Traffic Load Optimization for Multi-Satellite Relay Systems in Space Information Network: A Proportional Fairness Approach

Related theories and techniques of SIN are still in progress, and there are no certain models and structure design standards for SIN. Thus, an SIN architecture based on DSC is constructed in this paper according to the model in [21,22] and Definition 1. The architecture is shown in Figure 1.

SIN is divided into two layers from the networking aspect: the access network and the backbone network. The DSC architecture is adopted for the backbone network, and the DSC consists of multiple satellite clusters to construct a distributed satellite network with inter-cluster links (ICLs). Each satellite cluster consists of multiple GEO satellite on the same orbit, and there are inter satellite links (ISLs) connecting satellites to realize different topologies for each cluster with specific function for backbone networking task. Each cluster contains a primary satellite (PS), which realizes the connections with different clusters by ICLs. The ICLs and the ISLs are laser links. The access network is an integrated network with heterogeneous systems and platforms, which includes task and communication platforms distributing on the medium earth orbit and the low earth orbit, in the near space, the high altitude and the low altitude space, or on the ground. These heterogeneous systems and platforms achieve inter-connectivity and integration through the backbone network based on DSC.

In the SIN with a backbone network based on DSC, the systems and the platforms in the access network connect to the backbone network to access data and service; hence, they can be regarded as satellite users for the DSC. Then, the inter-connection between the access network and the backbone network can be described as a DSCN model. According to the payload diversity of the satellites and platforms and the features of links between satellites and users, the characteristics of the DSCN model are described as follows.

As can be seen from the above features, the wireless resource management of an SIN is a heterogeneous network resource configuration problem with high complexity due to the distributed characteristics of the network. At the same time, it needs to deal with the impact of network dynamic change and delay. The wireless resource optimization of SIN requires a unified resource management architecture to depict and plan resources, so as to adapt to various features caused by the distributed heterogeneous architecture of SIN. Thus, before the resource optimization of SIN, a reasonable resource management model should be designed according to the network characteristics of DSCN for effective network control.

In the previous subsection, SIN is described as a DSCN, and the main characteristics of SIN are given. It is necessary to design a reasonable resource management architecture for resource optimization and the management of such a complex comprehensive information service network. In order to adapt to characteristics of heterogeneity, high dynamic and long delay for SIN, and thus realize the rapid discovery and calculation of resources in the whole network, the reconstruction and configuration of local resources, and the central scheduling capability of SIN, SIN needs to have the ability of central control of wireless resources. At the same time, in order to improve the efficiency of resource allocation and reduce the processing delay of wireless resource allocation for some application scenarios, the resource control model of SIN needs to be capable of distributed resource optimization and allocation. In order to realize the centrally controllable and distributed optimization of resources to meet various comprehensive service requirements of SIN, this subsection proposes a hybrid resource management architecture combining distributed and central resources control schemes according to the main characteristics of SIN, which is shown in Figure 2.

As can be seen from Figure 2, the resource state information of the whole network is divided into local resource state information, regional resource state information, collaborative resource state information and global resource state information. A hierarchical structure is formed through user level, satellite level, satellite cluster level and DSCN global level. The resource state information contains parameters such as category, number and availability of resources, and link conditions. Related application protocols and software can be added through software-defined interfaces to realize information sharing and instruction transfer.

