1 Impact and Analysis of Space-Time Coupling on Slotted MAC in UANs

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Impact and Analysis of Space-Time Coupling on

Slotted MAC in UANs

Yan Wang, Quansheng Guan, Senior Member, IEEE, Fei Ji, Member, IEEE, Weiqi Chen

Abstract—The propagation delay is non-negligible in under-

water acoustic networks (UANs) since the propagation speed

is ﬁve orders of magnitude smaller than the speed of light.

In this case, space and time factors are strongly coupled to

determine the collisions of packet transmissions. To this end,

this paper analyzes the impact of space-time coupling on slotted

medium access control (MAC). We ﬁnd that a sending node has

speciﬁc location-dependent interference slots and slot-dependent

interference regions. Thus, the collisions may span multiple slots,

leading to both inter-slot and intra-slot collisions. Interestingly,

the slot-dependent interference regions could be an annulus inside

the whole transmission range. It is pointed out that there exist

collision-free regions when a guard interval larger than a packet

duration is used in the slot setting. In this sense, the long slot

brings spatial reuse within the transmission range. However,

we further ﬁnd that the guard interval is not larger than a

packet duration, which is much shorter than the existing slot

setting in typical Slotted-ALOHA, to reach a peak successful

transmission probabilities and throughput. Simulation results

verify our ﬁndings, and also show that the performance of

vertical transmissions is more sensitive to the spatial impact than

horizontal transmissions in UANs.

Index Terms—Collision analysis, space-time coupling, medium

access control, underwater acoustic networks.

I. INTRODUCTION

Underwater acoustic networks (UANs) are envisioned

widely for oceanic monitoring, ﬁsheries activities, ecological

protection, and other commercial or scientiﬁc applications,

which require multiple underwater sensors to sense the un-

derwater environment and report the data to the sink on the

water surface [1–3]. When the data packets from sensor nodes

arrive at a receiver simultaneously, these packets will collide

with each other and their reception fails. In this case, medium

access control (MAC) is critical to enable multiple sensor

nodes to share the open channel in UANs.

The low speed of acoustic waves and the long propagation

delay make the MAC in UANs different from that in terrestrial

radio networks (TRNs) [4–6]. The speed of acoustic waves

(i.e., 1500 m/s) is ﬁve orders of magnitude slower than

the speed of radio waves (i.e., 3×108m/s). In TRNs, the

propagation delay of radio waves can be neglected, and the

arriving time of a packet is approximated to its sending time.

The principle for MAC is to avoid concurrent sending in

order to guarantee collision-free transmissions. However, the

Y. Wang, Q. Guan, and F. Ji are with the School of Electronic and

Information Engineering, South China University of Technology, Guangzhou

510640, China

W. Chen is with the School of Internet Finance and Information

Engineering, Guangdong University of Finance, Guangzhou 510000, China.

low speed of underwater acoustic waves introduces a non-

negligible propagation delay to packet transmissions in UANs

[5]. The packet arriving time at the receiver is then determined

by both spatial distances between nodes and sending times of

packets [7]. We can observe on the one hand that packets who

are simultaneously sent from different nodes might not arrive

at the same time at the receiver. On the other hand, the packets

might arrive at the same time although their sending times are

different. To this end, the MAC strategies in TRNs like CSMA,

CDMA, etc. are not sufﬁcient to avoid the space-time coupling

collisions in UANs [8–13].

Such space-time coupling in UANs introduces not only time

uncertainty but also space uncertainty to random access based

MAC [14]. Since the generation of the data is often considered

as a random process in communication networks [15, 16], the

sending time is uncertain to the destination node. Similarly, the

positions of nodes are not available in UANs due to the lack

of positioning systems, and the sensor nodes are considered

as randomly deployed [17, 18]. The spatial propagation delays

are also uncertain to the destination node. Due to the ignorable

propagation delay, the MAC protocols in TRNs consider only

the time uncertainty.

