Estimation of Ship-to-Ship Link Persistence in Maritime Autonomous Surface Ship Communication Scenarios

Abstract

Maritime Autonomous Surface Ships (MASSs) are expected to become vital participants in future maritime commerce and ocean development activities. This paper investigates a channel capacity-based scheme for estimating the persistence of ship-to-ship communication links in MASS communication scenarios. Specifically, this study presents a relative motion model for nodes within the network and estimates link persistence based on the dynamic characteristics of the links. Additionally, transmission modes tailored to maritime communication scenarios are proposed to optimize link capacity and reduce interference. Simulation results demonstrate that the proposed method can accurately estimate the duration and capacity of the links, thereby achieving higher network capacity. When used as a metric for routing protocols, the proposed link-persistence measure outperforms traditional metrics in terms of packet loss ratio, end-to-end delay, and throughput. Comparisons with other mobility models show that the proposed mobility model offers greater accuracy and reliability in describing the relative mobility of nodes.

1. Introduction

The growth in Maritime Autonomous Surface Ship (MASS) technologies and their journey towards commercialization have drawn global attention, with both showing robust upward trends [1]. The plans of the International Maritime Organization (IMO) to roll out the non-mandatory MASS CODE in 2025, coupled with accelerated research, development, and onboard trials in major maritime nations, signals a bright future for MASS technologies in maritime applications [2]. Among the many enabling technologies, communication stands out. It underpins remote ship control and autonomous navigation, covers data transfer and ship monitoring, and enables interaction between ships and shore-based systems.

At present, MASS communication scenarios are fraught with many challenges. Node mobility, link persistence, and routing adaptability are the most prominent of these issues, which also interact with each other [3,4]. The dynamic nature of nodes in a mobile communication network determines the network’s topology and fluctuations, thereby directly affecting link persistence. Poor link persistence can directly lead to link disruption. When a link within a path fails, the network must repair the path by finding an alternative link or establishing a completely new path. These re-routing operations consume precious radio resources, and re-routing delays may affect service quality and overall network performance. Therefore, evaluating communication link persistence directly impacts the quality of ship-to-ship communication services and is crucial for ensuring the continuity, stability, and efficiency of maritime communication services. In the resource-constrained MASS communication scenario, link persistence is the key indicator for selecting the optimal communication path. An accurate node mobility model is an essential prerequisite for the estimation of link persistence.

In the field of mobility modeling, as early as 1926, Einstein proposed the concept of Random Walk Mobility (RWM) to simulate Brownian motion [5]. Subsequently, researchers developed a series of simple and easy-to-implement mobility models, including random waypoint models and random direction models, based on this concept [6,7]. These models have been widely used, and they have become the most commonly used default mobility models in various network simulation platforms due to their simplicity and ease of theoretical analysis. However, such models do not take relative motion between nodes into consideration. In mobile ad hoc networks, the relative speeds and directions of nodes significantly affect link persistence. When two nodes move in opposite directions or at a high relative speed, link persistence tends to be poor. Therefore, to achieve a more accurate estimation of link persistence, it is crucial to design a mobility model that incorporates the relative motion between nodes, thereby better capturing their relative movement trends.

Regarding communication link states, researchers have also conducted extensive and in-depth studies. In [8], the characteristics of wireless links at sea and the factors influencing communication are examined through data obtained from sea-testing experiments conducted on multiple fishing vessels. In [9], the authors define transmission reliability to measure the transmission performance of mobile ad hoc networks, considering the impact of interference and utilizing genetic algorithms for optimization. In [10], a Bayesian tracking algorithm is proposed to predict the duration of links by learning the environment and predicting node motion parameters. In [11], the authors present a link availability analysis of two cities in India under fog weather conditions, and propose a novel model to calculate the link availability. The authors of [12] present random graph analysis that allows for advanced link models in which the probability of link failure decreases with increasing distance between nodes, helping to determine the network availability of planned mesh networks as a function of the number of redundant nodes. Studies like the aforementioned have involved in-depth research on the characteristics of wireless links, but researchers have not adequately considered the dynamic changes in network topology caused by node mobility, especially in networks with continuous topology changes. Generally, for networks with continuously moving nodes, network topologies of more than two hops are not applicable. This is because longer and more-complex routing paths in such networks make them vulnerable to changes in topology. These changes impair link connectivity and validity, forcing routing protocols to perform route discoveries, updates, and maintenance operations. Consequently, this not only drastically increases routing overhead and communication latency but also significantly degrades overall network performance and efficiency. The authors of [13] propose a link persistence estimation method which can effectively quantify link reliability and provide a basis for path selection. In another study [14], the link quality metric is improved to better reflect the quality of links, and the contribution of the improved metric in enhancing network performance is discussed. The authors of [15] propose the Outlier Bounded Exponential Weighted Moving Average as a metric for reliability-constrained routing. Although the above studies all provide solutions for link state evaluation, the node mobility models they are based on are not accurate enough. The nodes’ speed and direction are fixed and unchanging, which limits the applicability of these methods.

