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1000 = 0.9, OS/AMDR Mean Waiting Time (seconds) 100 = 0.9, MDR = 0.8, OS/AMDR 10 = 0.7, MDR 1 = 0.7, OS/AMDR 0 0 100 200 300 400 500 600 700 800 900 1000 System Bandwidth (Mbps) = 0.8, MDR
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Figure 7.8a Mean client waiting time versus system capacity
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10000 = 0.9, OS/AMDR
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1000 max waiting time(s)
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100 = 0.9, MDR 10 = 0.8, MDR = 0.7, OS/AMDR 0 0 200 400 600 capacity (Mbps) 800 1000 1200 = 0.8, OS/AMDR
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= 0.7, MDR
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Figure 7.8b Worst-case client waiting time versus system capacity
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the optimal smoothing scheduler are negligible even for extremely small system bandwidth (e.g., 100 Mbps). This shows that the overhead incurred in maintaining a MDR schedule in the AMDR scheduler is negligible.
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7.6 Summary
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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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Appendix Proof of MDR Scheduler s Monotonicity Property
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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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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)
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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)
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which violates the de nition of ri . This contradicts the assumption that the schedule S(t) is generated by the MDR scheduler and therefore the schedule with increasing rates cannot be generated by the MDR scheduler and the result follows.
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