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Task effectiveness, defined as the likelihood that a system will complete specific tasks under predefined conditions and within a given timeframe, is a crucial indicator of the system’s overall performance and fundamental quality attributes. This metric holds particular significance for the design and optimization of constellation networks. However, current research on multi-layer constellation networks has its shortcomings. Existing methodologies fail to accurately represent the stochastic processes of satellite constellations, neglect the differences in multitask requirements, focus solely on single tasks or static structures, and overlook the dynamic nature of satellite networks. To tackle these challenges, this paper introduces a quantitative framework for assessing the grey multitasking effectiveness of satellite communication constellations. The framework is developed through a systematic approach: firstly, the operational mechanisms of individual satellites and various user task types are analyzed, enabling the construction of Satellite Grey Graphical Evaluation and Review Technique signaling transmission diagrams. Secondly, the network structure is modeled using the Satellite Tool Kit, which aids in determining the completion times and variances for each functional module. Thirdly, the multitasking effectiveness of the constellation is quantified by combining the effectiveness and importance weights of each communication link task. Lastly, the proposed framework is validated through a case study involving a sample satellite communication constellation, and the evaluation results are analyzed to showcase the framework’s applicability and robustness.
Introduction
In large and sparsely populated regions, particularly in remote areas like islands, economic development often faces significant challenges. The lack of adequate ground-based communication infrastructure, due to low population density, hinders local economic growth. Moreover, the vulnerability of infrastructure in these areas is compounded by complex geographic conditions and frequent natural disasters, which cause extensive damage to existing communication networks such as fiber optics and mobile systems. Restoring these systems quickly is often impractical, exacerbating the difficulty of ensuring timely communication and emergency response.
In this context, space-based communication systems, with their extensive coverage, lower susceptibility to risk, and rapid deployment capabilities, have emerged as a vital solution for bridging the communication gap and supporting emergency operations1. Satellite communication systems can be broadly categorized into three types based on their orbital altitudes: Geostationary Earth Orbit (GEO), Medium Earth Orbit (MEO), and Low Earth Orbit (LEO) systems2.
However, as task requirements diversify and network complexity increases, traditional single-layer satellite communication models no longer suffice. This has led to growing research on multi-layer satellite networks, which have become an essential focus in the field of network theory3. Understanding the interactions between satellite modules and the structure of constellation networks has therefore become a critical area of research.
GERT network theory and its applications
In 1962, Eisner introduced a generalized network technology that incorporated the concept of a “Decision Box,” laying the groundwork for probabilistic branching network technologies. This concept was later refined into the Graphical Evaluation and Review Technique (GERT) by scholars such as Elmaghraby and Pritsker4. GERT was initially applied in the Apollo Moon Landing Program, demonstrating its practical value5.
The application of GERT network theory has expanded into various domains, including emergency network modeling6,7, new product development8, 9–10, project schedule management11,12, and equipment reliability assessments13,14. In particular, GERT has proven effective in evaluating the effectiveness of complex networks, such as satellite constellations, by incorporating probabilistic decision-making and network structure analysis. For instance, Shao et al. developed a GERT-based model for evaluating the effectiveness of GEO satellite communication systems, highlighting the technique’s utility in assessing constellation network performance15. Additionally, Fang et al. analyzed the structure and operational process of joint operating systems, establishing the ADC-GERT effectiveness evaluation model16.
Research on effectiveness evaluation methods
Effective evaluation is a crucial aspect of system design and optimization17. Methods for evaluating effectiveness can be categorized into three main types: empirical and analytical methods, simulation and big data methods, and network model evaluation methods.
Empirical and analytical methods rely on historical data and subjective judgment to assess system effectiveness. For example, the Analytic Hierarchy Process (AHP) and Data Envelopment Analysis (DEA) are widely used to evaluate systems in various industries, including unmanned reconnaissance aircraft18 and e-commerce websites19. However, these methods lack objectivity and are often unable to account for the dynamic nature of complex systems.
Simulation and big data evaluation methods, on the other hand, use modeling and data analysis techniques to simulate system behavior and evaluate effectiveness. For example, multi-agent modeling has been applied to evaluate equipment support systems20, while AI and machine learning techniques have been used to evaluate student career counseling systems21. Despite their advantages, these methods are still limited in their application to satellite communication systems.
Network model evaluation methods leverage network properties to evaluate system effectiveness. For instance, Wang et al. developed GERT-based models using combat chain theory to analyze system contribution rates through transition probabilities22, while Shao et al. applied similar approaches to assess stability and effectiveness in communication satellite constellations under information-poor conditions15. However, the explanation of the effectiveness transfer mechanism remained unclear, and the dynamic evolution of the system was not considered. Furthermore, critical task requirements such as real-time performance and reliability are often excluded from effectiveness evaluations, limiting their practical applicability in mission-driven scenarios.
Effectiveness evaluation of satellite communication systems
With the rapid development of aerospace technologies, satellite communication systems have become increasingly critical for military, civilian, and commercial purposes. Evaluating the effectiveness of these systems is essential to ensure their reliability and performance23. Existing evaluation methods have made significant strides, such as the use of three-layer index systems and improved ADC methods to assess the availability and dependability of satellite systems24. Shao et al. developed a state transition process for satellite communication systems based on birth-death processes, deriving the improved ADC comprehensive effectiveness of satellite communication systems by solving for availability, dependability, and capability of available states25. Additionally, Shao et al. integrated the Lz-transform and the ADC method, incorporating the probability requirement of system performance exceeding mission demands into the effectiveness-solving algorithm, and computed the effectiveness of satellite communication systems using computers while preventing state space explosion26. Fang et al. proposed an equivalent transfer function algorithm based on characteristic functions and transfer probabilities using Graphical Evaluation and Review Technology (GERT), evaluating the effectiveness of GEO constellations after an in-depth analysis of constellation structural relationships27.