The slotting technique could be used to alleviate the time

uncertainty. It divides the channel time into slots, and nodes

are only allowed to access the channel at the beginning of a

slot. By ignoring the propagation delay, the slot length is often

set to the duration time of one packet transmission in TRNs.

It has been proved that the slotting technique could almost

double the network throughput of TRNs [15].

Unfortunately, it was reported in literature that the slotting

technique does not improve the performance of MAC in UANs

[19]. Due to the long propagation delay, the transmission in

the current slot may arrive at the following several slots at the

receiver, which brings inter-slot collisions to UANs. Further-

more, the propagation delays between nodes are uncertain, i.e.,

space uncertainty. An extra guard interval is demanded in a

slot to accommodate this kind of space uncertainty. To ensure

inter-slot collision-free transmissions for all the nodes in the

network, the slot length has to be one packet duration plus the

maximal propagation delay between nodes as a guard interval

[11, 12, 16]. Considering the long communication range and

low propagation speed in UANs, the maximal propagation

delay is always multiple times of a packet duration [13] which

is much longer than the slot length in TRNs (i.e., the duration

of a packet transmission).

Although a long slot can decouple space and time, it

degrades the MAC performance. On the one hand, a longer

slot will accumulate more data packets that have to compete to

arXiv:2210.14500v2 [cs.NI] 6 Aug 2023

access the next slot, aggravating the channel competition. The

intra-slot collision probability will increase in this case. On

the other hand, a long slot means a long idle waiting, which

further decreases the sending opportunities and the throughput

of MAC.

It is noticed that the concurrent transmissions may not lead

to collisions at the receiver due to the space-time coupling

feature of UANs [8–10]. In this sense, the existing slot length

setting (i.e., one packet duration plus the maximal propagation

delay) is unnecessary protection from transmission collisions.

It is possible to shorten the slot length to increase the MAC

performance intuitively. However, the slot length setting and

collision relate strongly to the space-time coupling factors,

such as the guard interval, transmission duration, setting time,

node positions, etc. Therefore, this paper wants to understand

the impact of space-time coupling and understand the optimal

slot setting for slotted MAC.

The contributions and ﬁndings of this paper are summarized

as follows.

•Analysis of space-time coupling collisions: We derive the

collision span of slots and the slot-dependent interference

regions in UANs. It is found that the interference region

could be an annuls for a slot. Therefore, collision-free

regions may exist in the transmission range of a receiver,

and space reuse is possible to accommodate concurrent

transmissions.

•Closed-form expressions for successful transmission

probability and throughput: We establish the relationship

between successful transmission probability and the cou-

pling factors. We ﬁnd that the successful transmission

probabilities and throughput are the same for the slot

lengths of one packet duration and of two packet dura-

tions. Therefore, the maximizer of the slot length exists

in between as shown in the simulations. We also ﬁnd

that the spatial factor has a greater impact on vertical

transmissions than horizontal transmissions in UANs.

The remainder of this paper is arranged as follows. Section

II overviews the related work, and Section III presents the

system model and the motivation of this work. The space-time

coupling collisions are discussed in Section IV. The successful

transmission probability and throughput are then analyzed in

Section V. Simulation results using NS3 are used to verify

our analysis in Section VI. Finally, Section VII concludes this

paper.

II. RELATED WORK

The idle waiting is the major challenge brought by the

long propagation delay in the MAC design for UANs. Many

efforts tried to exploit the idle waiting to increase the channel

utilization. Handshaking is often used to reach a consensus

among a group of nodes to reuse idle waiting. Bidirectional

Concurrent MAC (BiC-MAC) enabled bidirectional concurrent

transmissions which allow the receiver to transmit packets

during idle waiting [8]. In the Reverse Opportunistic Packet

Appending (ROPA) protocol, the handshake’s initiator invites

its neighbors to opportunistically transmit their packets after

the initiator’s transmission [20].

