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A novel demand\u2013assignment <jats:italic>medium access control<\/jats:italic> (MAC) protocol, named DRAMA+ (<jats:italic>dynamic resource assignment multiple access\u2014enhanced version<\/jats:italic>), is proposed, where voice and Web traffic sources obtain transmission slots through requests sent by means of a random access phase. The <jats:italic>round\u2010trip propagation delay<\/jats:italic> (RTD) of LEO\u2010MSSs prevents an immediate feedback for each transmission attempt. Therefore, the main concern of the DRAMA+ scheme is to realize an efficient access phase. All the transmission requests successfully received at the satellite are managed by an on board <jats:italic>scheduler<\/jats:italic>. We have shown that DRAMA+ outperforms other techniques appeared in the literature in terms of voice quality, transmission delays for bursty data traffics and  resource utilization. Moreover, a performance analysis of an ideal version of the DRAMA+ scheme has permitted us to prove the potentialities of the proposed DRAMA+ technique. Stability issues have been addressed as well as the impact on the DRAMA+ performance of the LEO satellite constellation RTD value. 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