Despite some progress in satellite network effectiveness evaluation, significant shortcomings remain when dealing with complex multi-task scenarios and dynamic environments. Firstly, existing methods struggle to accurately describe the stochastic processes of satellite constellations when handling complex and uncertain network behaviors resulting from frequent interactions between multi-layer constellation networks and the environment. Secondly, there is a lack of consideration for differences in task requirements under multi-task scenarios, making it difficult to reasonably combine functional modules based on specific task requirements such as real-time performance and reliability. Thirdly, most current effectiveness evaluation methods focus on single tasks or static network structures, lacking the capability to address the computational complexity and uncertainties brought by multiple task types and scenarios. Lastly, existing evaluation methods have limitations in handling the dynamic characteristics of satellite networks, often neglecting the high-speed movement of satellites and the dynamic changes in network topology. This results in an inability to accurately reflect the actual operational state of the network during parameter-solving processes.
Contributions of this work
To address the limitations of existing methods, this paper proposes a multi-tasking effectiveness evaluation framework based on the Graphical Evaluation and Review Technique (GERT). The primary contributions of this work are as follows:
Innovation in Multi-layer Constellation Network Models: This study introduces a multi-layer satellite communication constellation model designed for multi-tasking scenarios. By integrating GERT with grey system theory, we offer a robust theoretical foundation for analyzing complex constellation networks and their interactions.
Innovation in Modular Functional Processing: We modularize the satellite network’s functional components, such as transmission and routing modules, to meet the specific requirements of different tasks. A modular workflow is proposed to facilitate task-specific network needs.
Innovation in Multi-tasking Effectiveness Evaluation: This work establishes a multi-tasking evaluation framework that balances communication delay with task requirements. By extending GERT elements to multi-tasking scenarios, we enable a more accurate evaluation of network performance across various tasks.
Innovation in Parameter Solving with STK Simulation: To address the challenges of network parameter solving in dynamic satellite environments, we employ the Systems Tool Kit (STK) to simulate network topology changes. The Dijkstra algorithm is utilized to determine the transmission paths of each functional module and calculate expected completion times by considering factors like transmission distance and satellite processing time.
Preliminarie
SG-GERT network construction
This paper structurally describes the communication satellite constellation, modeling its constituent elements and their transfer relationships. Considering the unpredictable transmission processes on satellites, the paper analyzes the probability distribution of delays based on historical information. Constructing a satellite grey graphical evaluation and review technique (SG-GERT) network effectively characterizes the network’s uncertainty and complexity. The basic components are shown in Fig. 1.
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Figure 1
Basic unit of the SG-GERT network.
In Fig. 1, i and j represent the communication entities, while the arrow (i, j) indicates the associated activities. denotes the execution probability, represents the random duration of the activity, and U signifies the transfer relationship between the activities.
Assume that the task completion time is a continuous random variable with a probability density function . The characteristic function of , , and its corresponding transfer function are given in15,28:
1
2
Where .Let be n independent random variables with corresponding characteristic functions .The characteristic function of the random variable Y is the product of the characteristic functions of the individual random variables. Therefore, the characteristic function of Y is given by 29.
The k-th order derivative of the characteristic function with respect to t, evaluated at is related to the k-th order moment of the random variable as shown in29:
3
Consider n series activities from the terminal transmitting node to the terminal receiving node in a communication satellite network, with transfer functions for . Additionally, suppose there are m parallel activities with transfer functions for . The equivalent transfer function from the terminal transmitting node to the terminal receiving node is then given by22:4
Let denote the equivalent transfer function between communication entity i and j. The equivalent transfer probability from i to j is obtained by15:5
Abbreviations and symbols
See Table 1.
Table 1. Notations and Abbreviations.