Transmission scheduling is another efﬁcient method to min-

imize the idle waiting. Hus et al. used the delay information

to construct a Spatial-Temporal Conﬂict Graph (ST-GC) to

describe the conﬂict delays among links. The MAC problem

was transformed into a vertex coloring problem of ST-GC

and the optimal scheduling could be obtained [9]. Anjangi

et al. formulated the scheduling into a Mixed Integer Linear

Fractional Programming (MILFP) problem, which adjusts the

sending time and packet duration to minimize the frame length

[4, 10]. Since a packet may propagate for several scheduling

cycles, Chen et al. further considered the collisions among

scheduling cycles in the formulated mixed-integer linear pro-

gramming problem [7].

Analysis was also conducted to understand the impact of

long propagation delay on underwater MAC. The simulation

results showed that the throughput of Slotted-ALOHA is the

same as Pure-ALOHA due to the long propagation delay in

UANs [19], while the throughput of Slotted-ALOHA is double

compared to that of Pure-ALOHA in TRNs [15]. Ahn et

al. pointed out that it is the space uncertainty introduced by

the varying propagation delays that degrade Slotted-ALOHA

in UANs, and analyzed the throughput in a case that the

packet transmission duration is larger than the propagation

delay in [16]. However, it is noticed that the propagation

delay over long-range underwater acoustic links is potentially

larger than the transmission duration in practical UANs, and

the propagation may span several slots.

Other MAC protocols, like ﬂoor-acquisition multiple access

(FAMA) and distance-aware collision avoidance protocol (DA-

CAP), tried to apply the strategy of multiple access with col-

lision avoidance (MACA) in TRNs into UANs. Shahabudeen

et al. used a Markov-chain model to analyze the impact of

batch queuing on the MACA and gave closed-form expressions

for the mean service time and throughput [21]. Zhu et al.

studied the impact of the low transmission rates and long

preambles of the real acoustic modem on the random access-

based MAC and handshake-based MAC. In order to capture

the effects of space-time uncertainty, Guan et al. purposed a

statistical physical interference model which is based on the

observed interferences over passed slots [22]. This statistical

model was validated by testbed experiments and used by each

node to distributively select an optimal access probability,

which increases the sum-throughput over multiple slots. Chen

et al. further revealed that the collisions of space-time coupling

transmissions in UANs are different from that in TRNs [7].

It was shown that there exist possibly multiple concurrent

successful transmissions due to the coupling in space and time.

This paper further discusses the spatiotemporal impact on

slotted-MAC in UANs. We try to answer the questions that

whether space-time coupling brings performance gain to slot-

ted MAC and what is the optimal slot length to maximize the

successful transmission probability and throughput.

III. SYSTEM MODEL AND MOTIVATION

This section describes the system model and the motivation

of our work.

A. System Model

We consider a network that has one sink (e.g., a buoy on

the water surface) to collect data from 𝑁underwater sensor

nodes. Let N={1,2, . . . , 𝑁}denote the set of sensor nodes.

Underwater sensor nodes adopt a slotted MAC to share the

open channel. Packets that arrive in current slot are stored in

the buffer and sent at the beginning of the next slot. Each slot

accommodates one packet duration and one additional guard

interval. Let 𝑡𝑠𝑙𝑜𝑡 denotes the duration of a slot. The expression

of the slot length is written by

𝑡𝑠𝑙𝑜𝑡 =𝑡𝑓+𝛽·𝜏, (1)

where 𝑡𝑓is the transmission duration for a packet, 𝜏denotes

the maximal propagation delay, and 𝛽denotes the guard

coefﬁcient, which is the ratio of the guard interval to the

maximal propagation delay.

B. Motivation

The length of a slot affects the network performance di-

rectly. Let us look at two cases at 𝛽≥1and 𝛽 < 1,

respectively.

When 𝛽≥1, the slot length is not shorter than the

transmission time (i.e., 𝑡𝑓) plus the maximum propagation

time (i.e., 𝜏). In this case, one packet can be transmitted from

the sender to its receiver within one slot. According to the

principle of slotted MAC that packets are only allowed to be

sent at the beginning of a slot, collisions are impossible for any

packets that are sent in different slots. Collisions may happen

only for packets sent in the same slots, which is called intra-

slot collisions.

