Task-Aware Distributed Inter-Layer Topology Optimization Method in Resource-Limited LEO-LEO Satellite Networks

Due to the increasing demand for real-time observation data, remote-sensing satellites can be considered as users to access mega communication constellations, which can constitute a Low Earth Orbit (LEO) Double-Layered Satellite Network (DLSN) composed of Remote-sensing Layer Satellites (RLSs) and Communication Layer Satellites (CLSs). In this LEO DLSN, a Distributed Inter-Layer Topology Optimization (DITO) method is proposed based on time-space relationships between two layers, which can maximize task-aware observation benefits with limited onboard resources. For practical purposes, we take link switching interval constraints into account for the recapture and track of laser transceivers. Besides, we put forward a new time-slot division method based on the link switching interval for the linear modeling of the time-sequential coupling problem. With the simulation results in four constellation systems, the proposed distributed interactive method can obtain an optimal solution close to the centralized global solution with much less time. Simulation results indicate that the proposed method can support onboard optimization for future space networks.


I. INTRODUCTION
R EMOTE-SENSING satellites are significant for acquir- ing earth observation data and are valuable in forest fire monitoring, earthquake rescue, etc [1].Compared to UAVs [2], [3], Low Earth Orbit (LEO) remote-sensing satellites have a broader vision and can achieve stable global coverage with not much difference in latency.Besides, the observation data from sun-synchronous orbits have similar time and angle conditions for the same Earth region, which has unique advantages in data correlation analysis [4].However, the traditional remotesensing systems can only transmit data in the visible range of ground stations, which have been unable to meet the increasing demand for real-time and a mass of observation data [5].Mega constellation networks consist of large amounts of small, lowcost satellites, which can provide powerful tools for reducing the long wait time for observation data transmission [6], [7].
In these constellation networks, satellites can communicate with each other through laser transceivers.Satellites at the same orbital altitude form a stable topology [11] and can be considered as a "layer" in the satellite network.In a layer, satellites communicate with their adjacent neighbors through intra-layer links [12].A constellation network could be composed of multi-layers, and the topology between different layers is highly dynamic and irregular.Visible satellites at different layers communicate with each other through interlayer links [13].These links make the constellation network highly connected.
With the help of intra-layer links and inter-layer links, observation data of remote-sensing satellites can realize timely transmission for urgent and real-time business needs.Therefore, data transmission optimization in constellation networks has become a hot topic [14], [15].On this issue, research efforts have been undertaken in two aspects.
The first aspect focuses on the single-layered satellite networks where data transmission is implemented through intra-layer links.Many researches [11], [16], [17] studied the routing algorithms in single-layered mega-constellations with inclined/polar orbits, and they applied a fixed network topology.As for unfixed topology, Yan et al. [18], [19] proposed a link allocation strategy to maximize throughput from non-anchor satellites to anchor satellites, where anchor satellites are those connected to ground stations, and nonanchor are the opposite.Zhou et al. [20] optimized intersatellite and satellite-earth links to maximize the network's weighted observation benefits, considering task differences.
However, these studies [11], [16]- [20] do not reflect functional differences in satellites, which means remote-sensing satellites are also responsible for routing data back to Earth.These approaches may be efficient in small-scale systems but may result in congestion and low efficiency in mega constellations consisting of various types of satellites.
Since communication mega-constellations have unique advantages in wide area access and coverage, multi-layered satellite networks based on functional classification will become an inevitable trend in the future development of space networks.Therefore, this paper focuses on the second aspect of data transmission optimization.
The second aspect concentrates on the inter-layer link planning of Double-Layered Satellite Networks (DLSNs).Yan et al. [21] analyzed the inter-layer topology and link establishment strategy of MEO-LEO DLSNs.Lu et al. [22] optimized LEO DLSNs with the same inclined orbit from the view of constellation structure.Huang et al. [23] proposed an interlayer link allocation scheme in MEO-LEO DLSNs, based on the weighted sum of multiple objectives.In a network where low-orbit satellites transmit the mission data to ground stations via high-orbit relay satellites, Liu et al. [24] proposed a dynamic link allocation scheme for maximizing flow routing.
Among these networks, LEO DLSNs have caused concern due to the development of Starlink [8] and LEO satellites' advantages of being small, lightweight, and low-cost.However, the existing research [21] and [22] focus on the constellations' structures but do not apply to remote-sensing satellites in solar synchronous orbits.Besides, to the best of the authors' knowledge, all existing link allocation research [18]- [20], [23], [24] judged potential links with rough connectable conditions.Especially in LEO DLSNs, the inter-layer communication challenge brought by the high-speed relative motion has not been considered.As pointed out in [25], [26], the relative velocity and position of satellites have significant effects on tracking and precision-pointing of laser communication.Besides, laser transceivers need a certain time for recapturing and tracking when performing link switching [27], [28].But existing studies all ignored the practical situation.The neglect of link switching time makes the optimization between time-slots independent, which significantly reduces the solving difficulty and further leads to simple centralized methods for the optimization in each time-slot.But the centralized methods are hard to support onboard optimization in reality.Furthermore, these studies do not reflect the difference in observation tasks at the inter-layer link allocation schemes.
In this paper, we consider remote-sensing satellites as users to access the communication mega constellation, and they can form an LEO DLSN composed of Remote-Sensing Layer Satellites (RLSs) and Communication Layer Satellites (CLSs).We present a distributed interactive mechanism between RLSs and CLSs involving limited laser transceivers.The distributed method can replace the limited transceiver constraints in a centralized model with much less solving time.To maximize the observation benefits of each RLS, we propose an interlayer link allocation scheme based on a time-space topology graph considering observation task differences.The proposed model satisfies the inter-layer connectable conditions with the realistic limits of laser communication and LEO satellites' fast relative motion.We also present a new time-slot division method based on the maximum link switching interval to deal with the nonlinear correlation between adjacent time-slots and achieve linear modeling of the link switching interval constraints.As for limited storage and energy resources on board, we also consider resource management for each RLS.
According to the presented state of the art, the main contributions of this paper can be summarized as follows: • The task importance difference is considered in both the topology optimization model and resource management model, which can maximize the observation benefits of each RLS and make sure the limited link, energy, and storage resources are applied to the data transmission of important tasks.The rest of this paper is organized as follows.The system architecture of the LEO DLSN, the topology graph, and the distributed mechanism are presented in Section II.The mathematical models are formulated in Section III, realizing the topology optimization and resource management.In Section IV, we introduce the solution approach of the proposed DITO algorithm.Section V verifies the effectiveness of the proposed models and algorithms through simulation results in four different scale constellation systems.Section VI concludes this paper and discusses future work.