Symbol/Abbreviation | Description |
|---|---|
SG-GERT | Satellite grey-graphical evaluation and review technique |
STK | Satellite tool kit |
GEO | Geostationary earth orbit |
MEO | Medium earth orbit |
LEO | Low earth orbit |
Lower limit of interval grey numbers | |
Upper limit of interval grey numbers | |
Transmission time | |
Processing time of satellite nodes | |
Propagation time between satellite nodes | |
Propagation time from ground to satellite node | |
Propagation time from satellite node to ground | |
Propagation time from satellite node to ground station | |
Orbital altitude of LEO satellites | |
Orbital altitude of GEO satellites | |
Speed of light | |
Total number of LEO satellites | |
Sampling period | |
Simulation runtime | |
Total number of time slots | |
The shortest path between satellite and satellite in time slot | |
The minimum path distance between satellite and satellite during time slot | |
The number of satellites contained within the shortest path | |
The inter-satellite distance between two GEO satellites | |
Functional module | |
The number of GEO satellites included in the inter-satellite links of functional module | |
Packet size | |
The broadband transmission rate of inter-satellite links in LEO satellite network | |
The narrowband transmission rate of inter-satellite links in LEO satellite network | |
The broadband transmission rate of inter-satellite links in GEO satellite network | |
The narrowband transmission rate of inter-satellite links in GEO satellite network | |
The distance from a satellite to the center of the Earth | |
The geocentric angle subtended by satellite and satellite | |
The quantity of GEO satellites | |
EMG | Emergency Communication |
RT1 | Real-time 1 communication |
RT2 | Real-time 2 communication |
NRT | Non-real-time Communication |
NB | Narrowband |
WB | Broadband |
AS | Access satellite |
RS | Relay satellite |
The mean of the satellite node processing time for satellite and satellite during time slot | |
The variance of the satellite node processing time for satellite and satellite during time slot | |
The mean of the processing level time distribution for access satellites | |
The mean of the processing level time distribution for relay satellites | |
The variance of the processing level time distribution for access satellites | |
The variance of the processing level time distribution for relay satellites | |
The mean of the normally distributed end-to-end completion time for functional module | |
The variance of the normally distributed end-to-end completion time for functional module | |
The completion time of communication link | |
The equivalent transfer function of communication link | |
The value of the equivalent transfer function of communication link at | |
The task effectiveness of communication link | |
The characteristic function of communication link | |
Task required completion time | |
Task ’s required completion time | |
The completion time of communication link in task | |
Communication link | |
imp() | The importance of communication link |
imp() | The importance of communication link in task |
imp() | The importance of task |
The equivalent transfer function of the two-layer satellite communication constellation | |
The equivalent transfer function of task in the two-layer satellite communication constellation | |
The set of all communication links from user equipment A to B | |
The task effectiveness of communication link in task | |
The effectiveness of task | |
The multi-task effectiveness of the two-layer satellite communication constellation |
Task segmentation and representation
Task segmentation
Task effectiveness is defined as the probability that a system completes a specific task within a designated timeframe and under specified conditions. To evaluate the multitasking effectiveness of a network, it is essential to account for both the operational environment and the application context of the system.
With the rapid growth of the global economy, there has been an increasing demand for higher data rates and more reliable wireless communication, which heavily relies on communication satellite constellations30. GEO satellites provide extensive coverage and simplified network architectures, but they face challenges such as communication delays and high deployment costs31. In contrast, LEO satellites offer advantages such as reduced communication delay, lower link loss, and lower investment costs32. This paper presents a GEO/LEO satellite communication constellation, as illustrated in Fig. 2.
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Figure 2
Structural configuration of the satellite communication constellation.
Satellite services are typically categorized into handheld communication, onboard communication, data collection, broadcast distribution, and information processing. Satellites follow the ISO seven-layer model for communication signaling and establish a hierarchical service processing framework based on service types. Specifically, LEO satellites are equipped with modules for reliable transmission, route switching, resource allocation, baseband processing, and RF forwarding. GEO satellites, in turn, can perform information processing using the LEO satellite’s processing module, referred to as the FogProcess module. The two-layer satellite communication constellation’s processing architecture and functional modules are illustrated in Fig. 3.
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Figure 3
Diagram of the satellite communication constellation processing structure and functional modules.
Given that ; ,then have33:
6
Communication services are categorized based on processing levels into three types: emergency communication (EMG), real-time communication (RT1, RT2), and non-real-time communication (NRT). The service delivery process involves the transmission of signals from user terminals, which are then relayed via the uplink to access satellites. Depending on the specific requirements, the satellites either select inter-satellite link transmission or perform processing through ground management and control centers, before ultimately transmitting the signals via the downlink to the destination users.The satellite communication constellation serves six primary application sectors: telecommunications, exploration, fisheries and agriculture, emergency relief, oil extraction, and civil disaster mitigation. This two-layer network performs essential functions such as communication, data collection, broadcasting, and fog processing, each supporting either narrowband (NB) or wideband (WB) transmission. In multilayer networks, LEO satellites provide real-time access for communication, data collection, and broadcasting, while GEO satellites, with enhanced computing and storage capabilities, manage information processing34.
Task-specific priorities and structures vary across different applications: (a) In telecommunications, the primary focus is on communication, with data collection and broadcasting serving as secondary functions. (b) In exploration, data collection takes precedence, while communication acts as a supportive function. (c) For fisheries and agriculture, broadcasting is the dominant function, with communication playing a secondary role. (d) During emergency rescue operations, communication is prioritized to facilitate coordination. (e) In oil extraction emergencies, data collection is critical for responding to incidents such as oil spills. (f) Civil disaster mitigation emphasizes broadcasting for the rapid dissemination of critical information and warnings. The task structure is illustrated in Fig. 4.
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Figure 4
Block diagrams of task-specific structural frameworks.
Task representation
Upon analyzing the task types within the satellite communication constellation, it is evident that each type of task is defined according to its specific requirements (see Fig. 4). The structural block diagram representing the six task types of this two-layer satellite communication constellation is now depicted as the SG-GERT signaling transport diagram, as shown in Fig. 5. The interpretation of the nodes in Fig. 5 is provided in Table 2.
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Figure 5
Signaling transmission diagrams of SG-GERT for different tasks.
Table 2. Interpretation of SG-GERT network nodes.