In the field of routing research, the authors of [16] propose a routing metric based on evolution, generated automatically using genetic programming, which incorporates traffic-related features and mobility for adaptive routing in Mobile Ad Hoc Networks. The authors of [17] propose a path selection method based on the Hierarchical Manta Ray Foraging Optimization Algorithm, which employs a bio-inspired optimization mechanism to balance the Quality of Service (QoS) metrics and ultimately output the optimal routing path. The authors of [18] propose a hybrid optimization methodology that involves construction of a dual-constraint dynamic clustering model based on mobility metrics and hop count, which selects cluster heads by comprehensively considering fuzzy parameters such as node mobility speed and neighbor density, and finally executes routing through a hybrid cellular automaton approach. The authors of [19] propose an enhanced resilient routing protocol based on traditional routing mechanisms, leveraging blockchain to store node QoS information and employing a Deep Neural Network to select optimal routing nodes. However, the abovementioned studies primarily focus on procedural aspects of routing discovery and maintenance, typically prioritizing the shortest path with the fewest hops or links traversed. As yet, there has not been sufficient emphasis on adequately assessing link persistence as a routing metric to accurately reflect routing reliability. Addressing this challenge is crucial for enhancing the robustness and efficiency of routing in MASS communication scenarios. The authors of [20] propose a design for a routing metric suitable for Flying Ad Hoc Networks, which can effectively control overhead, improve network lifetime, and enhance search success rate. The authors of [21] propose a hybrid interpretable model for land traffic congestion estimation and prediction, which can assist in making rerouting decisions. Although the proposals in the abovementioned papers exhibit differences from maritime communication networks with respect to physical layers and link layers, the methodologies adopted can still provide insights for related work.

Motivated by the above discussions, in this paper we study link persistence estimation based on channel capacity, and propose routing metrics for MASS communication scenarios. This study focuses on MASS communication scenarios in civilian fields such as maritime search and rescue, emergency communications, and ocean environment monitoring. Figure 1 illustrates two typical application scenarios. In Scenario 1, the destination node and the source node cannot communicate directly; therefore, the MASS acts as a relay communication node. In Scenario 2, the MASS can assist Node A and Node B in expanding their dynamic search areas. The persistence of the communication links between the MASS and Nodes A and B directly affects the efficiency of the rescue.

Considering the high latency and resource overhead associated with multi-hop topologies (as analyzed previously), this paper examines communication links within two hops. To the best of the authors’ knowledge, this study represents the first comprehensive investigation into the integration of node mobility models, link persistence estimation, and routing metrics within the context of MASS communication scenarios. In the proposed architecture, the node mobility model serves as the foundation, and the link persistence is the result. Based on a quantified metric of link persistence, we develop a routing metric for path selection to enhance routing performance. The primary contributions of this work may be summarized as follows:

• We propose a joint mobility model to address the limitations of existing mobility models which neglect the mutual movement of nodes within the network. The mobility model can characterize the mutual distance between two nodes within a given time period, thereby enabling more accurate descriptions of nodes’ movements in MASS communication scenarios.
• Considering the characteristics of maritime communication and the mobility traits of nodes, we propose a link persistence estimation method as well as channel capacity-based objective functions for different transmission modes, which can better describe the dynamic properties of links.
• We compare routing performance under link persistence, shortest path, and first found path. Simulation results demonstrate that routing based on the proposed link persistence metric demonstrates superior performance in packet loss ratio, end-to-end delay and throughput compared to routing based on the other two metrics.

The remaining parts of this paper are structured as follows: Section 2 provides an in-depth description of the node mobility model. Section 3 introduces the link persistence analysis model based on the mobility model. The objective function for link persistence based on channel capacity under different transmission modes is provided in Section 4. Section 5 presents and analyzes the simulation results. Section 6 concludes the article and discusses future research directions.

2. Node Mobility Model

Predicting the next-stage motion state of nodes based on historical data involves numerous challenges, such as instantaneous disturbances from ocean waves and abrupt shifts in long-term trends. Nevertheless, link estimation necessitates that node motion adheres to certain patterns rather than being entirely random. In the context of MASS communication scenarios, ships and vessels can acquire positions through various devices, such as the Global Navigation Satellite System, enabling effective prediction of network topology based on node movement patterns [22]. Because one of the most distinctive characteristics of wide-area maritime communication compared to terrestrial communication is sparsity, which means that the probability of a ship appearing per unit time and per unit area in the vast expanse of the ocean is relatively low, and because the traffic of ships at sea is relatively stable, this paper employs the Poisson Distribution to describe the distribution of maritime communication nodes [23,24]. Consequently, the breakdown of communication links also occurs independently.

