Average Peak Age of Information in Underwater Information Collection With Sleep-Scheduling

Abstract

We investigate the peak age of information (PAoI) in underwater wireless sensor networks (UWSNs), where Internet of underwater things (IoUT) nodes transmit the latest packets to the sink node, which is in charge of adjusting the sleep-scheduling to match network demands. In order to reduce PAoI, we propose active queue management (AQM) policy of the IoUT node, beneficially compresses the packets having large waiting time. Moreover, we deduce the closed-form expressions of the average PAoI as well as the energy cost relying on probability generating function and matrix-geometric solutions. Numerical results verify that the IoUT node relying on the AQM policy has a lower PAoI and energy cost in comparison to those using non-AQM policy. Index Terms-Age of information, underwater wireless sensor networks, active queue management, multiple vacation queueing model.

Full text

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 1
Average Peak Age of Information in Underwater
Information Collection with Sleep-scheduling
Zhengru Fang, Student Member, IEEE, Jingjing Wang, Senior Member, IEEE, Chunxiao Jiang, Senior Member,
IEEE, Xijun Wang, Member, IEEE, and Yong Ren, Senior Member, IEEE
Abstract—We investigate the peak age of information (PAoI) in
underwater wireless sensor networks (UWSNs), where Internet
of underwater things (IoUT) nodes transmit the latest packets
to the sink node, which is in charge of adjusting the sleep-
scheduling to match network demands. In order to reduce PAoI,
we propose active queue management (AQM) policy of the IoUT
node, beneficially compresses the packets having large waiting
time. Moreover, we deduce the closed-form expressions of the
average PAoI as well as the energy cost relying on probability
generating function and matrix-geometric solutions. Numerical
results verify that the IoUT node relying on the AQM policy
has a lower PAoI and energy cost in comparison to those using
non-AQM policy.
Index Terms—Age of information, underwater wireless sensor
networks, active queue management, multiple vacation queueing
model.
I. INTRODUCTION
WITH the emergence of Internet of underwater things
(IoUT) in both civilian and military applications [1],
[2], the metric of data timeliness has drawn substantially grow-
ing attention, since the timely update of sensing data is critical
in monitoring systems, e.g., underwater target tracking, real-
time sensing of currents and platforms, etc [3]. Therefore, age
of information (AoI) is proposed to benchmark the timeliness
of data in the face of the quality of experience (QoE), which
is defined as the time elapsed since the last received status
update packet is generated [4]. Besides, peak AoI (PAoI) can
be utilized to evaluate a worse case of AoI, i.e., the maximum
value of the age achieved before the latest update [5].
A range of studies have been carried out to minimise the AoI
relying on queueing theory. Specifically, in [6], the average
AoI and PAoI of different queueing models were investigated
This work was partly supported by National Natural Science Foundation of
China (Grant No. 62071268 and 62127801), partly supported by the Young
Elite Scientist Sponsorship Program by CAST (Grant No. 2020QNRC001),
partly supported by Guangdong Basic and Applied Basic Research Foundation
under grant 2021A1515012631, partly supported by the National Key R&D
Program of China under Grant 2020YFD0901000. (Corresponding author:
Jingjing Wang.)
Z. Fang and Y. Ren are with the Department of Electronic Engineering,
Tsinghua University, Beijing, 100084, China. Y. Ren is also with Network and
Communication Research Center, Peng Cheng Laboratory, Shenzhen 518055,
China. E-mail: fangzhengru@gmail.com, reny@tsinghua.edu.cn.
J. Wang is with the School of Cyber Science and Technology, Beihang
University, Beijing 100191, China. Email: drwangjj@buaa.edu.cn.
C. Jiang is with the Tsinghua Space Center, Tsinghua University, Beijing,
100084, China. E-mail: jchx@tsinghua.edu.cn.
X. Wang is with the School of Electronics and Information Technol-
ogy, Sun Yat-sen University, Guangzhou 510006, China. E-mail: wangxi-
jun@mail.sysu.edu.cn.
by considering GI/GI/1, M/G/1 and GI/M/1 queueing models1.