Nodes | Meaning | Nodes | Meaning |
|---|---|---|---|
UE | User equipment 1 | D | LEO_DataCollect_NB_RT2 |
C | LEO_Communicate_WB_RT1 | D | LEO_DataCollect_WB_EMG |
C | LEO_Communicate_NB_RT1 | D | LEO_DataCollect_NB_EMG |
C | LEO_Communicate_WB_NRT | B | LEO_Broadcast_WB_NRT |
C | LEO_Communicate_NB_NRT | B | LEO_Broadcast_NB_NRT |
C | LEO_Communicate_WB_EMG | B | LEO_Broadcast_WB_RT1 |
C | LEO_Communicate_NB_EMG | B | LEO_Broadcast_NB_RT1 |
C | LEO_Communicate_WB_RT2 | B | LEO_Broadcast_WB_RT2 |
C | LEO_Communicate_NB_RT2 | B | LEO_Broadcast_NB_RT2 |
D | LEO_DataCollect_WB_NRT | B | LEO_Broadcast_WB_EMG |
D | LEO_DataCollect_NB_NRT | B | LEO_Broadcast_NB_EMG |
D | LEO_DataCollect_WB_RT1 | F | GEO_FogProcess_WB |
D | LEO_DataCollect_NB_RT1 | F | GEO_FogProcess_NB |
D | LEO_DataCollect_WB_RT2 | UE | User equipment 2 |
The processing levels of the various functional modules in the two-layer satellite communication constellation are presented in Table 3.
Table 3. Functional module processing levels in the two-layer satellite communication constellation.
Function module | Module name | AS processing level | RS processing level |
|---|---|---|---|
1 | LEO_Communicate_WB_RT1 | LEO_WB_Switch | LEO_WB_Route |
2 | LEO_Communicate_NB_RT1 | LEO_NB_Switch | LEO_NB_Route |
3 | LEO_Communicate_WB_NRT | LEO_WB_PForward | LEO_WB_PForward |
4 | LEO_Communicate_NB_NRT | LEO_NB_PForward | LEO_NB_PForward |
5 | LEO_Communicate_WB_EMG | LEO_WB_Transport | LEO_WB_Route |
6 | LEO_Communicate_NB_EMG | LEO_NB_Transport | LEO_NB_Route |
7 | LEO_Communicate_WB_RT2 | LEO_WB_Route | LEO_WB_Route |
8 | LEO_Communicate_NB_RT2 | LEO_NB_Route | LEO_NB_Route |
9 | LEO_DataCollect_WB_NRT | LEO_WB_PForward | LEO_WB_PForward |
10 | LEO_DataCollect_NB_NRT | LEO_NB_PForward | LEO_NB_PForward |
11 | LEO_DataCollect_WB_RT1 | LEO_WB_Switch | LEO_WB_Route |
12 | LEO_DataCollect_NB_RT1 | LEO_NB_Switch | LEO_NB_Route |
13 | LEO_DataCollect_WB_RT2 | LEO_WB_Route | LEO_WB_Route |
14 | LEO_DataCollect_NB_RT2 | LEO_NB_Route | LEO_NB_Route |
15 | LEO_DataCollect_WB_EMG | LEO_WB_Transport | LEO_WB_Route |
16 | LEO_DataCollect_NB_EMG | LEO_NB_Transport | LEO_NB_Route |
17 | LEO_Broadcast_WB_NRT | LEO_WB_PForward | LEO_WB_PForward |
18 | LEO_Broadcast_NB_NRT | LEO_NB_PForward | LEO_NB_PForward |
19 | LEO_Broadcast_WB_RT1 | LEO_WB_Switch | LEO_WB_Route |
20 | LEO_Broadcast_NB_RT1 | LEO_NB_Switch | LEO_NB_Route |
21 | LEO_Broadcast_WB_RT2 | LEO_WB_Route | LEO_WB_Route |
22 | LEO_Broadcast_NB_RT2 | LEO_NB_Route | LEO_NB_Route |
23 | LEO_Broadcast_WB_EMG | LEO_WB_Transport | LEO_WB_Route |
24 | LEO_Broadcast_NB_EMG | LEO_NB_Transport | LEO_NB_Route |
25 | GEO_FogProcess_WB | GEO_WB_FogProcess | GEO_WB_Route |
26 | GEO_FogProcess_NB | GEO_NB_FogProcess | GEO_NB_Route |
Model construction
Parameter solving for functional modules
The satellite communication constellation encompasses four primary functions—communication, Data Collection, Broadcast, and Fog Processing—comprising 26 functional modules, each with a distinct processing flow and hierarchical structure. Identifying the transmission path for each functional module is essential for evaluating the network’s multitasking effectiveness.
This paper employs Dijkstra’s algorithm to determine the optimal transmission paths for functional modules35. The end-to-end completion time t for each module, which depends on network bandwidth and link types (user/inter-satellite/feeder links36), is decomposed as36,37: :
7
where:: Propagation time from ground station to satellite (uplink)
: Propagation time from satellite to ground station (downlink)
: Processing delay at satellite node
: Inter-satellite link (ISL) propagation time
: Data transmission duration (function of packet size & bandwidth)
: Propagation time from the satellite to the signal gateway station
(1) Solution for and
8
Where , are LEO and GEO orbital altitudes, and C is the speed of light.(2) Solution for
The two-layer satellite communication constellation consists of 26 functional modules. Functional modules 3 and 4 are responsible for facilitating two-way feeder links for non-real-time services, while functional modules 9 through 24 support one-way feeder links for data collection and broadcasting. The detailed computation of is provided in Eq. (9).
9
Where is LEO altitude and C is the light speed.(3) Solution for
The satellite propagation time, denoted as , is influenced by the distance between satellites and is further complicated by the dynamic nature of the satellite network and the effects of LEO inclination. To accurately quantify , the STK is employed to simulate the network, enabling the derivation of for functional module m throughout hours.This relationship is mathematically expressed in Eq. (10) and visually illustrated in Fig. 6.
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Figure 6
Spatial geometry of inter-satellite links at identical orbital altitudes.
