I Introduction
In the past decades, the rapid development of cellular communications [1] has triggered users’ increasing demand for highdatarate applications [2, 3], which in turn raise stringent requirements on achieving massive connectivity and high capacity in the 5G systems^{1}^{1}1Typical highdatarate applications include video streaming [3, 4], environment sensing, etc.. To meet such demands, the heterogeneous network [5, 4] has been considered as a well accepted architecture due to its flexible deployment and effective data offloading. However, there still remain several unsolved issues. For example, the network coverage cannot be guaranteed due to the sparse resources and intractable access point deployment [6]. Moreover, it is usually not practical to construct fiberequipped or stable wireless backhaul links for every small cell [7] due to the complex environments. Therefore, the limited backhaul capacity of small cells may lead to a degraded offloading performance.
Fortunately, recent advances in low earth orbit (LEO) satellite networks over the highfrequency band have provided an alternative solution for coverage extension and backhaul connectivity. Driven by SpaceX [8] and OneWeb [9], the ongoing LEO constellation projects plan to launch thousands of LEO satellites over the earth, aiming to deploy an ultradense (UD) constellation and cooperate with the traditional operators to support seamless and highcapacity communication services. With provisioning of the feasibility, the projects perform the pipeline production of small satellites to lower the manufacturing cost [10]. Instead of the traditional small cell base station (SBS), a dedicated terrestrialsatellite terminal (TST) equipped with steerable antennas acts as the access point in this case. Easy to install on the roof or eNodeB owing to its miniaturized antennas, each TST supports both the highquality TSTsatellite backhaul links over Kaband and the userTST links over Cband, enabling a terrestrial small cell coverage for the users. Compared to the traditional networks, the LEO network provides a great number of users with a highcapacity backhaul, vast coverage, and more flexible access technique, which is less dependent of real environments.
To fully exploit the LEO constellation technique, we propose a terrestrialsatellite architecture to integrate the LEO satellite networks and the terrestrial networks for traffic offloading in this paper. Each user is scheduled to upload its data via the macro cell, the traditional small cell (TSC), or the LEObased small cell (LSC). Each SBS and TST then upload the collected data to the core network via the traditional backhaul and LEObased backhaul, respectively. Benefited from the UDLEO constellation, we assume that each TST is allowed to connect to multiple satellites simultaneously [8, 11], thereby improving the resilience to the frequent handover of satellites [12]. Deployed by the same operator, all cells share the same Cband frequency resources for terrestrial communications. Multiple TSTsatellite backhaul links over Kaband are scheduled for each LSC. The network aims to maximize the sum rate of all users and accommodate as many users as possible. Therefore, the user association and resource allocation of multiple cells should be optimized subject to the backhaul capacity constraint of each cell. At the same time, the backhaul capacity of each LSC also needs to be maximized via the satellite selection and resource allocation.
Various techniques have been considered for traffic offloading in the heterogeneous networks, such as satellite access [13, 14], hybrid satelliteterrestrial relaying [15, 16, 17], devicetodevice (D2D) multicell interference management [18, 19, 20], and cloud radio access. In [13], the terrestrialsatellite backhaul network shares the Kaband and the terrestrial wireless links deploy the hybrid analogdigital transmit beamforming to mitigate interference. In [14], a single satellite covers a large area and the integrated terrestrialsatellite networks perform the cloudbased resource allocation. In [15, 16, 17], a hybrid terrestrialsatellite network sharing the same spectrum has been considered where either the satellite serves as a relay for one terrestrial sourcedestination pair or a terrestrial node relayes for the satellite – user pair. The average symbol error rate has been derived and the analytical diversity order has been obtained. In [18, 19], the D2D underlaying dense network has been considered and a distributed algorithm has been proposed to cope with the strong intracell and intercell interference. In [20], energyefficient user association and resource allocation has been investigated in a multicell downlink network under the constraints of the backhaul capacity.
Most of the existing works [13, 14, 15, 16, 17, 18, 19, 20] have assumed either ideal or fixed backhaul capacity. Differently, in this paper we aim to improve the terrestrialsatellite system performance by considering the influence of dynamically varying backhaul capacity determined by the satellite selection and Kaband resource allocation. Moreover, the multiconnectivity also provides new dimension for backhaul capacity optimization, but at the same time new challenges have thus been posed on the traffic offloading strategy design. On the one hand, the LEObased network induces long propagation delay even though it has highcapacity backhaul links. Combined with the backhaul transmission delay, the overall delay will influence the user scheduling strategy. Therefore, scheduling and resource allocation of both terrestrial and satellite networks are coupled. On the other hand, due to the anglesensitivity and the multiconnectivity, the UDLEO satellite networks may suffer from intractable intersatellite cochannel interference. The influence of angular separation between two TSTsatellite links should be carefully considered to improve the LEObased backhaul capacity.