In [7], Asvadi et al. studied the impact on PAoI of queue-
ing models imposed by blocking and preemptive strategies,
respectively. However, it is quite difficult to achieve timely
underwater information collection without adaptive sampling
and energy scheduling considering the hostile environment of
underwater wireless sensor networks (UWSNs). Given that
underwater sensors are battery-powered and costly to recharge,
sleep-scheduling based on the vacation queueing model is
necessary to apply to underwater sensors. As for AoI perfor-
mance in UWSNs, Fang et al. in [3] proposed an autonomous
underwater vehicle assisted underwater information collection
scheme based on a limited-service M/G/1 vacation queueing
system without work-sleep scheduling. In [8], Muhammad et
al. proposed a traversal algorithm for underwater trajectory
scheduling to improve the timeliness of data collected.
In this letter, we conduct an active queue management
(AQM) scheme for IoUT information collection [9] to discard
the low-timeliness packets relying on a multiple vacation
M/M/1 queueing model. Furthermore, we derive the closed-
form expressions of average PAoI in the context of both
having AQM and non-AQM strategies in a first-in, first-
out (FIFO) manner and infinite queue buffer, followed by a
thorough analysis of IoUT node’s energy cost. To the best of
our knowledge, this is the first work that analyzes the PAoI
performance of the multiple vacation queueing models.
The rest of this letter is organized as follows. Section
II describes the network scenario, AoI metric and energy
cost modelling. Then, the closed-form solutions of PAoI with
different queue schedule policies are derived in Section III and
IV, respectively. Numerical results are illustrated in Section V
and Section VI concludes this letter.
II. NE TWORK SCENAR IO A ND U ND ERWATER ACOUSTIC
CHANNEL MODEL
A. Network Scenario
Without loss of generality, we consider a simple underwater
environmental monitoring system consisting of 𝑁IoUT nodes
and one sink node as shown in Fig. 1, where the IoUT
nodes send the latest packets to the sink node by acoustic
communication units. The packets are generated and stored
at the IoUT nodes’ queues in a FIFO manner. Each packet
1Typically a queue can be described as three variables, i.e., A/S/K, where
A and S denote the arrival and service process, respectively. K represents the
number of servers. GI and G mean the general distribution, and M means that
the interval of arrivals and service times yields exponential distribution.

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 2
IoUT node
... Idle
AQM
Active
Sink node
Underwater acoustic channel
...
PAoI in the sink node
Fig. 1. Sleep-scheduling aided underwater information collection.
contains underwater environmental information and a times-
tamp recording the packet generation time. The generation of
each packet follows a Poisson process with the arrival rate of
𝜆. Besides, the service time of IoUT nodes is exponentially
distributed with the rate 𝜇. For the sake of both reducing AoI
and saving energy, we assume that lossy packets are discarded
without retransmission. Due to the limitation of batteries, it
is impossible to keep uninterrupted transmission or follow a
simple transmission mechanism, which may lead to excessive
waste of energy. Therefore, the IoUT nodes should determine
different modes, i.e., the active mode and the idle mode,
according to dynamic workload state. Specifically, whenever
the queue buffer of the IoUT node becomes empty, it switches
to the idle mode. Moreover, in this mode, the IoUT node
turns off the acoustic transmitter except in AQM procedure,
while it still keeps data collection. If the queue keeps empty
at the end of one idle period, the system continues to enter
into another idle period, where the duration of one idle period
is exponentially distributed with 𝜃. Otherwise, the IoUT node
switches to the active mode and transmits packets again. This
sleep-scheduling based queueing system is modeled as the
multiple vacation mechanism [10].
In the above scenario, we use a specific AQM for reducing
average PAoI as well as network congestion in the idle mode.
By using lossy techniques, the AQM is designed to compress
and transmit the packets which have waited for a long time in
idle mode. Specifically, lossy compression technique reduces
bits by removing unnecessary or less important information.
The launch of AQM obeys a Poisson process with the rate of
𝛾. It is noted that IoUT node does not process compression if
queue is empty. In the procedure of AQM, it processes packet
in the head of the queue one after another, and compresses
packet with a probability of 𝛼or keeps it with a probability of
𝛽=1−𝛼, which is modeled as geometric abandonments [11].
