On an Ultra-Dense LEO-Satellite-Based Computing Network Constellation Design

Engineering ›› 2025, Vol. 54 ›› Issue (11) : 103 -114. DOI: 10.1016/j.eng.2025.06.007

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Received	Accepted	Published
2024-11-25	2025-06-09	2025-06-16
Issue Date	Revised Date
2025-06-16	2025-05-24

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Abstract

Commercial ultra-dense low-Earth-orbit (LEO) satellite constellations have recently been deployed to provide seamless global Internet services. To improve the satellite network transmission efficiency and provide robust wide-coverage computing services for future sixth-generation (6G) users, growing attention has been focused on LEO-satellite-based computing networks, to which ground users can offload computation tasks. However, how to design a LEO satellite constellation for computing networks, while considering discrepancies in the computing requirements of different regions, remains an open question. In this paper, we investigate an ultra-dense LEO-satellite-based computing network to which ground user terminals (UTs) offload part of their computing tasks to satellites. We formulate the ultra-dense constellation design problem as a multi-objective optimization problem (MOOP) to maximize the average coverage rate, transmission capacity, and computational capability, while minimizing the number of satellites. In order to depict the connectivity characteristics of satellite-based computing networks, we propose a terrestrial–satellite connectivity model to determine the coverage rate in different regions. We design a priority-adaptive algorithm to design the optimal inclined-orbit constellation by solving this MOOP. Simulation results verify the accuracy of our theoretical connectivity model and show the optimal constellation deployment, given quality-of-service (QoS) requirements. For the same number of deployed LEO satellites, the proposed constellation outperforms its existing counterparts; in particular, it achieves 25%–45% performance improvements in the average coverage rate.

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Keywords

Low-Earth-orbit satellite constellation / Satellite-based computing network / Multi-objective optimization

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Yijing Sun, Boya Di, Ruoqi Deng, Lingyang Song. On an Ultra-Dense LEO-Satellite-Based Computing Network Constellation Design. Engineering, 2025, 54(11): 103-114 DOI:10.1016/j.eng.2025.06.007

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1. Introduction

As intelligent applications and data traffic continue to evolve, the sixth-generation (6G) concept aims to build a large-scale and autonomous network [1,2] that can provide more flexible services, such as the Internet of Remote Things, smart city scenarios, large-scale machine communication, and emergency rescue [2,3]. However, terrestrial networks may suffer from tremendous traffic pressure and cannot support the services described above, due to their limited transmission and computational resources, and restricted coverage [4]. As an emerging technology, the ultra-dense low-Earth-orbit (LEO) satellite network, which is characterized by wide coverage and high capacity [[5], [6], [7]], can provide ground users with differentiated service requirements as a significant part of 6G [8]. Nevertheless, traditional satellite networks that adopt a “bent-pipe” architecture, in which satellites receive and retransmit signals without processing them, are likely to break down at the scale of an ultra-dense constellation [9] because each satellite acts only as a communication link to transmit a large amount of raw data. Therefore, satellite-based computing networks have been proposed as an effective solution in which onboard edge servers are deployed on satellites to provide wide-coverage computation services to ground users.

1.1. Related works

Recent advancements in LEO satellites and satellite edge computing have enabled the development of satellite-based computing networks [10]. Consequently, in the literature, multiple aspects related to satellite-based computing networks based on fixed satellites and LEO satellite constellation designs have been considered. Considerable progress has been made in research on satellite–terrestrial integrated edge computing networks, especially in the context of end-to-end delay, energy efficiency, and resource allocation. In Refs. [[11], [12], [13], [14], [15]], various algorithms are proposed to minimize the energy consumption of the whole network or of Internet of Things (IoT) mobile devices with delay tolerance. Computational latency was also carefully considered [[16], [17], [18]] by optimizing the offloading strategy and resource allocation.

Nevertheless, these studies focus on performance optimization under the premise of determined satellite positions, with little research examining the impact of satellite layout on the overall performance of satellite-based computing networks. The positions of individual satellites are determined by the constellation design, and different constellation designs alter the relative positional relationships between satellites and terrestrial users, as well as the inter-satellite relationships. Thus, satellite layout affects the performance of the entire LEO satellite network. Objectives of LEO satellite constellation design vary according to different requirements, including global or regional coverage [[19], [20], [21]], IoT [22,23], lunar exploration [24], and backhaul capacity [25]. To determine the minimum number of LEO satellites needed for a survivable LEO satellite network, while addressing challenges such as deployment cost and space debris, Lai et al. [26] proposed a requirement-driven constellation optimization mechanism based on an existing constellation. Furthermore, rather than focusing on traditional uniform constellations, Deng et al. [25] designed a three-dimensional (3D) optimized non-uniform polar-orbit constellation with a focus on ground user backhaul capacity requirements.

However, a constellation design often involves the formulation of multi-objective optimization problems (MOOPs) to address the complex tradeoffs between various performance metrics, rather than focusing on optimizing a single objective such as capacity or survivability. Therefore, a variety of multi-objective optimization algorithms have been applied to constellation design. Chen et al. [19] proposed the use of an elite strategic genetic algorithm to design a constellation of very-low-Earth-orbit-based satellites for aviation and marine users. Kak and Akyildiz [20] and Tang et al. [22] respectively employed simulated annealing and multi-objective Manta ray foraging optimization algorithms for the design of CubeSats-based constellations. Wang et al. [21] and Dai et al. [23] took multiple quality-of-service (QoS) metrics of the satellite network into account and solved the MOOP using the non-dominated sorting genetic algorithm (NSGA)-II and multilayer tabu search algorithm. The NSGA-II algorithm can also be used for the lunar satellite constellation design problem [24], which focuses on communication and navigation capabilities.

While existing research provides valuable insights into LEO satellite constellation design, these studies’ algorithms for constellation design are mostly based on traditional communication LEO network objectives. It is important to consider the specific performance of the computation network, including the offloading performance metrics and differential requirements of computing tasks. Additionally, many studies do not reveal the impact of constellation parameters on different optimization objectives due to a reliance on intelligent optimization algorithms to solve the MOOPs. Thus, the constellation design problem for satellite-based computing networks remains insufficiently explored and requires further investigation, as does the analysis of how constellation parameters affect network performance.

2. Open problems and contributions

Existing LEO satellite constellation design research mainly focuses on communication satellites, with optimization objectives primarily targeting communication network performance or coverage performance. Therefore, the design of global satellite-based computing network constellations remains to be addressed. Designing a reliable and efficient LEO satellite constellation for computing networks is one of the most critical challenges for enhancing network performance. First, due to the characteristics of the satellite internet network, with high satellite mobility and the Earth’s rotation, the connectivity between ground user terminals (UTs) and satellites dynamically changes and involves complex geometric relationships. Second, in contrast to traditional satellite constellations, which aim at seamless global or regional coverage, computational satellite networks focus more on reducing latency and providing reliable computing capabilities through a rational constellation layout. Due to limited computing capabilities, it is non-trivial to design a constellation for computing networks that can satisfy the delay and capacity requirements of ground users. Third, since satellite constellations for computing must balance multiple aspects of performance, determining the optimal configuration of the constellation is a significant challenge.

