## 1 Introduction  
Satellite applications play a fundamental role in modern society, offering practical services, including television broadcasting, Earth observation, navigation and communication [1]. The last decade has witnessed the booming revolution of satellite communication from fixed communication and mobile communication to high-throughput communication [2]. Several large constellations of low Earth orbit (LEO) satellites with altitudes of 1500 km or less were proposed to provide global wideband with cutting-edge technologies, shorter propagation delay and rapid global deployment [3, 4]. Meanwhile, the control and operation of LEO constellations offers challenges to the ground stations. Due to their orbital periods of approximately 100 min, they can only contact a single ground station for a few minutes per pass [5]. Since the capability to transmit information with the ground station in time is crucial for stable in-orbit operations, a network of ground stations spread over the world would be considered to reduce the gaps between the contact times, which results in additional costs and complexity of the system [6]. To overcome the problem, space-based data relay satellite was invented to offer data relay service at geostationary Earth orbit (GEO). With an altitude of 35,786 km, a GEO satellite can cover up to 69% of LEO satellites. Uninterrupted in-time connection toward LEO satellites can be realized with two GEO data relay satellites to provide the data path to the ground station [7]. The existing data relay satellites were initially designed and optimized with target objects of the International Space Station, Earth observation satellites and other space shuttles or launch vehicles with relatively lower volume [8]. These kinds of classical data relay satellites include the Tracking Data Relay Satellite (TDRS) of America, the Luch satellite of Russia, and the European Data Relay Satellite (EDRS) [9; 10]. The principal communication link can be established based on the steerable antenna tracing the user spacecraft. The single data relay satellite can only provide concurrent access to less than 10 LEO objects in general [11]. So, the use of the conventional data relay service is mainly the high-value LEO spacecraft. In the last decade, several giant LEO satellite constellations were proposed, which spawned extensive demand of data relay between space and ground segments [12]. Novel types of data relay satellites should be considered to meet the pent-up demand for continuous, multi-user and cost-effective access to the growing volume of data from space [13; 14]. Several commercial companies raised feasible proposals to build a data relay system through multi-beam satellites. The Inter-satellite Data Relay System (IDRSTM) directly brings the capabilities of the Inmarsat Broadband Global Area Network (BGAN) system to space by integrating the IDRSTM terminal into a LEO spacecraft. While in orbit, the communication link to the Inmarsat BGAN Ground Network can be switched and maintained between beam spots of the Inmarsat-4 GEO satellites, which are multi-beam global mobile satellites [15]. On the one hand, leveraging upon the existing BGAN service portfolio, this proposal provides a range of always-on, on-demand, real-time connectivity, thus resolving the need to wait for the LEO satellite to pass above a ground station. On the other hand, the BGAN system was initially designed for ground and flight applications. There is no further optimization of the beam coverage and spot allocation focused on the LEO spacecraft application of the data relay service [16]. For another example, the data relay system, named SpaceLink, is customized toward spaceborne objects. The satellites also implement a fixed multi-beam antenna to realize the coverages of the LEO constellation [17]. The only difference is that the SpaceLink system selects medium Earth orbit (MEO), which helps reduce latency by about half, compared to a similar service of GEO.