The main contribution of this paper can be summarized as below:

We propose a scheme for data offloading in the heterogenous networks by utilizing the highcapacity LEObased backhaul. To perform the user scheduling in such a network, a joint optimization problem is formulated to maximize the sum rate and number of accessed users subject to the backhaul capacity constraints.

The formulated optimization problem is decomposed into two subproblems, i.e., the maximization of the sum rate and number of accessed users in the terrestrial networks and the total backhaul capacity maximization in the satellite networks, closely connected by the iteratively varying Lagrangian multiplexers. We then convert these two subproblems into two matching problems with externalities. For the first subproblem, a lowcomplexity modified GaleShapely matching algorithm [21] is proposed. For the second one, we develop a novel swap matching algorithm with power control and gradientbased pruning procedures, in which the anglesensitive nature is captured.

Based on computer simulation results, the performance of our proposed UDLEO based integrated terrestrialsatellite (LITS) scheme significantly outperforms the traditional nonintegrated networks. We also find that the traffic load and different LEO satellite constellations will influence the user scheduling strategy.
The rest of this paper is organized as follows. In Section II, we describe the model of integrated UDLEO based terrestrialsatellite networks. In Section III, we formulate a joint optimization problem for the terrestrialsatellite networks and propose a framework to iteratively solve two decoupled subproblems from Lagrangian dual decomposition. In Sections IV and V, we convert these two subproblems into two different matching problems and solve them. Simulation results are presented in Section VI, and finally, we conclude the paper in Section VII.
Ii System Model
In this section, we first introduce the UDLEO based integrated terrestrialsatellite network in which the users can access the network via a macro BS or the TSCs or the LSCs. We then provide the transmission models of both the terrestrial and satellite communications. For convenience, we summarize all nonstandard abbreviations in Table I.
Abbreviation  Paraphrase 

LITS  LEObased integrated terrestrialsatellite scheme 
LBCO  LEObased backhaul capacity optimization 
LSC  LEObased small cell 
PC  Power control 
NITS  Nonintegrated terrestrialsatellite networks 
SBS  Small cell base station 
SMPC  Swap matching algorithm with power control 
TSC  Traditional small cell 
TST  Terrestrialsatellite terminal 
TTH  Traditional terrestrial heterogeneous networks 
TTO  Terrestrial traffic offloading 
TUASA  Terrestrial user association and subchannel allocation 
UD  Ultradense 
Iia Scenario Description
Consider an UDLEO based integrated terrestrialsatellite network as shown in Fig. 1, where one macro BS and a large number of small cells are deployed to serve the uplink ground users^{2}^{2}2The proposed architecture is independent of the satellite altitude and constellation, and thus, can be directly extended to the MEO and GEO satellite systems. The data routing and signaling control problems need to be considered for the multilayer system, which is not the focus of this paper. Readers may find an initial work focusing on contact graph based routing in [22].. Each small cell assists the macro cell to offload the traffic and is connected to the core networks via either wired or wireless backhauls. Therefore, each user, such as a mobile device or a sensor, can access the network via one of the following three cells: 1) the macro cell with large backhaul capacity supported by fiber links from the macro BS directly to the core network; 2) the TSC with very limited backhaul capacity connected to the core network via multihop wired or wireless backhaul links; 3) the LSC with large backhual capacity supported by the Kaband transmission. For LSC backhaul, each TST uploads the user’s data to the LEO satellites, and then each satellite forwards it to either an earth gateway station (connected to the core network) or a TSTequipped macro BS.
The ultradense LEO topology ensures that multiple satellites fly over the area of interest at each time slot, providing a seamless coverage for the mobile users. Equipped with multiple independent antenna apertures [8, 9], each TST can connect to multiple satellites simultaneously, which further improves the backhaul capacity of the LSCs. Based on the current development of satellite technique [9], we assume that each satellite serves as a remote radio head of the BS with no onboard processing capacity. Therefore, the access control is located at the macro BS equipped with a TST [23]. For intercell interference management, we assume that the satellite operator and the traditional terrestrial operator cooperate to serve the users in a centralized manner.