II. SYSTEM ARCHITECTURE
In this section, we will introduce the system architecture from three aspects.Firstly, we demonstrate an LEO DLSN and state the focus of this paper.Secondly, we build a time-space inter-layer topology graph as the basis of the mathematical model.At last, we propose a distributed interactive mechanism supporting onboard optimization.
TABLE I lists key notations used in this paper.

A. LEO DLSN
Fig. 1 shows an LEO DLSN composed of CLSs and RLSs.In this network, RLSs can collect observation data and transmit data to CLSs by inter-layer laser links.Then, CLSs route data to ground stations through intra-layer links and satellite-earth links with routing algorithms proposed in [11], [16], [17].CLSs consist of many small satellites and can cover all RLSs so that ground stations can get observation data in real time.We assume that intra-layer links of RLSs are not used for sending observation data back to Earth.
In this paper, we focus on the optimization of inter-layer links between RLSs and CLSs, which are red lines in Fig. 1.We consider RLSs as users to access CLSs, and the optimization is from the perspective of RLSs.Since CLSs are responsible for communication and have the advantage of routing by lots of algorithms studied in existing research, RLSs do not need to care about what happens inside CLSs, and the data transmitted to CLSs can be routed properly and obtain observation benefits.In a word, the objective for RLSs is to find the optimal inter-layer links to send data in nearly realtime, considering limited onboard resources.Connectable adjacency matrix and its binary elements of RLS i with all CLSs at all time-slots.Φ i,j,t , ξ i,j,t , d i,j,t , P S i,j,t Elevation, geocentric angle, distance, transmitting power from RLS i to CLS j at time-slot t.
Re, h i , h j , ϵ min , P S max Earth radius, altitude of RLS i, altitude of CLS j, the minimum elevation, the maximum transmitting power.ϕ i,t , λ i,t , ϕ j,t , λ j,t Latitude and longitude of RLS i and CLS j at each time-slot.
Receive sensitivity of CLS j, receive gain of CLS j, transmission gain of RLS i, inter-layer communication frequency.
dΦmax, dΨmax, dP S max Maximum change rate of elevation, azimuth, transmitting power.e i,j,t , v i,j,t , p i,j,t Logic variables of constraints about elevation, visibility, transmitting power.de i,j,t , da i,j,t , dp i,j,t Logic variables of constraints about change rate in elevation, azimuth, transmitting power.δ i , δ i,t , δ i,j,t , δ i,0 Matrix, vector, elements and initial vector of the connection relationship from RLS i to all CLSs.dδ i,t , dδ i,j,t Vector and its elements of differences between adjacent time-slots of connection relationships between RLS i and CLSs.
Determined matrix and its elements of the connection relationships between RLS i and all CLSs at all time-slots.
Importance factor of observation task m, importance factor of data collected by RLS i at time-slot t. q i , q i,j,t Matrix and its elements represent the quantity of flow in link (i, j) during each time-slot.
Data volume in i's buffer at the end of time-slot t, maximum buffer size, initial data volume in buffer.ν or i , ν rc i,j , l i,t Data collection velocity of i, data transmission velocity of link (i, j), sunlight duration of i in time-slot t.
Vector and its elements represent the harvested energy of i during each time-slot.
Remaining energy of i at the end of each time-slot t, total consumed energy during t, transmission energy during t.
Maximum battery capacity, maximum discharge depth, the initial energy of RLS i. P H , P O , P R Constant power of energy collection, observation, and normal operation.

B. Inter-layer Time-space Topology Graph
The LEO inter-layer topology is denoted as a directed graph G(V, E), which is shown in Fig. 2.
We divide the time domain into slots by virtual topology method (or named as 'snapshot') [29], and let T = {1, 2, • • • , t, • • • , T } be the set of time slots.The duration of each slot is ∆τ .In each time-slot, the network topology is considered static.
Define V = ∪ t∈T V t be the set of vertices in time-slots.Let where V o t is the set of virtual observation targets (green points in Fig. 2 is the set of RLSs' nodes (orange points), and Define E = ∪ t∈T E t be the set of edges in time-slots.Let , where E or t represents data acquisition links (orange lines in Fig. 2).In this paper, we do not arrange observation tasks and assume RLSs can obtain data continuously through data acquisition links from virtual targets.Therefore, the position of edges in set E or t are unchanged, but the weight of edges can be different with tasks of varying importance.Edges set E rc t = {(i, j)|i ∈ V r t , j ∈ V c t , t ∈ T } represents the inter-layer connection relationship between RLSs V r t and CLSs V c t (blue lines in Fig. 2(a) and red lines in (b)).Edges set E rs t = {(V r t−1 , V r t )|t ∈ T } denotes the data storage arc of RLSs between time-slot t−1 and t (purple lines in Fig. 2).We do not consider the storage limits of CLSs and assume they can store or transmit data properly by themselves.
In Fig. 2(a), we can obtain the potential inter-layer topology by judging connectable conditions between V r t and V c t .For