Unlike the above works, this paper proposes an ultra-dense LEO satellite constellation design for satellite-based computing networks in which the satellites have computational capabilities in addition to traditional communication functions, and the variability of satellite coverage requirements in different regions is taken into account. The main contributions of this paper can be summarized as follows.

•We construct a terrestrial–satellite connectivity model based on the mathematical relationship between an inclined-orbit constellation and the Earth in 3D space. This theoretical model enables rapid and accurate computation of the multi-connectivity coverage rate to a UT on the ground, and its accuracy is verified through validation with Systems Tool Kit (STK)’s Figure of Merit.

•By considering both the computing and communication performance of a satellite network, we formulate an MOOP for constellation design. To address this problem, we design a priority-adaptive inclined-orbit constellation design (PA-ICD) algorithm, which enables the optimal constellation configuration to be obtained under different prioritized objectives, given QoS requirements. We also decouple the parameters to derive a near-optimal inclined constellation design criterion for the orbital altitude and inclination.

•The simulation results verify that our proposed constellation has a higher average coverage rate, transmission capacity, and computing capability compared with four representative state-of-the-art constellations, given the same total number of satellites. The simulation results are also consistent with the theoretical analysis.

The rest of this paper is organized as follows. In Section 2, the system model of the satellite edge-assisted computation network is presented. In Section 3, we describe relevant QoS metrics and formulate an MOOP for inclined-orbit constellation design. Then, the PA-ICD algorithm to solve the MOOP is introduced in Section 4. Section 5 presents our simulation results. Finally, the conclusion is drawn in Section 6.

3. System model

In this section, we first introduce an uplink LEO satellite edge-assisted computation network, where ground UTs offload their tasks to the LEO satellites for computing. We then present the inclined-orbit constellation, terrestrial–satellite connectivity, and offloading models.

3.1. Scenario description

Terrestrial servers struggle with processing the surge of intelligent tasks due to their limited computational capability, whereas satellites equipped with edge computing servers can collaborate with ground servers to assist in computations. Additionally, unlike the characteristics of terrestrial infrastructures, which are prone to damage and have restricted coverage, satellites can provide reliable computing services in remote areas, large-scale events, and emergency scenarios [2]. Therefore, as shown in Fig. 1, we consider an uplink offloading process in which LEO satellites offer computing services for ground UTs.

To achieve this purpose, we propose the use of an inclined-orbit LEO satellite constellation to serve ground users over middle–low latitude regions, since user offloading requirements are concentrated in those populated areas. For each UT, there are N types of terrestrial tasks with different latency requirements—that is, computational capability and transmission capacity requirements—that need to be offloaded to the satellites for computation. We assume that the UTs and LEO satellites are equipped with multiple antennas, so the UTs offload their tasks to multiple satellites over the Ka-band. Each satellite can serve UTs within its coverage area by allocating offloading resources. Therefore, the entire offloading process is divided into five steps: ① Each terrestrial task sends an offloading request to the connectable satellites. ② When the LEO satellites receive offloading requests from UTs, the satellites formulate dynamic resource-allocation strategies for computing and transmission based on the task-completion requirements of the UTs. These allocation strategies are then broadcast to all UTs within the coverage. ③ UTs access LEO satellites via the popular orthogonal frequency-division multiple access scheme to transmit their tasks. ④ Each satellite performs onboard computing using the allocated resources. ⑤ After completing the computation, the results are returned to the user over the wireless link.

3.2. Inclined-orbit LEO satellite constellation model

Due to the issues of redundancy in polar satellite numbers and scarcity in equatorial satellite numbers, a polar orbit constellation is not a suitable constellation model to aid in task offloading for ground users at middle–low latitude regions. However, inclined-orbit constellations can adjust the inclination of their orbit to change their overlapping coverage of areas of concern [27]. Such constellation models can be flexibly designed and provide better satellite revisit frequencies for populated areas [28]. Walker-delta is one of the most-used inclined-orbit models because it is relatively simple to deploy, manage, and maintain, and has a relatively low long-term system cost [24]. The satellites in this model are uniformly distributed on inertial circular orbits.

In our model, we focus on the Walker-delta inclined constellation design, which can be defined using the following notation: i = K/Q/F. Q and W are the number of orbital planes and the satellite numbers on each plane, respectively. K is the total number of satellites in the constellation, which is equal to QW. The orbit altitude and inclination are denoted as h and i, respectively. The phase difference between corresponding satellites on neighboring orbital planes is represented as the phase factor F, which acts to stagger satellites on different orbital planes from each other and is an integer in the range of [0, Q − 1]. We then define the satellite w (1 ≤w ≤ W) in the orbital plane q (1 ≤ q ≤ Q) as k_q,w, so the LEO satellites in the constellation can be defined as

K

= {k₁_,₁, …, k_q,w, …, k_Q,W}. The eccentricity and perigee angle of the circular orbit are 0 [19], and the right ascension of the ascending node (RAAN) Ω_q,w and true anomaly υ_q,w of satellite k_q,w are given as follows:

(1)

Ω_{q, w} = Ω_{ref} + \frac{2 π (q - 1)}{Q}

(2)

υ_{q, w} = υ_{ref} + \frac{2 π (w - 1)}{W} + \frac{2 π (q - 1) F}{QW}

where Ω_ref and υ_ref represent the RAAN and the true anomaly of the reference satellite, respectively.