The term $\mathrm{R}$ is used to denote the maximum radius of radio coverage for a mobile node. When one node falls within the transmission range of the other node, a two-way link is established between them. It should be noted that one-way communication is not taken into account. Within a given time interval, each node moves linearly at a constant speed and in a constant direction until the beginning of the next time interval. These time intervals are of equal length and are independently distributed, with a mean value of $1/\mathrm{\lambda }$. At the start of each time interval, the speed and direction of the node’s movement are randomly and independently altered. As the speed is influenced by factors such as waves, winds, ship performance, and loading conditions, when there are numerous influencing factors and a sufficient number of ships, the speed distribution approximates a Gaussian distribution, according to the Central Limit Theorem. Gaussian distribution with a mean of $\mathrm{\mu }$ and a variance of ${\mathrm{\sigma }}^2$ is employed to describe the speed distribution of nodes over the entire time period. The direction distribution is uniformly distributed over $\left[0, 2\mathrm{\pi }\right]$ with a fluctuation range of less than $\mathrm{\pi }/2$. On an abstracted two-dimensional plane, the starting positions and destinations of the mobile nodes are random. Let $\overrightarrow{{\mathrm{s}}_{\mathrm{I}}}$ and $\overrightarrow{\mathrm{s}}\left(\mathrm{t}\right)$ denote the displacement vector of the node in epoch $\mathrm{i}$ and the total time $\mathrm{t}$, respectively, $\overrightarrow{\mathrm{s}}\left(\mathrm{t}\right)=\sum _{\mathrm{i}=1}^{\mathrm{N}\left(\mathrm{t}\right)}\overrightarrow{{\mathrm{s}}_{\mathrm{i}}}$. Here, $\mathrm{N}\left(\mathrm{t}\right)$ denotes the number of epochs simulated within time $\mathrm{t}$. The magnitude equals the distance from $\left(\mathrm{X}\left({\mathrm{t}}_0\right), \mathrm{Y}\left({\mathrm{t}}_0\right)\right)$ to $\left(\mathrm{X}\left({\mathrm{t}}_0+\mathrm{t}\right), \mathrm{Y}\left({\mathrm{t}}_0+\mathrm{t}\right)\right)$ which can be resolved into displacement components in the X and Y directions. As the direction is uniformly distributed, the displacement components in the X and Y directions are symmetric and independent. By the Central Limit Theorem, when the direction is random, the displacement components in the X and Y directions approximate zero-mean Gaussian distributions with identical variances. Consequently, the magnitude conforms to a Rayleigh distribution with parameters of $\mathrm{\alpha }=\left({\displaystyle \frac{2\mathrm{t}}{\mathrm{\lambda }}}\right)\times \left({\mathrm{\mu }}^2+{\mathrm{\sigma }}^2\right)$. Therefore, the mobility characteristics of the node can be described by the parameter set $\left\{\mathrm{\lambda }, \mathrm{\mu }, {\mathrm{\sigma }}^2\right\}$. Given that $\mathrm{x}\left(\mathrm{t}\right)$ and $\mathrm{y}\left(\mathrm{t}\right)$ are uncorrelated, the joint probability density of the node is as follows:

$${\mathrm{f}}_{\mathrm{x},\mathrm{y}}\left(\mathrm{x},\mathrm{y}\right)={\displaystyle \frac{1}{\mathrm{\pi }\mathrm{\alpha }}}\exp \left[-{\displaystyle \frac{\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}{\mathrm{\alpha }}}\right].$$

(1)

Assume nodes M and N move according to mobility parameters $\left({\mathrm{\lambda }}_{\mathrm{M}}, {\mathrm{\mu }}_{\mathrm{M}}, {\mathrm{\sigma }}_{\mathrm{M}}^2\right)$ and $\left({\mathrm{\lambda }}_{\mathrm{N}}, {\mathrm{\mu }}_{\mathrm{N}}, {\mathrm{\sigma }}_{\mathrm{N}}^2\right)$, respectively. As the movements of the two nodes are independent and uniform, the joint motion distribution is investigated by considering node N as stationary and the movement of node N as the opposite movement of node M. In Figure 2, after time $\mathrm{t}$, nodes M and N reach positions M′ and N′, respectively, and the distance between them is denoted as $\mathrm{D}$. Figure 3 is equivalent to Figure 2, where the random mobility vector of node M relative to node N is derived by fixing node N. This is represented as ${\mathrm{s}}_{\mathrm{M}, \mathrm{N}}\left(\mathrm{t}\right)={\mathrm{s}}_{\mathrm{M}}\left(\mathrm{t}\right)-{\mathrm{s}}_{\mathrm{N}}\left(\mathrm{t}\right)$, and its amplitude approximately follows a Rayleigh distribution with parameter ${\mathrm{\alpha }}_{\mathrm{M}, \mathrm{N}}={\mathrm{\alpha }}_{\mathrm{M}}+{\mathrm{\alpha }}_{\mathrm{N}}$.

$${\mathrm{\alpha }}_{\mathrm{M}, \mathrm{N}}= ({\displaystyle \frac{2\mathrm{t}}{{\mathrm{\lambda }}_{\mathrm{M}}}})({\mu }_{\mathrm{M}}^2+{\mathrm{\sigma }}_{\mathrm{M}}^2)+({\displaystyle \frac{2\mathrm{t}}{{\mathrm{\lambda }}_{\mathrm{N}}}})({\mu }_{\mathrm{N}}^2+{\mathrm{\sigma }}_{\mathrm{N}}^2).$$

(2)

3. Link Persistence Analysis

The failure of any link within a path results in the failure of the entire path. Hence, the stability of a path is contingent upon the stability of all the links that constitute it [25]. In periods of non-breakage, the optimal path is characterized by maximal capacity and minimal interference. Therefore, an analysis of link persistence and interference is essential for enhancing network reliability, and an objective function is necessary to quantify topology performance.

In this model, link persistence is defined as the probability that an active link exists between two mobile nodes at the time ${\mathrm{t}}_0+\mathrm{t}$, provided that this link was available at the time ${\mathrm{t}}_0$ [13]. To illustrate the link connectivity between two nodes over a given time span, it is essential to analyze the motion distribution of individual initially. Subsequently, the connection probability of a node is calculated by considering the magnitude and direction of one moving node relative to another, which represents the impact of motion on link persistence.