IiB Transmission Model for Terrestrial Communications
Denote the set of cells as in which represents the macro BS, represents the TSCs, and indicates the LSCs. To describe the relationship between the positions of users and the coverage of each cell, a binary coverage matrix A of size is introduced where indicates that user lies in within the coverage of cell , and otherwise.
For frequency reuse, all cells share the same frequency resource pool, which can be divided into subchannels. To better depict the user association and subchannel allocation, we introduce a binary matrix X of size in which indicates that user is served by BS over subchannel , and otherwise. The received signal of BS sent by user over subchannel is then given by
(1) 
where is the transmit powers of each user, (or ) is the transmitted signal of user (or user ) with unit energy, and the corresponding channel coefficient is (or ). Specifically, we denote , where is a complex Gaussian variable representing Rayleigh fading,
follows lognormal distribution representing shadowing fading,
is the distance between users and , and represents the pathloss exponent. The additive white Gaussian noise (AWGN) at BS is denoted by , andis the noise variance.
The achievable rate of each user served by BS over subchannel can be expressed by
(2) 
and thus, the data rate of each cell can be obtained by
(3) 
In practice, the data rate of each cell is required to be no larger than the backhaul capacity of this cell, which also influences the traffic offloading scheme of the system. For convenience, we denote the backhaul capacity of each cell as , which is fixed when .
IiC Transmission model for LEObased Backhaul
Instead of a fixed value, the backhaul capacity of each LSC is related to the transmission model of the LEObased backhaul over Kaband. Note that the satellitetoground links usually occupy a wider bandwidth than the TSTtosatellite links and the transmit power of satellites is usually larger than that of the TSTs [8]. The LEO backhaul capacity is thus constrained by the TSTtosatellite links, as illustrated below.
We assume that there are satellites flying over the area of interest. Due to the preplanned orbit of each satellite, its altitude, speed, and position information are known to all cells in each time slot. For convenience, we adopt a qausistatic method to split a time period into multiple time slots^{3}^{3}3To save the signaling cost, we adopt the semipersistent scheduling where the LEO backhaul links are updated every few time slots. during each of which the position of a satellite is unchanged. We divide the available bandwidth over the Kaband spectrum as a set of subchannels . A binary matrix depicting the TSTLEO association and subchannel allocation is defined as B where the element indicates that TST is associated with satellite over subchannel and otherwise. The received signal of satellite sent by TST over subchannel can be given by
(4) 
where (or ) is the transmit power of TST to satellite (or TST to satellite ) over subchannel , (or ) is the transmitted signal of TST to satellite (or TST to satellite ). The channel gain of the TST – satellite link over subchannel is denoted by , with both the largescale fading and the shadowedRician fading [24] taken into consideration^{4}^{4}4
The channel state information (CSI) here can be obtained by adopting the widely utilized training data based CSI estimation techniques
[25, 26, 27]. For reference, the p.d.f. of the channel coefficient is given as , where the parameters , , , and can be found in [17].. As shown in Fig. 2, is the antenna gain of TST towards satellite and is the offaxis antenna gain of TST towards the direction of satellite when the target direction of TST is towards satellite . Denote the angular separation between the TST – satellite link and the TST – satellite link as . The item is a function of , as shown in [28] (Attachment III, Appendix 8). AWGN at satellite is .The achievable rate of the TST – satellite link over subchannel can be obtained by
(5) 
Note that each TSTsatellite link suffers the propagation delay due to the long distance compared to the terrestrial communications. Therefore, given the traffic load of each TST, i.e., , the equivalent backhaul capacity can be given by
(6) 
where is the capacity of the TST – satellite link, is the traffic load over the TST – satellite link, and is the propagation delay calculated by with being the altitude of satellite and being the speed of light. The values of can be obtained by solving the following equations:
(7) 
where is the set of associated satellites for TST . In practice, the traffic load is set as , where is the set of users associating to cell and is the amount of data generated by each user . For convenience, we denote the total backhaul capacity without considering the propagation delay as .
Iii Problem Formulation and Decomposition
In this section, we aim to maximize the sum rate and number of accessed users by jointly optimizing the terrestrial data offloading, TSTsatellite association, and resource allocation.