C. Distributed Interactive Mechanism
The interactive mechanism of RLSs and CLSs is distributed, and the optimization algorithms can be executed onboard.We assume each RLS can only connect to one CLS at each time-slot, and each CLS can connect up to L c RLSs at each time-slot due to the limited laser transceivers.The specific optimization and interaction process is shown in Fig. 3 and can be summarized as follows: 1.Each RLS makes a connection scheme with all CLSs of a given time range T based on the topology optimization model.Before every task execution period T , a planning process is required to determine the inter-layer data transmission scheme for the whole T period.A cycle comprises the planning process and the task execution period T .Since the planning process takes much less time than T , the time spent by the planning process is negligible, and the cycle can be considered as T .After one cycle ends, the system executes the planning process again for the next cycle.The status variables at the end of the previous cycle are substituted into the next cycle as initial values.

III. MATHEMATICAL MODELS
In this section, we will introduce the potential inter-layer link model, topology optimization model, and resource management model in turn, corresponding to Fig. 2(a), (b), (c).

A. Potential Inter-layer Link Model
t , t ∈ T }, ∀i ∈ V r t be the connectable adjacency matrix of RLS i with all CLSs at all time-slots, where S i (j, t) = s i,j,t .Let s i,j,t = 1 when i and j are connectable at time-slot t, otherwise s i,j,t = 0.
Fig. 4 shows the geometric relationship between RLS i and CLS j.There are six kinds of constraints for S i as follows.
1) Elevation Constraints: Define Φ i,j,t be the elevation from i to j at time-slot t.It should not exceed the minimum communication angle ϵ min if i and j can connect.Let e i,j,t be a logical variable and be '1' when the elevation constraint is satisfied, which can be described as: where the operator '&' represents the logical AND.
In Fig. 4, by applying the Sine theorem in ∆OIJ, we can obtain the following formula: where R e is the Earth radius, h i and h j are satellites' orbit altitudes, ξ i,j,t is the geocentric angle between satellites, d i,j,t is the inter-satellite distance.Then, the elevation from i to j can be calculated by: where ( 4) is the trigonometric formula for calculating the intersatellite distance; (5) is the formula for geocentric angle calculated by the satellites' latitudes ϕ i,t and ϕ j,t , and longitudes λ i,t and λ j,t .
2) Visible Relationship Constraint: Both satellites are visible only when their connecting line is not obscured by the Earth.Therefore, when the link between two satellites is tangent to Earth, they are connected to the farthest distance [23].Let v i,j,t be the logical variable of this constraint and describe it as: 3) Transmitting Power Range Constraints: Define P S i,j,t be the transmitting power from i to j at t, it should not exceed the maximum transmitting power P S max of laser transceiver.Let p i,j,t be the logical variable of this constraint and describe it as: p i,j,t = P S i,j,t ≤ P S max (7) where P S i,j,t can be expressed in decibels language as [23], [30]: where R j and G r j are receive sensitivity and gain of CLS j, and G t i is the transmitting gain of RLS i, and F SL is the free space loss expressed by communication frequency f (in MHz) and distance d i,j,t (in km).
4) Elevation Change Rate Constraint: LEO satellites with different altitudes have rapid relative motion, but the laser alignment devices have limited tracking speed, so the elevation change rate should be restricted [25]- [27].Define dΦ max be the maximum elevation change rate.Let de i,j,t be the logical variable of this constraint and describe it as: where the elevation change rate can be calculated from equations (3)-(5).5) Azimuth Change Rate Constraint: Similarly, the azimuth change rate should also be considered.Define Ψ i,j,t be the azimuth from i to j at t, and dΨ max be the maximum change rate of azimuth.Let da i,j,t be the logical variable of this constraint and describe it as: (11) where the azimuth calculation equations can be found in [31], and the azimuth range constraints can also be added to this model.6) Transmitting Power Change Rate Constraint: Due to the rapid change of satellites' distances, the transmitting power has to be adjusted rapidly [25], [26].But the change rate is limited by the laser transceiver.Define dP S max be the maximum change rate of transmitting power.Let dp i,j,t be the logical variable of this constraint and describe it as: where the change rate can be calculated from ( 8)- (9).Consequently, the connectable matrix S i can be calculated by setting s i,j,t = 1 when the above constraints 1)-6) are all satisfied, which can be expressed as: It can be noticed that the connectable conditions are related to the satellites' relative position, relative velocity, and the tracking limits of laser transceivers.All factors should be taken into account for the stability of laser communication between LEO layers.With this model, each RLS i can obtain potential inter-layer links at each time-slot, as shown in Fig. 2(a).

