Streaming Variable Bit-Rate Media Streams - Scalable Continuous Media Streaming Systems - page 126

contradiction. Let Y ( t ) be a feasible schedule with zero start-up delay:

Y ( t ) ≥ A ( t ) (7.13)

and has bit-rate no larger than the MDR schedule S ( t ) before the first bit-rate reduction point
T 1 , i.e.,

Y ( t ) S ( t )

≤ , ≤ <

for 0 t T 1 (7.14)

,

T 1 ), equality in equation (7.14) holds only if Y ( t )
is equivalent to S ( t ). As we assumed they are different, that implies

As S ( t ) is a straight line in the range (0

Y ( t ) S ( t ) < , for 0 ≤ t < T 1 (7.15)

Integrating equation (7.14) on both sides with respect to t :

T 1 T 1

Y ( t ) dt S ( t ) dt

< (7.16)

0 0

and we obtain

Y ( T 1 ) < S ( T 1 )

= A ( T 1 ) (7.17)

which implies there will be a buffer underflow at the point T 1 . This contradicts our assumption
that Y ( t ) is a feasible schedule and thus the result follows.

7.3.4 Client Buffer Requirement
Similar to video smoothing, the MDR scheduler also requires the client to buffer video data
ahead of their playback schedule. Given a MDR schedule S ( t ), the buffer requirement is the
maximum difference between the transmission curve S ( t ) and the data consumption curve
A ( t ):

B = max { S ( t ) − A ( t ) |∀ t ≥ 0 } (7.18)

As discussed in Section 7.3.1, there are infinitely many feasible transmission schedules that
are also monotonic decreasing. The one defined in Section 7.3.1 however, has the minimum
client buffer requirement as stated in Theorem 7.2:

Theorem 7.2. The MDR scheduler generates schedules with the minimum buffer requirement
among all feasible monotonic decreasing rate schedules.

Proof. We will prove by contradiction. Let X ( t ) be a feasible monotonic decreasing rate
schedule, i.e.,

X ( t ) ≥ A ( t ) (7.19)