Iiia Angular constraints for LEO backhaul
To further depict the characteristics of UDLEO networks such as angular sensitivity on satellite selection, the angular constraint on the terrestrialsatellite links are illustrated below.
As shown in Fig. 2, due to different altitudes of the satellites, the elevation angles of two satellites serving the same TST may be the same. When a TST transmits to such two satellites over the same subchannel, they appear to be “in line” in this case, thereby leading to transmission failure. To cope with this issue, we define the angular separation as within which two satellites cannot provide service to the same TST over the same subchannel. Based on the preplanned satellite orbits, we introduce an angle matrix , where the elevation angle of the TST – satellite link is denoted as . The angle constraint can then be presented as
(8) 
IiiB Problem Formulation
We consider two performance metrics in this work. First, we aim to maximize the sum rate of all cells over the Cband spectrum subject to the backhaul capacity constraints. Second, to extend the coverage of the traditional terrestrial cellular networks, it is necessary to maximize the number of active users in the network. The problem formulation can be shown as below.
where is the conversion parameter. Constraint implies that each user can only access a cell within its reach, i.e., user is within the range of its subscribed cell. For fairness, we assume that each user can only access one cell and be assigned one subchannel, as shown in constraint . As it is typical for the uplink in 3GPP, we consider orthogonal use of frequency resources within each cell, which is guaranteed by constraint . The relationship between the data rate of each cell and its backhaul capacity is shown in , constructing the coupling between the terrestrial and satellite networks. Without loss of generality, we assume that each subchannel of satellite can only be assigned to at most one TST and each TST is associated to at most satellite links simultaneously for backhaul, as presented in constraints and . The transmit power constraint is presented in where is the maximum transmit power of each TST for backhaul.
IiiC Lagrangian Dual Decomposition
As shown in problem , the terrestrial data offloading and the satellite selection are coupled with each other via the backhaul capacity constraint. Therefore, we decompose the original problem into two subproblems connected by the Lagrangian multiplexers.
Denote
as a vector containing the Lagrangian multiplexers associated with constraint
. The Lagrangian associated with problem is then defined as(10) 
Correspondingly, the dual optimum is given by
(11) 
For given , the first item in is only determined by X and the second item is only determined by B and . Therefore, we can divide the original problem into two subproblems, i.e., the terrestrial traffic offloading problem
(12) 
and the LEObased backhual capacity optimization problem
(13) 
where the summation is counted from since is fixed.
The optimization process consists of multiple iterations and in each iteration two steps are performed: i) given , the TTO problem and LBCO problem are solved, respectively; ii) update by in which is a monotonically decreasing exponential function of . The iterations will not stop until .
For those optimization problems with continuous variables, the outcome obtained from the Lagrangian dual method always satisfies the constraints [29]
. However, this cannot be guaranteed for our formulated problem since there are only binary variables such that the data rate of each cell
does not vary continuously. Therefore, we adjust the user scheduling strategy after step one if the constraint is violated. For those cells violating , the associating users are removed one by one following the order of increasing data rate of each userBS link until is satisfied.Iv Algorithm Design for Terrestrial Data Offloading
In this section, we formulate the TTO problem in as a onetoone matching with externalities and propose a modified GaleShapley algorithm with redefined preference relations.
Iva Matching Problem Formulation for Terrestrial Traffic Offloading
Note that the TTO problem is a threedimensional integer programming problem with a nonconvex objective function. Aiming at solving this problem by a lowcomplexity algorithm, we recognize that the user association and subchannel allocation can be regarded as a multivariate matching process. Specifically, the users, BSs, and the subchannels are three sets of players to be matched with each other to maximize the utility, while the interdependencies exist among the users due to the cochannel interference. This enables us to solve the problem by utilizing the matching theory, as shown below.
IvA1 Definitions
Consider the set of users, BSs, and the subchannels as , and , which are disjoint with each other. By associating each BS with each subchannel, we construct a new set of size , where each BSsubchannel unit can be represented by . A matching is defined as a mapping from the set into itself such that for each user and each BSsubchannel unit , we have if and only if . In other words, if , then user is associated with BS over subchannel . Such a onetoone matching naturally satisfies constraints and .
To construct a matching , each BSsubchannel unit selects a user from to match with such that the weighted sum rate in can be maximized. For convenience, when , we denote the utility of the BSsubchannel unit and user as .
IvA2 Preference Relation
Due to the intercell cochannel interference, there exist externalities [30] in this matching problem. That is, the utility of is influenced by other users matched with . Therefore, each BSsubchannel unit actually has preference over the matching pairs due to different interferences levels brought by them.