B. Topology Optimization Model
This model is from each RLS' point of view, and the objective is to make an optimal connection scheme, corresponding to Step 1 in Fig. 3.
Define binary matrix δ i ∈ R Nc×T = {δ i,j,t |j ∈ V c t , t ∈ T } for ∀i ∈ V r t be decision variables of each RLS' topology optimization model, where δ i (j, t) = δ i,j,t .Let δ i,j,t be '1' when i selects to connect with j at time-slot t, and '0' when the opposite.There are three kinds of constraints as follows, and the optimization problem formulation is described at last.
1) Connectable Conditions Constraint: Decision variables δ i can be defined with S i being the upper bounds.It means only when two satellites are potentially connectable, the interlayer links can be optimized.It is described as: 2) Link Number Constraints: For each RLS, we assume they can only connect to one CLS at each time-slot, which is as: Note that link number limits for CLSs are not considered in this model since each RLS does not know how many other RLSs are also sending connection requests.
3) Link Switching Interval Constraints: For each RLS i, when switching the connected CLS, it takes time to recapture and track, so a certain link switching interval is required [27], [28].As shown in Fig. 5, RLS i does not link to any CLSs during the transition from j to j ′ .However, if we choose the length of each time-slot optionally, such as 30 seconds, 1 minute, etc., the link switching interval has to Vector norm require several time-slots.This leads to complex nonlinear relationships between time-slots since decision variables at t6 must consider the value of those at t4 (as shown in the upper part of Fig. 5).Therefore, we merge time-slots based on the switching interval, which means the length ∆τ of each timeslot should be as long as the link switching interval.Since RLSs have little difference in the needed switching time, the interval is considered a constant value of the maximum needed time.
With the new time-slot division method, we can list the different situations between adjacent time-slots to model the time-sequential nonlinear relationship in a linear form.
Define Nc,t ]⊤ be the difference vector between δ i,t and δ i,t−1 , then dδ i,t is expressed as: where δ i,0 is the link status with CLSs at the last time-slot of the previous cycle.Different cases of dδ i,t are listed in TABLE II.There are four cases: • Case 1: RLS i connects to the same CLS at time-slot t−1 and t, which means ∃j ∈ V c t−1 ∪ V c t , δ i,j,t−1 = δ i,j,t = 1.• Case 2: RLS disconnects with a CLS at time-slot t, which means ∃j ∈ V c t−1 , δ i,j,t−1 = 1, ∀j ∈ V c t , δ i,j,t = 0. • Case 3: RLS rebuilds link with a CLS at time-slot t, which means ∀j ∈ V c t−1 , δ i,j,t−1 = 0, ∃j ∈ V c t , δ i,j,t = 1.
• Case 4: RLS switches link from one CLS to another CLS with no time interval, which means ∃j ∈ V c t−1 , δ i,j,t−1 = 1, ∃j ′ ̸ = j, j ′ ∈ V c t , δ i,j ′ ,t = 1.Among the four cases, Case 4 should be avoided since it does not have enough link switching time.As summarized, the switching interval constraint can be formulated as follows: The norm operation of the difference vector dδ i,t is linear.Consequently, with the new time-slots division method based on link switching interval and the inequality (18), we can model the nonlinear link switching constraint in a linear form.
4) Topology Optimization Problem Formulation: Define ω i,m be the importance factor of observation task m executed by i, satisfying m ω i,m = 1.Let ω i,t = {ω i,m |t ∈ [t s m , t e m ]} be the importance factor of observation data collected by RLS i at time-slot t, where t s m , t e m are the start and end time of task m.The objective for each RLS is to maximize the total interlayer network benefits within T , which can be formulated as follows: subject to: where ( 1)-( 13) are used to calculate parameter S i , and ( 14)-( 18) are linear constraints of integer variable δ i , so it is an Integer Linear Programming (ILP) model.

C. Resource Management Model
After the inter-layer time-space topology is determined by CLSs (Steps 3-4 in Fig. 3), each RLS can manage flow and energy under limited storage and energy resources (Step 5 in Fig. 3).Define represents the connection relationship of (i, j) at t.There are two kinds of constraints on flow and energy.1) Flow-related Constraints: Define q i,j,t (in Gbit) be the quantity of flow in link (i, j) ∈ E rc t during time-slot t, which are decision variables.Define b i,t (in Gbit) be the data volume in the buffer of RLS i at the end of time-slot t, and b max i be the maximum buffer size of i.Let ν or i (in Gbit/s) denote the data collecting velocity of i, and ν rc i,j (in Gbit/s) be the data transmission velocity of link (i, j).
The flow-related constraints are as follows: where b i,0 is the initial data volume in RLS i's buffer.Inequality (20) constrains the data path through established links and lets data volume not exceed the maximum link transmission capacity.Equation ( 21) is the flow balance constraint, which means the data amount change in buffer at a time-slot is equal to the incoming data amount minus transmitting data amount during this time-slot.This equation determines E rs t in graph G. Inequality (22) let the stored data not exceed the maximum buffer size.
2) Energy-related Constraints: Define M H i,t (in kJ) be the harvested energy by the solar panel of RLS i during time-slot t, which are decision variables.Define M C i,t as the consumed energy during t, and M i,t be the remaining energy at the end of time-slot t.Let M max i be the maximum battery capacity, and θ is the maximum discharge depth of the battery.
The energy-related constraints are as follows: where P H (in kJ/s) is the energy collection rate; l i,t is RLS i's duration in sunlight during time-slot t, whose values belong to the interval [0, ∆τ ]; M i,0 (in kJ) is the initial energy of RLS i.
Inequality (23) indicates that RLS i can only charge in sunlight duration, and the harvested energy in t should not exceed the maximum energy amount that can be collected in sunlight.Equation ( 24) is the energy balance constraint.Inequality (25) let the energy use be in the range of the battery limits.
The consumed energy amount M C i,t can be calculated by the formula below: M S i,t = j:(i,j)∈E rc t where M S i,t is the energy consumed by transmitting data to CLSs, P O is the constant power for observing and collecting data, and P R is the constant power for normal operation and other activities.Here, P S i,j,t is the transmitting power as introduced in (8).
3) Resource Management Problem Formulation: In this problem, buffer variables b i,t are determined by q i,j,t with (21), and energy variables M i,t are determined by M H i,t and q i,j,t with ( 24), ( 26)- (27).Therefore, the decision variables are q i,j,t and M H i,t .Define q i ∈ R Nc×T = {q i,j,t |j ∈ V c t , t ∈ T } be flow decision matrix of ∀i ∈ V r t , where q i (j, t) = q i,j,t , and |t ∈ T } be harvest energy decision vector of i, where M H i (t) = M H i,t .The objective for each RLS is to maximize the observation benefits by sending the largest amount of weighted observation data to CLSs within T , which can be formulated as: max qi,M H i t∈T j:(i,j)∈E rc t ω i,t q i,j,t ∀i ∈ V r t (28) subject to: where elements of δ ′ i are taken as known parameters in the resource management model.

