a,\frac{\sqrt{{{h}^{2}}+{{d}^{2}}/4}}{2},\
-\frac{d}{4},\frac{h}{2},\arctan(\frac{2h}{d}),1\right),\arctan(\frac{2h}{d})\in[\frac{\pi}{2}-\arctan\frac{b}{a},\
\frac{\pi}{2}+\arctan\frac{b}{a}]\\\
&x=g\left(y,a,\frac{\sqrt{{{h}^{2}}+{{d}^{2}}/4}}{2},-\frac{d}{4},\frac{h}{2},\arctan(\frac{2h}{d}),1\right),\arctan(\frac{2h}{d})\in[0,\
\arctan\frac{b}{a}]\cup[\pi-\arctan\frac{b}{a},\ \pi].\\\ \end{split}\right.$
(39)
$C_{S_{1},S_{3}}:\left\\{\begin{split}&y=f\left(x,a,\frac{\sqrt{{{h}^{2}}+{{d}^{2}}/4}}{2},\frac{d}{4},\frac{h}{2},\pi-\arctan(\frac{2h}{d}),-1\right),\
\pi-\arctan(\frac{2h}{d})\in[\frac{\pi}{2}-\arctan\frac{b}{a},\frac{\pi}{2}+\arctan\frac{b}{a}]\\\
&x=g\left(y,a,\frac{\sqrt{{{h}^{2}}+{{d}^{2}}/4}}{2},\frac{d}{4},\frac{h}{2},\
\pi-\arctan(\frac{2h}{d}),1\right),\pi-\arctan(\frac{2h}{d})\in[0,\arctan\frac{b}{a}]\cup[\pi-\arctan\frac{b}{a},\pi].\\\
\end{split}\right.$ (40)

(a) $info_{[2,\ 1,\ 3]}$

(b) $info_{[2,\ 1,\ 3]}({{s}_{3}})$

(c) $info_{[2,\ 1,\ 3]}({{s}_{3}})$

(d) $info_{[2,\ 1,\ 3]}({{s}_{3}})$

Figure 5: Spatio-temporal information distribution maps. (a) The typical
information residual distribution map when the AoI vector is [2,1,3]. (b) The
typical information map when the AoI vector is [2,1,3] and node $s_{3}$ is
activated at this time, which can be abbreviated as the map of
$info_{[2,1,3]}(s_{3})$. (c) The special case that boundary curve
$C_{S_{1},S_{2}}$ is not conveniently expressed. (d) The information map of
$info_{[2,1,3]}(s_{3})$ when the node distance is small, and the boundary
curve existence condition is not met.

The intersection of the three curves can be solved by combining the equations
of the three curves, namely $C_{S_{1},S_{2}}$, $C_{S_{1},S_{3}}$ and
$C_{S_{2},S_{3}}$. However, a quartic equation can be obtained by combining
two of the curve formulas. Since the simplification process and the solving
formula are complicated, they are not detailed here. In addition, since the
incremental information formula cannot calculate the exact analytical result,
the numerical calculation can also be used to find the approximate numerical
solution when solving intersections. The multiple solutions are then verified
using another curve formula, and the solution that satisfies the condition is
selected.

When node $s_{3}$ is activated at this moment, the conditions regarding the
region where no new valuable incremental information is obtained, i.e., the
set $D^{\prime\prime}$, can be determined by the previous analysis method as
follows:

$\left\\{\begin{split}&|p-{{p}_{{{s}_{3}}}}|-|p-{{p}_{{{s}_{1}}}}|\geq\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot
AoI_{s_{1}}=\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot 2\Delta t\\\
&|p-{{p}_{{{s}_{3}}}}|-|p-{{p}_{{{s}_{2}}}}|\geq\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot
AoI_{s_{2}}=\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot\Delta
t.\end{split}\right.$ (41)

Then combining the constraints of Eq. (33), when the conditions
${{d}_{{{s}_{1}},{{s}_{3}}}}\geq\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot
2\Delta t$ and
${{d}_{{{s}_{2}},{{s}_{3}}}}\geq\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot
2\Delta t$ are met, two boundary curves of $D^{\prime\prime}$ can be obtained
as shown in Fig. 5(b), where the rectangular, upper triangular, star, and
circular marker regions represent the sets $D^{\prime\prime}$,
$S_{1}^{\prime}$, $S_{2}^{\prime}$, and $S_{3}^{\prime}$, respectively, and
the set $S_{j}^{\prime}$ is defined as shown in Eq. (26). Moreover, the
boundary curve formula $C_{S_{2}^{\prime},\ D^{\prime\prime}}$ of the
rectangular marked region near node $s_{2}$ can be obtained from Eq. (34) as

$C_{S_{2}^{\prime},D^{\prime\prime}}:x=h(y,a,\frac{d}{2},-1)\text{=}-a\cdot\sqrt{1+\frac{{{y}^{2}}}{\frac{{{d}^{2}}}{4}-{{a}^{2}}}},$
(42)

and the boundary curve $C_{S_{1}^{\prime},D^{\prime\prime}}$ of the
rectangular marked area near node $s_{1}$ can be obtained from Eq. (36). The
coordinates of the intersection points of curves can be obtained by combining
the formulas of the curves, as shown in the Fig. 5(b), which are set as
$(x_{1},y_{1})$, $(x_{2},y_{2})$, respectively. Let the valuable incremental
information that can be obtained at the position $(x,y)$ be

$\begin{split}&info_{{{s}_{e}},{{s}_{p}}}(x,y)\\\
=&-\frac{1}{2}\log\frac{1-{{e}^{-{{2\lambda}_{d}}\cdot\sqrt{{{(x-{{x}_{{{s}_{e}}}})}^{2}}+{{(y-{{y}_{{{s}_{e}}}})}^{2}}}}}}{1-{{e}^{-{{2\lambda}_{d}}\cdot\sqrt{{{(x-{{x}_{{{s}_{p}}}})}^{2}}+{{(y-{{y}_{{{s}_{p}}}})}^{2}}}-{{2\lambda}_{t}}\cdot
AoI_{{s}_{p}}}}}\\\ \end{split}$ (43)

where $s_{e}$ is the currently active node, and $s_{p}$ is the node with
information residuals at that location. Then the valuable information
increments that can be acquired in regions $S_{1}^{\prime}$, $S_{2}^{\prime}$
and $S_{3}^{\prime}$ are shown in Eq. (44), Eq. (45) and Eq. (46),
respectively.