In order to shorten the delay of control signaling transmission between the satellites and various platforms accessing to satellites, the main network control functions are carried out by the satellite-borne network control center (NCC), while the ground NCC only uploads necessary update information, codes, data and manual intervention instructions into the satellite through the communication station to achieve network maintenance. The center-distributed hybrid structure is adopted to manage network resources, and each satellite in the backbone network carries a satellite-borne NCC. Data transmission among clusters is achieved by the PS of each cluster, and the primary satellite is the most idle satellite chosen from a cluster. NCCs of multiple PSs can cooperate together to realize collaborative resource management for SIN. Meanwhile, each satellite-borne NCC can also work independently to optimize resource allocation for subnetworks in its coverage area. When collaborative resource management is performed for the global optimization of SIN, satellite users under each backbone satellite sense and collect local resource state information. Through signaling channels, satellite users interact with the backbone satellite, and then, local resource state information is aggregated with resource state information on satellites to form regional resource state information. Regional resource state information is aggregated through ISLs among satellites within the cluster to form the collaborative resource state information of each cluster. The PS of each cluster is in charge of forming global resource state information through ICLs among each other, and they share the information with the whole network as a reference for resource allocation and calculation. Sub-nets of SIN start their own satellite-borne NCC according to the global resources status information (including network demands, capacity, etc.), and distributed computing is adopted to reduce the consumption of resource calculation. The satellite base station on the ground uploads resource allocation algorithms to the satellite-borne NCCs with an interface supplied by virtual network embedding and software definition technology. Then, the NCC calculations generate the respective resource configuration schemes according to the network demands. The resource configuration schemes guide the resource configuration of satellite users, the on-board resource reconstruction of backbone satellites, and the topology reconstruction of cluster links to realize network optimization. Meanwhile, after the resource configuration and reconstruction, the resource configuration and reconstruction results are shared through the internal information interface to realize the update of the resource state information at all levels of SIN. In the meantime, with updating instructions and codes uploaded from ground, the decision generation and calculation algorithms of satellite-borne NCCs can be upgraded, which realize the evolution of global and local decisions for SIN so as to ensure that the network has the ability of dynamic evolution according to the changes of network conditions, user behavior, electromagnetic environment and so on. Through the combination of multi-party distributed computing and central decision making, using virtual network embedding and software-defined technology to achieve different functions, the computing efficiency and flexibility of resource management can be effectively improved. At the same time, each network element in the above architecture can be decoupled and coordinated according to the needs of networking so as to meet the multi-scene and asymmetric resource optimization requirements in SIN.

In SIN, when there is a link failure or no direct link between the source node and destination node, data transmission can be achieved by multi-satellite relay through a backbone network. Relay transmission through the backbone satellite mainly exists in the following two situations. (1) The destination node (DN) and the source node (SN) are located in the same coverage area of a cluster, and the data are forwarded by the destination satellite (DS) directly covering the destination node. (2) The DN and the SN of the data are located in the coverage areas of different satellite clusters; then, the data can only reach the DS which covers the destination node by crossing multiple clusters through multiple PSs. Therefore, two multi-satellite relay scenarios for SIN are considered in this paper, and their models are shown in Figure 3. For scenario 1, the source node of the data is the communication platform (such as LEO and medium earth orbit (MEO) satellite) operating in non-geosynchronous orbit. This scenario describes a scenario in which non-geostationary communication platforms transmit data packets to the destination node through backbone satellites. For scenario 2, the source node of data is the PS of a cluster, which represents the scenario in which the PS of a cluster receives the data packets sent by PSs of other clusters and forwards them to the destination satellite [23].