IV. SOLUTION APPROACH
In this section, we first introduce three algorithms and a time sequential decoupling method in turn to show the preprocessing and solution approach of the main steps in Fig. 3.Then, we present the DITO algorithm to connect all the proposed algorithms and methods.The DITO algorithm contains the entire planning process of the LEO DLSN.At last, we analyze the time complexity of the proposed algorithms.

A. Preprocesing to Reduce Complexity
Considering the connectable adjacency matrix S i and the selected connection matrix δ i , δ ′ i are very sparse, we preprocess these matrices for each RLS i in two ways to meet the practical cases and reduce the solving complexity.
1) Remove Connection Windows that Have a Too Short Duration: In reality, too short inter-layer windows are impractical for laser communication.Define w min be the minimum effective inter-layer window.Then, before we solve the topology optimization model, we can preprocess the connectable matrix S i by removing the connectable windows shorter than w min so that fewer variables of δ i need to be optimized because of the constraint (14).
We call this preprocessing function RSW (Remove Short Windows) and describe it as Algorithm 1.
In Algorithm 1, Steps 3-7 can obtain the start and end time of connectable windows from RLS i to a certain CLS j.In Step 3, let St ∈ R T +1 be the vector to record link status changes (from '0' to '1', or '1' to '0').In St, the element '1' indicates that the status changes at this timeslot, and '0' indicates that the status keeps the same as the previous time-slot.Since a connectable window has two '1' in St (represent the window's start and end status change, respectively), the number of connectable windows can be obtained in Step 4.Then, windows' start time ts corresponds to the odd-positioned '1' in St, as stated in Step 6.Similarly, even-positioned '1' represent the time-slots after the windows' end time te, as stated in Step 7.
Steps 8-12 are applied to remove windows shorter than w min .Note that short windows covering the first or last timeslots will not be removed since they might form longer windows with the previous or next cycle.The 'remove' operation is realized by rewriting the corresponding elements to '0' in S i during short windows .
Algorithm 1 instructs removing short windows in S i .But the optimized linking topology δ i may still have short windows.Therefore, to guarantee the inter-layer windows be practical, the RSW function should also be applied to δ i after the topology optimization process.
Note that both RSW (S i ) and RSW (δ i ) are applied in Step 1 of Fig. 3. Function RSW (S i ) helps save solving time for making connection schemes.Function RSW (δ i ) makes sure the requested links have practical duration of connection windows.
2) Remove CLSs with No Effective Connection Window: Since each RLS need to determine the connection relationship with all CLSs at all time-slots, there are N c • T decision variables in the topology optimization model (19).But for Algorithm 1 Function RSW: remove short connection windows for each RLS.Require: Matrix S i or δ i to be preprocessed, here takes S i as an example.
St ← xor ([S i (j, :), 0], [0, S i (j, :)]), take the XOR of S i (j, :) at adjacent time-slots to get the change status of link (i, j); , where [•] is the rounding operation, St is the sum of non-zero elements in St, and K is the number of connectable windows;  end for 13: end for 14: return S i (or δ i ) Algorithm 2 Function RNC: remove CLSs with no effective window to reduce the number of optimized variables.Require: Matrix S i or δ ′ i to be preprocessed, here takes S i as an example.
) a certain RLS, there are some CLSs whose total connectable duration in T is less than the minimum effective window w min , so there is no need to optimize the topology with these CLSs.Therefore, we delete these CLSs in {j| t∈T s i,j,t < w min } to reduce the number of optimized variables in (19).We call this preprocessing function RNC (Remove Noeffective CLSs) and describe it as Algorithm 2. In Step 2, let tv ∈ R Nc be the vector of total connectable duration from RLS i to CLSs.If an element of tv is larger than w min , it indicates that the connection relationship with this CLS is worth optimizing, then the index of this CLS can be stored in vector j opt ∈ R Nc , as stated in Step 3. In Step 4, the sum of non-zero elements in j opt is the number of CLSs for optimization, defined as N opt c .For each RLS, it only needs to optimize the connection relationship with N opt c CLSs.Algorithm 2 instructs removing no-effective CLSs for the topology optimization model (19).This function can also be applied to the resource management model (28) to reduce the optimized variables by replacing S i with δ ′ i in Algorithm 2. Note that RN C(S i ) is applied in Step 1 of Fig. 3 and  RN C(δ ′ i ) is applied in Step 5 of Fig. 3.

B. Selection and Reply Process of each CLS
The selection and reply process of each CLS corresponds to Steps 3-4 in Fig. 3.As introduced in Section II-C, each CLS only has L c inter-layer laser transceivers for RLSs.So if more RLSs send requests to a CLS for a certain time-slot, the CLS needs to select at most L c RLSs to establish inter-layer links at that time-slot.The purpose of the selection and reply process for each CLS is to replace the laser transceiver limit constraint as follows: The selection and reply process of each CLS j ∈ V c t is described in Algorithm 3. Define N j r be the number of RLSs sending connection requests to j.These RLSs belong to the set {i| t∈T δ i,j,t ̸ = 0}, which means these RLSs sent requests for at least one time-slot in T .