The position of satellite k_q,w underlying the Cartesian coordinate frame can be expressed as

(3)

(\begin{matrix} x_{q, w} \\ y_{q, w} \\ z_{q, w} \end{matrix}) = (h + R_{e}) (\begin{matrix} \cos υ_{q, w} \cos Ω_{q, w} - \sin υ_{q, w} \cos i \sin Ω_{q, w} \\ \cos υ_{q, w} \sin Ω_{q, w} + \sin υ_{q, w} \cos i \cos Ω_{q, w} \\ \sin υ_{q, w} \sin i \end{matrix})

where R_e is the mean radius of the Earth^†¹. x_q,w, y_q,w, and z_q,w denote the satellite’s positions along the X, Y, and Z axes, respectively. By deriving the 3D coordinate relationship between the satellites and the Earth, the latitude of the subsatellite point ψ_q,w—which is on the Earth’s surface directly below satellite k_q,w—can be calculated as follows:

(4)

ψ_{q, w} = \arcsin (\sin υ_{q, w} \sin i)

For each satellite, the angular radius of the satellite coverage circle φ can be given by

(5)

φ = \arccos (\frac{R_{e}}{R_{e} + h} \cos θ_{\min}) - θ_{\min}

where θ_min represents the minimum elevation angle at the UTs, and the corresponding coverage area S is

(6)

S = 2 π R_{e}^{2} (1 - \cos φ)

3.3. Terrestrial–satellite connectivity model

Terrestrial–satellite connectivity can be determined based on the inclined orbital structures and geometric relationships in spherical trigonometry. More specifically, the number of connectable satellites varies considerably at different latitudes [29] for the inclined-orbit constellation. Considering the long-term averaged state for the dense LEO satellite constellation analysis, the number of connectable satellites can be transformed into the multi-connectivity coverage rate [21]. The multi-connectivity coverage rate refers to the capability of a UT to establish communication links with multiple satellites within the constellation. We divide the target area, where the maximum latitude value is l_tar, into M disjoint spherical zones based on latitude. The upper and lower bounds of latitude for region m are lmax m and lmin m, respectively. For a specific region m, the multi-connectivity coverage rate can be determined using the ratio of the satellite coverage area to the surface area of the region. Therefore, the coverage rate σ_m of each UT within region m can be expressed as

(7)

σ_{m} = \sum_{q = 1}^{Q} \sum_{w = 1}^{W} \frac{S_{c}^{m, q, w}}{S_{g}^{m}}

where Sm,q,w c and Sm g represent the coverage area provided by satellite k_q,w and the surface area of the Earth within region m, respectively. The coverage rate of the latitude region m can be determined by traversing the coverage status of all satellites.

The coverage contributed by satellite k_q,w to the latitude region m is shown in Fig. 2. The surface area of the Earth within region m is

(8)

S_{g}^{m} = 2 π R_{e}^{2} (\sin l_{m}^{\max} - \sin l_{m}^{\min})

Corresponding to solving for the area covered by satellite k_q,w in region m, the irregularly shaped area of the covered region can be determined by subtracting the overlapping areas of the upper and lower spherical caps. Therefore, Sm,q,w c can be expressed as

(9)

S_{c}^{m, q, w} = \bar{S} (l_{m}^{\min}, φ, ψ_{q, w}) - \bar{S} (l_{m}^{\max}, φ, ψ_{q, w})

where

\bar{S} (l_{m}^{\min}, φ, ψ_{q, w})

and

\bar{S} (l_{m}^{\max}, φ, ψ_{q, w})

are the satellite coverage areas beyond the latitude of lmin m and lmax m, respectively.

Proposition 1

\bar{S} (l_{m}^{\min}, φ, ψ_{q, w})

can be calculated as Eq. (10):

(10)

\begin{matrix} \bar{S} (l_{m}^{\min}, φ, ψ_{q, w}) = 2 R_{e}^{2} \cdot Re [π - 2 \sin l_{m}^{\min} \arcsin (\sqrt{\frac{\sin (\frac{1}{2} (φ + l_{m}^{\min} - ψ_{q, w})) \sin (\frac{1}{2} (φ + ψ_{q, w} - l_{m}^{\min}))}{\cos ψ_{q, w} \cos l_{m}^{\min}}}) \\ - 2 \cos φ \arcsin (\sqrt{\frac{\cos (\frac{1}{2} (φ + l_{m}^{\min} + ψ_{q, w})) \sin (\frac{1}{2} (φ + ψ_{q, w} - l_{m}^{\min}))}{\sin φ \cos ψ_{q, w}}}) \\ - 2 \arcsin (\sqrt{\frac{\cos (\frac{1}{2} (φ + l_{m}^{\min} + ψ_{q, w})) \sin (\frac{1}{2} (φ + l_{m}^{\min} - ψ_{q, w}))}{\sin φ \cos l_{m}^{\min}}})] \end{matrix}

where Re represents real part.

Proof: See Section S1 in Appendix A.

The computation of

\bar{S} (l_{m}^{\max}, φ, ψ_{q, w})

follows the same process as that of

\bar{S} (l_{m}^{\min}, φ, ψ_{q, w})

3.4. Offloading model

Task offloading refers to the process of transferring computational tasks from ground UTs to one or more satellites within the constellation. Terrestrial offloading tasks are distinguished according to different latency sensitivities, where latency-sensitive tasks require higher computation and transmission rates than latency-tolerant tasks. The requirement of the nth (1 ≤ n ≤ N) type of task is defined as (c_n, r_n), where c_n and r_n denote the computational capability and transmission data rate requirements of type n tasks, respectively. We consider the distribution of tasks on each UT to conform to the homogeneous Poisson point process, where the average UT density of the target area is ρ, and the arrival number of type n tasks on each UT is modeled with the intensity λ_n.

3.4.1. Computing model

Each UT that offloads its tasks to the satellite for computation has limited access to computational resources due to the limited onboard computational capability of each satellite. We assume that the maximum computational capability on each satellite is f_max, and the computational capability allocated to type n tasks is f_n, which denotes the number of central processing unit (CPU) operations per unit time. The allocated computational capability should satisfy the constraint

\sum_{n = 1}^{N} f_{n} \leq f_{\max}

. According to the expectation calculation of the Poisson distribution, the probability of v tasks existing at each UT (f(x_n = v)) can be expressed as

(11)

f (x_{n} = v) = {(λ_{n})}^{v} \frac{e^{- λ_{n}}}{v!}

where x_n is the number of type n tasks. We define the per-satellite computational capability available to type n tasks in area u as C_u_,_n. Then, the expected value of C_u_,_n, denoted as

E (C_{u, n})

, can be calculated as

(12)

\begin{matrix} E (C_{u, n}) = f_{n} E ({\bar{C}}_{u, n}) = \frac{f_{n}}{ρ_{u} S} \sum_{v = 1}^{\infty} {(λ_{n})}^{v} \frac{e^{- λ_{n}}}{v!} \frac{1}{v} \\ = \frac{f_{n}}{ρ_{u} S} (Ei (λ_{n}) - \ln (λ_{n}) - γ) e^{- λ_{n}} \end{matrix}

where

Ei (λ_{n}) = \int_{- \infty}^{λ_{n}} \frac{e^{t}}{t} d t

is the exponential integral function, γ is the Euler constant, and ρ_u is the density of UTs of area u. We define

{\bar{K}}_{u, n}^{CO}

as the number of satellites within the unit region of area u that satisfy c_n. Consequently,

{\bar{K}}_{u, n}^{CO}

can be formulated as follows:

(13)

{\bar{K}}_{u, n}^{CO} = \frac{c_{n}}{E (C_{u, n}) S} = \frac{c_{n} ρ_{u}}{f_{n} (Ei (λ_{n}) - \ln (λ_{n}) - γ) e^{- λ_{n}}}