Moreover, the AQM stops when the first packet is kept, or all
packets are compressed and transmitted. After processing data
compression, the compression ratio of data up to 55%-98%,
where it saves energy cost up to 88%-97%. Thereby, relying
on AQM policy, we ignore the energy cost of transmission
during idle mode [12]. Considering the limited computational
resource of IoUT node, we model the energy consumption of
data compression in Section III.
B. The general definition of PAoI
In the considered scenario, we use PAoI to indicate the
worse case of the packet AoI, which is influenced by packet
generation rate, IoUT node transmit power and underwater
channel delay. Furthermore, each IoUT node delivers packets
by frequency-division multiple access (FDMA), and the sink
node stores each packet in different storage locations according
to its associated IoUT node. For simplicity, therefore, we only
consider the 𝑖-th IoUT node’s PAoI in the following section.
Let 𝛼𝑛and 𝛽𝑛denote the packet generation time in the 𝑖-th
IoUT node and received time in the sink node, respectively. 𝑛
denotes the index of the status update. According to [6], the
𝑛-th average PAoI 𝐴𝑝
𝑛is given by:
E𝐴𝑝
𝑛=E[𝛽𝑛−𝛼𝑛−1]=E[𝐺𝑛]+E[𝑆𝑛],(1)
where 𝐺𝑛=𝛼𝑛−𝛼𝑛−1denotes the interval between the (𝑛−1)-
th and the 𝑛-th generated packets in the IoUT node. 𝑆𝑛=𝐷𝑛+
𝜚𝑛represents the sum of system delay 𝐷𝑛and propagation
latency 𝜚𝑛. As the packet transmission for each IoUT node is
independent of each other, the subscript “𝑛” of the notations
is omitted.
C. Underwater acoustic channel model
In this subsection, we analyze the network capacity of un-
derwater acoustic communication channel in terms of transmit
power, carrier frequency and aquatic environmental factors.
Furthermore, the propagation latency 𝜚of the underwater
channel can be modeled as distance function. The attenuation
of the channel over a distance 𝑙for the subchannel carrier
frequency 𝑓is formulated as:
10 log [𝐴(𝑙 , 𝑓 )/𝐴0]=𝑘·10 log 𝑙+𝑙·log 𝑎(𝑓),(2)
where 𝐴0denotes a unit-normalizing constant, 𝑘represents the
spreading factor, and 𝑎(𝑓)is the absorption coefficient. The
first term of (2) denotes the spreading loss, while the second
term represents the absorption loss. By using Thorp’s model
in [3], the absorption coefficient in dB/km for several kilohertz
is formulated as:
10 log 𝑎(𝑓)=0.11 𝑓2
1+𝑓2+44 𝑓2
4100 +𝑓2+2.75 ·10−4𝑓2+0.003.(3)
The ambient noise of underwater acoustic channel 𝑁(𝑓)can
be formulated as the sum of four sources’ power spectral
density, i.e. turbulence 𝑁t(𝑓), shipping 𝑁s(𝑓), waves 𝑁w(𝑓)
and thermal noise 𝑁th (𝑓). Therefore, the signal-to-noise ratio
(SNR) for the IoUT node considered can be given by:
𝐶(𝑙, 𝑓 )=𝐵log21+𝜂 𝑃𝑡 𝑟 𝜁(𝑙, 𝑓 )
2𝜋𝐻 · (1𝜇Pa) · 𝐵,(4)
where 𝜂is the overall efficiency of circuit, the channel
attenuation coefficient is 𝜁(𝑙, 𝑓 )=[𝐴(𝑙 , 𝑓 )𝑁(𝑓)]−1, and 𝑃𝑡 𝑟
denotes the transmit power. 𝐻and 𝐵indicate the depth and
sub-bandwidth of the IoUT node, respectively. Since the sink
node only transmits packets at the active mode, we define E[E]
as the average energy cost of the active mode as follows:
E[E] =𝑝𝑎𝑃𝑡𝑟 𝑇𝑡,(5)

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 3
1,1 2,1
1,0 2,0
0,0
…
…
Active Mode
Idle Mode
Fig. 2. The two-dimensional Markov chain of the AQM policy.
