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In this chapter, we investigate an alternative approach to disk-scheduler design soft scheduling. Speci cally, by designing the disk scheduler with statistical performance guarantees instead of deterministic performance guarantees, we can use the disk I/O bandwidth more ef ciently and at the same time, still be able to satisfy the continuity requirement with high probability. In addition, by placing data randomly instead of sequentially across the disk surface, we can make use of the higher transfer rates of the outer zones in multi-zone disks to achieve higher disk throughput. To further increase the usable disk capacity, we present a Dual-Round Scheduling technique to schedule disk rounds in pairs so that over ows in the next service round can be absorbed in the current service round, and an Early-Admission Scheduling technique to enable the use of larger media block size for better disk ef ciency without adversely increasing the system response time. Finally, we present methods for detecting and recovering from service round over ow. Results from a detailed simulation of ve disk drives will be used to explore the potential performance gains of the presented techniques over hard-scheduling approaches.

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To avoid division by 0, we overwrite the differential equation for u[1] with the correct differential equation

The worst-case dimensioning technique in hard scheduling enables the disk to provide deterministic performance guarantee. However, as with any worst-case techniques, the trade-off would be lower disk utilization in practice as the worst-case scenario occurs very sparingly. For example, ignoring rotational latency for the moment, the worst-case seek time under CSCAN for a disk with a total of N tracks occurs with probability Pr n i = N 1 , i = 1, 2, . . . , k = k+1 1 N 1

(4.1)

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d/dt u[1] = 2 * uxx[1]

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where n i denotes seek distance for the ith request (see Table 4.1 for a summary of notations). For a disk with N = 5,001 tracks and k = 10, this computes into a probability of 1.024 37 . This is clearly negligible in practice and this motivates us to investigate soft scheduling to provide statistical rather than deterministic performance guarantee. In statistical capacity dimensioning, the objective is to nd an operating point that provides higher usable disk capacity than deterministic capacity dimensioning, subject to a given over ow probability constraint. Let Fround (t, k) be the cumulative distribution function (CDF) for the disk service round length, i.e., Fround (t, k) = Pr {tround (k) t} (4.2)

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derived by l H spital s rule (ux/x uxx/1 as x 0) Figure 7-8 shows solution time histories of the temperature u for the n uniformly spaced radius values defined by Eq (7-5)

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We can then de ne an over ow probability constraint that speci es the maximum acceptable occurrence probability for violating the continuity condition in equation (3.4). Using this constraint and equation (4.2), we can then compute the usable disk capacity, denoted by C( ), from C( ) = max {k| (1 Fround (Tr , k)) , k = 0, 1, . . .} (4.3)

where Tr = Q/R is the maximum length of a service round. This is the maximum number of requests that can be served in each service round with an over ow probability no greater than . Note that storage allocation for a media object must be pseudo-randomized under soft scheduling, i.e., available disk blocks are randomly selected to store a media title. This

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7-13 Generalizations Our space-derivative submodels apply to many systems of parabolic linear or nonlinear partial differential equations with one space dimension, but problems

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