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However, the objective metrics in both  and  do not directly have effect on the three main aspects of high UPQ (i.e., high PSNR, smooth playback, and continuous playback), nor the SCR, which are used to evaluate the performance of replication strategies in video streaming applications.
These studies might satisfy continuous playback, but a high UPQ requirement could not be ensured because PSNR and smooth playback were not considered.
Furthermore, enhancing startup latency is not adequate for high UPQ.
Clearly, we can see the problem that all considered performance metrics of the proposed techniques in VoD system do not directly reflect the subjective human perception , the most important application metric, which can be characterized by the three aspects of UPQ as we mentioned.
But, the other two metrics of UPQ as well as SCR were not considered in .
In this section, we introduce the problem formulations to express some important metrics associated with UPQ and SCR, which enable us to propose optimal replication density in our strategy.
For the first metric of UPQ, high PSNR, we aim at minimizing average maximum reconstructed distortion (AMRD) of received video streaming which comes from source distortion model under the effect of transmission error as follows.
In case of the second metric of UPQ, that is smooth playback, our objective is to reduce the reconstructed distortion fluctuation among the frames of received video streaming.
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.
In this section, we first propose a method for solving an unconstrained optimization problem in order to improve UPQ. And then, a closed-form of optimal replication density is given by applying Zipf-like distribution.
Mathematically, based on the mentioned expressions of UPQ and SCR, we can formulate a nonlinear constrained optimization problem in which (5) is considered as the objective function while the metrics in (6), (7), and (8) are used as constraints.
In this subsection, we aim at proving that high encoding rate of video streaming does not always provide users with high UPQ in the presence of inherent problems of wireless environment (e.g., relatively unreliable connectivity, dynamically varying network topology, and limited resources), and meanwhile high SCR is consumed.
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