$\begin{split}\iint\limits_{{{S}_{1}}^{\prime}}{{info_{{{s}_{3}},{{s}_{1}}}(x,y)}}dxdy=&\int_{{{x}_{1}}}^{{{x}_{2}}}{dx\int_{C_{S_{1}^{\prime},S_{2}^{\prime}}}^{C_{S_{1}^{\prime},D^{\prime\prime}}}{info_{{{s}_{3}},{{s}_{1}}}(x,y)}dy}\\\
+&\int_{{{x}_{2}}}^{\inf}{dx\int_{C_{S_{1}^{\prime},S_{3}^{\prime}}}^{C_{S_{1}^{\prime},D^{\prime\prime}}}{info_{{{s}_{3},{{s}_{1}}}}(x,y)}dy},\\\
\end{split}$ (44)
$\begin{split}\iint\limits_{{{S}_{2}}^{\prime}}{{info{{}_{{{s}_{3}},{{s}_{2}}}}(x,y)}}dxdy=&\int_{-\infty}^{{{y}_{2}}}{dy\int_{C_{S_{2}^{\prime},D^{\prime\prime}}}^{C_{S_{2}^{\prime},S_{3}^{\prime}}}{info{{}_{{{s}_{3}},{{s}_{2}}}}(x,y)}dx}\\\
+&\int_{{{x}_{1}}}^{{{x}_{2}}}{dx\int_{y_{2}}^{C_{S_{1}^{\prime},S_{2}^{\prime}}}{info{{}_{{{s}_{3}},{{s}_{2}}}}(x,y)}dy}\\\
-&\int_{{{y}_{2}}}^{{{y}_{1}}}{dy\int_{{{x}_{1}}}^{C_{S_{2}^{\prime},D^{\prime\prime}}}{info{{}_{{{s}_{3}},{{s}_{2}}}}(x,y)}dx},\\\
\end{split}$ (45)
$\begin{split}\iint\limits_{{{S}_{3}}^{\prime}}{{info{{}_{{{s}_{3}},{{s}_{3}}}}(x,y)}}dxdy=&\int_{-\infty}^{{{y}_{2}}}{dy\int_{C_{S_{2}^{\prime},S_{3}^{\prime}}}^{+\infty}{info{{}_{{{s}_{3}},{{s}_{3}}}}(x,y)}dx}\\\
+&\int_{{{x}_{2}}}^{\inf}{dx\int_{{{y}_{2}}}^{C_{S_{1}^{\prime},S_{3}^{\prime}}}{info{{}_{{{s}_{3}},{{s}_{3}}}}(x,y)}dy},\\\
\end{split}$ (46)

where

$\left\\{\begin{split}&C_{S_{1}^{\prime},S_{2}^{\prime}}:f(x,a,\frac{\sqrt{{{h}^{2}}+{{d}^{2}}/4}}{2},-\frac{d}{4},\frac{h}{2},\arctan(\frac{2h}{d}),1)\\\
&C_{S_{1}^{\prime},D^{\prime\prime}}:f(x,2a,\frac{\sqrt{{{h}^{2}}+{{d}^{2}}/4}}{2},\frac{d}{4},\frac{h}{2},\pi-\arctan(\frac{2h}{d}),1)\\\
&C_{S_{1}^{\prime},S_{3}^{\prime}}:f(x,a,\frac{\sqrt{{{h}^{2}}+{{d}^{2}}/4}}{2},\frac{d}{4},\frac{h}{2},\pi-\arctan(\frac{2h}{d}),-1)\\\
&C_{S_{2}^{\prime},S_{3}^{\prime}}:h(y,2a,\frac{d}{2},1)\\\
&C_{S_{2}^{\prime},D^{\prime\prime}}:h(y,a,\frac{d}{2},-1).\\\
\end{split}\right.$ (47)

The above results are then summed to obtain the expression for the total
incremental information of activating node $s_{3}$ at this moment. However, it
should be noted that the curve formulas in the above expressions,
$C_{S_{1}^{\prime},S_{2}^{\prime}}$, $C_{S_{1}^{\prime},S_{3}^{\prime}}$ and
$C_{S_{1}^{\prime},D^{\prime\prime}}$ all take the first expression form
$y=f(x)$, which is applicable to most of the node layout cases, but not to
all. For example, when the curve $C_{S_{1}^{\prime},\ D^{\prime\prime}}$ is
only conveniently expressed in the form $x=g(y)$, the information integral
expression for the region $S_{1}^{\prime}$ can be rewritten according to the
same method by changing the order of integration. In addition, there are a few
special layouts where a curve cannot be uniquely expressed as either of the
two forms $y=f(x)$ and $x=g(y)$. For example, when
$\frac{\sqrt{{{h}^{2}}+{{d}^{2}}/4}}{2}<\sqrt{2}\cdot 2a$, the regional
boundary curve $C_{S_{1}^{\prime},D^{\prime\prime}}$ cannot be written as a
unique functional expression, as schematically shown in Fig. 5(c). Since the
curve has a large radian and the two asymptotic lines of the curve have at
most two intersections with the horizontal or vertical lines, it cannot be
uniquely expressed as either of the two functional expressions in the current
coordinate system. It is one solution to adjust the angle of the coordinate
system to facilitate the integration expression. For example, for Fig. 5(c),
set the direction of the line where nodes s1 and s3 are located as the
horizontal or vertical axis of the right angle coordinate system, and then
choose the appropriate order of integration. Moreover, if the distance between
nodes is small and does not meet the conditions for the existence of boundary
curves, the information map is relatively simplified as exemplified in Fig.
5(d) when
${{d}_{{{s}_{1}},{{s}_{3}}}}<\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot
2\Delta t$ and
${{d}_{{{s}_{2}},{{s}_{3}}}}<\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot
2\Delta t$.

Theoretically, if the distance between nodes $s_{2}$ and $s_{3}$ is gradually
reduced and other parameters remain unchanged, the incremental information
that can be obtained by activating node $s_{3}$ after activating node $s_{2}$
should gradually decrease due to the increase of spatial correlation of nodes
$s_{2}$ and $s_{3}$. Meanwhile, the incremental information that can be
obtained by activating node $s_{1}$ gradually increases because the effective
area jointly covered by nodes $s_{2}$ and $s_{3}$ is gradually reduced with
the shortening of their distance. Thus theoretically, there should be a
critical case where the information increments obtained by activating node
$s_{1}$ and node $s_{3}$ are the same, i.e., the condition
$info_{[2,1,3]}({{s}_{3}})=info_{[2,1,3]}({{s}_{1}})$ is met. If the distance
between nodes $s_{2}$ and $s_{3}$ continues to be reduced in this critical
case, it is clear from the previous analysis that this will lead to optimal
scheduling of 1213… cycles at this moment under the single-step decision
mechanism.

For the convenience of curve expression, the straight line formed by node
$s_{1}$ and $s_{2}$ can be set as the $x$ axis, and the midpoint of $s_{1}$
and $s_{2}$ is the origin to establish a rectangular coordinate system.
According to the same analysis process, the information increment formula that
can be obtained by activating node $s_{1}$ at this time can be written. The
distribution of the different areas where activation node $s_{1}$ can obtain
information increments is shown in Fig. 6(a).

(a) $info_{[2,\ 1,\ 3]}({{s}_{1}})$

(b) $info_{[inf,\ 2,\ 1]}({{s}_{2}})$

(c) $info_{[inf,\ 2,\ 1]}({{s}_{1}})$

(d) $info_{[inf,\ 2,\ 1]}({{s}_{1}})$

Figure 6: Spatio-temporal information distribution maps. (a) The typical
information map of $info_{[2,\ 1,\ 3]}({{s}_{1}})$. (b) The typical
information map of $info_{[inf,\ 2,\ 1]}({{s}_{2}})$. (c) The information map
of $info_{[inf,\ 2,\ 1]}({{s}_{1}})$ with three curves having an intersection
point. (d) The information map of $info_{[inf,\ 2,\ 1]}({{s}_{1}})$ with the
asymptotes of three curves being parallel to each other.

Obviously, it is very similar to Fig. 5(b), and the Eq. (48) and Eq. (49) can
be obtained from the previous analysis.