At time $\mathrm{t}$, the relationship between the distance separating two nodes and link persistence is established. The distance ${\overrightarrow{\mathrm{D}}}_{\mathrm{M}, \mathrm{N}}\left(\mathrm{t}\right)$ is the result of the sum of the random mobility vector $\overrightarrow{\mathrm{s}}\left(\mathrm{t}\right)$ and the constant vector $\overrightarrow{\mathrm{c}}$, as illustrated in Figure 4. The solution corresponds exactly to the general problem of determining the sum of a Rayleigh-distributed vector and a constant. By decomposing $\overrightarrow{\mathrm{s}}\left(\mathrm{t}\right)$ into its X and Y components from the Gaussian distribution and adding these to the ${\mathrm{x}}_{\mathrm{c}}$ and ${\mathrm{y}}_{\mathrm{c}}$ components of the initial distance, the link persistence distance probability distribution function is obtained:

$$\begin{array}{cc}\hfill {\mathrm{f}}_{{\mathrm{D}}_{\mathrm{M}, \mathrm{N}},{\mathrm{\theta }}_{\mathrm{D}}}\left(\mathrm{d},\mathrm{\theta }\right)& ={\mathrm{f}}_{\mathrm{x},\mathrm{y}}\left(\mathrm{x},\mathrm{y}\right)\left(\begin{array}{cc}\cos \mathrm{\theta }& -\mathrm{d}\sin \mathrm{\theta }\\ \sin \mathrm{\theta }& \mathrm{d}\cos \mathrm{\theta }\end{array}\right)\hfill \\ & ={\displaystyle \frac{\mathrm{d}}{\mathrm{\pi }{\mathrm{\alpha }}_{\mathrm{M}, \mathrm{N}}}}\exp \left[-{\displaystyle \frac{(\mathrm{d}\cos \mathrm{\theta }-{\mathrm{x}}_{\mathrm{c}}{)}^{2 }+(\mathrm{d}\sin \mathrm{\theta }-{\mathrm{y}}_{\mathrm{c}}{)}^2}{{\mathrm{\alpha }}_{\mathrm{M}, \mathrm{N}}}}\right]\hfill \\ & ={\displaystyle \frac{\mathrm{d}}{\mathrm{\pi }{\mathrm{\alpha }}_{\mathrm{M}, \mathrm{N}}}}\exp \left[-{\displaystyle \frac{\left({\mathrm{d}}^2+{\mathrm{C}}^2\right)-2\mathrm{d}\left({\mathrm{x}}_{\mathrm{c}}\cos \mathrm{\theta }+{\mathrm{y}}_{\mathrm{c}}\sin \mathrm{\theta }\right)}{{\mathrm{\alpha }}_{\mathrm{M}, \mathrm{N}}}}\right],\hfill \end{array}$$

(3)

where ${\mathrm{C}}^2={\mathrm{x}}_{\mathrm{c}}^2+{\mathrm{y}}_{\mathrm{c}}^2$.

When node N reaches the edge of node M’s communication range, the angle between line M′N′ and the X-axis can be determined using the law of cosines, resulting in an angle of

$${\mathrm{\theta }}_{\mathrm{D} }={ \mathrm{\theta }}_{\mathrm{Z} }+{ \mathrm{\theta }}_{\mathrm{Z}, \mathrm{D}}={ \mathrm{\theta }}_{\mathrm{Z}}+\arcsin \left\{{\displaystyle \frac{\mathrm{C}}{\mathrm{R}}}\sin \left[{\mathrm{\theta }}_{\mathrm{Z} }+\left(180°-{ \mathrm{\theta }}_{\mathrm{c}}\right)\right]\right\}.$$

(4)

The link persistence ${\mathrm{A}}_{\mathrm{M}, \mathrm{N}}$ corresponds to the probability ${\mathrm{P}}_{\mathrm{M}, \mathrm{N}}$ that ${\mathrm{D}}_{\mathrm{M}, \mathrm{N}}\left({\mathrm{t}}_0\right) \le \mathrm{R}$, and remains ${\mathrm{D}}_{\mathrm{M}, \mathrm{N}}({\mathrm{t}}_0+\mathrm{t}) \le \mathrm{R}$, which can be calculated by integrating the probability distribution function of distance. The probability distribution function of distance is shown in Equation (3). Hence, the link persistence is expressed as follows:

$$\begin{array}{cc}\hfill {\mathrm{A}}_{\mathrm{M}, \mathrm{N} }& ={ \mathrm{P}}_{\mathrm{M}, \mathrm{N}}\left[ {\mathrm{D}}_{\mathrm{M}, \mathrm{N}}({\mathrm{t}}_0+\mathrm{t}) \le \mathrm{R} | {\mathrm{D}}_{\mathrm{M}, \mathrm{N}}\left({\mathrm{t}}_0\right) \le \mathrm{R}\right]\hfill \\ & ={\int }_0^{\mathrm{R}}{\int }_0^{2\mathrm{\pi }}{\mathrm{f}}_{{\mathrm{D}}_{\mathrm{M}, \mathrm{N},{\mathrm{\theta }}_{\mathrm{D}}}}\left(\mathrm{d},\mathrm{\theta }\right)\mathrm{d}\mathrm{\theta }\mathrm{dd}\hfill \\ & ={\displaystyle \frac{\mathrm{d}}{\mathrm{\pi }{\mathrm{\alpha }}_{\mathrm{M}, \mathrm{N}}}}{\int }_0^{\mathrm{R}}{\int }_0^{2\mathrm{\pi }}\exp \left[-{\displaystyle \frac{\left({\mathrm{d}}^2+{\mathrm{C}}^2\right)+2\mathrm{d}\left(\mathrm{C}\cos {\mathrm{\theta }}_{\mathrm{C} }+ \cos \mathrm{\theta } +\mathrm{C}\sin {\mathrm{\theta }}_{\mathrm{C}}\sin \mathrm{\theta }\right)}{{\mathrm{\alpha }}_{\mathrm{M}, \mathrm{N}}}}\right]\mathrm{d}\mathrm{\theta }\mathrm{dd},\hfill \end{array}$$

(5)

where the distance $D_{M, N}\left(\mathrm{t}\right) \ge 0$. The link persistence reaches its maximum value when $\mathrm{R}$ reaches to infinite.