In the two scenarios above, data packets of SN can be transmitted to DS by two ways: (1) constructing stable ISLs between SN and DS and (2) constructing relay ISLs to multiple backbone satellites with stable ISLs to DS. Suppose there is a failure or outage of ISL between SN and DS (shown by the gray links in Figure 3); at the same time, there exists M backbone satellites which have stable ISLs to DS. Then, M backbone satellites can be regarded as DRSs, and packets of SN can be forwarded to DS through these DRS. However, the capacity of a relay channel for one backbone satellite is limited. Hence, in order to improve the efficiency of transmission and reduce the transmission delay, relay data can be forwarded by the cooperative transmission of M DRSs. Since ISLs among DRSs are laser links, through satellite orbit position controlling and attitude adjustment, there can be no physical shielding and blocking among satellites. Therefore, the ISL channel for the line-of-sight signal between two satellites can be modeled as a Rician Fading channel with additive white Gaussian noise (AWGN), and the influence of rain attenuation can be ignored. Then, the received signal at time T of each DRS can be denoted by [24]. where $y_{s,m,t}$ is the received signal of DRS m at time t from SN s, $P_{s,m,t}$ is the transmit power of the signal that SN s is sending to DRS m at time t, $d_{s,m}$ denotes the distance between SN s and DRS m, $\gamma$ denotes the path-fading coefficient, $n_{s,m,t}$ denotes the AWGN at time t of the ISL between SN s and DRS m, and $h_{s,m,t}$ is a cyclosymmetry complex Gaussian random variable, which denotes the channel fading coefficient at time t of the ISL between SN s and DRS m. $n_{s,m,t}$ and the channel fading ${\left|h_{s,m,t}\right|}^2$ are independently and identically distributed on the ISL between SN s and DRS m. The mean value and variance of $n_{s,m,t}$ are 0 and $N_0$, respectively. The probability density function of ${\left|h_{s,m,t}\right|}^2$ is denoted by [24] where $s^2={\mu }_1^2+{\mu }_2^2$ is the power of the line of sight (LoS) signal, ${\sigma }^2$ is the power of the scattering signal, and $I_0\left(•\right)$ is the first kind of zeroth order modified Bessel function.

Hence, the channel-power-gain to noise ratio at time t on the ISL from SN s to DRS m can be expressed as

The received signal of each DRS must be higher than the signal to noise ratio (SNR) threshold of the satellite antenna $\mathrm{t}{\mathrm{h}}_m$ [24]; otherwise, the signal cannot be received and forwarded accurately. Then, we have

For scenario 1 in Figure 3, due to the existing of relative motion due to differences in orbital position and operating period, DRSs are periodically visible to SN s. Hence, there is a window time for relay, which can be denoted by $T_a$. Since the bandwidth of SN in scenario 1 is limited, the bandwidth of ISL between SN and DRS is divided into $T_a$ time-slots. Then, we use ${\delta }_{s,m,t}$ to denote the occupancy of time-slots t in the bandwidth of SN s for DRS m, where ${\delta }_{s,m,t}=1$ and ${\delta }_{s,m,t}=0$ where the mean time-slot t is and is not occupied by DRS m, respectively. Use $C_{s,m}$ to denote the capacity of received relay data from SN s for DRS m during $T_a$, which can be formulated as follows according to the Shannon formula [13,24].

As for scenario 2, SN and DRSs are both on stationary orbit; hence, the transmitting time is not limited by the visibility between SN and DRSs. The relay period $T_b$ is used to keep the stability of the data queue for SN and avoid congestion of the queue cache caused by a long queue length. The data have not been sent during $T_b$ and would be deleted after a relay period; then, the SN as well as PS in the cluster would inform the PS of the cluster where the data came from for retransmission through its ICL. The bandwidth of SN is relatively wider compared to scenario 1; hence, the bandwidth B can be divided into multiple sub-bandwidths and allocated as required to improve resource utility. Use $B_{s,m,t}$ to denote the bandwidth allocated by SN s to DRS m at time-slot t; similarly, according to the Shannon formula, the capacity of the signal at DRS m received from SN s which is denoted by $C_{s,m}$ can be formulated as

Since the main function of the SN in scenario 1 is not communication, the payload is limited and the ability of transmission is relatively low. Hence, a reasonable assumption can be made that the SN in scenario 1 only has one laser antenna; then, the laser antenna can only be aligned with one DRS during a time-slot. Suppose that the antenna adjustment algorithm is running by a specialized software module with an independent calculation unit, which can be accomplished synchronously with signal transmission; then, the time of adjustment can be ignored. To transmit data as much as possible during $T_a$, the occupancy of time-slots should be optimized according to the queue length and the link conditions of ISLs between SN and DRSs.

For scenario 2, the SN as well as PS has a powerful payload, and it can be assumed that the number of laser antennas is lager than M, which means the SN in scenario 2 can adjust laser antennas to aim at M DRSs simultaneously. Similarly, in order to improve the transmission capacity during $T_b$, bandwidths allocated to M DRSs should be optimized according to the queue length and the link conditions of ISLs between SN and DRSs.