Algorithm 3
The selection and reply process of each CLS j.Require: δ i,j,t , ∀i ∈ {i| t∈T δ i,j,t ̸ = 0}, ∀t ∈ T , each CLS j gets the connection variables with RLSs who sent requests for at least one time-slot.1: for t = 1 : T do 2: δ ′ i,j,t ← δ i,j,t ∀i ∈ {i| t∈T δ i,j,t ̸ = 0}, initialize the determined connection variables; 3: i sort ← sort(ω i,t ), sort i with the value of ω i,t in descending order, get a vector i sort ∈ R N j r ; 5: i ′ ← i sort (L c + 1 : N j r ), get the index vector of abandoned RLSs which are not in the top L c , where ∀i ∈ i ′ , δ ′ i,j,t ← 0, let the connection variables with abandoned RLSs be 0; Send REJECT to i with δ ′ i = 0, since all requested time-slots are rejected; Send PARTIAL and δ ′ i to i, since part of the requested time-slots are approved, and the approved time-slots should be replied; 16: end if 17: end for Since we expect more important mission data can be transmitted, each CLS can sort the RLSs by their importance factor ω i,t in descending order and select the top L c RLSs at each time-slot.
In Algorithm 3, Steps 1-8 correspond to the selection process for all time-slots, and Steps 9-17 correspond to the reply process to all RLSs that sent requests.With the whole process, the determined connection variables δ ′ i,j,t with a specific CLS j can be obtained.When an RLS receives replies from all CLSs, its determined connection matrix δ ′ i can be obtained.

C. Time Sequential Decoupling of the Resource Management Problem
The time sequential decoupling operation is applied in Step 5 of Fig. 3 to help solve the resource management problem (28).
In the resource management model, variables b i,t and M i,t are coupled between adjacent time-slots, as formulated in ( 21) and (24), which result in the constraints of this model can not be described in a linear form with decision variables q i,j,t .To reduce the solving complexity, we rewrite (21) in a recursive way as: By substituting (30) into (22), we can obtain inequality constraints of q i,j,t as: Consequently, we can replace ( 21) and ( 22) with inequalities ( 31) and ( 32), so there are only linear constraints of q i,j,t .Similarly, we rewrite the energy-related constraint (24) in a recursive way as: By substituting ( 26) and ( 27) into (33), and then into (25), we can obtain inequality constraints: Therefore, we can replace ( 24)-( 27) with linear constraints (34)- (35) of decision variables q i,j,t and M H i,t .In the resource management model for each RLS i, inequalities ( 20) and (23) determine lower and upper bounds for variables, and the number of inequality constraints is 4T with ( 31)-( 32), ( 34)- (35).It is a Linear Programming (LP) model.

D. Distributed System Solution Approach
In this subsection, we connect all the proposed algorithms and methods and give the DITO algorithm containing the entire planning process of Steps 1-5 in Fig. 3.The DITO algorithm of the LEO DLSN is illustrated in Algorithm 4.
In DITO, Steps 1-8 correspond to the topology optimization of each RLS.This part represents Steps 1-2 in Fig. 3. Before solving the ILP model (19), functions RSW and RNC are both used for meeting reality and reducing decision variables.With RNC, (N c − N opt c ) • T decision variables are reduced.After the topology optimization, function RSW is used again in Step 6 to modify the optimization results for the sake of practical windows' duration.
Steps 9-11 correspond to the selection and reply process of each CLS.This part represents Steps 3-4 in Fig. 3.
Steps 12-15 correspond to the resource management optimization of each RLS with δ ′ i received from CLSs.This part represents Step 5 in Fig. 3. Before solving the LP model (28), function RNC is used to reduce (N c − N opt ′ c ) • T decision variables.In Step 14, each RLS can obtain the maximum taskaware observation benefits.
With the proposed DITO algorithm, the linking and data transmission plan of all time-slots during this cycle T are settled.Then, at the whole task execution period T , RLSs Algorithm 4 DITO: A Distributed Inter-layer Topology Optimization algorithm of an LEO DLSN.
1: for each RLS i ∈ V r t do 2: Calculate the connectable matrix S i by ( 1)-( Solve (19) with S i , j opt , N opt c to obtain δ i ; 6: Send requests to corresponding CLSs with δ i ; 8: end for 9: for each CLS j ∈ V c t do 10: Send replies to RLSs with Algorithm 3; 11: end for 12: for each RLS i ∈ V r t do 13: Solve (28) subject to ( 20), ( 23), ( 31)-( 32), ( 34)-( 35) to obtain q i , M H i , taking δ ′ i as parameters; 15: end for can send data to CLSs as planned, which corresponds to Step 6 of Fig. 3.
In practice, synchronization issues are critical for distributed scenarios of large-scale DLSNs, so frequent interaction between different layers may result in low efficiency and poor robustness [32].The proposed DITO asks RLSs and CLSs to interact only once for a relatively long cycle so that network efficiency can be guaranteed.Besides, the interactions between RLSs and CLSs are not subject to rigorous clock synchronization.Because RLSs can send requests to CLSs with little time deviation due to their different topology optimization time.CLSs can set a time tolerance (e.g., 1 second) and wait for a while to make sure all requests of RLSs are received.The time deviation of CLSs' replies is also allowed.Once an RLS receives all replies from the requested CLSs, it can run the resource management program and send data to corresponding CLSs during T as planned.Consequently, the proposed DITO can ensure the efficiency and robustness of the LEO DLSN by the limited inter-layer interactions and appropriate time deviation tolerance.• T + T ) 3 ) [34].Since the algorithm is distributed, the total complexity of DITO is upper bounded by O(2