$\begin{split}inf{{o}_{[2,1,3]}}({{s}_{1}})=&\int_{-\infty}^{{{y}_{1}}}{dy\int_{-\inf}^{C_{S_{1}^{\prime},S_{2}^{\prime}}}{info{{}_{{{s}_{1}},{{s}_{1}}}}(x,y)}dx}+\int_{-\inf}^{{{x}_{1}}}{dx\int_{{{y}_{1}}}^{C_{S_{1}^{\prime},S_{3}^{\prime}}}{info{{}_{{{s}_{1}},{{s}_{1}}}}(x,y)}dy}\\\
+&\int_{-\infty}^{{{y}_{1}}}{dy\int_{C_{S_{1}^{\prime},S_{2}^{\prime}}}^{C_{S_{2}^{\prime},D^{\prime\prime}}}{info{{}_{{{s}_{1}},{{s}_{2}}}}(x,y)}dx}+\int_{{{x}_{1}}}^{{{x}_{2}}}{dx\int_{{{y}_{1}}}^{C_{S_{2}^{\prime},S_{3}^{\prime}}}{info{{}_{{{s}_{1}},{{s}_{2}}}}(x,y)}dy}-\int_{{{y}_{1}}}^{{{y}_{2}}}{dy\int_{C_{S_{2}^{\prime},D^{\prime\prime}}}^{{{x}_{2}}}{info{{}_{{{s}_{1}},{{s}_{2}}}}(x,y)}dx}\\\
+&\int_{-\inf}^{{{x}_{1}}}{dx\int_{C_{S_{1}^{\prime},S_{3}^{\prime}}}^{C_{S_{3}^{\prime},D^{\prime\prime}}}{info{{}_{{{s}_{1}},{{s}_{3}}}}(x,y)}dy}+\int_{{{x}_{1}}}^{{{x}_{2}}}{dx\int_{C_{S_{2}^{\prime},S_{3}^{\prime}}}^{C_{S_{3}^{\prime},D^{\prime\prime}}}{info{{}_{{{s}_{1}},{{s}_{3}}}}(x,y)}dy}\\\
\end{split}$ (48)
$\left\\{\begin{split}&C_{S_{1}^{\prime},S_{2}^{\prime}}:h(y,a,\frac{\sqrt{{{h}^{2}}+{{d}^{2}}/4}}{2},-1)\\\
&C_{S_{1}^{\prime},S_{3}^{\prime}}:f(x,a,\frac{\sqrt{{{h}^{2}}+{{d}^{2}}/4}}{2},-\frac{{{d}^{2}}}{2\sqrt{4{{h}^{2}}+{{d}^{2}}}},\frac{hd}{\sqrt{4{{h}^{2}}+{{d}^{2}}}},\arctan(\frac{4hd}{4{{h}^{2}}-{{d}^{2}}}),1)\\\
&C_{S_{2}^{\prime},S_{3}^{\prime}}:f(x,2a,\frac{d}{2},\frac{4{{h}^{2}}-{{d}^{2}}}{4\sqrt{4{{h}^{2}}+{{d}^{2}}}},\frac{hd}{\sqrt{4{{h}^{2}}+{{d}^{2}}}},\pi-\arctan(\frac{2h}{d}),1)\\\
&C_{S_{2}^{\prime},D^{\prime\prime}}:h(y,a,\frac{\sqrt{{{h}^{2}}+{{d}^{2}}/4}}{2},1)\\\
&C_{S_{3}^{\prime},D^{\prime\prime}}:f(x,3a,\frac{\sqrt{{{h}^{2}}+{{d}^{2}}/4}}{2},-\frac{{{d}^{2}}}{2\sqrt{4{{h}^{2}}+{{d}^{2}}}},\frac{hd}{\sqrt{4{{h}^{2}}+{{d}^{2}}}},\arctan(\frac{4hd}{4{{h}^{2}}-{{d}^{2}}}),1).\\\
\end{split}\right.$ (49)

It is also important to note that the above integral formula may not be
applicable to all layout conditions, which is subject to the following
restrictions as Eq. (50).

$\left\\{\begin{split}&\arctan\frac{4hd}{4{{h}^{2}}-{{d}^{2}}}\in[\frac{\pi}{2}-\arctan\frac{\sqrt{\frac{4{{h}^{2}}+{{d}^{2}}}{16}-9{{a}^{2}}}}{3a},\frac{\pi}{2}+\arctan\frac{\sqrt{\frac{4{{h}^{2}}+{{d}^{2}}}{16}-9{{a}^{2}}}}{3a}]\\\
&\pi-\arctan\frac{2h}{d}\in[\frac{\pi}{2}-\arctan\frac{\sqrt{\frac{4{{h}^{2}}+{{d}^{2}}}{16}-4{{a}^{2}}}}{2a},\frac{\pi}{2}+\arctan\frac{\sqrt{\frac{4{{h}^{2}}+{{d}^{2}}}{16}-4{{a}^{2}}}}{2a}].\\\
\end{split}\right.$ (50)

There are also cases where the curve formulas cannot be uniquely represented,
and the method to deal with this is to adjust the coordinate system angle and
rewrite the integral formula.

The boundary situation of the above two scheduling situations meets the
condition as

$info_{[2,1,3]}({{s}_{3}})-info_{[2,1,3]}({{s}_{1}})=0.$ (51)

Obviously, due to the cumbersome integral formula, it is challenging to seek
the exact analytical solution, and only the approximate numerical solution can
be obtained by numerical calculation. The specific solution algorithm can use
the dichotomy method, that is, first determine the approximate range of the
solution, second take the midpoint of the range and calculate the value of the
equation at this point, then select the range where the value of the equation
on two sides of the range with different signs, which means that one is
greater than zero and the other is less than zero. Repeat this operation until
the error meets certain limits, then the midpoint of the interval can be set
as the approximate solution. The specific calculation results will be given in
Section V. The obtained discrete values of $d$ and $h$ correspond to the
boundary distributions of the periodic scheduling sequences $123$ and $1213$.

On the contrary, if the distance between node $s_{2}$ and $s_{3}$ gradually
increases, the incremental information obtained by alternate activation
between them gradually increases due to the gradual weakening of spatial
correlation. While the distance between node $s_{1}$ and node $s_{2}$, $s_{3}$
remains unchanged and the effective coverage area of node $s_{2}$ and $s_{3}$
gradually expands with the increase of distance, it is possible that the
proportion of node $s_{2}$ and $s_{3}$ being activated gradually increases and
the proportion of node $s_{1}$ being activated gradually decreases. First, a
quite extreme case is considered where the distance between node $s_{2}$ and
$s_{3}$ is very large, which is equivalent to the case where the base of an
isosceles triangle is much larger than the height. Since node $s_{2}$ and
$s_{3}$ are far away from each other, the incremental information that node
$s_{2}$ and $s_{3}$ can obtain when they are alternately activated is less
than or equal to $inf{{o}_{[\inf,1,2]}}({{s}_{3}})$ or
$inf{{o}_{[\inf,2,1]}}({{s}_{2}})$. When the residual information of node
$s_{1}$ is completely covered by the new data of node $s_{2}$ or $s_{3}$, the
equal sign is taken.

