We analyse the performance of an adaptive random-reservation medium access control (MAC) protocol which can support all ATM service classes while providing the required quality of service (QoS). Our study focuses on parameter optimisation of the multiple access schemes for ATM over a GEO satellite with on-board processing capabilities, considering various traffic mixes of constant bit rate (CBR), real-time VBR (RT-VBR), non-real time VBR (NRT-VBR) and unspecified bit rate (UBR). It is shown that maximum throughput can be achieved by using this access scheme. A TDMA access protocol combining both random access and demand assignment multiple access (DAMA) is particularly suited for a scenario with a high number of terminals with very bursty UBR traffic (e.g. Web browsing). UBR sources with short burst length access the slots remaining after the reservation procedure by random access which drastically reduces the slot access delay, at the expense of lower utilisation. However for UBR sources with burst sizes consisting of several ATM cells, reservation access provides higher throughput but the access delay is considerable longer.