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4.3.2 Performance Modeling
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To quantify the improvement, reconsider the round lengths for any two consecutive disk service rounds. As data placement is random, the service round lengths for any two disk service rounds are independent and identically distributed according to Fr ound (t, k). Let f round (t, k) be the (2) density function of Fround (t, k) and letFround (t, k) be the distribution of the sum of two service round lengths, which is the auto-convolution of Fround (t, k):
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(2) Fround (t, k) =
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Fround (t x, k) f round (x, k)d x
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(4.4)
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Now consider an arbitrary service round i. With DRS, round i can over ow under two conditions. First, if round (i 1) does not over ow, then round i will over ow only if the combined round lengths are longer than 2Tr : Pr {(ti + ti 1 ) > 2Tr |ti 1 Tr } Pr {(ti + ti 1 ) > 2Tr }
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(2) = 1 Fround (2Tr , k)
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(4.5)
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Second, if round (i 1) does over ow, then it will be truncated to at most Tr (see Section 4.5.2). In this case, round i over ows if it is longer than Tr : Pr {ti > Tr |ti 1 > Tr } = 1 Fround (Tr , k) Hence the over ow probability of round i, denoted by (k) = Pr {ti > Tr } = Pr {(ti + ti 1 ) > 2Tr |ti 1 Tr } Pr {ti 1 Tr } + Pr {ti > Tr |ti 1 > Tr } Pr {ti 1 > Tr }
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(2) 1 Fround (2Tr , k) Fround (Tr , k) + (1 Fround (Tr , k))2
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(4.6)
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(k), can be computed from
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Scalable Continuous Media Streaming Systems
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Finally, we can compute the usable disk capacity under DRS from CDRS ( ) = max {k| (k) , k = 0, 1, . . .} (4.8)
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4.3.3 Buffer Requirement
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To achieve the capacity gains, there is also a trade-off in DRS the additional buffers used to store the early-retrieved media blocks. To obtain an upper bound for the extra buffer requirement, we note that in the worst case, the second disk service round will have a length of max tround (k) as given in equation (3.5). To prevent over ow, the server will have to start the service round earlier by
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max Tearly = tround (k) Tr
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Note that DRS cannot compensate for over owed rounds with length longer than 2Tr as the slack time for the previous round is bounded by Tr . Now the time to retrieve a media block of size Q bytes is bounded from below by
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min tread =
Q rmax
(4.10)
where rmax is the maximum transfer rate (e.g., at the outer-most zone). Hence during the time interval Tearly , we need at most Bearly = min Tearly , CDRS ( ) min tread (4.11)
extra buffers to store the early-retrieved media blocks. To be fair, the extra buffers may also be used to increase the media block size Q, which also increases disk ef ciency, instead of using DRS. However, increasing the media block size will result in longer service round length and, consequently, will increase the admission delay for new streams. In practice, a system is likely to have been dimensioned to use the largest media block size for maximum disk ef ciency and hence increasing the block size further will not be feasible. By contrast, DRS does not affect the scheduling delay as the media-block size is unchanged and hence we can employ DRS to further increase the usable disk capacity in a system with an already optimized media block size.
4.4 Early-Admission Scheduling
In conventional round-based scheduler such as SCAN and CSCAN, the media block size is one of the key parameters in determining the achievable disk utilization. As current memory costs continue to drop due to rapid increases in memory density, it may appear that one can keep increasing disk utilization simply by choosing larger block sizes. However, in addition to memory cost, the usable block size is also limited by the admission delay, as discussed in Section 3.3. For example, in conventional round-based scheduler a new request arriving mid-way in round i will receive service beginning in the next round (i + 1). Due to double-buffering, the retrieved block will be transmitted in round (i + 2). Hence, the worst-case admission delay is