3.4.2. Communication model

We define the transmission resource (i.e., bandwidth) of each satellite allocated to type n tasks within the satellite coverage area as ϕ_n, and the available transmission resource over the Ka-band for each satellite is ϕ_max, which is divided into J orthogonal subchannels. Without loss of reasonableness, the transmission resource should satisfy the constraint

\sum_{n = 1}^{N} ϕ_{n} \leq ϕ_{\max}

Let us define P as the transmission power for a UT connected to one LEO satellite and G as the power gain factor the antenna amplifier provides. To describe the terrestrial–satellite channel, α denotes the path loss exponent, and σ² is the additive white Gaussian noise (AWGN) variance. In addition, given that LEO satellites assign subchannels to tasks randomly, there is a probability that

ρ_{u}^{'} = ρ_{u} \sum_{n = 1}^{N} λ_{n} / J

might be using the same subchannel [25]. Therefore, according to Campbell’s theorem [30] (Appendix A in Ref. [25]), the expected value of interference I_u within area u for task offloading link, denoted as

E (I_{u})

, is

(14)

E (I_{u}) = \{\begin{matrix} \frac{ρ_{u}^{'} π P G R_{e}}{R_{e} + h} [\ln (h^{2} + 2 R_{e} h) - \ln (d_{\max}^{2})], α = 2 \\ \frac{2 π R_{e} ρ_{u}^{'} P G}{(α - 2) (R_{e} + h)} [d_{\max}^{2 - α} - {(2 R_{e} h + h^{2})}^{\frac{2 - α}{2}}], α > 2 \end{matrix}

where d_max represents the maximum distance from the Earth’s surface to the satellite within its coverage area and can be calculated by

(15)

d_{\max} = \sqrt{R_{e}^{2} + {(R_{e} + h)}^{2} - 2 R_{e} (R_{e} + h) \cos φ}

Then, the achievable capacity R_u,n for task n assigned to the UT in area u can be given by

(16)

R_{u, n} = ϕ_{u, n} E (\log_{2} (1 + \frac{PG d^{- α}}{σ^{2} + E (I_{u})}))

where

E

represents the expected value operator and d is the distance between UT and the satellite. The spectral efficiency [25] can be calculated by

(17)

\begin{matrix} E (\log_{2} (1 + \frac{PG d^{- α}}{σ^{2} + E (I_{u})})) = \frac{1}{2 \ln 2 R_{e} (R_{e} + h) (1 - \cos φ)} \{d_{\max}^{2} [- \frac{α}{2}_{2} F_{1} (1, - \frac{2}{α}; 1 - \frac{2}{α}; - \frac{PG d_{\max}^{- α}}{σ^{2} + E (I_{u})}) \\ + \ln (1 + \frac{PG d_{\max}^{- α}}{σ^{2} + E (I_{u})}) + \frac{α}{2}] - h^{2} [- \frac{α}{2}_{2} F_{1} (1, - \frac{2}{α}; 1 - \frac{2}{α}; - \frac{PG h^{- α}}{σ^{2} + E (I_{u})}) \\ + \ln (1 + \frac{PG h^{- α}}{σ^{2} + E (I_{u})}) + \frac{α}{2}]\} \end{matrix}

where ₂F₁ is the hypergeometric function of the first kind. Therefore, we define

E (R_{n, u})

as the average capacity provided by each satellite for type n tasks in area u, and it can be denoted as

(18)

\begin{matrix} E (R_{n, u}) = ϕ_{n} E ({\bar{R}}_{u, n}) = \frac{ϕ_{n}}{ρ_{u} S} \sum_{v = 1}^{\infty} {(λ_{n})}^{v} \frac{e^{- λ_{n}}}{v!} \frac{1}{v} E (\log_{2} (1 + \frac{PG d^{- α}}{σ^{2} + E (I_{u})})) \\ = \frac{ϕ_{n}}{ρ_{u} S} (Ei (λ_{n}) - \ln (λ_{n}) - γ) e^{- λ_{n}} E (\log_{2} (1 + \frac{PG d^{- α}}{σ^{2} + E (I_{u})})) \end{matrix}

Then, the number of satellites

{\bar{K}}_{u, n}^{TR}

satisfying r_n within the unit region of coverage area u can be formulated as follows:

(19)

{\bar{K}}_{u, n}^{TR} = \frac{r_{n}}{E (R_{u, n}) S} = \frac{r_{n} ρ_{u}}{ϕ_{n} (Ei (λ_{n}) - \ln (λ_{n}) - γ) e^{- λ_{n}} E (\log_{2} (1 + \frac{PG d^{- α}}{σ^{2} + E (I_{u})}))}

4. Inclined-orbit constellation design problem formulation

In this section, with the aim of designing an inclined-orbit constellation for satellite-based computing networks, we first minimize the number of satellites required to meet the computational and transmission requirements by optimizing the resource-allocation scheme. Then, based on the optimal result, an MOOP concerning the constellation design is presented, in which multiple QoS metrics are integrated.

4.1. Resource-allocation scheme

We define the computational and transmission resource-allocation vectors as F = [f₁, …, f_n, …, f_N] and Φ = [ϕ₁, …, ϕ_n, …, ϕ_N], respectively. According to the offloading model described in Section 2, the computational capability allocation optimization sub-problem

P

1 can be formulated as

(20a)

P 1 : \min_{F} \max_{n} {\bar{K}}_{u, n}^{CO}

(20b)

s . t . \sum_{n = 1}^{N} f_{n} \leq f_{\max}

(20c)

f_{n} \geq 0, \forall n \in {1, . . ., N}

where the optimization objective of

P

1 is to minimize the maximum number of satellites for computation within the target unit area. After introducing an auxiliary variable and transforming

P

1 into a new optimization problem,

P

1 can be solved by utilizing the Karush–Kuhn–Tucker (KKT) conditions.

Proposition 2

The optimal computational resource-allocation scheme for task offloading

f_{n}^{*}

(21)

f_{n}^{*} = f_{\max} \frac{c_{n}}{E ({\bar{C}}_{u, n}) \sum_{n = 1}^{N} \frac{c_{n}}{E ({\bar{C}}_{u, n})}}

and the minimum required number of satellites for computation

{\bar{K}}_{u}^{CO}

can be expressed as

(22)

{\bar{K}}_{u}^{CO} = \frac{\sum_{n = 1}^{N} \frac{c_{n}}{E ({\bar{C}}_{u, n})}}{f_{\max} S}

Proof: See Section S2 in Appendix A.