where 𝑝𝑎denotes the probability of the active mode in the
IoUT node. 𝑇𝑡is defined as the average time from the existing
packets being generated to departure from the IoUT node
during the active mode. Additionally, the propagation latency
of acoustic channel 𝜚is formulated as 𝜚=𝑙
𝑉𝑠, where 𝑉𝑠
denotes the velocity of sound in water. For obtaining the
closed-form expression of the average PAoI in Eq. (1), we
derive the expectations of the interval of packet generation
E[𝐺𝑛]and the system delay E[𝐷𝑛]in the following section.
III. PAOIOF M ULTIPLE VACATI ON QUEUE WITH AQM
In this section, we derive the average PAoI of multiple
vacation queue relying on the AQM policy. According to the
aforementioned assumptions, a continuous-time Markov chain
models the considered system states as {𝑁(𝑡), 𝐽 (𝑡)}, which
has a state space 𝛩={(0,0)} ∪{(𝑘 , 𝑗):𝑘>1, 𝑗 =0,1}.
Let 𝑁(𝑡)and 𝐽(𝑡)denote the amount of packets and the
mode of the IoUT node, respectively. 𝐽(𝑡)=1represents the
active mode, while 𝐽(𝑡)=0denotes the idle mode. Then, we
utilize 𝑸𝐴to describe the instantaneous rate for the Markov
chain state transition, and its element 𝑞(𝑖, 𝑗 ),(𝑘, 𝑧)denotes the
departing rate from state (𝑖, 𝑗)to (𝑘, 𝑧). The diagram for the
two-dimensional Markov chain based on vacation queueing
model with AQM policy is portrayed in Fig. 2. Let 𝑓𝑖, 𝑗
denote the transition rate caused by AQM at idle mode. When
𝑖>1, 𝑗 =0, we have 𝑓𝑖 , 𝑗 =𝛾𝛼𝑖+1, while 𝑖 > 𝑗 > 0, we have
𝑓𝑖, 𝑗 =𝛾𝛼𝑖−𝑗𝛽, otherwise, 𝑓𝑖, 𝑗 =0. Therefore, the transition
rate matrix 𝑸𝐴can be formulated as:
𝑸𝐴=©«
𝑩0𝑪0
𝑩1𝑨1𝑨0
𝑩2𝑨2𝑨1𝑨0
𝑩2𝑨3𝑨2𝑨1𝑨0
.
.
..
.
..
.
..
.
..
.
....
ª®®®®®®¬
(6)
where 𝑨0=diag(𝜆, 𝜆 ),𝑪0=(0, 𝜆),𝑩0=(𝜇, 𝛾𝛼)Tand 𝑩𝑖=
0, 𝛾𝛼𝑖+1Twith 𝑖>1. Additionally, 𝑨1=−(𝜇+𝜆)0
𝜃−(𝛾 𝛼+𝜃+𝜆),
𝑨2=diag(𝜇, 𝛾 𝛼𝛽)and 𝑨𝑖=diag(0, 𝛾𝛼i+1𝛽)with 𝑖>3.
Because we only consider the behavior of the stable queue,
the workload of queue yields 𝜌=𝜆/𝜇 < 1for satisfying
the balance condition (Theorem 1.7.1 in [13]). Then, the
equilibrium distribution of the queueing system is defined
as 𝝅=(𝝅0,𝝅1,···), where 𝝅𝑖=𝜋𝑖, 1, 𝜋𝑖,0,(𝑖≥0).
The balancing equation is 𝝅𝑸 𝐴=0and the normalization
equation is 𝝅𝒆 =1, in which 𝒆=(1,1,· · ·)T. Moreover,
the partial PGFs of the active mode and the idle mode are
𝛷𝑎(𝑧)=Í∞
𝑛=0𝜋𝑛,0𝑧𝑛and 𝛷𝑖(𝑧)=Í∞
𝑛=1𝜋𝑛,1𝑧𝑛, respectively.