$\begin{split}inf{{o}_{[\inf,2,1]}}({{s}_{1}})<inf{{o}_{[\inf,2,1]}}({{s}_{2}}).\end{split}$
(52)

If the constraint of Eq. (52) is met, whenever node $s_{1}$ is activated, the
amount of incremental information node it can obtain does not exceed that of
alternate activation of $s_{2}$ and $s_{3}$. Thus, node $s_{1}$ will never be
activated in the system with the single-step decision mechanism. The critical
condition for this situation can be obtained by making the above inequality
take an equal sign and finding the layout constraints that meet the condition
of the equation. Specifically, according to the previous analysis ideas as
shown in Fig. 6(b), the right-hand side of the inequality can be written as
follows:

$\begin{split}inf{{o}_{[\inf,2,1]}}({{s}_{2}})=&\int_{-\infty}^{+\infty}{dy}\int_{-\infty}^{C_{S_{2}^{\prime},S_{3}^{\prime}}}{info{{}_{{{s}_{2}},{{s}_{2}}}}(x,y)dx},\\\
+&\int_{-\infty}^{+\infty}{dy\int_{C_{S_{2}^{\prime},S_{3}^{\prime}}}^{C_{S_{3}^{\prime},D^{\prime\prime}}}{info{{}_{{{s}_{2}},{{s}_{3}}}}(x,y)dx}}\\\
\end{split}$ (53)

where

$\left\\{\begin{split}&C_{S_{2}^{\prime},S_{3}^{\prime}}:h(y,a,\frac{d}{2},-1)\\\
&C_{S_{3}^{\prime},D^{\prime\prime}}:h(y,a,\frac{d}{2},1).\\\
\end{split}\right.$ (54)

Meanwhile, the left-hand side of the inequality can be written as

$\begin{split}inf{{o}_{[\inf,2,1]}}({{s}_{1}})=&\int_{{{y}_{1}}}^{+\infty}{dy\int_{C_{S_{2}^{\prime},D^{\prime\prime}}}^{C_{S_{2}^{\prime},S_{3}^{\prime}}}{info{{}_{{{s}_{1}},{{s}_{2}}}}(x,y)dx}}\\\
+&\int_{{{y}_{1}}}^{+\infty}{dy}\int_{C_{S_{2}^{\prime},S_{3}^{\prime}}}^{C_{S_{3}^{\prime},D^{\prime\prime}}}{info{{}_{{{s}_{1}},{{s}_{3}}}}(x,y)dx}\\\
\end{split}$ (55)

where

$\left\\{\begin{split}&C_{S_{2}^{\prime},S_{3}^{\prime}}:h(y,a,\frac{d}{2},-1)\\\
&C_{S_{2}^{\prime},D^{\prime\prime}}:g(y,2a,\frac{\sqrt{{{h}^{2}}+{{d}^{2}}/4}}{2},-\frac{d}{4},\frac{h}{2},\arctan(\frac{2h}{d}),-1)\\\
&C_{S_{3}^{\prime},D^{\prime\prime}}:g(y,a,\frac{\sqrt{{{h}^{2}}+{{d}^{2}}/4}}{2},\frac{d}{4},\frac{h}{2},\pi-\arctan(\frac{2h}{d}),1),\\\
\end{split}\right.$ (56)

and $y_{1}$ is the vertical coordinate of the intersection point of the three
curves as illustrated in Fig. 6(c). However, the intersection point does not
exist in all cases. When there is no intersection of the three curves, the
$y_{1}$ in the integral limit is taken as negative infinity. When the
asymptotes of the three curves are parallel, the relevant angle relationship
can be used to deduce the critical condition of whether there is an
intersection, as shown in Fig. 6(d) and Eq. (57).

$\arctan\left(\frac{\sqrt{\frac{{{h}^{2}}+{{d}^{2}}/4}{4}-4{{a}^{2}}}}{2a}\right)={{\theta}_{2}}={{\theta}_{1}}-{{\theta}_{3}}=2\arctan\left(\frac{d}{2h}\right)-\arctan\left(\frac{\sqrt{\frac{{{h}^{2}}+{{d}^{2}}/4}{4}-{{a}^{2}}}}{a}\right).$
(57)

If the left-hand side of the Eq. (57) is greater than the right-hand side,
there is an intersection, and vice versa.

The layout constraints for the critical case can be determined by combining
Eq. (53) and Eq. (55). Again the above integral expressions apply to most
layout cases, which means that the above integral expression may not be fully
applicable due to the few exceptional layout cases.

On the other hand, if the inequality in Eq. (52) does not hold, node $s_{1}$
is bound to be activated again after certain moments, and the frequency of
node $s_{1}$ being activated gradually increases as the distance of nodes
$s_{2}$, $s_{3}$ shrinks. Using the properties of the exponential correlation
function, it can be analyzed how many times the information of the data of
node $s_{1}$ will be covered by the new data of node $s_{2}$ or $s_{3}$, and
it can be obtained that the maximum time interval of the activation frequency
of node $s_{1}$ does not exceed
$\left\lceil\frac{{{\lambda}_{d}\cdot{d}_{{{s}_{1}},{{s}_{2}}}}}{{{{{\lambda}_{t}}}}{{}}\cdot\Delta
t}\right\rceil$.

Regarding the critical situation between various cases with different
activation frequencies of nodes $s_{1}$, the main solution idea is the same as
the previous ones, which let the two incremental information integral formulas
be equal to find the relationship between node layout distances.

From the above analysis, it can be seen that if the information increment is
calculated by dividing the integral equation into different regions, although
an analytical integral equation is obtained, the process is tedious and the
exact analytical results cannot be obtained. In addition, when traversing the
calculation for different node layouts, due to the coordinate system and curve
angle, it leads to that the integral equation need to be rewritten according
to different situations. Therefore, when solving the equation, the information
incremental calculation equation can be written as Eq. (58), and the
approximate numerical solution can be solved with the help of computer for
numerical calculation.

$\begin{split}{{I}_{gain}}=\int_{-\infty}^{+\infty}{\int_{-\infty}^{+\infty}{\max\left\\{-\frac{1}{2}\log\frac{1-{{e}^{-2{{\lambda}_{d}}\cdot\sqrt{{{(x-{{x}_{{{s}_{e}}}})}^{2}}+{{(y-{{y}_{{{s}_{e}}}})}^{2}}}}}}{\min\left\\{1-{{e}^{-2{{\lambda}_{d}}\cdot\sqrt{{{(x-{{x}_{{{s}_{i}}}})}^{2}}+{{(y-{{y}_{{{s}_{i}}}})}^{2}}}-2{{\lambda}_{t}}\cdot
Ao{{I}_{{{s}_{i}}}}}}\Big{|}i=1,2,3\right\\}},0\right\\}dxdy}}\\\ \end{split}$
(58)

In summary, the different scheduling results and corresponding conditions with
three nodes presenting an isosceles triangle layout are shown in Table II,
which shows the typical location layout of the nodes corresponding to various
scheduling results, and the conditions of the associated boundary
distribution.

TABLE II: Different scheduling situations with the single-step mechanism when three nodes present an isosceles triangle layout.  Typical layout | Scheduling results | Conditions
---|---|---
| $23\ 23\cdots$ | $inf{{o}_{[\inf,\ 2,\ 1]}}({{s}_{2}})>inf{{o}_{[\inf,2,1]}}({{s}_{1}})$
| boundary | $inf{{o}_{[\inf,2,1]}}({{s}_{2}})=inf{{o}_{[\inf,2,1]}}({{s}_{1}})$
$\vdots$ | $\vdots$
$1\ 232\ 1\ 323\cdots$ |  | $inf{{o}_{[3,2,1]}}({{s}_{1}})<inf{{o}_{[3,2,1]}}({{s}_{2}})$
---
$inf{{o}_{[2,1,3]}}({{s}_{3}})>inf{{o}_{[2,1,3]}}({{s}_{1}})$
$inf{{o}_{[1,3,2]}}({{s}_{2}})>inf{{o}_{[1,3,2]}}({{s}_{3}})$
$inf{{o}_{[4,2,1]}}({{s}_{1}})>inf{{o}_{[4,2,1]}}({{s}_{2}})$
boundary | $inf{{o}_{[3,2,1]}}({{s}_{1}})=inf{{o}_{[3,2,1]}}({{s}_{2}})$
|  |
$123\ 123\cdots$ |  | $inf{{o}_{[3,2,1]}}({{s}_{1}})>inf{{o}_{[3,2,1]}}({{s}_{2}})$
---
$inf{{o}_{[2,1,3]}}({{s}_{3}})>inf{{o}_{[2,1,3]}}({{s}_{1}})$
$inf{{o}_{[1,3,2]}}({{s}_{2}})>inf{{o}_{[1,3,2]}}({{s}_{3}})$
boundary | $info_{[2,1,3]}({{s}_{3}})=info_{[2,1,3]}({{s}_{1}})$
|  |
$1213\ 1213\cdots$ |  | $info_{[2,1,3]}({{s}_{3}})<info{{}_{[2,1,3]}}({{s}_{1}})$
---
$info_{[1,2,4]}({{s}_{3}})>info{{}_{[1,2,4]}}({{s}_{2}})$
$info_{[1,4,2]}({{s}_{2}})>info{{}_{[1,4,2]}}({{s}_{3}})$
$info_{[2,3,1]}({{s}_{2}})<info{{}_{[2,3,1]}}({{s}_{1}})$

