With the last metric of UPQ, besides providing the users with minimum AMRD for high PSNR and small reconstructed distortion fluctuation for smooth playback, optimal replication strategy should ensure that every frame of a video streaming arrives in time for continuous playback.
Unlike , ,  and , our objective is to minimize AMRD, [bar.D], by determining optimal replication density, [r.sub.i], for frame i.
Theorem 1: To minimize AMRD, [bar.D], the replication density of the i-th frame of a video streaming is proportional to [([p.sub.i] [e.sub.i] [d.sup.el.sub.i]).sup.2/3].
Proof: From (5), the problem of minimizing AMRD, [bar.D], is now equivalent to minimizing the amount of
By replacing (14) with (9), (10) and (15), and because of the constraint [[summation].sup.n.sub.i=1] [r.sub.i] = 1, we finally obtain the closed-form of optimal replication density to minimize AMRD as below
And the minimum value of AMRD, [bar.D], can be readily calculated by substituting (5) with (1), (15), and (16).
It is interesting that during generating optimal values of [r.sub.i] for minimum AMRD, the reconstructed distortion fluctuation among frames is also kept balanced for smooth playback.
Finally, according to the above considered parameters, a computation will be executed to find out optimal replication density for each frame so that the AMRD is minimized.
In this section, we simulate to demonstrate that our method (named minimum AMRD, MAD) outperforms other optimal replication strategies, for example: minimum access cost (MAC, [r.sub.i] [varies] [p.sup.2/3.sub.i]) , minimum query cost (MQC, [r.sub.i] [varies] [p.sup.1/2.sub.i]) , and standard proportional strategy (PRO, [r.sub.i] [varies] [p.sub.i]) which is often used as the reference strategy for comparison.