4. Objective Function Under Different Transmission Modes

The transmission mode is one of the primary factors influencing outage capacity and interference. The objective function is designed to represent the network capacity during connection. During transmission, signals attenuate from one terminal to another, limiting direct communication between terminals to nearby ones. If terminals are distant from each other, intermediate terminals are needed to relay information until the message reaches the destination terminal. This paper considers two transmission modes: direct transmission and two-hop transmission.

4.1. Direct Transmission

According to the initial positions depicted in Figure 4, the initial positions of the nodes determine whether the source S and destination D are adjacent. Specifically, when the distance between S and D is less than the effective communication radius R, the nodes are deemed adjacent. Here, the source S is centrally located within the circle. For normal point-to-point direct transmission, the destination D must lie within the effective communication radius of the source S. In wireless communication, signal attenuation occurs during transmission. The received signal power ${\mathrm{P}}_{\mathrm{r}}$ can be denoted as ${\mathrm{P}}_{\mathrm{r}}={ \mathrm{P}}_{\mathrm{t}} \times {\mathrm{d}}^{-\mathrm{\lambda }}$, where ${\mathrm{P}}_{\mathrm{t}}$ represents the transmitted power from the antenna, $\mathrm{d}$ represents the distance between nodes, and $\mathrm{\lambda }$ is the path loss exponent. The Signal-to-Noise Ratio (SNR) of the received signal is calculated as

$${\mathrm{\gamma }}_{\mathrm{S}, \mathrm{D}}\left(\mathrm{t}\right)={\displaystyle \frac{{\mathrm{P}}_{\mathrm{t}} \times { \mathrm{d}}_{\mathrm{S}, \mathrm{D}}^{-\mathrm{\alpha }}\left(\mathrm{t}\right)}{{\mathrm{N}}_0}}.$$

(6)

The outage capacity is employed to characterize the link rate, representing the achievable link capacity under a specified low outage probability $\epsilon$. For direct transmission, the outage capacity is given by

$${\mathrm{C}}_{\mathrm{DT}}^{\mathrm{\epsilon }}\left(\mathrm{t}\right)=\mathrm{B}{\log }_2\left[1+{\displaystyle \frac{1.5\mathrm{\epsilon } \times { \mathrm{\gamma }}_{\mathrm{S}, \mathrm{D}}\left(\mathrm{t}\right)}{0.2-\mathrm{\epsilon }}}\right].$$

(7)

To guarantee the transmission quality, the interference along the transmission path should account for the cumulative effect of all nodes involved in the transmission process. Specifically, in the case of direct transmission between two nodes, the interfering nodes comprise those encompassed by both the source and destination nodes, so that

$${\mathrm{I}}_{\mathrm{DT} }=\mathrm{I}\left(\mathrm{S}\right) \cup \mathrm{I}\left(\mathrm{D}\right).$$

(8)

Node mobility introduces variability into their motion states, impacting link connectivity and system stability, and thereby influencing network capacity. The relative motion speed between the source and terminal nodes, denoted as ${\mathrm{V}}_{\mathrm{S}, \mathrm{D}}$, can be represented as the effective value resulting from the sum of the randomly moving velocity vectors ${\mathrm{V}}_{\mathrm{S}}$ and ${\mathrm{V}}_{\mathrm{D}}$. As the destination node D approaches the communication range boundary of the source node S, the relative motion distance between nodes, governed by the law of cosines, becomes

$$\mathrm{Z}=\mathrm{R} \times {\displaystyle \frac{\sin \left[180°-{ \mathrm{\theta }}_{\mathrm{Z}}-\left(180°-{ \mathrm{\theta }}_{\mathrm{C}}\right)-{\mathrm{\theta }}_{\mathrm{Z}, \mathrm{D}}\right]}{\sin \left[{\mathrm{\theta }}_{\mathrm{Z}}+\left(180°-{ \mathrm{\theta }}_{\mathrm{C}}\right)\right]}},$$

(9)

and the theoretical communication time is ${\mathrm{T}}_{\mathrm{S}, \mathrm{D}}={\displaystyle \frac{{\mathrm{Z}}_{\mathrm{S}, \mathrm{D}}}{\left|{\mathrm{V}}_{\mathrm{S}, \mathrm{D}}\right|}}$. The link persistence can then be extrapolated along the path. Given the nonlinear nature of link connectivity in MASS communication scenarios, individual links operate independently. Therefore, when a route is on the verge of failure, alternative routes should be preemptively selected based on link persistence. The estimated communication time that takes into account the possibility of link disruption can then be denoted as follows:

$${\mathrm{T}}_{\mathrm{P}}={\mathrm{T}}_{\mathrm{S}, \mathrm{D} }\times {\mathrm{A}}_{\mathrm{S}, \mathrm{D}}\left(\mathrm{t}\right).$$

(10)