Substituting 𝑸𝐴into the balancing equation, we can obtain
the equilibrium distribution:
For active mode:
𝜋𝑛,1=









0, 𝑛 =0,
𝜃 𝜋1,0+𝜇 𝜋2,1
𝜇+𝜆, 𝑛 =1,
𝜃 𝜋𝑛,0+𝜆 𝜋𝑛−1,0+𝜇 𝜋𝑛+1,1
𝜇+𝜆, 𝑛 >2.
(7)
For idle mode:
𝜋𝑛,0=



𝜇 𝜋1,1+𝛾Í∞
𝑖=0𝛼𝑖𝜋𝑖,0
𝜆+𝛾, 𝑛 =0,
𝜆𝜋𝑛−1,0+𝛾𝛽 Í∞
𝑖=𝑛𝛼𝑖−𝑛𝜋𝑖,0
𝜃+𝜆+𝛾, 𝑛 >1.
(8)
Multiplying both sides of 𝜋𝑛,0by 𝑧𝑛and summing for all
𝑛≥1, we can get the PGF for the idle mode:
𝛷𝑖(𝑧)=(𝜃+𝜆+𝛾) (𝑧−𝛼)𝜋0,0−𝛾 𝛽𝑧𝛷𝑎(𝛼)
(𝑧−𝛼) (𝜃+𝜆+𝛾−𝜆𝑧)−𝛾 𝛽𝑧 ,(9)
where 𝑧≠𝛼. Substituting 𝜋0,0into (9), we get 𝛷𝑖(𝛼)=
𝛾−1(𝜆+𝛾)𝜋0,0−𝜇𝜋1,1. Then, let 𝛬1(𝑧)and 𝛬2(𝑧)be
the numerator and the denominator of 𝛷𝑖(𝑧), respectively.
Moreover, let 𝑧0be the root of 𝛬1(𝑧). Let 𝑧1and 𝑧2be
the roots of 𝛬2(𝑧). Since 𝛬2(0)<0and 𝛬2(1)>0, we
have 0< 𝑧1<1< 𝑧2. If the queue system is stable,
𝛷𝑖(𝑧)is convergent with |𝑧|61. Therefore, according to
Rouché’s theorem [13], we have 𝑧0=𝑧1,𝛬1(𝑧1)=0
and 𝛷𝑖(𝑧)=𝑧2
𝑧2−𝑧𝜋0,0, where 𝑧1(𝑧2)=−𝛾 𝛽+𝜃+𝜆+𝛾+𝛼𝜆−(+)√𝛥
2𝜆
and 𝛥=[(𝜃+𝜆+𝛾)+𝛼𝜆 −𝛾 𝛽]2−4𝛼𝜆 (𝜃+𝜆+𝛾). Ac-
cording to 𝛬1(𝑧1)=0and 𝑧1𝑧2=𝜆−1𝛼(𝜆+𝜃+𝛾)and
𝑧1+𝑧2=𝜆−1(𝜆+𝜆𝛼 +𝜃+𝛾𝛼), we can derive that 𝜋1,1=
(𝛽𝜇)−1[𝜆𝑧2−(𝜃+𝛼𝜆 +𝛼𝛾)𝜋0,0]. Likewise, we can derive
𝛷𝑎(𝑧)=𝑧 𝜃𝛷𝑖(𝑧)−𝑧[𝜃 𝜋0,0+𝜇 𝜋1,1]
(𝜆𝑧−𝜇) (1−𝑧)by using the same steps above.
Substituting 𝜋1,1and 𝛷𝑖(𝑧)back to 𝛷𝑎(𝑧), the PGF for the
active mode can be expressed as:
𝛷𝑎(𝑧)=𝑧𝜋0,0𝑧2[𝜆(1−𝑧1)𝑧−(𝜆 𝑧2−𝜆𝑧1𝑧2−𝜃 𝛽)]
𝛽(𝑧2−𝑧) (𝜆𝑧 −𝜇) (1−𝑧).(10)
According to Rouché’s theorem [13], the numerator of 𝛷𝑎(𝑧)
has a root 𝑧=1. Hence, we can obtain 𝛷𝑎(𝑧)=𝑧 𝜋0,0𝑧2𝜆(1−𝑧1)
𝛽(𝑧2−𝑧)(𝜇−𝜆𝑧).