In the same case, we can formulate the optimal transmission resource-allocation scheme for task offloading

ϕ_{n}^{*}

(23)

ϕ_{n}^{*} = ϕ_{\max} \frac{r_{n}}{E ({\bar{R}}_{u, n}) \sum_{n = 1}^{N} \frac{r_{n}}{E ({\bar{R}}_{u, n})}}

The minimum required number of satellites for transmission

{\bar{K}}_{u}^{TR}

is given by

(24)

{\bar{K}}_{u}^{TR} = \frac{\sum_{n = 1}^{N} \frac{r_{n}}{E ({\bar{R}}_{u, n})}}{ϕ_{\max} S}

4.2. Problem formulation

It is essential to consider the number of satellites in a constellation, given the substantial costs associated with satellite manufacturing, launch, and maintenance. Therefore, one of our goals is to minimize the total number of LEO satellites. In addition, we describe four key QoS metrics that indicate the validity and reliability of the satellite-based computing network. Then, the MOOP is formulated.

4.2.1. QoS metrics description

(1) Minimum coverage rate

\bar{σ}

. To quantitatively assess the reliability of the proposed LEO satellite constellation, we introduce the parameter

\bar{σ}

, which represents the minimum coverage rate within the target area.

\bar{σ}

can be calculated as

(25)

\bar{σ} = \min_{m} \sum_{q = 1}^{Q} \sum_{w = 1}^{W} \frac{S_{c}^{m, q, w}}{S_{g}^{m}}

This metric provides a basic guarantee within the specified latitude range to avoid a scenario in which the UT cannot connect to enough satellites.

(2) Average coverage rate

\bar{σ}

. When UTs offload tasks to satellites, densely populated areas have more computational and transmission demands for offloading compared with sparsely populated areas. Consequently, a greater satellite coverage rate—that is, sufficient connectable satellites—is required for densely populated areas. Moreover, a constellation deployment that accounts for variations across different regions can prevent task queuing or latency. Therefore, we introduce the average coverage rate

\bar{σ}

when designing the inclined-orbit constellation. It can be denoted as follows:

(26)

\bar{σ} = \sum_{m = 1}^{M} \sum_{q = 1}^{Q} \sum_{w = 1}^{W} \frac{β_{m} S_{c}^{m, q, w}}{S_{g}^{m}}

where β_m denotes the population proportion within the latitude of region m. Maximizing the multi-connectivity coverage rate can enhance the reliability and flexibility of the satellite-based computing network, preventing UTs from being unable to offload because some visible satellites fail. In addition, an average coverage rate is essential to ensure consistency between the connectivity coverage rate and the requirements of UTs and thereby improve the efficient utilization of network resources.

(3) Average computational capability

\bar{C}

and transmission capacity

\bar{R}

. From our derivation in Proposition 2, it follows that the required number of satellites for computation

{\bar{K}}^{CO}

can be further calculated as

(27)

{\bar{K}}^{CO} = \int_{u \in S_{tar}} {\bar{K}}_{u}^{CO} d u

where S_tar represents the target area. The number of satellites

\bar{K}

that the UTs can offload to can be calculated by

(28)

\bar{K} = \frac{\sum_{m = 1}^{M} \sum_{q = 1}^{Q} \sum_{w = 1}^{W} S_{c}^{m, q, w}}{S}

which is proportional to the satellite coverage area within the target region. By means of the inequality transformation

\bar{K} \geq {\bar{K}}^{CO}

, we define

\bar{C}

as denoting the average computational capability for each UT; it can be given by

(29)

\bar{C} = \frac{\sum_{m = 1}^{M} \sum_{q = 1}^{Q} \sum_{w = 1}^{W} S_{c}^{m, q, w}}{S \int_{u \in S_{tar}} ρ_{u} d u} f_{\max}

The computational capability requirement for each UT is

\sum_{n = 1}^{N} c_{n} e^{λ_{n}} / (E i (λ_{n}) - \ln (λ_{n}) - γ)

.In the same way, we define

\bar{R}

as the average transmission capacity for each UT; it is given by

(30)

\bar{R} = \frac{\sum_{m = 1}^{M} \sum_{q = 1}^{Q} \sum_{w = 1}^{W} S_{c}^{m, q, w}}{S \int_{u \in S_{tar}} \frac{ρ_{u}}{E (\log_{2} (1 + \frac{PG d^{- α}}{σ^{2} + E (I_{u})}))} d u} ϕ_{\max}

The transmission data rate requirement for each UT is

\sum_{n = 1}^{N} r_{n} e^{λ_{n}} / (E i (λ_{n}) - \ln (λ_{n}) - γ)

. The average computational capability and transmission capacity can then be derived, where ρ_u is aligned with the population proportion.

4.2.2. MOOP formulation for constellation design

To begin with, we focus on minimizing the number of satellites and maximizing the three QoS evaluation metrics by jointly optimizing the constellation parameters (Q,W,h,i,F). It should be noted that F is set to 1, as the same configuration in Ref. [20,21,31] has the ability to simplify the constellation design while providing better coverage performance. Consequently, the constellation design MOOP

P

2 is formulated as follows:

(31a)

P 2 : MOOP ≜ \{\begin{matrix} \min_{(Q, W, h, i)} K \\ \max_{(Q, W, h, i)} \bar{σ} \\ \max_{(Q, W, h, i)} \bar{C} \\ \max_{(Q, W, h, i)} \bar{R} \end{matrix}\}

(31b)

s . t . \bar{C} \geq \sum_{n = 1}^{N} c_{n} e^{λ_{n}} / (Ei (λ_{n}) - \ln (λ_{n}) - γ)

(31c)

\bar{R} \geq \sum_{n = 1}^{N} r_{n} e^{λ_{n}} / (Ei (λ_{n}) - \ln (λ_{n}) - γ)

(31d)

\bar{σ} \geq {\bar{σ}}_{\min}

(31e)

\bar{σ} \geq {\bar{σ}}_{\min}

(31f)

h_{\max} \geq h \geq h_{\min}

(31g)

i_{\max} \geq i \geq i_{\min}

where

(\sum_{n = 1}^{N} c_{n} e^{λ_{n}} / (Ei (λ_{n}) - \ln (λ_{n}) - γ), \sum_{n = 1}^{N} r_{n} e^{λ_{n}} / (Ei (λ_{n}) - \ln (λ_{n}) - γ), {\bar{σ}}_{\min}, {\bar{σ}}_{\min})

denotes the given QoS requirements, while

{\bar{σ}}_{\min}

and

{\bar{σ}}_{\min}

respectively denote the minimum and average coverage rate thresholds. The orbital configuration is constrained by altitude h∈[h_min, h_max] and inclination i∈[i_min, i_max], with h_min and h_max representing the altitude bounds, and i_min and i_max defining the inclination range for satellite deployment.