10
Where : path distance between LEOs, : GEO spacing38, and : number of GEO satellites. (4) Solution forThe processing time of a satellite node, denoted as , is primarily determined by bandwidth and processing level. Due to the rapid movement of satellites, access satellites frequently change between time slots, rendering static network topology inadequate for accurately calculating . Instead, is determined through simulations constructed using the STK over several hours.
In a two-layer satellite communication constellation, assuming functional uniformity within the same orbital altitude, the processing times for different levels follow a normal distribution with mean and variance , where .
For LEO modules (m=1,2,...,24)39:
11
12
For GEO modules ():13
Where : number of satellites on the path, : access satellite, : relay satellite, : Total number of LEOs, : Number of time slots,: Mean/variance of access satellite, : Mean/variance of relay satellite, and :umber of GEO satellites. (5) Solution forThe transmission time for a given task packet size is determined by the link rate and the number of satellites in the transmission path. Due to the high-speed movement of the LEO satellite network, both routing and the number of satellite nodes in the path undergo frequent changes. Consequently, calculating based on a single-moment topology is inherently inaccurate. Instead, a -hour simulation using STK is employed to determine the transmission time for mission module , as expressed in Eq. (14).
14
Where : packet size, : LEO wide/narrowband rate, : GEO wide/narrowband rate.(6) End-to-end completion time
The total end-to-end completion time for each functional module follows a normal distribution :
15
Single task effectiveness
Solving for the task effectiveness of each communication link
The SG-GERT network provides equivalent transfer functions for communication satellite activities and equivalent transfer probabilities between communication entities. Building on traditional GERT theory29 and the Grey GERT network model40, it characterizes the completion time of each communication link.
The completion time for each communication link is is derived as follows:
16
Where : equivalent transfer function of link l, : Lower and upper bounds on completion time.Proof: If the equivalent transfer function of communication link l in a communication satellite network is , its characteristic function is given by:
17
According to the properties of the characteristic function, the value of its derivative of each order at t=0 is functionally related to the origin moments of the random variable at each order. Thus, the completion time of each communication link is given by:18
The task effectiveness of each communication link is measured by the probability that the required completion time meets or exceeds the actual completion time , expressed as:19
Where :task requirement completion time interval, :actual completion time interval of the link.If the equivalent transfer function of communication link l in the communication satellite network is , the completion time of each communication link is given by:
20
Proof : From , we obtain:21
Then, it follows that:22
Thus, the completion time is:23
Communication link importance analysis
The equivalent transfer function of the communication satellite network is derived from the equivalent transfer function of link l. This derivation provides the derivative function of the effectiveness parameter for link l, where the first-order origin moment represents the degree of influence of link l on the network’s effectiveness. The influence degree of link l on the overall task effectiveness of the network, denoted as imp(l), is defined as the proportion of the importance of link l, expressed as:
24
Where : communication link , : All link sets from user devices A to B, and : Network equivalent transfer function.Each of the six task types consists of distinct functional modules. Let denote a task, and yl represent link l in task y. The mean and variance for each functional module can be obtained by Eqs. (7–15). Given the activity probability and the mean time between different functional modules, the equivalent transfer function can be derived from Eq. (2). Subsequently, the equivalent transfer function for each link in task y can be calculated, and the completion time for link l in task y can be determined using Eq. (20). If the task required completion time for task y is , the task effectiveness for each link in each task type can be expressed using Eq. (25) in conjunction with Eq. (19).
25
Combining Eq. (24), the importance of links in each task type, denoted as imp(yl) (in %), can be expressed as shown in Eq. (26).26
Where : the -th link of task , : the equivalent transfer function of task .The task effectiveness of each communication link is weighted by its importance imp(yl) to derive the effectiveness of task y.
27
Multitasking effectiveness
The importance of task y on the task effectiveness of the satellite communication constellation, denoted as imp(y), represents the proportion of the influence of all tasks on the effectiveness of the communication satellite constellation. It is defined as:
28
By applying the GERT parameter transfer algorithm to calculate the task effectiveness for each task type and weighting each task type’s importance imp(y), the multitasking effectiveness E of the satellite communication constellation is given by:29
Where : task effectiveness of link for task , : importance of link for task , : the global importance weight of task y, and : single-task effectiveness.Algorithm for evaluating multitasking effectiveness in satellite communication constellation
To systematically evaluate the multitasking effectiveness of a satellite communication constellation, this paper proposes the following algorithmic framework:
Functional and User Analysis
Analyze the functions and user types supported by the satellite communication network.
Task-Based Module Division
Parse the working mechanism of the satellite constellation based on task types and divide its processing modules (Fig. 3).
Task-Specific Structural Analysis
Construct structural block diagrams for each task type, considering functional and user requirements (Fig. 4).
SG-GERT Network Construction
Develop SG-GERT signaling transmission diagrams for the six task types, incorporating demand modules and processing hierarchies (Fig. 5).
Constellation Modeling in STK
Simulate the satellite constellation using STK to capture dynamic network behavior.
Functional Module Parameterization
Apply Dijkstra’s algorithm to determine transmission paths for each functional module.
Compute the expected completion time and variance for each module using Eqs. (7)–(15).
Link-Level Effectiveness Calculation
Derive the equivalent transfer function between modules (Eq. 2).
Compute the completion time bounds for each link (Eqs. 17, 20).
Evaluate link importance imp(yl) (Eq. 26) and task effectiveness (Eq. 27).
Task Importance Quantification
Determine the global importance imp(y) of each task type using Eq. 28.
Multitasking Effectiveness Aggregation
Compute the overall multitasking effectiveness E by weighting task effectiveness with importance (Eq. 29).