V. SIMULATION AND ANALYSIS
We simulate the proposed LEO DLSN with four constellation systems consisting of 33, 50, 136, and 500 satellites, and the orbit information is listed in TABLE III, referred to Starlink [8] and OneWeb [10].RLSs are in solar synchronous orbits, and CLSs are in inclined or polar orbits.The duration ∆τ of each time-slot is equal to the link switching interval, which is set to 120s.For the sake of reality, the minimum effective connection window w min is set to two time-slots, which is 240s.We assume each observation task of RLS takes 6-10 minutes.The tasks' importance factor m ω i,m = 1 and ω i,t are generated by ω i,m .Other simulation parameters are listed in TABLE IV, referring to the data in [20], [26], [28].
The system is solved on a computer Dell 7080MT Optiplex with Intel(R) Core(TM) i7-10700 CPU@ 2.90GHz, 32GB RAM, Windows 11, and MATLAB R2022b.Software STK is used to generate constellations and obtain satellites' position data and lighting information.We use the modeling tool Yalmip [35] and solver Gurobi [36] to solve the ILP problem (19).We use the interior point algorithm of linprog [37] to solve the LP problem of Step 14 in Algorithm 4.

A. Results and Indicator Analysis
Taking System-50 as an example, the connection situations of several RLSs are shown in Fig. 6.It can be seen that the link switching interval of each RLS is well satisfied, which is consistent with the actual situation of inter-layer link switching.
Comparisons of indicators in four systems are listed in TABLE V.The cycle T is set to the orbital period of RLSs, which is 96 minutes with 48 time-slots.Indicators are defined as follows: • Consent Rate: for each RLS, the ratio of the final total link duration to the requested link duration, calculated by ( t j δ ′ i,j,t )/( t j δ i,j,t ).• RLSs' Connection Rate: the proportion of total link duration in a cycle, calculated by ( t j δ ′ i,j,t )/T .• CLSs' Connection Rate: the proportion of total link duration in a cycle, calculated by ( t i δ ′ i,j,t )/T .• Benefits: the total weighted data volume sent to CLSs, which is the objective value of (28).From TABLE V, some conclusions can be summarized as follows.1) Average Consent Rate of all systems can be 94% or above, which means most link requests of RLSs can be accepted.In System-50, the consent rate is close to 100% because there are significantly more CLSs than RLSs.Here, we set CLSs' link number limits L c be 2.If CLSs have more laser transceivers, the consent rate of all systems can be 100%.
2) Average Connection Rate of RLSs and CLSs are both the largest in System-136 because CLSs are in polar orbits with larger covered areas.In System-50, CLSs' connection rate is relatively poor since the fact that more CLSs than RLSs makes many CLSs idle.Therefore, a close number of satellites in two layers makes the connection rate higher.
3) The higher RLSs' average connection rate, the higher the average benefits.Since more time connect to CLSs for each RLS, more observation data can be sent.

B. Algorithm Performance
In this subsection, we evaluate the solving time performance of the proposed DITO algorithm with different system scales and different cycles.Algorithm 4 has three steps: RLSs' topology optimization, CLSs' selection, and RLSs' resource management.We assume each RLS optimizes by itself concurrently (same as each CLS), so the solving time for each step is considered as the maximum time needed by one RLS (or CLS).Total solving time is the sum of three steps, and the interaction communication time between steps is considered negligible.
The relationship between solving time and system scale is shown in Fig. 7. Most time of the planning process is used for RLSs' topology optimization.CLSs' selection and RLSs' resource management take a very short time.Larger systems have a larger variance of solving time since the gap between average time and maximum time becomes quite different in System-136 and System-500.When setting T to be 30 minutes, the total solving time of System-500 is only 1.49s, which is far less than the cycle T .Therefore, the efficiency of the proposed algorithm can support onboard optimization.
The total solving time for the planning process under different settings of T in four systems is shown in Fig. 8.In System-33, System-50, and System-136, the solving time does not increase significantly with T , and all can be within 5s.But in System-500, when setting a large value for T , the solving time increase sharply.Especially when T is 96 minutes, the solving time comes to 3007s.Consequently, for onboard optimization, T needs to be set properly for largescale systems.