It is clear that this MOOP is challenging to solve directly, for the following reasons: ① Since the four objectives conflict, there is no optimal constellation deployment; and ② all objective functions and constraints demonstrate nonlinear relationships with the optimization parameters, and the coupling among the parameters results in a problem that is highly non-convex. Traditional weighted combination methods can be used to solve the MOOP problem, but such methods are susceptible to subjective choices of weighting parameters and may fail to capture nonlinear relationships between different optimization objectives [31]. In addition, some multi-objective algorithms transform certain objectives into constraints in order to form a single-objective optimization problem [32]; however, this method can only optimize individual objectives and consider optimization objectives independently. Instead of using a traditional constraint-driven algorithm, we propose the PA-ICD algorithm, by which all objectives can be optimized simultaneously within a single framework. In addition, the PA-ICD algorithm can exploit the relationships of the constellation parameters concerning each objective, which may enable more robust constellation configurations of all optimization objectives.

5. LEO satellite constellation design for computing networks

In this section, we introduce the proposed PA-ICD algorithm and provide theoretical analyses that offer an intuitive understanding of rational constellation deployment.

5.1. LEO satellite constellation design algorithm

The whole process is designed in two steps to prioritize minimizing the number of satellites, including the initialization and determination. First, utilizing an iterative search with K_max = 2K_min, where [K_min, K_max] is the search interval, we continue the process until the range of the minimum total number of satellites satisfies the constraint conditions. Then, we adopt the bisection method to find the minimal number of satellites given K_min and K_max in order to reduce the computational complexity of the algorithm.

For the other three objectives, we set different priorities for

\bar{σ}

\bar{C}

, and

\bar{R}

under the determined number of satellites, and relevant designated algorithms based on the priority setting are introduced. First of all, we adopt a constraint-driven search method to address the optimization problem with optimization variables and obtain updated feasible configurations. The search space corresponding to each constellation parameter is {

Q

W

H

I

}, where

Q

and

W

are derived from the factorization of K, representing the search space of Q and W, respectively. {

H

I

} signifies a collection of alternative values for the orbital altitude and inclination.

According to satellite coverage band theory, the minimum number of satellites required for a satellite ring with seamless coverage [23] is

(32)

W \geq ceil (π / φ)

Furthermore, to ensure that there are no coverage gaps between inclined satellite bands, the number of orbital planes Q should satisfy the following [33]:

(33)

Q \geq ceil (\frac{π}{\arcsin (\sin Δ / \sin i)})

where Δ = arccos(cosφ/cos(π/W)). Next, we can update the search space by utilizing these constraints with the specific steps shown in Algorithm 1.

Algorithm 1

PA-ICD

Input: QoS requirements $(\sum_{n = 1}^{N} c_{n} e^{λ_{n}} / (Ei (λ_{n}) - \ln (λ_{n}) - γ), \sum_{n = 1}^{N} r_{n} e^{λ_{n}} / (Ei (λ_{n}) - \ln (λ_{n}) - γ), {\bar{σ}}_{\min}, {\bar{σ}}_{\min})$ ; limitation of the constellation parameters configuration (h_min, h_max) and (i_min, i_max);
Output: Optimal constellation configuration {Q, W, h, i};
1. Initialize: K_min; priority selection;
2. repeat
3. Construct search space { $Q$ , $W$ , $H$ , $I$ } for K_min;
4. Update the search space by eliminating configurations that do not satisfy the requirements of Eqs. (31f), (31g) and the constraints of $P$ 2;
5. Set K_min = 2K_min;
6. until search space { $Q$ , $W$ , $H$ , $I$ } is not empty;
7. Set K_max = K_min and K_min = K_min/2;
8. repeat
9. Set K = $⌊(K_{max} + K_{min}) / 2⌋$ ;
10. Construct search space { $Q$ , $W$ , $H$ , $I$ } for K;
11. Update the search space by eliminating configurations that do not satisfy the requirements of Eqs. (31f), (31g);
12. if search space { $Q$ , $W$ , $H$ , $I$ } is empty then
13. Set K = K – 1;
14. if K ≤ K_min then
15. K_min = $⌊(K_{max} + K_{min}) / 2⌋$ and return to step 9;
16. end if
17. Return to step 10;
18. else
19. Update the search space by eliminating configurations that do not satisfy the constraints of $P$ 2;
20. end if
21. if search space is empty then
22. Set K_min = K;
23. else
24. Set K_max = K;
25. end if
26. until K_max − K_min ≤ 1;
27. {Q, W, h, i} is obtained by Eqs. (34)−(36) according to the priority selection;

5.1.1. priority

Suppose the average coverage rate is the priority optimization objective. Since the computational capability and transmission capacity constraints are well derived according to the offloading requirement, Eq. (34) can express the optimal constellation parameters.

(34)

(Q^{*}, W^{*}, h^{*}, i^{*}) = \arg \max_{(Q, W, h, i)} \bar{σ}, Q \in Q, W \in W, h \in H, i \in I

where {

Q

W

H

I

} represents the updated feasible constellation configurations after the constraint-driven update.

5.1.2. priority and $\bar{R}$ priority

Since the average coverage rate can reflect the validity of the satellite network, it is crucial to ensure that (h, i) in the search space facilitates a balanced optimization between the coverage rate and the offloading requirements before we obtain the optimal constellation parameters. Therefore, the optimal constellation parameters for

\bar{C}

priority can be expressed as follows:

(35)

(Q^{*}, W^{*}, h^{*}, i^{*}) = \arg \max_{(Q, W, h, i_{Q, W, h})} \bar{C}, Q \in Q, W \in W, h \in H, i \in I ⊖

The optimal constellation parameters for

\bar{R}

priority can be expressed as follows:

(36)

(Q^{*}, W^{*}, h^{*}, i^{*}) = \arg \max_{(Q, W, h, i_{Q, W, h})} \bar{R}, Q \in Q, W \in W, h \in H, i \in I

where

i_{Q, W, h} = \arg \max_{i} \bar{σ} (Q, W, h)

5.2. Theoretical analysis

The variability in the constellation parameters leads to significant differences in the coverage rate among different constellations, despite all being dense constellations with inclined orbits. We consider the impact of two primary constellation parameters, orbital inclination i and altitude h, on the coverage rate over different regions; this offers an alternative perspective to illustrate rational constellation deployment. We explore the approximate optimal values of h and i that maximize the coverage rate of latitude l_exp, where l_exp corresponds to the latitude region with the maximal offloading requirements.

To analyze the monotonicity and extremum of the coverage rate in different regions, we establish a universal mathematical relationship and derive the coverage rate at latitude l.

Proposition 3

The coverage rate at latitude l, denoted as

σ_{l}

, can be expressed as

(37)

σ_{l} = \sum_{q = 1}^{Q} \sum_{w = 1}^{W} Re [\arccos (\frac{\cos φ - \sin ψ_{q, w} \sin l}{\cos ψ_{q, w} \cos l})] / π

Proof: See Section S3 in Appendix A.