[See PDF for image]
Algorithm 1
Grey Multitasking Effectiveness Evaluation Framework
Case study
A two-layer satellite communication constellation comprises 72 LEOs and 3 GEOs. The LEOs operate at an 86° inclination over 24 hours, evenly distributed across 6 orbital planes, with 12 satellites per plane at an altitude of 1100 km. Each satellite establishes links with 4 neighboring satellites, excluding cross-seam connections. The 3 GEOs are uniformly spaced in geosynchronous orbit at approximately 80°E, 200°E, and 320°E, with an orbital altitude of 35,786 km. Two gateway stations are located in Xi’an and New York. The simulation spans 24 hours with 1-minute sampling intervals, and the transmitted packet size is set to 1000 bytes. Table 5 provides the bandwidth parameters for wide-beam (WB) and narrow-beam (NB) links between LEO and GEO satellites. The two-layer satellite communication constellation implements six mission categories: telecommunication, exploration survey, fisheries and agriculture, emergency relief, oil extraction, and civil disaster reduction. Each task (detailed in Fig. 5) corresponds to distinct processing hierarchies across functional modules (specified in Table 3). The completion times for different processing levels follow normal distributions, with statistical parameters systematically documented in Table 4. Under the condition of a uniform transmission probability of 0.5 between network functional modules (Table 5) across all six task types (y=1 to y=6), Table 6 specifies the required completion time ranges for each task. Given the speed of light as , the multitasking effectiveness of this satellite network is calculated.
Table 4. Time parameters for different processing levels (unit: milliseconds).
Number | Process level | Time distribution mean | Variance |
|---|---|---|---|
1 | LEO_WB_PForward | [217,256] | 30 |
2 | LEO_WB_Switch | [332,378] | 27 |
3 | LEO_WB_Route | [426,450] | 32 |
4 | LEO_WB_Transport | [549,561] | 43 |
5 | GEO_WB_Route | [470,487] | 20 |
6 | GEO_WB_FogProcess | [585,622] | 10 |
7 | LEO_NB_PForward | [289,324] | 48 |
8 | LEO_NB_Switch | [405,443] | 34 |
9 | LEO_NB_Route | [500,517] | 12 |
10 | LEO_NB_Transport | [630,660] | 27 |
11 | GEO_NB_Route | [536,552] | 19 |
12 | GEO_NB_FogProcess | [724,759] | 15 |
Table 5. Satellite communication constellation link bandwidth parameters.
Rate(kbps) | WB | NB |
|---|---|---|
LEO inter-satellite link | 11 | 6.8 |
GEO inter-satellite link | 4 | 2.4 |
Table 6. Required completion times for the six tasks (unit: seconds).
y=1 | [25,40] |
y=2 | [20,24] |
y=3 | [20,27] |
y=4 | [30,38] |
y=5 | [20,22] |
y=6 | [23,25] |
Model results
Parameter solving for functional modules
When the running time ts is 24 hours and the sampling interval r is 1 minute, a total of 1441 time slots are available for the network topology connections. The Dijkstra algorithm is employed to determine transmission paths for functional modules in the LEO layer, identifying the shortest path , its distance , and the number of satellites within the shortest path between satellite node pairs ij across different time slots. For functional modules , GEO satellites are utilized. These satellites are evenly distributed along the equator, resulting in consistent path lengths. Given and d = 42164km the distance between any two GEO satellites is D=73030km. Since GEO satellites are geostationary, their topological connections remain unaffected by time slots. In practical information processing, GEO satellites are suitable for long-distance transmission, and it is generally assumed that user terminals are within the coverage of at most two GEO satellites (). A larger K indicates more GEO satellites involved, leading to longer transmission distances, increased end-to-end completion times, and reduced efficiency. To conservatively assess the multitasking effectiveness of this two-layer satellite network, we set .
Through 24-hour network topology simulations (using STK) and calculations with Eqs. (7)–(15), the and for each functional module were derived, as summarized in Table 7. The data, rooted in detailed network simulations and precise formula derivations, accurately reflect the operational efficiency of functional modules across different time slots.
Table 7. Functional module completion time and variance (unit: seconds).
Function modules | Time Distribution mean | Variance | Function modules | Time Distribution mean | Variance |
|---|---|---|---|---|---|
Communicate_WB_RT1 | [8.4306,8.5253] | 0.0954 | DataCollect_NB_RT2 | [6.1348,6.1865] | 0.0365 |
Communicate_NB_RT1 | [5.9218,5.9942] | 0.0582 | DataCollect_WB_EMG | [8.7639,8.8251] | 0.0757 |
Communicate_WB_NRT | [7.8945,8.0132] | 0.0913 | DataCollect_NB_EMG | [6.2629,6.3275] | 0.0513 |
Communicate_NB_NRT | [5.3806,5.4871] | 0.1461 | Broadcast_WB_NRT | [8.0065,8.1252] | 0.0913 |
Communicate_WB_EMG | [8.6446,8.7058] | 0.0757 | Broadcast_NB_NRT | [5.4926,5.5991] | 0.1461 |
Communicate_NB_EMG | [6.1436,6.2082] | 0.0513 | Broadcast_WB_RT1 | [8.5499,8.6446] | 0.0954 |
Communicate_WB_RT2 | [8.5233,8.5963] | 0.0974 | Broadcast_NB_RT1 | [6.0411,6.1135] | 0.0582 |
Communicate_NB_RT2 | [6.0155,6.0672] | 0.0365 | Broadcast_WB_RT2 | [8.6426,8.7156] | 0.0974 |
DataCollect_WB_NRT | [8.0065,8.1252] | 0.0913 | Broadcast_NB_RT2 | [6.1348,6.1865] | 0.0365 |
DataCollect_NB_NRT | [5.4926,5.5991] | 0.1461 | Broadcast_WB_EMG | [8.7639,8.8251] | 0.0757 |
DataCollect_WB_RT1 | [8.5499,8.6446] | 0.0954 | Broadcast_NB_EMG | [6.2629,6.3275] | 0.0513 |
DataCollect_NB_RT1 | [6.0411,6.1135] | 0.0582 | GEO_FogProcess_WB | [5.6520,5.7260] | 0.0200 |
DataCollect_WB_RT2 | [8.6426,8.7156] | 0.0974 | GEO_FogProcess_NB | [8.5970,8.6670] | 0.0300 |
Single task effectiveness solution
In the case of task 1 (), the equivalence transfer function is calculated using Eq. (2), which describes the performance characteristics of the links between different functional modules. The completion time and importance imp(1l) for task 1 are determined based on Eqs. (20) and (26), reflecting the actual performance of each link in the task transmission.
