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(4.16)
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+ f seek (N vu+1 ) + tresidual where the rst term is the time to retrieve data blocks for the existing streams; the second term is the time to retrieve data block for the new stream; the third term is the head-repositioning time; and the last term is the time to complete retrieving the current data block w. Similarly, we can compute the residual service time for preemptive schedule from
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= f seek (vw vu+1 ) +
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+ f seek (N vu ) + tresidual where the rst term is the time to backtrack and retrieve data block for the new stream; the second term is the time to retrieve data block for stream (w + 1); the third term is the time to retrieve data blocks for the remaining streams; the fourth term is the head-repositioning time; and the last term is the time to complete retrieving the current data block w. By comparing max{ nps } with max{ ps }, the scheduler can choose the method with shorter delay and then check for over ow using method similar to Eqs. (4.14) and (4.15). If the round does not over ow, then the new stream can be admitted into the current round, thereby shortening the scheduling delay by one complete service round of Tr seconds.
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4.4.2 First-Block Replication
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To further improve the chance of successfully admitting a new stream into the current round and to simplify the disk scheduler, we can use a technique called First-Block Replication (FBR) which replicates the rst data block of a media stream to both the innermost track and the outermost track. With this technique, we can guarantee that request for a new stream will always be located downstream in the disk head scanning direction. Moreover, as the disk head has to be repositioned to the platter edge at the end of a service round, seek time for the new request is eliminated as well. The residual service time with FBR can be computed from
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Q j+1 + tlatency r j+1 (4.18)
and the scheduler can admit the new stream immediately if the round does not over ow. The signi cance of FBR is that it guarantees that the data block of the new stream will be located downstream in the scanning direction. Consequently, rather than using the worst-case to estimate round over ow as in equation (4.14), we can approximate the probability of round over ow by the round-length distribution = Pr {over ow} = 1 Fround (Tr , u + 1) (4.19)
which is a conservative measure as the disk head has already scanned past some tracks. Using equation (4.19) we can then compute the average scheduling delay under EAS+FBR from D = 1.5Tr + 0.5Tr (1 ) 0.5Tr (4.20)
for small . Therefore, together Early-Admission Scheduling and First-Block Replication can reduce the scheduling delay by two-thirds for small . The trade-off is the additional storage required to replicate the rst data block of each media stream, which is relatively insigni cant, given the size of typical media streams. For example, the overhead for a one-hour MPEG1 video of bit-rate 150KB/s and block size 128KB is only 0.024%. Note that we can extend this FBR technique to replicate additional blocks within a media stream, such as the rst block in the beginning of a seekable chapter to achieve similar admission-delay reduction when performing interactive playback controls (e.g., chapter selection). To quantify the gain of FBR, we assume that an admission delay constraint Dmax is given as part of the service requirement. Then we must ensure that the expected admission delay is
Soft Scheduling
smaller than Dmax : Q (0.5 + ) Dmax R Rearranging gives the largest block size that can be used: Q R Dmax 0.5 + (4.22) (4.21)
Note that we also need to round Q computed from equation (4.22) to integral multiples of disk sector size.