For an ultra-dense-LEO satellite constellation, based on the relationship ψ = arcsin(sinisin(2πw/W)), we can determine that w = Warcsin(sinψ/sin i)/(2π) when −i ≤ψ ≤ i. The deformed expression can then be given by

(38)

\begin{matrix} σ_{l} = \lim_{Q, W \to \infty} \sum_{q = 1}^{Q} \sum_{w = 1}^{W} Re [\arccos (\frac{\cos φ - \sin ψ_{q, w} \sin l}{\cos ψ_{q, w} \cos l})] / π \\ = \frac{QW}{π^{2}} \int_{ψ = - i}^{ψ = i} \frac{\cos ψ}{\sqrt{\sin^{2} i - \sin^{2} ψ}} Re [\arccos (\frac{\cos φ - \sin ψ \sin l}{\cos ψ \cos l})] d ψ \end{matrix}

Since the constellations are symmetrical, we only consider the northern hemisphere. For the inclined constellation, we define f₁(ψ) as the satellite distribution at different latitudes, which denotes

(39)

f_{1} (ψ) = \frac{\cos ψ}{\sqrt{\sin^{2} i - \sin^{2} ψ}}

The monotonicity of the function f₁(ψ) can be explored by taking the derivative of ψ, as follows:

(40)

\frac{\partial f_{1} (ψ)}{\partial ψ} = \frac{\sin ψ \cos^{2} i}{{(\sin^{2} i - \sin^{2} ψ)}^{\frac{3}{2}}}

Therefore, f₁(ψ) increases monotonically with ψ.

Let f₂(ψ, l) denote the half angle covered by satellites at latitude l:

(41)

f_{2} (ψ, l) = \arccos (\frac{\cos φ - \sin ψ \sin l}{\cos ψ \cos l})

where f₃(ψ, l)=(cosφ-sinψsinl)/(cosψcosl). Assuming that l₁ < l₂ and 0 < l₁ – ψ₁ = l₂ – ψ₂ < φ, where l₁ and l₂ are two different latitudes, ψ₁ and ψ₂ are corresponding latitudes of subsatellite points, the difference between f₃(ψ₁, l₁) and f₃(ψ₂, l₂) can be rewritten as

(42)

f_{3} (ψ_{1}, l_{1}) - f_{3} (ψ_{2}, l_{2}) = \frac{\sin (l_{2} - l_{1}) \sin (ψ_{1} + l_{2}) (\cos (l_{1} - ψ_{1}) - \cos φ)}{\cos ψ_{1} \cos l_{1} \cos ψ_{2} \cos l_{2}}

which implies that the function f₃(ψ, l) monotonically decreases and f₂(ψ, l) monotonically increases with l for a certain difference of l – ψ.

Hence, we can observe that, when 0 < l < i – φ, σ_l monotonically increases, but when l = i, σ_l will decrease as l increases because the satellite number decreases in high-latitude regions. Although we cannot determine the accurate value of the latitude with the maximum coverage rate, we know that it is within the interval (i – φ,i). Therefore, we can set i – φ/2 as the approximate value. Based on the above analysis, the orbital altitude and inclination should be adjusted to align the value of i – φ/2 closely with the latitude l_exp, thereby enhancing the utilization of resources in the satellite-based computing network. However, to ensure that the satellite network provides reliable computing services for the entire target area, the constellation coverage area i + φ [33] should be congruent with the target region (i.e., i + φ should tend toward l_tar).

Therefore, from the perspective of the monotonicity analysis, the rational orbital altitude and inclination should satisfy the condition of minimizing |i – φ/2 – l_exp| and |i + φ – l_tar| for an inclined constellation design aimed at providing offloading service for middle–low latitude regions. These findings almost align with the simulation results.

6. Simulation results

In this section, we evaluate the performance of the proposed algorithm in a LEO satellite edge-assisted computation network and verify the consistency of the theoretical analysis and simulation results. Regarding the simulation parameter settings, we consider the average UT density to be 4×10^–6 km^–2 and the minimum elevation angle of the UTs to be 10°. Each UT has three types of tasks to be offloaded to the satellite for edge computing; the relevant parameters are shown in Table 1. The computational capability and transmission resources of each satellite are f_max = 30 Gcycles·s^–1 and ϕ_max = 800 MHz, respectively. For the terrestrial–satellite channel configuration, the transmit power of each UT is P = 2 W, with an antenna gain power factor of G = 43.3 dBi. The noise density for Ka-band communications is σ² = –174 dBm·Hz^–1, with the path loss factor α = 2. These main parameters and their default values are set as 3rd Generation Partnership Project (3GPP) R-15 [34].

We define the geographical area from 56°N to 56°S as our target area, whose total population exceeds 99% of the global population. The target area is equally divided into disjoint latitudinal bands, with lmax m – lmin m= 1° for each latitude region. Population distribution data was obtained from the LandScan Global Population Dataset [35], developed by the Oak Ridge National Laboratory, and β_m was calculated accordingly. Regarding the basic properties of the LEO satellite constellation, the range of the orbital latitude and the inclination are (300 km,1500 km) and (30°,60°), respectively.

Table 2 demonstrates that the analytical modeling proposed in Section 2.3 almost fits STK’s Figure of Merit simulation. We compared the results of

\bar{σ}

calculations for a fixed altitude of 600 km with different orbital inclinations and STK grid divisions of longitude and latitude. The operational duration was set as the orbital period of the satellites in the STK simulation to capture the satellites’ dynamic characteristics. Our computational results exhibit a discrepancy of approximately 1% when compared with the results obtained from the STK geographic grid division with a precision of 1°, but the computational speed is improved by more than 10³ orders of magnitude, illustrating the reliability and validity of our proposed connectivity model.

Fig. 3 shows the number of deployed LEO satellites versus the orbital altitude and compares the effect of coverage rate on the optimization of the total number of satellites. A comparison of Figs. 3(a) and 3(b) reveals that the coverage rate constraint will increase the number of satellites for lower orbital altitude constellations but have less influence on the higher orbit constellations. This is mainly because a low orbital altitude results in smaller satellite coverage areas, leading to insufficient coverage. The results also indicate that the total number of satellites that satisfy the average computational capability is robust as the orbital altitude increases when coverage rate constraints are not considered. The reason for this phenomenon is that our algorithm can optimize other constellation parameters at different orbital altitudes to ensure the robustness of the average computational capability. However, regarding transmission capacity, the number of satellites first decreases and then increases with increasing altitude because an increase in orbital altitude contributes to severe path loss.