Furthermore, for the above-mentioned two scenarios, the transmit power needs to be adjusted reasonably under different link conditions to satisfy the capacity requirements of each queue and constrains of the receiving SNR threshold for DRSs. Meanwhile, when there comes a bulk data flow, the traffic load at each DRS should be well-balanced to avoid data overload for each DRS, which would cause data congestion or even packet loss. In fact, the traffic load optimization problem can be regarded as a fairness issue for resource allocation, which means fairness resource allocation among multiple DRSs with optimal capacity. In this paper, the PF criterion is adopted to formulate the capacity fairness among DRSs, which can be denoted by [25]. where $U_m$ is the utility function for DRS m.

In scenario 1, a time-slot can only allocated to one DRS, which can be denoted by ${\displaystyle \sum _{m=1}^M}{\delta }_{s,m,t}\le 1,\forall t$. In scenario 2, the bandwidth allocated to one DRS should be no more than the total bandwidth, which can be denoted by ${\displaystyle \sum _{m=1}^M}{B}_{s,m,t}\le B,\forall t$. For these two scenarios, the total capacity of M DRSs cannot exceed the total capacity of the packets that need to be transmitted, which can be denoted by $\frac{1}{T_a}{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{t=1}^{T_a}}{\delta }_{s,m,t}\mathrm{lb}\left(1+P_{s,m,t}{r}_{s,m,t}\right)\le C_{s1}$ and $\frac{1}{T_{b}B}{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{t=1}^{T_b}}{B}_{s,m,t}\mathrm{lb}\left(1+P_{s,m,t}{r}_{s,m,t}\right)\le C_{s2}$ for scenarios 1 and 2, respectively, where $C_{s1}$ and $C_{s2}$ are the total capacity of packets that need to be transmitted for two scenarios. The transmit power at each time-slot should be no more than the total power, which can be denoted by $P_{s,m,t}\le P_{1,\mathrm{total}},\forall m,t$ and ${\displaystyle \sum _{m=1}^M}{P}_{s,m,t}\le P_{2,\mathrm{total}},\forall t$, where $P_{1,\mathrm{total}}$ and $P_{2,\mathrm{total}}$ represent the total power in scenario 1 and 2, respectively. In addition, the received signal of each DRS must be higher than the SNR threshold of the satellite antenna $\mathrm{t}{\mathrm{h}}_m$, which is shown in Equation (4).

Let $U_m=C_{s,m}$, in order to obtain a fair allocation of bandwidth and power resource, two joint bandwidth (time-slots) and power allocation problems with the constrains mentioned above for traffic load optimization in two scenarios can be formulated as follows.

Scenario 1:

$C1$ is the constrain of total capacity, which ensures that the total capacity of M DRSs is not more than $C_{s1}$. $C2$ denotes the constrain of the receiving SNR threshold, which ensures that the transmit power can satisfy the receiving SNR threshold. $C3$ is the constrain of total power, which ensures that the transmit power at each time-slot is less than or equal to total power. $C4$ is the constrain of time-slot occupation, which ensures that one time-slot can only be occupied by one DRS; in other words, the laser antenna of SN can only align with one DRS.

Scenario 2:

Similar to scenario 1, $C1$ is the constrain of total capacity, $C2$ denotes the constrain of the receiving SNR threshold, $C3$ is the constrain of total power, and $C4$ is the constrain of bandwidth occupation, which ensures the summation of allocated bandwidths for DRSs to be not more than the total bandwidth B.