###  III-B Three-node general triangular layout

The previous subsection preliminarily analyzes the possible optimal scheduling
results under the single-step information-optimal mechanism when the three
nodes exhibit an isosceles triangle distribution. This subsection extends the
above results to cases where the nodes exhibit a general triangle
distribution. When three nodes present a general triangular distribution, at
least three variables are required to describe a general triangle. Due to the
unintuitiveness of the three-dimensional diagram and the difficulty of
theoretical analysis, the scheduling analysis in this subsection focuses on
keeping the value of one of the variables unchanged and changing the values of
the other two variables to traverse and obtain the optimal scheduling results.
The specific analysis idea in this subsection is mainly to fix the maximum
node distance and keep the positions of the two nodes with the largest
distance unchanged, traversing the different positions of the other node and
analyzing the possible scheduling results. For example, it is assumed that the
distance $d$ between nodes $s_{2}$ and $s_{3}$ is the largest, and they are
located at $(0,0)$ and $(d,0)$ respectively in the 2D rectangular coordinate
system. The traversed area can be reduced by half according to the horizontal
symmetry of the triangle. Then node $s_{1}$ needs to traverse the area as
illustrated in the Fig. 4(b), where the shaded area in the figure is the
optional position of node $s_{1}$.

When the position of node $s_{1}$ is selected on the boundary of the traversal
region, the three nodes present an isosceles triangle. According to the
previous analysis, the possible scheduling situation can be roughly obtained.
For example, as the vertical height of node $s_{1}$ increases on the
perpendicular bisector of nodes $s_{2}$ and $s_{3}$, the final activation
proportion of node $s_{1}$ may gradually increase from $0$ to $\frac{1}{3}$.
In addition, if traversing from the top to the lower left on the arc boundary,
the scheduling situation may change from alternating activation at the
beginning to a situation where node $s_{3}$ activation accounts for a half and
the proportion of the other two nodes is one fourth under the minimum node
distance constraint, as shown in the previous analysis.

Based on the above analysis, the possible scheduling results under the general
triangular layout can be obtained, as shown in Table III.

TABLE III: Different scheduling situations with the single-step mechanism when three nodes present an general triangle layout.  | Typical
---
layout
| Scheduling
---
results
Conditions
| $23\ 23\cdots$ | $inf{{o}_{[\inf,2,1]}}({{s}_{2}})>inf{{o}_{[\inf,2,1]}}({{s}_{1}})$
boundary | $inf{{o}_{[\inf,2,1]}}({{s}_{2}})=inf{{o}_{[\inf,2,1]}}({{s}_{1}})$
$\vdots$ | $\vdots$
$1232\ 1323\cdots$ |  | $inf{{o}_{[3,2,1]}}({{s}_{1}})<inf{{o}_{[3,2,1]}}({{s}_{2}})$
---
$inf{{o}_{[2,1,3]}}({{s}_{3}})>inf{{o}_{[2,1,3]}}({{s}_{1}})$
$inf{{o}_{[1,3,2]}}({{s}_{2}})>inf{{o}_{[1,3,2]}}({{s}_{3}})$
$inf{{o}_{[4,2,1]}}({{s}_{1}})>inf{{o}_{[4,2,1]}}({{s}_{2}})$
boundary | $inf{{o}_{[3,2,1]}}({{s}_{1}})=inf{{o}_{[3,2,1]}}({{s}_{2}})$
|  |
$123\ 123\cdots$ |  | $inf{{o}_{[3,2,1]}}({{s}_{1}})>inf{{o}_{[3,2,1]}}({{s}_{2}})$
---
$inf{{o}_{[2,1,3]}}({{s}_{3}})>inf{{o}_{[2,1,3]}}({{s}_{1}})$
$inf{{o}_{[1,3,2]}}({{s}_{2}})>inf{{o}_{[1,3,2]}}({{s}_{3}})$
boundary | $inf{{o}_{[1,3,2]}}({{s}_{3}})=inf{{o}_{[1,3,2]}}({{s}_{2}})$
|  |
$3132\ 3132\cdots$ |  | $inf{{o}_{[2,4,1]}}({{s}_{2}})>inf{{o}_{[2,4,1]}}({{s}_{1}})$
---
$inf{{o}_{[1,3,2]}}({{s}_{3}})>inf{{o}_{[1,3,2]}}({{s}_{2}})$
$inf{{o}_{[4,2,1]}}({{s}_{1}})>inf{{o}_{[4,2,1]}}({{s}_{2}})$
$info_{[3,1,2]}({{s}_{3}})<info{{}_{[3,1,2]}}({{s}_{1}})$

Therefore, the node positions can be traversed in the positive direction from
the horizontal axis, and the vertical values can be traversed sequentially in
the case of fixed values of each horizontal axis, then the numerical solutions
of the following equations can be found respectively to provide the basis for
judging the optimal scheduling of different regions as follows:

$\left\\{\begin{split}&inf{{o}_{[\inf,2,1]}}({{s}_{2}})=inf{{o}_{[\inf,2,1]}}({{s}_{1}})\\\
&inf{{o}_{[3,2,1]}}({{s}_{1}})=inf{{o}_{[3,2,1]}}({{s}_{2}})\\\
&inf{{o}_{[1,3,2]}}({{s}_{3}})=inf{{o}_{[1,3,2]}}({{s}_{2}}).\\\
\end{split}\right.$ (59)

In Eq. (59), the first equation represents the boundary of whether node
$s_{1}$ will be activated in the long term; the second equation represents the
critical case with three nodes alternately activated and the proportion of
node $s_{1}$ less than $\frac{1}{3}$; the third equation represents the
critical case with the proportion of node $s_{3}$ as $\frac{1}{2}$, i.e., the
boundary layout with the periodic scheduling of $3132$ and three nodes
alternately activated in equal proportion.

The integral information expressions in the general triangular layout are
similar to the previous one, whose results can also be obtained by numerical
calculation using Eq. (58) in order to obtain the scheduling results and the
boundary distribution case. The details of the obtained results will be
expanded in Section V.

##  IV Long-term optimal mechanism

The core of the single-step decision mechanism is to precompute the amount of
information and activate the node that can obtain the most information
increment at the current moment. However, in terms of the long-term operation
of the system, the single-step decision mechanism only concerns the current
gain for each decision, and it does not consider the possible impact of the
current decision on the future information gain. Therefore, this section
focuses on analyzing the optimal scheduling strategy under different node
layouts when the average information acquisition for the long-term operation
is taken as the optimization goal.

The main scheme adopted in this section is to model the process as a Markov
decision process, taking the current information residual map of the system as
the state. The information gain obtained by the node activated at the current
moment is only related to the current information residual map, independent of
the residual information map at all previous moments. Since the currently
analyzed system contains three nodes and activates a node periodically, the
number of system states is finite. Then the Q-learning algorithm can be used
to eventually converge to the optimal scheduling result after a finite number
of training steps.