As described above, link persistence, rate, and interference constitute the three primary factors determining network capacity. To optimize network capacity during communication, it is essential to enhance link persistence and rate while minimizing interference. The traffic exchanged between nodes during communication time can be derived by integrating the outage capacity. Consequently, the derived objective function is

$${\mathrm{g}}_{\mathrm{DT}}\left(\mathrm{t}\right)={\int }_0^{{\mathrm{T}}_{\mathrm{S}, \mathrm{D}}\times {\mathrm{A}}_{\mathrm{DT}}\left(\mathrm{t}\right)}{\displaystyle \frac{{\mathrm{C}}_{\mathrm{DT}}^{\mathrm{\epsilon }}\left(\mathrm{t}\right)}{{\mathrm{I}}_{\mathrm{DT}}}}\mathrm{dt}.$$

(11)

4.2. Two-Hop Transmission

The two-hop transmission scheme improves link quality by replacing long-distance direct transmission with two sequential phases. This process requires two time slots. In the first time slot, the source transmits data to the relay. The relay then forwards the data to the destination in the second time slot. The received SNRs for the two time slots are denoted, respectively, as ${\mathrm{\gamma }}_{\mathrm{S}, \mathrm{R}}\left(\mathrm{t}\right)={\displaystyle \frac{{\mathrm{P}}_{\mathrm{t}} \times {\mathrm{d}}_{\mathrm{S}, \mathrm{R}}^{-\mathrm{\alpha }}\left(\mathrm{t}\right)}{{\mathrm{N}}_0}}$ and ${\mathrm{\gamma }}_{\mathrm{R}, \mathrm{D}}\left(\mathrm{t}\right)={\displaystyle \frac{{\mathrm{P}}_{\mathrm{t}} \times {\mathrm{d}}_{\mathrm{R}, \mathrm{D}}^{-\mathrm{\alpha }}\left(\mathrm{t}\right)}{{\mathrm{N}}_0}}$. The maximum achievable instantaneous mutual information for the link is determined using the following equation:

$${\mathrm{C}}_{\mathrm{MT}}^{\mathrm{\epsilon }}\left(\mathrm{t}\right)=\min \left\{{\mathrm{C}}_{\mathrm{S}, \mathrm{R}}, {\mathrm{C}}_{\mathrm{R}, \mathrm{D}}\right\}=\min \left\{\mathrm{B}{\log }_2\left(1+{\displaystyle \frac{1.5\mathrm{\epsilon } \times {\mathrm{\gamma }}_{\mathrm{S}, \mathrm{R}}\left(\mathrm{t}\right)}{0.2-\mathrm{\epsilon }}}\right), \mathrm{B}{\log }_2{\displaystyle \frac{1.5\mathrm{\epsilon } \times {\mathrm{\gamma }}_{\mathrm{R}, \mathrm{D}}\left(\mathrm{t}\right)}{0.2-\mathrm{\epsilon }}}\right\}.$$

(12)

The protocol interference model mandates that all nodes within the transmitter’s coverage area remain silent during ongoing transmissions. This defines the interference region of a communication link as the union of active nodes within the transmission range. Each transmission phase generates distinct interference patterns. Because the two hops operate in orthogonal time slots, their interference regions are defined as ${\mathrm{I}}_{\mathrm{S}, \mathrm{R} }=\mathrm{I}\left(\mathrm{S}\right) \bigcup \mathrm{I}\left(\mathrm{R}\right)$ and ${\mathrm{I}}_{\mathrm{R}, \mathrm{D}}=\mathrm{I}\left(\mathrm{R}\right) \bigcup \mathrm{I}\left(\mathrm{D}\right)$, respectively. As the two phases are temporally separated, the effective interference for the complete two-hop transmission is governed by the maximum interference region:

$${\mathrm{I}}_{\mathrm{MT}}=\max ({\mathrm{I}}_{\mathrm{S}, \mathrm{R}}, {\mathrm{I}}_{\mathrm{R}, \mathrm{D}}).$$

(13)

Figure 5 illustrates the mobility relationships. In the figure, ${\mathrm{V}}_{\mathrm{S}, \mathrm{R}}$ represents the relative velocity between source and relay, and ${\mathrm{V}}_{\mathrm{R}, \mathrm{D}}$ denotes the relay-destination relative velocity. The link lifetimes are ${\mathrm{T}}_{\mathrm{S}, \mathrm{R}}={\displaystyle \frac{{\mathrm{Z}}_{\mathrm{S}, \mathrm{R}}}{\left|{\mathrm{V}}_{\mathrm{S}, \mathrm{R}}\right|}}$ and ${\mathrm{T}}_{\mathrm{R}, \mathrm{D}}={\displaystyle \frac{{\mathrm{Z}}_{\mathrm{R}, \mathrm{D}}}{\left|{\mathrm{V}}_{\mathrm{R}, \mathrm{D}}\right|}}$. The persistence of a link can be extrapolated to the path, as the persistence of the path is contingent upon the persistence of links from the source to the relay and from the relay to the destination. Communication can only occur when both links are simultaneously maintained. If either link is disconnected, the entire path fails. Therefore, the anticipated communication time along the path is limited by the shorter-lasting of the two links, denoted as

$${\mathrm{T}}_{\mathrm{p}}=\min \left({\mathrm{T}}_{\mathrm{S}, \mathrm{R}} \times {\mathrm{A}}_{\mathrm{S}, \mathrm{R}}\left(\mathrm{t}\right), {\mathrm{T}}_{\mathrm{R}, \mathrm{D}} \times { \mathrm{A}}_{\mathrm{R}, \mathrm{D}}\left(\mathrm{t}\right)\right).$$