The optimization problem in this paper can be solved by minimizing its dual problem. By introducing Lagrange multipliers ${\lambda }_1$, $\left\{{\alpha }_{1,m,t}\right\}$ and $\left\{{\beta }_{1,t}\right\}$, the Lagrangian function of the problem in scenario 1 can be denoted by    where ${\lambda }_1\ge 0$, ${\alpha }_{1,m,t}\ge 0,\forall m,t$, and ${\beta }_{1,t}\ge 0,\forall t$. ${\mathbf{P}}_1={\left[P_{s,m,t}\right]}_{M\times T_a}$ is a power allocation matrix, which denotes the power values of SN s in ISLs to M DRSs at each time-slot. ${\Delta }_1={\left[{\delta }_{s,m,t}\right]}_{M\times T_a}$ is the time-slot allocation matrix, which denotes time-slots occupation during $T_a$. Thus, the dual function is denoted by

Hence, the problem can transferred into a dual problem, which can be expressed as

By simplifying, decomposing, merging, etc., Equation (18) can be rewritten as where $L_1^{*}\left({\lambda }_1,\left\{{\alpha }_{1,m,t}\right\},\left\{{\beta }_{1,t}\right\},P_1,{\delta }_1\right)$ is the component including ${\mathbf{P}}_1$ and ${\Delta }_1$, which can be denoted by

In fact, $\underset{{\mathbf{P}}_1,{\Delta }_1}{\max }{L}_1\left({\lambda }_1,\left\{{\alpha }_{1,m,t}\right\},\left\{{\beta }_{1,t}\right\},{\mathbf{P}}_1,{\Delta }_1\right)$ is equivalent to $\underset{{\mathbf{P}}_1,{\Delta }_1}{\max }{L}_1^{*}({\lambda }_1,\left\{{\alpha }_{1,m,t}\right\},\left\{{\beta }_{1,t}\right\}$, ${\mathbf{P}}_1,{\Delta }_1)$; hence, we have

According to Equation (22), the cumulative sum of components for M DRSs and the maximization of the dual function are decoupled; therefore, the above problem can be broken down to M subproblems. With given ${\lambda }_1$, $\left\{{\alpha }_{1,m,t}\right\}$ and $\left\{{\beta }_{1,t}\right\}$, the first derivative of $L_1^{*}\left({\lambda }_1,\left\{{\alpha }_{1,m,t}\right\},\left\{{\beta }_{1,t}\right\},{\mathbf{P}}_1,{\Delta }_1\right)$ with respect to $P_{s,m,t}$ can be denoted by and the first derivative of $L_1^{*}\left({\lambda }_1,\left\{{\alpha }_{1,m,t}\right\},\left\{{\beta }_{1,t}\right\},{\mathbf{P}}_1,{\Delta }_1\right)$ with respect to ${\delta }_{s,m,t}$ can be denoted by For Equation (24), when ${\delta }_{s,m,t}=0$, the right-hand side of the formula does not make sense. This is due to the coupling relationship between time-slots and power, when a time-slot is not occupied by one DRS, the power in the ISL between the DRS and SN is 0. Hence, we only consider the condition of ${\delta }_{s,m,t}=1$, according to the Karush–Kuhn–Tucker (KKT) condition [28]. Let Equation (24) be equal to 0, then, we have

After transposing and combining, we have

Take the exponent of 2 from both sides of Equation (27); then, we have

Let $\phi =1+P_{s,m,t}{r}_{s,m,t}$; then, the equation above can be rewritten as

When the total packet quantity is greater than the capacity, constrain $C_1$ in scenario 1 is always true, which means that in this case, ${\lambda }_1$ can be considered as 0. When the total packet quantity is less than the capacity, $C_{s1}$ can be regarded as equal to the total packet quantity. Hence, according to convex optimization theory, ${\lambda }_1$ satisfies the following complementary slackness conditions. where $P_{s,m,t}^{*}$ and ${\delta }_{s,m,t}^{*}$ are optimal solutions. Let ${\lambda }_1>0$, if ${\delta }_{s,m,t}$ is relaxed to a continuous number between 0 and 1, then, when $\frac{\partial L_1^{*}\left({\lambda }_1,\left\{{\alpha }_{1,m,t}\right\},\left\{{\beta }_{1,t}\right\},{\mathbf{P}}_1,{\Delta }_1\right)}{\partial {\delta }_{s,m,t}}=0$, ${\delta }_{s,m,t}$ reaches the maximum value, which is 1. Hence, according to Equation (25), we have put it into Equation (29); then, we have