Specifically, given a matched BSsubchannel unit , it prefers to be cohabitated with a new pair with highquality user – BS link and poorquality user – BS interference link over subchannel . To depict such mutual effect, we construct a preference matrix of size for each BSsubchannel unit evaluating all possible matching pairs over subchannel . Each element in is defined as
(14) 
where and are preference parameters. We then say that a BSsubchannel unit prefers to if , i.e.,
(15) 
where is the set of unmatched users. Each BSsubchannel unit’s attitude towards the potential matching pairs is affected by the preference parameters and in . When , the unit has a pessimistic attitude, i.e., it only cares to minimize the interference brought by a potential matching pair. When , the unit has an optimistic attitude, i.e., it only cares to maximize the benefit brought by the new matching pair. Accordingly, reflects a neutral attitude.
The traditional preference of each subchannel over different matchings is also introduced. Define the utility of each subchannel as the sum rate of all BSs over this subchannel, i.e., . The preference of subchannel is then given by
(16) 
where is the utility obtained by subchannel under .
Due to the logarithmicform utility function, the externalities in our formulated problem do not share the additive characteristic with other welldefined onetoone matchings with externalities [31].
IvB Algorithm Design
IvB1 Initialization
We adopt a greedy algorithm to initialize the matching where each subchannel is matched with the most preferred combination of a BS and a user that can accept it. Specifically, each unmatched subchannel proposes to an unmatched user and BS to match with satisfying
(17) 
If user is proposed by more than one subchannel, then it selects a unit with the largest channel gain from the candidates and rejects the others.
IvB2 ProposeandReject Operation
Based on the defined preference relationship, we introduce the key operation of each matching pair, consisting of one proposing phase and two rejecting phases. Different from the traditional matchings, each matched BSsubchannel unit is allowed to propose to potential matching pairs instead of users.

BSsubchannel unit proposing: Each matched BSsubchannel unit in the current matching selects its most preferred matching pair satisfying
(18) where is the set of unmatched BSs over subchannel . A set of candidate pairs consisting of is then constructed for each subchannel .

Subchannel Rejecting: Each subchannel selects one candidate matching pair from such that
(19) i.e., the matching pair that brings the highest positive utility to subchannel . Other matching pairs in are then rejected.

User Rejecting: If an unmatched user is proposed more than once, it only accepts the BSsubchannel unit with the highest utility and rejects all the other proposals. Once a user is matched, it is removed from .
IvB3 Algorithm Description
The whole matching algorithm for terrestrial user association and subchannel allocation (TUASA) is presented in detail in Table 1. In the initialization step (line 212), each subchannel is matched with a combination of one user and one BS with the best channel condition that it can achieve. The following matching process (line 1433) consists of multiple iterations in each of which the proposeandreject operation (line 1629) is performed. The iterations will not stop until no matched BSsubchannel unit would like to propose to the users any more.
IvC Algorithm Analysis
IvC1 Remark on the number of accessed users
The proposeandreject operation offers the optimal strategy to maximize the number of accessed users in the TTO problem since each user is compulsively served by a BS as long as there exist an unmatched BSsubchannel unit satisfying . In other words, the proposed TUASA algorithm guarantees the maximum number of accessed users while improving the sum rate.
IvC2 Stability and Convergence
As proved in Appendix A, our proposed TUASA algorithm is guaranteed to converge to a final matching after a limited number of iterations.
Due to the complicated interaction relationship between different pairs brought by the cochannel interference, the preference in our formulated problem does not satisfy the substitutability condition, which is usually a sufficient condition for the existence of a stable matching. Therefore, the traditional solution concepts of blocking pair and stability cannot be applied in this case. We then introduce the concept of a blocking pair, which is stricter than the traditional version.
Definition 1: Given a matching , where and ( and are allowed to be equal), we denote the matching where user is matched with while other pairs are unchanged as . The pairs and are individualrational blocking pairs if i) , , ; ii) and . In other words, the blocking pairs and can bring higher utility to all existing pairs with respect to subchannels and in .
Given the above definition, we discuss the stability and equilibrium based on different attitudes of the BSsubchannel units. The following statements are proved in Appendix B.

When , we have the group stability concept in the sense that , , , and are considered as a group. In other words, there does not exist individualrational blocking pairs satisfying condition ii) in Definition 1.