Furthermore, the PGF of the equilibrium distribution 𝝅can
be given by 𝛷(𝑧)=𝛷𝑎(𝑧)+𝛷𝑖(𝑧). Thus, we obtain the
probabilities of active mode 𝑝𝑎=𝛷𝑎(1)and of idle mode
𝑝𝑖=𝛷𝑖(1), respectively. Relying on the normalization equa-
tion 𝛷(1)=𝝅𝒆 =1, the steady-state probability 𝜋0,0is
formulated as:
𝜋0,0=𝛽(𝑧2−1) (𝜇−𝜆)
𝑧2[𝛽(𝜇−𝜆)+𝜆(1−𝑧1)] .(11)
Taking derivative and substituting 𝑧=1, the average queue
length with AQM is derived as:
E[𝐿𝐴]=𝛷0
𝑖(1)+𝛷0
𝑎(1).(12)
After some algebra, the average queue length in the IoUT node
relying on the AQM policy is formulated as:
E[𝐿𝐴]=𝑧2𝜋0,0
(1−𝑧2)2+𝜌(1−𝑧1)𝑧2𝜋0,0(𝑧2−𝜌)
𝛽(𝑧2−𝜌𝑧2−1+𝜌)2.(13)
The packets arriving at the IoUT node follows the Poisson
process with the arrival rate of 𝜆. Therefore, the average inter-
arrival time of packets is E[𝐺]=𝜆−1. According to the Little’s

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 4
law [10], E[𝐷𝐴]=𝜆−1E[𝐿𝐴]. Assuming the stationary and
i.i.d transmission process, we have E[𝐺𝑛]=E[𝐺]and
E[𝑆𝑛]=E[𝐷𝐴]+𝜚, where 𝜚denotes the propagation latency
delay in underwater acoustic channel. According to Eq. (1) and
(13), the average PAoI of the packet generated in the receiver
of the sink node yields:
Eh𝐴𝑝
𝐴i=E[𝐺]+E[𝐷𝐴]+𝜚
=1
𝜆"1+𝑧2𝜋0,0
(1−𝑧2)2+𝜌(1−𝑧1)𝑧2𝜋0,0(𝑧2−𝜌)
𝛽(𝑧2−𝜌𝑧2−1+𝜌)2#+𝜚.
(14)
The energy cost is formulated as the average power con-
sumption during the unit time 𝑇𝑡in the associated IoUT
node. Relying on Eqs. (5) and (10), the energy cost of data
transmission is obtained as follows:
E𝑡𝑟
𝐴=𝑝𝑎𝑃𝑡𝑟 𝑇𝑡=𝑧𝜋0,0𝑧2𝜆(1−𝑧1)
𝛽(𝑧2−1) (𝜇−𝜆)𝑃𝑡𝑟 𝑇𝑡.(15)
While the energy consumption for data compression by the
IoUT node is given by
E𝑐
𝐴=(1−𝑝𝑎)𝜅[𝜌𝑐𝜆𝐿 ]3𝑇𝑡,(16)
where 𝜅and 𝜌𝑐denote the effective switched capacitance and
the processing density, respectively. Relying on AQM policy,
the sum of energy cost in the IoUT node can be formulated
as E𝐴=E𝑡𝑟
𝐴+ E𝑐
𝐴.
IV. PAOIOF MULTIP LE VACATI ON Q UE UE W IT H NO N-AQM
In this section, we derive the average PAoI for multiple
vacation queue without AQM (i.e., non-AQM policy), i.e. the
IoUT node does not compress and transmit outdated packets
in its idle mode. Substituting 𝛽=1and 𝛼=0(i.e., 𝑓𝑖 , 𝑗 ≡0.)
back to Eq. (6), we achieve the transition rate matrix 𝑸𝑁 𝐴 for
the system relying on non-AQM policy, and its status update
is formulated as a quasi-birth-and-death (QBD) process. Since
the system is ergodic, the probabilities 𝝅𝑖yield the recursive
relationship, i.e., 𝝅𝑖+1=𝝅𝑖𝑹and 𝝅𝑘=𝝅0𝑹𝑘with 𝑘≥0.