###  IV-A States, Actions, and Rewards

The state-space $State$ mainly records the current information residual map of
the system and can directly take the AoI of the latest sensed data of each
node as follows:

$State=AoI=\left[AoI\left({{s}_{1}}\right),AoI\left({{s}_{2}}\right),AoI\left({{s}_{3}}\right)\right].$
(60)

Since the decision time interval is fixed as $\Delta t$, the AoI can be
abbreviated only as its coefficient about $\Delta t$.

From the node spatio-temporal correlation and the previous analysis, it is
known that the system takes the same action continuously with poor information
gain. Thus, it can be stipulated by default that the system will not take the
same action twice and more consecutively, which can reduce the total number of
states and improve the training efficiency.

The survival time of data information of each node is limited for the reason
that it will always be covered by new data of adjacent nodes after a certain
time. Thus the AoI of each node data has a maximum, greater than which the
data contain no valuable information, and the AoI can be abbreviated as $inf$.
The maximum AoI of each node can be calculated as Eq. (62), where

${{K}_{{{s}_{i}},{{s}_{j}}}}=\left\lfloor\frac{{{d}_{{{s}_{i}},{{s}_{j}}}}}{\frac{{{\lambda}_{t}}}{{{\lambda}_{d}}}\cdot\Delta
t}\right\rfloor,$ (61)

and $i,\ j,\ k\in\\{1,2,3\\}$, $i\neq j\neq k$. In addition, $s_{j}$ and
$s_{k}$ represent the two nodes whose data AoI takes the values of 1 and 2,
respectively.

$AoI{{({{s}_{i}})}_{\max}}=\max\left\\{\min\left({{K}_{{{s}_{i}},{{s}_{j}}}}+AoI({{s}_{j}}),{{K}_{{{s}_{i}},{{s}_{k}}}}+AoI({{s}_{k}})\right)\Big{|}AoI({{s}_{j}}),AoI({{s}_{k}})\in\\{1,2\\},AoI({{s}_{j}})\neq
AoI({{s}_{k}})\right\\}$ (62)

The total number of states of the system is shown in Eq. (63),

$\begin{split}&N_{State}=\sum\limits_{i=1}^{3}{N_{State}\Big{|}_{AoI_{max}=AoI(s_{i})_{max}}}=\sum\limits_{i=1}^{3}\sum\limits_{m=1}^{AoI(s_{i})_{max}}{N_{State}\Big{|}_{AoI_{max}=AoI(s_{i})=m}}\\\
&=\sum\limits_{i=1}^{3}\left\\{1+\sum\limits_{m=2}^{AoI(s_{i})_{max}}\Bigg{[}u\Big{(}K_{s_{i},s_{j}}-(m-1)\Big{)}\cdot
u\Big{(}K_{s_{i},s_{k}}-(m-2)\Big{)}+u\Big{(}K_{s_{i},s_{j}}-(m-2)\Big{)}\cdot
u\Big{(}K_{s_{i},s_{k}}-(m-1)\Big{)}\Bigg{]}\right\\}\end{split}$ (63)

where $u(.)$ is a step function as follows:

$u(x)=\left\\{\begin{aligned} 1\quad x\geq 0\\\ 0\quad x<0\\\
\end{aligned}\right.$ (64)

The set of actions is

$Action=\\{s_{1},s_{2},s_{3}\\},$ (65)

where $s_{1}$, $s_{2}$ and $s_{3}$ represent the activation of the
corresponding nodes, respectively.

The state transfer process is as follows. Firstly, the system performs an
action each time and activates the corresponding node, and the AoI value of
the corresponding node data is set to 0. Secondly, the AoI value of all node
data plus 1 elapsed time ${\Delta t}$. Then determine whether the AoI value of
the node data exceeds the maximum value, and if it exceeds the maximum value,
it is recorded as $inf$.

The immediate reward for each action is the incremental amount of information
acquired, that is

$Reward=\frac{I_{gain}(State,s_{i})}{I_{gain}([inf,\ inf,\ inf],s_{1})},\
i\in\left\\{1,2,3\right\\},$ (66)

where the denominator is the information increment that the system can obtain
by activating the node for the first time, whose role is to normalize the
reward.

###  IV-B Q learning algorithm

The Q table in Q learning algorithm represents the quality of an action in a
particular state [35]. The output of the algorithm is the Q table that tends
to converge after finite training, based on which the system can know the
optimal decision action in all states.

The training process mainly adopts a greedy strategy, which means that the
agent mainly takes random actions to explore the environment in the initial
stage, and gradually increases the greedy coefficient as the number of
training steps increases, i.e., the agent tends to choose the action with a
larger Q value. The optimal long-term scheduling of the system is obtained by
waiting for the almost complete convergence of the Q-table. The process of
updating the Q value for each training is as follows[35]

$Q_{new}(s,a)\leftarrow
Q(s,a)+\alpha\left(R\left(s^{\prime}\right)+\gamma\max_{a^{\prime}}Q\left(s^{\prime},a^{\prime}\right)-Q(s,a)\right),$
(67)

where $s$ denotes the current state, $a$ denotes the action, and $Q(s,a)$ is
the previous Q value. Meanwhile, $\alpha$ is the learning rate, and $\gamma$
is the discout factor whose value determines the different importance attached
to the gain at all future moments. The $R$ is the immediate reward observed in
the new state $s^{\prime}$, and the
$max_{a^{\prime}}Q\left(s^{\prime},a^{\prime}\right)$ represents the estimate
of the optimal future reward from the next state $s^{\prime}$. The long-term
optimal scheduling results obtained by traversing different layouts are
developed specifically in the next section.

##  V Numerical and simulation results

###  V-A Scheduling results with single-step optimal mechanism

This subsection gives the scheduling results under single-step decision
mechanism and numerical approximation bounds from theoretical analysis.
Without loss of generality, the value of $\Delta t$ is taken as $1$ in this
section, and the variation of the time parameter is mainly reflected in
$\lambda_{t}$.

####  V-A1 Three-node isosceles triangle layout

When three nodes present an isosceles triangle layout and traversal step is 5
with parameters $\lambda_{d}=0.01$ and $\lambda_{t}=0.3$, the main scheduling
results are illustrated in Fig. 7(a),

(a) $\lambda_{d}=0.01$, $\lambda_{t}=0.3$.

(b) $\lambda_{d}=0.025$, $\lambda_{t}=0.5$.

Figure 7: Scheduling results of single step optimal decision mechanism with
three nodes presenting an isosceles triangle layout. (a) Scheduling results
with $\lambda_{d}=0.01$, $\lambda_{t}=0.3$. (b) Scheduling results with
$\lambda_{d}=0.025$, $\lambda_{t}=0.5$.

where the horizontal axis represents the values of the base of isosceles
triangle, and the vertical axis represents the height of isosceles triangle.
The scatter points of different shapes in the Fig. 7(a) represent different
type of scheduling situation, and their corresponding periodic scheduling
sequences are shown in the legend of the figure, respectively. The three
curves are the approximate numerical bounds on the critical scheduling case
obtained from the previous theoretical analysis, i.e., the critical $h$
corresponding to each $d$ is obtained by using the approximate solution
algorithm.