C. Comparison with Other Strategies
Since existing studies do not consider the link switching interval as shown in Fig. 6, these studies' optimization of each time-slot is independent.Therefore, these works mostly adopt a centralized approach to maximize benefits per timeslot, as in [18]- [20], [23], [24], and neglect the difference of observation tasks, as in [18], [19], [23], [24].Without the link switching interval, strategies' comparison is meaningless since it is out of reality.Therefore, for the sake of comparison, link switching constraints ( 17)-( 18) and the proposed timeslot division method are contained in all compared strategies.The comparison with three other strategies is to highlight the solving time improvement of the distributed method and the preprocessing algorithms, and also the benefits improvement of the tasks' importance consideration.The compared three strategies are as follows: • COP: a centralized optimization problem to maximize the total network benefits in T , which can be expressed as: COP: max δi,j,t,qi,j,t,M H i,t t∈T i∈V r t j∈V c t ω i,t q i,j,t (36) subject to: (1) − (18), ( 20), ( 23), ( 29), ( 31) − (32), (34) − (35) where link number limits of CLSs are directly considered in the optimization problem, taken as the constraint (29).
It is a Mixed Integer Linear Programming (MILP) model, and the number of variables is (2N c + 1)N r T .• DITO-nodif: a distributed model with the main idea of the proposed DITO, but does not consider the different importance of observation tasks by setting ω i,m be equal for ∀i, ∀m.For normalization, i m ω i,m = N r is satisfied.
• DITO-nopre: a distributed model with the main idea of DITO, but does not apply the preprocessing algorithms RSW and RNC (Algorithm 1 and 2).Fig. 9 shows the solving time and benefits of four strategies under different system scales.The cycle is set to 30 minutes.From this figure, several conclusions can be obtained: 1) The solving time gap between COP and DITO is significant, while the optimal solution of DITO is very close to the centralized solution of COP.In System-500, COP can not obtain an optimal solution within 12 hours only for 30 minutes cycle since there are 1875600 variables!The problem is too huge for the computer to do centralized optimization work.While for the same problem, the solving time of DITO is only 1.49s.It is clear even with computing resources on the ground, centralized algorithms would not work for large constellation systems, and onboard calculations are even more impossible.Therefore, it can be concluded that DITO has good solution results with a short solving time.
2) DITO-nodif has a similar solving time to DITO, but the network benefits are less than DITO.Especially when the system gets larger, the benefits' difference gradually increases.When there are 136 satellites, DITO's benefits value is 12.4% larger than that of DITO-nodif.With the consideration of task importance factor in both topology optimization and resource management, DITO not only makes inter-layer links connect to important RLSs at important time-slots but also leaves limited storage and energy resources to important time-slots.The result verifies the benefits improvement of the tasks' importance consideration.
3) DITO-nopre has similar network benefits to DITO, but the solving time is longer.The solving time gap will be larger and larger with the increase of satellites and the cycle T .When there are 500 satellites, the solving time of DITO-nopre is about 16 times longer than that of DITO.Therefore, the preprocessing algorithms of DITO are effective for reducing optimization complexity.

VI. CONCLUSION
In this paper, we study the inter-layer time-space topology optimization problem in an LEO DLSN composed of communication satellites and remote-sensing satellites.A distributed interactive method DITO is proposed for onboard optimization and real-time observation data collection, which can achieve the goal of maximizing observation benefits considering task importance differences.From the perspective of each RLS, an ILP model based on realistic connectable conditions and link switching interval is proposed, with a new time-slots division method.An LP model for resource management is put forward with the time-sequential decoupling of variables.
Two preprocessing methods are introduced to meet reality and reduce complexity.We simulate the proposed DITO in four constellation systems with different scales.The feasibility of onboard optimization is verified as follows: for a system with 500 satellites and 30 minutes cycle, the total solving time is only 1.49s.In addition, we compare the proposed method with three other strategies, the results show that DITO can obtain an optimal solution close to the centralized global solution with much less time, and the consideration of task difference helps utilize limited storage and energy resources.
In the future, we will add RLSs' observation mission planning in the proposed model to achieve efficient collaboration of observation and communication.We will also do some experiments on semi-physical simulation systems to verify the effectiveness of the proposed distributed approach.
N j r , w min Number of RLSs sending requests to j, minimum duration of effective connection window.N opt c , j opt Number of CLSs for optimization, index vector of the optimized CLSs.

Fig. 2 .
Fig. 2. LEO inter-layer time-space topology graph: (a)Potential inter-layer topology.(b)Optimal inter-layer topology based on limited laser transceivers.(c)Data transmission scheduling based on limited resources.

2 .
Each RLS requests to establish inter-layer links with the selected CLSs during corresponding time-slots in T .3. Each CLS receives requests from RLSs.If there are more than L c requests at a time-slot, then only L c RLSs can be selected based on their data importance.4. Each CLS sends three types of messages to RLSs who made a request: AGREE (all requested time-slots are approved), REJECT (all requested time-slots are rejected), or PARTIAL (part of requested time-slots are approved, and the approved time-slots should also be sent).5.According to the determined topology scheme, each RLS makes a flow and energy scheme based on the resource management model.6.During T , each RLS sends data to CLSs based on the planned topology, flow, and energy scheme.

Fig. 4 .
Fig. 4. Geometric relationship diagram of RLS i and CLS j.

Theorem 1 .
The time complexity of the DITO algorithm is approximately upper bounded by O(2 N opt c •T ).Proof. 1) Complexity of Steps 2-7 in Algorithm 4 for each RLS making connection schemes.Function RSW of Algorithm 1 has a complexity of O(N c • K), where K has the maximum of T /2, so its complexity is upper bounded by O(N c •T /2).Function RN C of Algorithm 2 has a complexity of O(1).In Step 5 of Algorithm 4, the ILP problem can be solved by the Branch and Bound (B&B) algorithm, which has exponential time complexity related to the number of integer variables[33].Since there are N opt c •T integer decision variables of δ i in(19), the complexity of the B&B algorithm is O(2 N opt c •T ).Therefore, the total complexity of Steps 2-7 is2O(N c • T /2) + O(1) + O(2 N opt c •T ) ≈ O(2 N opt c •T ).2) Complexity of Step 10 in Algorithm 4 for each CLS making selections and sending replies.The complexity of Algorithm 3 is upper bounded by O(T ) + O(N r ) ≈ O(max{T, N r }). 3) Complexity of Steps 13-14 in Algorithm 4 for each RLS managing resources.The LP problem (28) has N opt ′ c •T continuous variables of q i and T continuous variables of M H i .It can be solved by the interior algorithm with the complexity of O((N opt ′ c

Fig. 7 .
Fig. 7. Solving time for different scale systems, setting T be 30 minutes.

Fig. 9 .
Fig. 9. Performance evaluation of four strategies under different system scales with T of 30 minutes.(a)Total solving time.(b)Total network benefits.

TABLE II DIFFERENT
CASES OF dδ i,t S i ← S i (j opt , :), update the new connectable matrix by removing no-effective CLSs. 6:

TABLE V COMPARISONS
OF INDICATORS IN FOUR SYSTEMS.