According to the matrix-geometric solutions in [13], the square
matrix 𝑹is a nonnegative solution for the following matrix-
quadratic equation:
𝑹2𝑫0+𝑹 𝑨 +𝑪=0,(17)
where 𝑫0=diag(𝜇, 0),𝑨=−(𝜇+𝜆)0
𝜃−(𝜃+𝜆),𝑪=diag(𝜆, 𝜆 )
are lower triangular matrices. Then, we define 𝑹=𝑟11 0
𝑟21 𝑟22 as
a lower triangular matrix accordingly. Substituting them back
to (17), we obtain the equations about the elements of 𝑹:








𝜇𝑟2
11 −(𝜇+𝜆)𝑟11 +𝜆=0,
(𝑟21𝑟11 +𝑟22𝑟21 )𝜇−(𝜇+𝜆)𝑟21 +𝜃𝑟22 =0,
𝜆−(𝜃+𝜆)𝑟22 =0.
(18)
Thus, we find the minimal nonnegative solution as follows:
𝑟11 =𝜌,𝑟22 =𝜆/(𝜃+𝜆)and 𝑟21 =𝜃𝑟22/[𝜇(1−𝑟22 )], where
𝜌=𝜆/𝜇denotes the workload of the queue. Relying on the
normalization condition 𝝅0(𝑰−𝑹)−1𝒆=1[13], we can get
𝝅0. Furthermore, the probability of the active mode in the
IoUT node relying on non-AQM policy is formulated as:
𝑝𝑁 𝐴,𝑎 =∞
Õ
𝑘=1
𝜋𝑘, 1=𝑟21
1−𝜌+𝑟21
.(19)
1.53
3.07
4.60
6.14
7.68
9.22
10.8
12.29
1
1.2
1.4
1.6
1.8
2
2.2
(a)
1.53
3.07
4.60
6.14
7.68
9.22
10.8
12.29
1
1.2
1.4
1.6
1.8
2
2.2
(b)
Fig. 3. The average PAoI under three policies over different packet generation
rate 𝜆𝐿𝑠.
1
2
10
3
10
4
5
88
66
44
22
00
non-AQM policy
AQM policy
Fig. 4. The average PAoI under AQM policy and non-AQM policy over
different packet generation rate 𝜆𝐿𝑠and the idle mode parameter 𝜃.
Then, according to Little’s law [13], the average system delay
of packet in the associated IoUT node’s queue is obtained as:
E[𝐷𝑁 𝐴]=𝜆−1+∞
Õ
𝑘=1
𝑘𝝅0𝑹𝑘𝒆
=𝜆−1𝝅0(𝑰−𝑹)−2𝑹𝒆 .
(20)
According to Eq. (1), the expression of average PAoI for
multiple vacation queue with non-AQM policy can be obtained
as follows:
E𝐴𝑝
𝑁 𝐴=E[𝐺]+E[𝐷𝑁 𝐴]+𝜚
=𝜆−11+𝝅0(𝑰−𝑹)−2𝑹𝒆 +𝜚. (21)
According to Eqs. (5) and (19), the energy cost in the
associated IoUT node relying on non-AQM policy is:
E𝑁 𝐴 =𝑝𝑁 𝐴,𝑎 𝑃𝑡 𝑟 𝑇𝑡=𝑟21
1−𝜌+𝑟21
𝑃𝑡𝑟 𝑇𝑡.(22)
V. NUMERICAL RESULTS
In this section, we present numerical results to show the
PAoI of the considered system in terms of different system
parameters and policies, i.e., AQM, non-AQM and benchmark.