It would be much more complicated for the case of 𝛽 < 1.

A slot length having 𝛽 < 1cannot accommodate the complete

transmission of a packet, and a packet transmission may prop-

agate across several slots. In addition to intra-slot collisions,

packets from different slots may also collide with each other,

i.e., inter-slot collisions, as shown in Fig. 1b, which is the

main reason that degrades Slotted-ALOHA to Pure-ALOHA

[19].

A relatively long slot, e.g., 𝛽≥1, eliminates inter-slot

collisions by the long guard interval. However, Fig. 1a shows

that the long slot length may still decrease MAC performance

in two aspects. Since transmissions are only allowed at the

beginning of a slot, a long guard interval means a long idle

waiting between packet transmissions, which decreases the

transmission rate. In addition, although the long slot elim-

inates inter-slot collisions, it aggravates intra-slot collisions.

More packets may arrive during a longer slot and will be

accumulated to compete for channel access at the next slot.

Thus, the intra-slot collision probability will increase for

these backlogged packets. The low transmission rate and the

high collision probability will ﬁnally lead to low network

throughput.

Although a slot having 𝛽≥1could eliminate space

uncertainty and inter-slot collisions, the space-time coupling

slot setting having 𝛽 < 1may achieve better performance. The

space-time coupling feature in UANs motivates our study in

this paper. The notations in Tab. I are used in our discussions.

(a) Slotted-MAC with 𝛽≥1

(b) Slotted-MAC with 𝛽 <1

Fig. 1: Transmission collisions in slotted-MAC. (a) When 𝛽≥

1, only intra-slot collisions exist for packets sent from two

senders at distances of 𝑑1and 𝑑2, respectively). However, more

packets will be accumulated in one slot, compared to the case

of 𝛽 <1. (b) Both intra-slot and inter-slot arrivals may not

exist without collision when 𝛽 <1.

TABLE I: The notations in the discussion.

Symbols Meaning

𝑁The number of sensors

NThe set of sensors

𝑡𝑓The packet duration

𝑡𝑠𝑙𝑜𝑡 The slot duration

𝜏The maximal propagation delay in the horizontal plane

𝛽The guard coefﬁcient

𝑡𝑖The sending time of sensor 𝑖

𝑑𝑖The distance from sensor 𝑖to the sink

𝑣The propagation speed of underwater acoustic wave

Δ𝑚The sending slot of an interference node

𝑀The maximal number of interference slots

MThe set of slot difference to possible interfering slots

𝑅The maximal communication range in the horizontal plane

𝐾The amount of distance segments

DkThe 𝑘-th distance segment

DThe set of distance segments

CkThe set of interference slots for a tagged node in Dk

𝜆The packet arrival rate per node

𝛼The expanding coefﬁcient for the vertical communication range

IRΔ𝑚

𝑖The interference region of slot Δ𝑚for a node 𝑖in Dk

𝑆Δ𝑚

1,𝑘 The area of IRΔ𝑚

𝑖for node 𝑖

𝑆Δ𝑚

2,𝑘 The area of DIR of Slots Δ𝑚and Δ𝑚+1for node 𝑖

𝑃𝑧,𝑘 The probability that an interference node falls in IRs of node 𝑖

𝑃𝑜,𝑘 The probability that an interference node falls in DIRs of node 𝑖

𝑃𝑡The sending probability of a node

𝑃𝑘The probability that a node locates in Dk

IV. SPACE-TIME COUPLING COLLISIONS IN SLOTTED

MAC

We adopt the protocol transmission model to describe

whether packet transmissions collide [7, 18, 23]. Suppose node

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1ImpactandAnalysisofSpace-TimeCouplingonSlottedMACinUANsYanWang,QuanshengGuan,SeniorMember,IEEE,FeiJi,Member,IEEE,WeiqiChenAbstract—Thepropagationdelayisnon-negligibleinunder-wateracousticnetworks(UANs)sincethepropagationspeedisfiveordersofmagnitudesmallerthanthespeedoflight.Inthiscase,spaceandtimef...

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