Fig. 4 presents the optimal constellation configurations for the objective of minimizing the number of satellites, given QoS requirements. It can be seen that, as the offloading tasks increase, the total number of satellites and the inclination increase, while the satellite altitude decreases with the same coverage rate requirements. This phenomenon occurs because more satellites generate a larger coverage rate, which allows the UT connectivity requirements to be met at lower satellite altitudes and larger inclinations. As the connectivity demand continues to increase, it leads to an increase in the number of satellites and orbital altitude.

The performance of the proposed system model and the PA-ICD algorithm for inclined-orbit constellation design was further explored by comparing the proposed constellation with the inclined orbital layers of other representative LEO satellite constellations. The configurations of four constellations, including Kuiper, Telesat, OneWeb, and SpaceX, are shown in Table 3 [29]. The table mainly shows the satellite network performance of our proposed constellation compared with those of other constellations with the same number of satellites for different priority objectives. ^†² The proposed constellation exhibits significant improvements in the average coverage rate, computational capability, and transmission capacity of over 45%, 13%, and 20%, respectively. The proposed constellation optimizes the average coverage rate by increasing the orbital altitude and decreasing the inclination compared with the representative constellations. However, the proposed constellation achieves enhanced channel capacity by operating at lower orbital altitudes compared to the representative constellations, thereby optimizing the average transmission capacity.

A more specific example is demonstrated through a comparative analysis of the Kuiper and proposed constellation based on a priority optimization of the average transmission capacity. We assume the latitude region with the maximal offloading requirements is 26˚. Then, the theoretical rational deployment criteria of these two constellations for |i – φ/2 – l_exp| are 17.7 and 11.3, and those for |i + φ – l_tar| are 12.2 and 4.5. Therefore, the proposed constellation has a greater average coverage rate, average computational capability, and transmission capacity than Kuiper, suggesting that the theoretical analysis effectively clarifies the rationale for constellation optimization.

Fig. 5 illustrates the coverage rate of each latitude region and population. The constellation parameters presented in the figure are optimized for four representative constellations by prioritizing the average coverage rate as the optimization objective. Furthermore, the theoretical approximate latitude i – φ/2 with the maximum coverage rate, as derived in Section 4.2, of Kuiper, Telesat, OneWeb, and SpaceX, is about 43.7°, 38.2°, 43.0°, and 45.5°, respectively. It is clear that the theoretical results are close to the results presented in the figure, illustrating the consistency of the theoretical analysis and simulation. We also observe that the regions with a high concentration of coverage rate have deviated from the populated areas for Kuiper, Telesat, OneWeb, and SpaceX, causing an imbalance in the allocation of offloading resources. Our proposed constellation is more in line with the population density trend.

To provide reliable offloading services for global UTs while enhancing satellite coverage in densely populated middle-low latitude regions, we extended the proposed constellation design algorithm to the design of dual-layer constellations. More specifically, we first designed a high-inclination orbital layer constellation to meet the coverage requirements of UTs in high-latitude regions, including polar regions. Then, based on the specific QoS requirements of UTs in middle-low latitude regions and the designed high-inclination layer constellation, the proposed PA-ICD algorithm was used to design the optimal low-inclination layer constellation. Fig. 6 shows that the proposed dual-layer constellation achieves the same average coverage rate as the complete Telesat constellation [29] with fewer satellites. Our proposed constellation^†³ can provide about 14 connectable satellites for UTs in the polar region, indicating that the constellation is capable of supporting computational tasks for global users. Furthermore, compared with the complete Telesat constellation, the proposed constellation exhibits a higher coverage rate in densely populated equatorial and middle-low latitude regions.

Table 4 shows the multi-layer constellation deployment of each constellation. The constellation deployment cost (C_total) can be calculated as follows [33]:

(43)

C_{total} = C_{manufacture} + C_{launch} + C_{insurance} + C_{maintenance}

which consists of the manufacturing cost C_manufacture, launch cost C_launch, insurance cost C_insurance, and maintenance cost C_maintenance. Those costs are determined as follows:

(44)

C_{manufacture} = (1064000 + 35.5 \cdot m_{sat}^{1.261}) \cdot K^{1 - \log_{2} (\frac{100 %}{κ})}

(45)

C_{launch} = C_{LV} \cdot Q

(46)

C_{insurance} = β_{insur} (C_{manufacture} + C_{launch})

(47)

C_{maintenance} = (C_{manufacture} + C_{launch} + C_{insurance}) / Y_{life}

where m_sat is the weight of each satellite and κ is the learning curve slope. The cost of a single launch C_LV is directly proportional to the orbital altitude h and orbital inclination i. Moreover, β_insur and Y_life represent the proportion of insurance cost and the satellite lifetime, respectively. Although an increase in orbital altitude results in a higher launch cost, it also contributes to a longer lifetime. Based on the relevant deployment parameters [33], the proposed multi-layer constellation has a cost of 841 million USD, which is 24.8% lower than the 1.119 billion USD cost of the Telesat constellation.

7. Conclusions

In this paper, we considered an ultra-dense LEO-satellite-based computing network where UTs can offload local tasks to satellites for reliable computing. This was done by deriving a theoretical connectivity model for ground users and satellites. Then, we formulated a multi-objective problem to maximize the performance of the constellation for computing networks, constrained by the given requirements. We also determined the optimal orbital inclination and altitude to minimize the number of satellites. To achieve the optimal constellation configuration, we designed the PA-ICD algorithm to address this highly non-convex optimization problem; we also analyzed the influence of orbital parameters on constellation deployment.

Finally, three conclusions can be drawn from the simulation results. First, our proposed constellation for a satellite-based computing network is robust regarding the average computational capability that UTs can obtain in terms of varying orbital parameters. Second, an optimal orbital altitude exists that minimizes the number of LEO satellites when considering QoS metrics of the satellite-based computing network. Third, increasing the satellite altitude and decreasing the inclination allows the coverage rate of the constellations to be concentrated in low-latitude regions.

In this work, by minimizing the total number of satellites, our proposed constellation can effectively reduce deployment costs, in comparison with the Telesat constellation. The simulation results validate the feasibility of the proposed constellation in practical deployment scenarios. However, it should be mentioned that the cost of constellation deployment is not only related to the number of satellites but also influenced by different orbital parameters; this can lead to tradeoffs in various cost components, including manufacturing, launch, insurance, and maintenance costs. Therefore, the constellation design problem can be extended further to integrate the performance of the satellite-based computing network with the actual constellation deployment cost.

CRediT authorship contribution statement

Yijing Sun: Writing – original draft, Visualization, Validation, Software, Methodology, Investigation. Boya Di: Writing – review & editing, Methodology, Conceptualization. Ruoqi Deng: Methodology, Investigation, Conceptualization. Lingyang Song: Writing – review & editing, Supervision, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors acknowledge partial support from the GuangDong Basic and Applied Basic Research Foundation (2023B0303000019) and partial support from the National Natural Science Foundation of China (62322101).

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