Task effectiveness is derived by comparing the actual completion time of the task with the required completion time. Specifically, for task 1 (), the task effectiveness is calculated using Eq. (27), based on the required completion time range provided in Table 6, along with the actual completion time and variance data for each link. Task effectiveness reflects the contribution of each link to the task completion, i.e., whether the link can meet the task’s requirement for completion within the specified time.
The completion time and variance for each link are derived from statistical results based on simulations. These data not only reflect the actual performance of each link but also reveal the potential fluctuations that may occur in the completion of the task.
Table 8 presents the completion time ,its importance imp(1l), and task effectiveness for task 1 (). Similarly, the task effectiveness values for tasks 2 through 6 () are calculated through analogous procedures and summarized in Table 9.
Table 8. Task effectiveness at y=1 (unit: seconds).
Link number | Equivalent transfer function | Completion time | Required completion time | Link task effectiveness | Importance | Task Effectiveness | |
|---|---|---|---|---|---|---|---|
l=1 | [30.0956,30.5017] | 0.298 | [25,40] | 0.6429 | 0.0563 | 0.8144 | |
l=2 | [33.0406,33.4427] | 0.308 | 0.4518 | 0.1247 | |||
l=3 | [27.5817,27.9756] | 0.3528 | 0.8067 | 0.0137 | |||
l=4 | [30.5267,30.9166] | 0.3628 | 0.6156 | 0.067 | |||
l=5 | [27.5817,27.9756] | 0.3528 | 0.8067 | 0.0137 | |||
l=6 | [30.5267,30.9166] | 0.3628 | 0.6156 | 0.067 | |||
l=7 | [25.0677,25.4495] | 0.4076 | 0.9708 | 0.0975 | |||
l=8 | [28.0127,28.3905] | 0.4176 | 0.7795 | 0.0009 | |||
l=9 | [27.5868,27.9706] | 0.2608 | 0.8069 | 0.0139 | |||
l=10 | [30.5318,30.9116] | 0.2708 | 0.6156 | 0.0668 | |||
l=11 | [25.0728,25.4445] | 0.3156 | 0.9711 | 0.0978 | |||
l=12 | [28.0178,28.3855] | 0.3256 | 0.7797 | 0.0011 | |||
l=13 | [25.0728,25.4445] | 0.3156 | 0.9711 | 0.0978 | |||
l=14 | [28.0178,28.3855] | 0.3256 | 0.7797 | 0.0011 | |||
l=15 | [22.5589,22.9184] | 0.3704 | 1.000 | 0.2002 | |||
l=16 | [25.5039,25.8594] | 0.3804 | 0.944 | 0.0824 |
Table 9. Task importance and multitasking effectiveness in the two-layer satellite communication constellation.
y=1 | 0.8144 | 0.1542 | 0.7501 |
y=2 | 0.6887 | 0.1861 | |
y=3 | 0.7903 | 0.1861 | |
y=4 | 0.8583 | 0.2048 | |
y=5 | 0.5161 | 0.1352 | |
y=6 | 0.7676 | 0.1352 |
Table 10. Effectiveness evaluation criteria.
Efficiency value | Evaluate |
|---|---|
Very good | |
Good | |
Average | |
Bad | |
Very bad |
Multitasking effectiveness solution
By analyzing the proportion of various tasks in determining the multitasking effectiveness of the communication satellite network and combining this with Eq. (28), the importance of each task type, denoted as imp(y), is calculated. The multitasking effectiveness of this two-layer satellite communication constellation is then determined by weighting the task effectiveness of each task type based on its importance, as expressed in Eq. (29). The specific results are presented in the fourth column of Table 9.
Based on the analysis of effectiveness evaluation results, the criteria for assessing the effectiveness of the satellite communication constellation are outlined in Table 10. In summary, the multitasking effectiveness of this two-layer satellite communication constellation is 0.7501, indicating overall effectiveness of the constellation is ”Good.”
Discussion
Task 4 has the most significant impact on constellation effectiveness
By analyzing the sensitivity of each task’s effectiveness in a two-layer satellite network, we compare , which represents the slope in the graph. The relationship between task effectiveness and multitasking effectiveness is shown in Fig. 7. As observed in Fig. 7, the changes in task effectiveness are associated with network multitasking effectiveness in varying degrees. Specifically, analyzing the slope is crucial, as it reflects how changes in task effectiveness influence multitasking effectiveness. The steeper the hill, the greater the impact a small change in the effectiveness of that task will have on multitasking effectiveness. For example, the curve corresponding to task 4 shows a relatively steep slope, indicating that task 4 has a considerable impact on multitasking effectiveness. In contrast, the curves for task 5 and task 6 have gentler slopes, suggesting their impacts on multitasking effectiveness are minimal. This result further reinforces the critical role of task 4 in the network’s multitasking effectiveness.
