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As one of the most important convergence layer (CL) protocol for delay/disruption-tolerant networking (DTN), Licklider transmission protocol (LTP) is designed for deep space communications, but it has rarely been considered for near earth applications. In this paper, LTP is proposed instead of TCP as CL with Bundle protocol (BP) for future application in GEO-relayed space networks (GRSN). Analytical models are built to estimate the file delivery time of LTP in GRSN. Experiments are also conducted on our computer based testbed in emulation of the basic scenarios during data transmission from LEO satellite to a ground station in GRSN. The results show that in transmission efficiency BP with LTPCL outperforms other protocols, such as BP with TCPCL, direct terrestrial TCP (TCP Cubic) and TCP variants (TCP Hybla) for space segments in most scenarios. It could be envisioned that DTN with LTPCL for space segment is currently the best choice for future GEO-relayed space internetworking. Moreover, performance of two different networking schemes,i.e. the on-board DTN routers and the bent-pipe relays are compared. It is observed that bent-pipe relays outperform the on-board DTN routers scheme when the channel condition is good, while the on-board DTN routers scheme performs better with high data loss ratio. [ABSTRACT FROM AUTHOR]