(14)

Therefore, the objective function is

$${\mathrm{g}}_{\mathrm{MT}}\left(\mathrm{t}\right)={\int }_0^{{\mathrm{T}}_{\mathrm{P}}}{\displaystyle \frac{{\mathrm{C}}_{\mathrm{MT}}^{\mathrm{\epsilon }}\left(\mathrm{t}\right)}{{\mathrm{I}}_{\mathrm{MT}}}}\mathrm{dt}.$$

(15)

The system selects the suitable transmission mode and relay node while minimizing interference to optimize link capacity. Therefore, the overall objective function for the path is as follows:

$$\mathrm{g}\left(\mathrm{i}\right)=\max \left({\mathrm{g}}_{\mathrm{DT}}, {\mathrm{g}}_{\mathrm{MT}}\left(\mathrm{i}\right)\right).$$

(16)

Given the relay candidate set ${\mathrm{r}}_{\mathrm{S}, \mathrm{D}}=\left\{1, 2, \dots , \mathrm{m}\right\}$ residing within the coverage range of both source and destination, the objective function for each relay node can be computed. By selecting the relay node with the maximum value, the transmission mode and relay node that optimize network capacity can be determined. Because the objective function is divided into several independent sub-targets, parallel computing can expedite the computation process.

$$\mathrm{i}=\arg \underset{\mathrm{i}\in {\mathrm{\nu }}_{\mathrm{i}}}{\max }{\mathrm{g}}_{\mathrm{MT}}\left(\mathrm{i}\right), \mathrm{for} \mathrm{all} \mathrm{i}\in {\mathrm{r}}_{\mathrm{S}, \mathrm{D}}.$$

(17)

The selection process of the transmission mode and the relay node is illustrated in Figure 6.

5. Simulation Results

This section presents simulations that evaluate dynamic topology characteristics and compare different routing metrics. Wireless channels adhere to the Rayleigh distribution of slow fading, with focus solely on the impact of network topology on capacity. In the simulation scenario, 30 nodes are randomly dispersed across a 1000 × 1000 area. It should be noted that the simulations in this section are intended to evaluate link quality, and no physically meaningful units, such as m or m/s, are set for distance or speed. Nodes follow the proposed mobility model, generating velocities for their subsequent movements randomly. Figure 7a illustrates the initial positions of each node, while Figure 7b illustrates the positions at time 100 s. The x-axis and y-axis represent the coordinate positions of the nodes within the two-dimensional area (units are not defined).

Figure 8 illustrates a comparison of the proposed method with conventional direct transmission in terms of topological performance under different coverage radii. The x-axis represents the size of the coverage radius, while the y-axis indicates the channel capacity performance. Statistical validation was carried out through 5000 Monte Carlo trials for each parameter configuration. The maximum speed is set to 1, and the path loss exponent is configured to 3.5. As can be seen from Figure 8, when the coverage radius is small, the opportunities for nodes in the network to establish communication links are limited, and the number of communicable adjacent nodes is restricted, resulting in poor network performance. As the coverage radius gradually increases, the channel capacity performance significantly improves. This is because more nodes can enter each other’s communication range, increasing the number of links and thereby enhancing overall network performance. However, as the coverage radius continues to increase further, the performance begins to deteriorate. This is because, as the number of links increases, interference also significantly rises. Under a large coverage radius, each node may experience signal interference from multiple other nodes, leading to a decline in network performance. Furthermore, it can be observed from the figure that the proposed method outperforms conventional direct transmission under almost all coverage radii. This optimization is attributed to the joint design of the transmission mode and link persistence estimation. In this experimental scenario, both methods achieve optimal performance when the coverage radius is 100, but the proposed method’s performance is improved by 17% compared to direct transmission. This result indicates that there may exist an optimal coverage radius for such communication scenarios, where the number of links and interference are balanced. At this radius, each node can establish communication links with a sufficient number of other nodes, ensuring the existence of multiple available paths in the network while keeping the interference within an acceptable range, thus guaranteeing high channel capacity.

Figure 9 shows the comparison curves between the estimated results and theoretical values of link persistence under simulation conditions of 30 nodes, communication radius R = 100, and path loss exponent α = 3.5. The horizontal axis represents the maximum node mobility speed V_max, while the vertical axis indicates the channel capacity performance. The green line represents the estimated results, and the purple line represents the theoretical results. As can be seen from the figure, although there are deviations between the estimated and theoretical values due to the randomness of node mobility, their trends of change are highly consistent. This indicates that the proposed estimation method can capture the fundamental dynamic behavior of link persistence. Additionally, when the speed is relatively low, the deviation between the predicted and theoretical values is small. With increased speed, this deviation gradually increases. This is because, in the simulations, nodes are confined within a given space, and the higher the node speed, the more susceptible the link is to disruption. In addition, although a lower level of node activity results in greater accuracy in link estimation, the proposed method can fully reflect the dynamic characteristics of the network topology during rapid changes even when node activity is high.