The length of each packet is 𝐿𝑠=1.5kb, the service rate
of the IoUT node is 𝜇=10, the capacity of the acoustic
channel is 𝐶=15kbps and the overall efficiency of the circuit
is 𝜂=20%. The depth of IoUT node is 𝐻=100m, the acoustic
frequency of carrier is 𝑓=20kHz, the transmission distance
is 𝑙=1km and the bandwidth of sub-channel is 𝐵=1kHz.

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 5
6.14
6.91
7.68
8.45
9.21
9.98
10.75
11.52
12.29
13.05
13.82
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
(a)
2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
(b)
Fig. 5. The average energy cost of the IoUT node under three policies over
different packet generation rate 𝜆𝐿𝑠and the idle mode parameter 𝜃.
Besides, an approximate value for underwater sound speed
𝑉𝑠is 1.5×103m/s [3]. Additionally, the amount of packets
transmitted in our network simulator is 104.
Figs. 3(a) and 3(b) depict the average PAoI of the system
considered as a function of packet generation rate (𝜆𝐿𝑠) for
different idle mode parameter 𝜃, the AQM rate 𝛾and the
discarding probability 𝛼, respectively. Specifically, simulation
results are obtained through a network simulator, and closed-
form results are analytical expressions. As the benchmark,
IoUT nodes relying on age-optimal scheme generate and
transmit data without the latency of idle period. Hence, it is
obvious that such system can achieve the lowest peak AoI,
which serves as the benchmark. When packet generation rate
increases, the PAoI dramatically decreases firstly and increases
after 𝜆𝐿 𝑠>7.68 kbps, because the excessive update interval
and heavy system load aggravate the timeliness of packets. In
Fig. 3(a), when the duration of the idle mode (𝜃−1) increases,
the average PAoI increases. This is because when 𝜃decreases,
the frequency of status update is reduced. In addition, the
IoUT node relying on AQM is superior to non-AQM, because
AQM processes the packets waiting a long term. Similarly, in
Fig. 3(b), increasing the rate of AQM 𝛾and the probability 𝛼
contributes to the mitigation of the average PAoI.
Fig. 4 illustrates the average PAoI with different policies
versus the idle mode parameter 𝜃, and the packet generation
rate, respectively. We set 𝛾=10 and 𝛼=0.8. As the figure
shows, the average PAoI first decreases then increases with
the growth of the packet generation rate, because the lack
of update aggravates data timeliness when throughput is low.
Then, the long system delay becomes the dominant factor with
the excessive rate 𝜆𝐿𝑠. However, the PAoI is not very sensitive
to the idle mode parameter 𝜃, because the larger 𝛾and 𝛼
frequently eliminate the packets with high PAoI.
Fig. 5(a) and (b) illustrate the energy cost of the IoUT
node as a function of the packet generation rate 𝜆𝐿𝑠and the
idle mode parameter 𝜃. It is obvious that the benchmark has
the worst energy-efficiency, and the reason is instinctive. In
underwater environment, the lack of sleep-scheduling causes
excessive power dissipation. Fig. 5(a) shows that the energy
cost increases with the growth of packet generation rate. Fig.
5(b) depicts the influence of the idle mode duration on the
energy cost of the IoUT node. As it shows, with the growth
of 𝜃, if 𝛾becomes large, the energy cost decreases, whereas
when 𝛾is small (i.e., 𝛾=10), the energy cost converges to
a constant. It is noted that a larger sleep duration results in
more packets transmitted and power dissipation. However, an
arbitrarily large number cannot set 𝛾or 𝛼because it leads to
low network throughput.
VI. CONCLUSION
In this letter, we analytically evaluated the average PAoI
and energy cost for an underwater information collection
system relying on a pair of sleep-scheduling policies. Both
the active mode and idle mode are considered to save energy
consumption. Also, we derived closed-form expressions of
average PAoI and energy cost in terms of both AQM and non-
AQM policies, which were verified by sufficient numerical
simulations. Simulation results indicate that to reduce the
packets’ PAoI, the IoUT node should choose an appropriate
packet generation rate and compress packets having been
waiting for a long time relying on AQM policy. Furthermore,
the growth of the idle mode parameter mitigates energy cost
and prolongs the lifetime of the system.
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