Streaming Variable Bit-Rate Media Streams - Scalable Continuous Media Streaming Systems - page 137

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.

7.6 Summary
By scheduling the transmission of video data in amonotonic-decreasingmanner, we can deliver
VBR videos in a mixed-traffic 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
32MB. Thus, using the AMDR scheduler one can provide performance guarantee in streaming
VBR videos over mixed-traffic networks with no trade-off in terms of admission complexity,
network utilization, client waiting time, and client buffer requirement.

Appendix
Proof of MDR Scheduler's Monotonicity Property
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 r i and r i + 1 be the transmission rate of
the i th 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 ( T i − 1 ) and S ( T i + 1 ), where T i − 1 and T i + 1
are the ( i

+

− 1)th and ( i + 1)th rate reduction points.

Assume r i < r i + 1 , i.e., the rate allocated are not monotonic decreasing. Then we have:

S ( T i ) − S ( T i − 1 ) S ( T i + 1 ) − S ( T i )

< (7.29)

T i − T i − 1 T i + 1 − T i

or

[ S ( T i ) − S ( T i − 1 )]( T i + 1 − T i ) < [ S ( T i + 1 ) − S ( T i )]( T i − T i − 1 ) (7.30)

We expand equation (7.30) to obtain.

S ( T i ) T i + 1 − S ( T i ) T i − S ( T i − 1 ) T i + 1 + S ( T i − 1 ) T i

<

S ( T i + 1 ) T i − S ( T i + 1 ) T i − 1 − S ( T i ) T i +

S ( T i ) T i − 1 (7.31)

We cancel the S ( T i ) T i term on both sides and after rearranging we obtain:

S ( T i ) T i + 1 − S ( T i − 1 ) T i + 1 − S ( T i ) T i − 1 < S ( T i + 1 ) T i − S ( T i + 1 ) T i − 1 − S ( T i − 1 ) T i (7.32)