Specifically, the curve marked by circles represents the boundary node
distribution for the cycle scheduling sequence of $1213$ and $123$. The curve
marked by triangles represents the boundary node distribution for the cycle
scheduling sequence of 123 and node $s_{1}$ activation percentage less than
$\frac{1}{3}$, i.e., the scheduling result of $1232\ 1323\cdots$. In addition,
the curve marked with pentagrams represents the boundary node distribution of
whether node $s_{1}$ will be activated or not, in other words, it represents
the critical distribution of the scheduling sequence $23$ and $23\cdots 1$,
and the interval between two node $s_{1}$ activations is not greater than
$\left\lceil\frac{{{\lambda}_{d}\cdot{d}_{{{s}_{1}},{{s}_{2}}}}}{{{{{\lambda}_{t}}}}{{}}\cdot\Delta
t}\right\rceil=\left\lceil\frac{{{\lambda}_{d}\cdot\sqrt{h_{th}^{2}+d_{th}^{2}/4}}}{{{{{\lambda}_{t}}}}{{}}\cdot\Delta
t}\right\rceil$, where $h_{th}$ and $d_{th}$ correspond to the values taken on
the boundary curve.

It can be observed that the theoretical numerical approximation bounds match
the simulation, and the scheduling results of the simulation are consistent
with the previous theoretical analysis. In addition, when traversal step is 4
with $\lambda_{d}=0.025$ and $\lambda_{t}=0.5$, the result is shown in the
Fig. 7(b).

####  V-A2 Three-node general triangular layout

This subsection mainly presents the theoretical analysis and simulation
results when the three nodes exhibit a general triangular layout. When
$\lambda_{d}$ is 0.01 and $\lambda_{t}$ is 0.3, the main scheduling results
with the maximum distance between nodes $d_{s_{2},\ s_{3}}=280$ and
$d_{s_{2},\ s_{3}}=450$ are respectively illustrated in Fig. 8(a) and Fig.
8(b), where the curves are the boundary cases between different scheduling
cases obtained by numerical approximation according to theoretical analysis.
Specifically, curve bound A marked by pentagrams corresponds to the first
equation in Eq. (59), meanwhile, curve bound B marked by triangles corresponds
to the second equation, and curve bound C marked by circles corresponds to the
third equation. According to the discriminant conditions in Table III, the
scheduling situation in different regions can be judged, which coincides with
the simulation results.

(a) $d_{s_{2},\ s_{3}}=220$.

(b) $d_{s_{2},\ s_{3}}=330$.

Figure 8: Scheduling results of single step optimal decision mechanism with
three nodes presenting a general triangle layout and parameters
$\lambda_{d}=0.01$ and $\lambda_{t}=0.3$. (a) Scheduling results with the
maximum distance between nodes $d_{s_{2},\ s_{3}}=220$. (b) Scheduling results
with the maximum distance $d_{s_{2},\ s_{3}}=330$.

###  V-B Scheduling results with long-term optimal mechanism

This subsection mainly presents the long-term optimal scheduling results
obtained by modeling the system scheduling as a markov decision process. In
order to facilitate the comparison with the single-step decision results, some
parameter values is the same as the previous single-step decision results. In
addition, the learning rate $\alpha$ is 0.1, and the discount factor $\gamma$
is 0.9 in the Q-learning algorithm.

When three nodes present an isosceles triangular layout with parameters
$\lambda_{d}=0.01$ and $\lambda_{t}=0.3$, the final obtained scheduling
results are shown in Fig. 9(a), where the optimal cycle scheduling sequence
for each layout is shown in the legend. Moreover, the scheduling results are
shown in Fig. 9(b) with parameters $\lambda_{d}=0.025$, $\lambda_{t}=0.5$.

Meanwhile, the scheduling results are shown in Fig. 10(a) when three nodes
present a general triangular layout with parameters $\lambda_{d}=0.01$,
$\lambda_{t}=0.3$, and the longest node distance $d_{s_{2},\ s_{3}}=280$. In
addition, the the scheduling results are shown in Fig. 10(b) with the longest
node distance $d{s_{2},\ s_{3}}=450$. The comparison of the scheduling results
of the two mechanisms is developed in the next subsection.

(a) $\lambda_{d}=0.01$, $\lambda_{t}=0.3$.

(b) $\lambda_{d}=0.025$, $\lambda_{t}=0.5$.

Figure 9: Scheduling results of long-term average information acquisition
optimal decision mechanism with three nodes presenting an isosceles triangle
layout. (a) Scheduling results with $\lambda_{d}=0.01$, $\lambda_{t}=0.3$. (b)
Scheduling results with $\lambda_{d}=0.025$, $\lambda_{t}=0.5$.

(a) $d_{s_{2},\ s_{3}}=220$.

(b) $d_{s_{2},\ s_{3}}=330$.

Figure 10: Scheduling results of long-term optimal decision mechanism with
three nodes presenting a general triangle layout and parameters
$\lambda_{d}=0.01$, $\lambda_{t}=0.3$. (a) Scheduling results with the maximum
distance $d_{s_{2},\ s_{3}}=220$. (b) Scheduling results with the maximum
distance $d_{s_{2},\ s_{3}}=330$.

###  V-C Performance comparison of two mechanisms

This subsection analyzes the different performance of the scheduling results
obtained by two mechanisms. The scheduling results given in the previous
subsection show that the results with the single-step decision mechanism and
the final convergence results with the long-term optimization are the same in
quite a few regions, which can be observed specifically in combination with
Fig. 7 and Fig. 9, or Fig. 8 and Fig. 10. In other words, the results of the
single-step optimal decision are also the results that meet the long-term mean
optimum in these regions. However, there are some regions where the results of
the two mechanisms are not the same, that is, the results of the single-step
optimal decision in these regions are not the results meet the long-term mean
optimum, which indicates that each selection of the local optimum in these
regions cannot lead to the long-term optimum. Therefore, in the long-term
decision-making process, it is necessary to select a specific moment to select
the second-best decision, which may be able to bring more future benefits.
Meanwhile, the fluctuation of the incremental information obtained under the
long-term mean-optimal mechanism should be slightly larger than the
incremental information obtained under the single-step optimal decision. In
other words, although the mean value of the incremental information under the
long-term mean-optimal mechanism is larger than that of the single-step
decision mechanism, the instability of the information obtained under the
long-term mean-optimal mechanism, i.e., the standard deviation, is also
slightly larger than that of the single-step decision.

In the following, the above analysis is verified by the specific different
simulation results obtained. When the three nodes show isosceles triangular
distribution and traversal step is 5 with $\lambda_{d}=0.01$,
$\lambda_{t}=0.3$, the highest mean value of information increment obtained
with the long-term mechanism is about $2.5\%$ higher than that with the
single-step mechanism. The mean value of information obtained under the long-
term mechanism is about 0.8% higher than that of the single-step mechanism in
the term of node layouts with different results obtained by the two
mechanisms. In addition, as for the inclusion of all layout cases, the average
value of the long-term mechanism information is slightly higher by about 0.1%.
The standard deviation of the incremental information acquisition with the
long-term mechanism is also higher, up to about 300% higher than that of the
single-step, and it is on average about 60% higher in the situation of node
layouts for which the two mechanisms obtain different results.

Similarly, when the three nodes are distributed in a general triangle and
traversal step length is 4 with $d_{s_{2},\ s_{3}}=220,\lambda_{d}=0.01$, and
$\lambda_{t}=0.3$, the mean value of information increment obtained by the
long-term mechanism is up to 2.1% higher than that of the single-step
mechanism, which is about 0.6% higher than that of the single-step for the
situation of node layouts with different results obtained by the two
mechanisms. As for the inclusion of all layout cases, the average value of the
long-term mechanism information is slightly higher by about 0.1%. On the other
hand, the standard deviation of information acquisition for the long-term
mechanism is up to about 200% higher than that of the single-step mechanism,
and it is on average about 80% higher in the node layout cases with different
scheduling results for the two mechanisms. In summary, the information
acquisition performance of the two scheduling results is consistent with the
analysis that the information mean value of the long-range mechanism is
slightly larger, but at the same time the standard deviation of information
acquisition is also slightly larger.