[See PDF for image]
Figure 7
Relationship of task effectiveness dynamics to network multitasking effectiveness.
Furthermore, task time plays a significant role in constellation effectiveness. Adjusting task time from 0 to 10 demonstrates its substantial impact on the constellation’s multitasking effectiveness. As shown in Fig. 8, longer task times lead to increased effectiveness, approaching 1, while shorter task times result in a linear decrease in effectiveness. Notably, an increase in task time by 6 units leads to a marked improvement in effectiveness, while a reduction in task time brings effectiveness closer to 0. Therefore, considering marginal effects is essential for optimizing task effectiveness.
[See PDF for image]
Figure 8
Impact of tasking time variations on network multitasking effectiveness.
High impact of the function module 25 and 26 on constellation effectiveness
By analyzing the sensitivity of the satellite communication constellation’s multitasking effectiveness to changes in the mean and variance of activity time distribution for each functional module, the impact of each module on network effectiveness is evaluated. Using the original network effectiveness in Fig. 9 as a baseline, the mean activity time for each module is increased by 5 seconds to observe deviations from the original value. Figure 10 reveals the following: (a) Functional modules 25 and 26 exhibit the greatest impact on network effectiveness, with functional modules 4, 7, 15, and 16 also playing significant roles. (b) Network multitasking effectiveness is primarily influenced by changes in the mean activity time, showing a negative correlation with task completion time, particularly when the increment is less than 0.4. (c) Changes in activity time variance have a comparatively weaker impact, with task effectiveness increasing by approximately 0.05 under the same variance trend, confirming the stability of network functional module effectiveness.
[See PDF for image]
Figure 9
Impact of mean activity time distribution variations on network effectiveness.
[See PDF for image]
Figure 10
Relationship between incremental mean and variance of activity time and network effectiveness.
Compressed expectation time has a significant effect on constellation effectiveness
Figure 11 analyzes the relationship between changes in task importance and network multitasking effectiveness. It examines the effects of increasing the activity time by 5 seconds and the activity time variance by 20 seconds across different processing levels. The horizontal axis at 0 represents the current task importance and network multitasking effectiveness, while vertical axes 1, 2, 3, and 4 correspond to changes at the EMG, RT1, RT2, and NRT processing levels, respectively. The figure demonstrates that increasing activity time significantly influences task importance and network multitasking effectiveness, particularly when activity time is reduced. In contrast, increasing the activity time variance by 5 seconds has negligible effects, and even a 20-second increase results in minimal impact. This consistency across different functional modules further validates the accuracy of the model.
[See PDF for image]
Figure 11
Impact of activity time increments at different processing levels on network multitasking effectiveness.
Conclusion
Satellite communication constellations involve diverse task types, often requiring multiple functions to fulfill them, necessitating a detailed analysis to establish inter-functional relationships. However, the complexity, cost, operational constraints, and uncertainties associated with these constellations pose significant challenges for effectiveness evaluation. To address these challenges, this study constructs the SG-GERT network, analyzing user service requirements and establishing complex relationships between functional modules. The GERT method is adopted to address uncertainty through probabilistic branching and network logic. Parameter-solving methods for each functional module are defined, along with a modular functional flow and a grey multitasking effectiveness evaluation framework. This framework uniquely enables the calculation of multitasking effectiveness despite complex configurations and varied task demands, while accounting for uncertain transmission processes. Focusing on practical requirements facilitates batch task evaluation and enhances efficiency.
Acknowledgements
This research was funded by the Youth Program for Humanities and Social Sciences Research of the Ministry of Education of China under Grant No. 23YJCZH180, titled "Expediting the Development of China’s Communication Satellite System: Dynamic Efficiency Measurement and Strategic Breakthrough Selection." It also received support from the Natural Science Foundation of Jiangsu Higher Education Institutions in China under Grant No. 23KJD120003, titled "Investigation into Multi-task Efficiency Evaluation and Optimization Model of Agent-GERT Network for Dual-layer Maritime Satellite Communication Systems." Additionally, this study aligns with the Key Project of the Jiangsu Provincial Education Science Planning under Grant No. B-b/2024/01/42, titled "Strategic Design and Practical Exploration for High-quality Development of Jiangsu Provincial Universities: Bottleneck Breakthroughs, Comprehensive Evaluation, and Paradigm Construction." The authors also acknowledge partial support from the National Natural Science Foundation of China under Grant No. 62206114 for investigating learning methods assisted by noisy label selection for real-world noisy datasets.
Author contributions
Ruirui Shao: framework development, analyzing task mechanisms, manuscript writing.Ximing Wang: contributed to quantitative methods and statistical analysis.Su Gao: used STK to build the network model and improve the efficiency of multitasking.Yuhong Li: responsible for data collection, preprocessing and validation.Weiqing You: provided theoretical insights on data randomness.Qian Zhang: provided expert feedback on the study design and findings.
Data availibility
The data (source code) supporting this study’s findings are available in the supplementary material of this article. The latest source code is available at https://doi.org/10.5281/zenodo.14869951.
Competing interests
The authors declare no competing interests.
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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