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By scheduling the transmission of video data in a monotonic-decreasing manner, we can deliver VBR videos in a mixed-traf c network with deterministic performance guarantee. This enables the service provider to exploit the available bandwidth to support other non-delay-sensitive data services and thus improves network utilization. Extensive simulations using 274 real-world VBR video bit-rate traces showed that the MDR scheduler can achieve good performance in terms of waiting time under the same network utilization, and is comparable to that of Optimal Smoothing, while still be able to guarantee playback continuity. For applications that require a bounded client buffer requirement, the AMDR scheduler can be applied and results showed that the performance is nearly identical to Optimal Smoothing even for a buffer size as small as 32 MB. Thus, using the AMDR scheduler one can provide performance guarantee in streaming VBR videos over mixed-traf c networks with no trade-off in terms of admission complexity, network utilization, client waiting time, and client buffer requirement.



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computes the vector produced by Vector y = vector expression and then sets all but its largest component to 0 Such vectors are particularly useful as

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Theorem 7.3. Transmission schedules generated by the MDR scheduler are guaranteed to comprise monotonic decreasing rates. Proof. We prove the theorem by contradiction. Let ri and ri+1 be the transmission rate of the ith and (i + 1)th segments of a feasible schedule S(t) generated by the MDR scheduler. ` Graphically, let r be the slope of the line connecting S(Ti 1 ) and S(Ti+1 ), where Ti 1 and Ti+1 are the (i 1)th and (i + 1)th rate reduction points. Assume ri < ri+1 , i.e., the rate allocated are not monotonic decreasing. Then we have: S(Ti ) S(Ti 1 ) S(Ti+1 ) S(Ti ) < Ti Ti 1 Ti+1 Ti or [S(Ti ) S(Ti 1 )](Ti+1 Ti ) < [S(Ti+1 ) S(Ti )](Ti Ti 1 ) We expand equation (7.30) to obtain. S (Ti )Ti+1 S(Ti )Ti S(Ti 1 )Ti+1 + S(Ti 1 )Ti < S(Ti+1 )Ti S(Ti+1 )Ti 1 S(Ti )Ti + S(Ti )Ti 1 We cancel the S(Ti )Ti term on both sides and after rearranging we obtain: S(Ti )Ti+1 S(Ti 1 )Ti+1 S(Ti )Ti 1 < S(Ti+1 )Ti S(Ti+1 )Ti 1 S(Ti 1 )Ti (7.32) (7.31) (7.30) (7.29)

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pattern selectors in neural-network simulations (Section 6-4) To find the value ymax of the largest vector component of vector expression, use

Add S(Ti 1 )Ti 1 to both sides and we obtain: S (Ti )Ti+1 S(Ti 1 )Ti+1 S(Ti )Ti 1 S(Ti 1 )Ti 1 < S(Ti+1 )Ti S(Ti+1 )Ti 1 S(Ti 1 )Ti S(Ti 1 )Ti 1 Then factorize equation (7.33) to get: [S(Ti ) S(Ti 1 )](Ti+1 Ti 1 ) < [S(Ti+1 ) S(Ti 1 )](Ti Ti 1 ) which is equivalent to S(Ti ) S(Ti 1 ) S(Ti+1 ) S(Ti 1 ) < =r Ti Ti 1 Ti+1 Ti 1 (7.35) (7.34) (7.33)

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Since the transmission schedule coincides with the data consumption curve at bit-rate reduction points, i.e., S(Ti ) = A(Ti ) for all i, we have A(Ti ) A(Ti 1 ) A(Ti+1 ) A(Ti 1 ) < Ti Ti 1 Ti+1 Ti 1 Now according to the MDR scheduler (cf. equation (7.6)): ri = max A(t) A(Ti 1 ) t > Ti 1 t Ti 1 (7.37) (7.36)

Vector y^ = vector expression | DOT ymax = y * 1

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From equation (7.36), we must have ri equal to ri = A(Ti+1 ) A(Ti 1 ) Ti+1 Ti 1 (7.38)

The index I of the largest vector component y[i] can be determined with a small loop in the experiment-protocol script:

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