##  VI Experimental evaluation

In this section, the relative humidity data in the homogeneous grid dataset of
China land surface observation[36]-[40] are used to evaluate our proposed
scheduling mechanism. In the experiments, it is assumed that the grid data are
identical to the corresponding random variables in the real world. In other
words, the data are considered entirely accurate. In the first part of this
section, the data are analyzed to extract the scaling parameters of the
covariance model described earlier. In the second part, four scheduling
methods are compared in terms of their two-dimensional sense performance based
on the extracted covariance scaling parameters in conjunction with the
spatiotemporal scope information model. In the third part, a summary and
discussion are presented, illustrating the model’s performance and
limitations.

###  VI-A Data Analysis

The data set is dense grid data of relative humidity for China in recent
decades with a temporal resolution of one month. In order to fully reflect the
stochastic variation among the data, a down-sampling operation with a step
size of 3 is performed on the grid data, then a circular grid area data with a
radius of 10 is selected to compare the global sensing performance of
different scheduling methods for the three nodes in different layouts.

The Pearson correlation coefficient formula is used to calculate the spatio-
temporal correlation of the grid data in the dataset. Since the modeling
mentioned above in this paper adopts a spatio-temporal separable covariance
function, the spatio-temporal correlation parameters can be extracted
separately in the experiment. Specifically, firstly, the correlation
coefficients of the data about the spatial distance between the same moments
of grid data are calculated, and then the distance correlation parameter
$\lambda_{d}$ can be fitted. Secondly, the time correlation between different
moments of the same grid data is calculated, and the time correlation
parameter $\lambda_{t}$ can be fitted. The joint spatio-temporal correlation
between the data is then verified by calculating the correlation coefficient
when both the distance and time difference between different grid data are not
zero, and comparing it with the fitting results to verify whether the
correlation in the data conforms to the assumption of separable covariance
function. If the fitting results deviate from the actual results, it will
affect the performance of the subsequent scheduling. It is also clear from
this that the accuracy of the parameter fitting is one of the possible
limitations of the scope information model.

###  VI-B Performance comparison

The specific implementation steps of the experiment are as follows: within the
circular region mesh data, an equilateral triangle of suitable size is
selected, and two of the vertices are set as the location of the node, while
the traversal region of the other node, remains as shown in Fig. 4(b). That
is, the position of the longest edge of the node layout is determined, and the
feasible region of the third point is traversed to compare the performance of
the four different scheduling methods.

The specific metric of the experiment is the mean absolute error over the
entire two-dimensional circular grid area. The effective coverage area of each
node can be determined by the previous analysis, combined with the data
spatiotemporal distance. Within the coverage area of the respective node, the
data at the node position represent all the data. The four scheduling methods
include the single-step and long-term mechanisms previously proposed in this
paper, and the other two are the ideal scheduling and uniform alternation,
respectively. The ideal scheduling method is a system that has all the grid
data in real time and makes the scheduling decision that brings the minimum
total scope error. The uniform alternation method is a strategy that activates
nodes in equal proportional alternation regardless of the layout of the three
nodes.

In the experiments, the time span of each evaluation is nine years, which are
1982-1990,1991-1999,2000-2008,2009-2017, respectively. Moreover, the model
parameters were extracted mainly using the data from the first three years
used in each experiment. Then the extracted parameters are used for the entire
period of the experiment.

The experimental results are shown in Fig. 11. Obviously, the ideal scheduling
method has the best performance with the minimum scope mean absolute error.
Furthermore, the uniform alternation method has the maximum mean absolute
error, about 23% higher than the ideal scheduling method. The performance of
the single-step and long-term mechanisms lies between the above two, and the
mean absolute error of both methods is about 21% higher than that of the ideal
scheduling method. Since the single-step and long-term mechanisms get the same
scheduling results in most node layout situations, and the global average
difference in the information mean of the scheduling results obtained by the
two mechanisms is less than 1% in simulation, the global performance mean
difference caused by the two methods in the experiment is also minimal, as
shown in Fig. 11. In other words, the mean absolute error of the long-term
mechanism is only slightly higher than that of the single-step mechanism.

Figure 11:

If the reasonableness of node layout within the information sense region is
considered when traversing node locations, it is not very reasonable to
uniformly examine the total scope error in the circular region. Since if there
are many areas where nodes have difficulty sensing effective information, it
may lead to more fluctuations in the total scope error. For example, when two
nodes are very close to each other, the effective coverage of three nodes may
converge to an ellipse. For this consideration, in the experiments, when
traversing different node layouts, circles of appropriate size are
respectively selected with each node position as the center of the circle,
then the joint coverage of the three circles is set as the data evaluation
scope for examining the model performance.

Under the above experimental operation, the performance comparison of the four
scheduling methods is obtained as shown in Fig. 12. Samely, the ideal
scheduling method has the minimum mean absolute error, the single-step
decision and the long-term mechanism have about 19% higher mean error than the
ideal scheduling, and the performance difference between them is very little.
The error of the uniform alternation scheduling method is about 28% higher
than that of the ideal scheduling method.

Figure 12:

###  VI-C Summary and discussion

From the above experiments, the two proposed scheduling mechanisms, single-
step decision and long-term mechanisms, basically meet the expectations, and
the global average performance of long-term mechanisms is slightly higher than
that of single-step mechanisms. The maximum program complexity of single-step
mechanisms is $O(3n)$, where $n$ represents the total number of AoI states of
the system. Moreover, the complexity of the algorithm for long-term mechanisms
depends mainly on the speed of convergence, but in general, its complexity is
higher than that of single-step mechanisms.

In addition, it is known experimentally that the correctness of the
correlation parameter fit is the primary factor in determining the model
performance. In the case of stochastic processes, the correlation model may
not be stationary, i.e., the correlation coefficients may be time-varying. In
addition, whether the adopted covariance function model fits the distribution
of the data set is also an important factor in determining the model
performance. In future research, time-varying correlation models and multi-
node cooperative perception situations under different covariance function
models may be major research interests.

##  VII Conclusion

In this paper, a Spatio-temporal Scope Information Model (SSIM) is developed
to quantify the valuable information of sensor data, which decays with space
and time. Periodically activating one node to obtain sensing data, a sensor
monitoring system containing three sensor nodes is considered. For more
efficient access to information, two optimal scheduling mechanisms are
proposed. One is the single-step optimal decision mechanism, and the
approximate numerical bounds for the node layout between partial scheduling
results are obtained by theoretical analysis and numerical calculation, which
coincide with the simulation results. The other one is the long-term optimal
mechanism, which is modeled as a Markov decision process. The optimal
scheduling results for long-term information with different node layouts are
obtained using the Q-learning algorithm. Through simulation and experiments,
the scheduling results of both mechanisms are the same in many node layout
cases. In few node layouts, the mean value of the incremental information
obtained by the long-term mechanism is higher than that of the single-step
mechanism, but the standard deviation of the incremental information obtained
by the long-term mechanism is also higher than that of the single-step
mechanism. The average performance of the two mechanisms is similar under all
node layouts, and the long-term mechanism is slightly higher.

In future work, we may focus on spatio-temporal scope information models with
time-varying and different covariance functions, and study the cooperative
scheduling of multiple nodes to improve energy efficiency and extend lifetime.

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