According to the formal characteristic of Equation (32), the Lambert-W function can be introduced to simplify the equation, which is denoted by where $W(·)={\displaystyle \sum _{i=1}^{+\infty }}\left({(-i)}^{i-1}/i!\right){(·)}^i$ is the Lambert-W function. Put $\phi =1+P_{s,m,t}{r}_{s,m,t}$ into Equation (33), through basic operations such as transposition, the closed-form solution of optimal power $P_{s,m,t}^{*}$ can be derived. Since $P_{s,m,t}^{*}$ is greater or equal to 0, it can be denoted by    where ${\left(x\right)}^{+}=\max (0,x)$.

Equation (25) decreases with ${\delta }_{s,m,t}$, and when ${\delta }_{s,m,t}$ is relaxed to a continuous number between 0 and 1, 1 is the maximum value on the domain of ${\delta }_{s,m,t}$. Hence, Equation (25) reaches its minimum value at ${\delta }_{s,m,t}^{*}=1$. Then, the closed-form solution of optimal time-slots allocation indexes ${\delta }_{s,m,t}^{*}$ can be denoted by

Similar to scenario 1, by introducing Lagrange multipliers ${\lambda }_2$, $\left\{{\alpha }_{2,m,t}\right\}$, $\left\{{\beta }_{2,t}\right\}$ and $\left\{{\pi }_{2,t}\right\}$, the Lagrangian function of the problem in scenario 21 can be denoted by where ${\lambda }_2\ge 0$, ${\alpha }_{2,m,t}\ge 0,\forall m,t$, ${\beta }_{2,t}\ge 0,\forall t$ and ${\pi }_{2,t}\ge 0,\forall t$. $L_2^{*}\left({\lambda }_2,\left\{{\alpha }_{2,m,t}\right\},\left\{{\beta }_{2,t}\right\},\left\{{\pi }_{2,t}\right\},{\mathbf{P}}_{\mathbf{2}},{\mathbf{B}}_{\mathbf{2}}\right)$ is the component including $P_2$ and $B_2$, which can be denoted by where ${\mathbf{P}}_2={\left[P_{s,m,t}\right]}_{M\times T_b}$ is the matrix of power allocation, and ${\mathbf{B}}_2={\left[B_{s,m,t}\right]}_{M\times T_b}$ is the matrix of bandwidth allocation, which denote the power and bandwidth allocated in each time-slot of ISLs between SN and M DRSs, respectively. Then, the dual function can be formulated as

Hence, the original problem can be transferred into a dual problem as follows.

According to Equation (37), the dual problem in Equation (39) can be decoupled into M subproblems. Similar to the last subsection, with given ${\lambda }_2$, $\left\{{\alpha }_{2,m,t}\right\}$, $\left\{{\beta }_{2,t}\right\}$ and $\left\{{\pi }_{2,t}\right\}$, let $\frac{\partial L_2^{*}\left({\lambda }_2,\left\{{\alpha }_{2,m,t}\right\},\left\{{\beta }_{2,t}\right\},\left\{{\pi }_{2,t}\right\},{\mathbf{P}}_{\mathbf{2}},{\mathbf{B}}_{\mathbf{2}}\right)}{\partial P_{s,m,t}}=0$ and $\frac{\partial L_2^{*}\left({\lambda }_2,\left\{{\alpha }_{2,m,t}\right\},\left\{{\beta }_{2,t}\right\},\left\{{\pi }_{2,t}\right\},{\mathbf{P}}_{\mathbf{2}},{\mathbf{B}}_{\mathbf{2}}\right)}{\partial B_{s,m,t}}=0$. Then, Equations (40) and (41) can be obtained according to the KKT condition.

With the transposition of Equation (41), the optimal bandwidth allocation solutions $B_{s,m,t}^{*}$ with given optimal power allocation solutions $P_{s,m,t}^{*}$ can be denoted by

Substituting Equation (42) into Equation (40), we have

With the transposition of Equation (43), it can be rewritten as