Figure 10 further introduces different interruption probabilities for comparison based on Figure 8. It can be seen in the figure that when communication radii are the same, the channel capacity corresponding to a high interruption probability is always higher than that of a low interruption probability. This indicates that allowing a higher interruption probability can enhance the network performance. This is because under the scenario of interruption probability, the constraint on link reliability can be relaxed in exchange for higher capacity. For the high interruption probability, the channel capacity reaches its peak when the communication radius is around 110, and then decreases; the peak of the interruption probability occurs when the communication radius is around 80. This is because when the interruption probability is higher, the system can accept more link interruptions and thus chooses a longer transmission distance (increased radius). There is a positive correlation between interruption probability and the optimal radius.

Table 1. Coverage radii for given (ɛ, α) pairs.

	α = 2.5	α = 3	α = 3.5	α = 4
ɛ = 10−4	160	133	79	78
ɛ = 10−3	199	160	113	101

lists the optimal coverage radii under different combinations of interruption probability and path loss exponent. It can be observed from the table that when the path loss exponent increases from 2.5 to 4, the optimal radius corresponding to the same interruption probability decreases significantly.

In the following section, we present simulations illustrating performance variations observed with the Dynamic Source Routing (DSR) protocol, utilizing the routing metric described above alongside classic routing metrics. In this setup, the source node transmits data packets at a fixed packet rate. The following routing metrics are employed:

• Link persistence: as discussed above.
• Shortest path: the path with the minimum number of hops is chosen.
• First found path: the path carried by the initial routing reply packet received by the source node is chosen.

Table 2. Comparison of different metrics.

	Packet Loss Ratio	End-to-End Delay	Throughput
Link persistence	0.6456	12.38	18.98
Shortest path	0.7105	25.62	15.78
First found path	0.6562	42.03	16.47

compares the three routing metrics (link persistence, shortest path, and first found path) in terms of network performance. The performance metrics include packet loss ratio, end-to-end delay, and throughput. It can be seen that link persistence outperforms the other two metrics in packet loss ratio, end-to-end delay, and throughput. This indicates that link persistence can better balance link stability and efficiency during path selection, thereby enhancing overall network performance. However, it is crucial to note that the delay measurement only encompasses received packets, leading to shorter delays as the packet loss ratio reaches certain thresholds. This occurs because the routing protocol discards buffered packets that have surpassed a specific threshold. In instances where a path becomes unavailable, rerouting can significantly contribute to queuing delay, leading to the dropping of more packets and the experiencing of prolonged delays.

In general, the reliability of a path tends to decrease as the number of hops or links it traverses increases. Consequently, link persistence emerges as superior to other routing metrics in terms of path reliability. The demonstrated performance indicates that the proposed link persistence estimation can effectively assist routing protocols in selecting more reliable paths.

Finally, we present a comparison between the mobility model proposed in this paper, the mobility model outlined in [7], and a mobility model derived from actual vessel trajectories. Because of disparities in simulation experiment configurations, we were unable to directly correlate our findings with those of [7]. Therefore, we replicated their methodologies and assessed them within the same simulation environment. Employing the DSR protocol and adopting the routing metric proposed in this paper—link persistence—we were able to synthesize the results which are shown in

Table 3. Comparison of different mobility models.

	Packet Loss Ratio	End-to-End Delay	Throughput
Proposed model	0.6456	12.38	18.98
Model in [7]	0.8964	45.34	15.62
Actual vessel trajectories	0.6116	16.86	17.23

.

The results in Table 3 clearly demonstrate that the proposed mobility model more closely aligns with the performance of the mobility model derived from actual vessel trajectories than with the mobility model presented in [7]. This highlights the accuracy and reliability of the proposed mobility model in characterizing node mobility. An accurate mobility model is fundamental to the effectiveness of routing metrics. The proposed mobility model shows superior performance to the model in [7] in terms of packet loss ratio, end-to-end delay, and throughput. This indicates that the proposed model has a higher degree of accuracy and reliability in describing node mobility and predicting link persistence.

6. Conclusions

In MASS communication scenarios, the mobility of nodes results in dynamic changes in network topology. This paper proposes an outage capacity-based link persistence estimation scheme tailored for dynamic maritime environments. The scheme initially accounts for mutual node movement and predicts link persistence based on relative node distances. Subsequently, an objective function is formulated for maritime transmission modes to optimize channel capacity. Simulation results demonstrate the efficacy of the proposed method in accurately predicting link duration and capacity, thereby enabling higher network capacity compared to direct transmission. Moreover, an increase in interruption probability leads to further enhancements in topological performance. Using the proposed link persistence as a metric for routing protocol can result in better performance in terms of packet loss ratio, end-to-end delay and throughput, compared to traditional metrics. The proposed mobility model exhibits better accuracy and reliability in describing the mobility of nodes, compared with other mobility models. However, there are limitations arising from the scene assumptions and simulation settings used in this paper. Due to the absence of global information, we resorted to initial path interference. Future research endeavors could explore the utilization of machine learning algorithms for interference prediction, or create distributed cooperation mechanisms between network nodes. This would enable nodes to share local interference information, resulting in a more accurate estimation of link persistence. Future research should also focus on combining environmental perception technologies or probabilistic models to supplement the description of environmental factors and quantify the impact of environmental factors on link persistence. Finally, in our own future work, we will incorporate more parameters and constraints related to actual maritime communication, to better reflect real-world application scenarios.

This study strictly focuses on civilian MASS communication technologies, with all proposed models and algorithms designed solely to enhance public service capabilities. The research explicitly excludes any military applications, and fully complies with civilian standards, including the International Convention for the Safety of Life at Sea and the MASS Code.