When , instead of the group stability, we have the concept of equilibrium. When there is no individualrational blocking pair in a matching, any new matching pair cannot improve the utility of the unit without compromising the utility of other matched BSsubchannel units. The final matching then reaches an equilibrium point.
IvC3 Computational Complexity
The maximum number of iterations in the initialization phase is , and the number of outer iterations in the matching phase is also proportional to . For the worst case, in each outer iteration all matched pairs propose to the same user such that only one new matching pair is formed. The total number of outer iterations in this case is . For the best case, in each iteration a new matching pair is accepted over a subchannel, i.e., new pairs are recorded. Since there are users in total, the number of required iterations is . In practice, the iteration number varies between and .
V Algorithm Design for LEObased Backhaul Capacity Maximization
In this section, we convert the LBCO problem in into a manytoone matching problem with externalities. To depict the multiconnectivity and anglesensitive nature of the UDLEO networks, a novel swap matching algorithm with continuous power control (SMPC) is proposed.
Va Matching Problem Formulation
VA1 Definitions
Consider the set of TSTs, satellites, and available subchannels as , , and . We assume that each satellite and subchannel form a satellitesubchannel (SS) unit . A matching is defined as a mapping between and such that for each TST and each SS unit , we have: i) if and only if ; ii) and ; iii) and . Specifically, each matching pair is denoted by , where is the transmit power of TST over this link, satisfying constraint (9g). We aim to find a matching such that the weighted capacity can be maximized. The equivalent capacity can then be obtained by solving equations in .
VA2 Preference Relation
Following Section IVA2, the influence brought by a potential pair with respect to the existing matched SS unit is redefined as
(20) 
Therefore, the preference relation can be given by
(21) 
Each subchannel’s preference over different matchings is similar to that in Section IVA2. The utility of subchannel is defined as in this case.
VA3 Remark on the New Problem
Two new challenges have been posed compared to the TTO problem, rendering most existing matching algorithms [32, 33, 34] as well as Algorithm 1 not suitable any more, as illustrated below.
First, there lacks a mechanism for transmit power adjustment in each projectandreject operation as shown in Section IVB2. In addition, the complete preference list based on over all possible power levels is very difficult to construct.
Second, due to the angular sensitivity, the group stability in Algorithm 1 cannot be guaranteed any more. Typically, in the matching obtained from Algorithm 1, the interference caused by user to a given BS is irrelevant of its matched BS. However, in the LBCO problem, as long as TST switches its matched satellites from to , the cochannel interference brought to any other satellite varies due to different offaxis antenna gains, i.e., . By comparing equations and , we can infer that the group stability does not hold in the LBCO problem even if , implying that the proposeandreject operation fails to depict such dynamic matching structure change over each subchannel in the LBCO problem.
VB Algorithm Design
We introduce the swap matching performed by each TST as below. Generally speaking, in a swap operation, a TST tends to swap its matches with another TST while keeping other TSTs’ strategies unchanged.
Definition 2: Given a matching with two existing matching pairs and , a swap matching is defined as
(22) 
Both the TSTs and SS units are allowed to be virtual. There is no physical meaning for a virtual TST or SS unit and the corresponding utility is zero.
a) Power Allocation in Swap Matchings:
Based on this generalized definition, we discuss the power control strategy for five typical types of swap matchings. Other cases not mentioned here can be classified into one of the following cases.
Type1 (TST and SS unit are virtual): The swap matching degrades to TST proposing to the unmatched SS unit and . Note that by allocating power to the TST – satellite link, the SS units in may be influenced since the allocated power needs to be adjusted to satisfy the power constraint. Therefore, different from the traditional swap matching, more than two matching pairs may be involved during the swap matching in the LBCO problem.
To evaluate how the power control influences other SS units in , we consider a pair with . The utility of subchannel is rewritten as a function of ,
(23) 
where (or ) is the interference received by satellite (or ) over subchannel and is the signal strength of the TST – satellite link. All these three items are irrelevant of .
Define the negative gradient of as . It is observed that the larger is, the smaller influence suffers from the reduction of its transmit power. Therefore, we select a pair from the least affected by the swap matching, i.e.,
(24) 
To perform the swap matching, we consider maximizing the total utility of the involved subcahnnels via the power control of TST . For convenience, we denote the available power budget of this swap matching as , where is the unallocated power of TST . The power control problem for can be formulated as
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