Fault-Tolerant Algorithms for the Concurrent-Push Architecture - Scalable Continuous Media Streaming Systems - page 214

For the second case, playback is not delayed by the failure. Hence we have

min { F ( i + l − 2) }≥ max { P After ( i ) } (11.67)

or

( i

f − > i ( N S − T L

(11.68)

+ l − 1)( N S − K ) T a v g + t 0 + K ) T a v g + max { F N ( y − 1) } +

Substituting the upper bound of equation (11.51) into equation (11.68), we have
( i

f − > iN S T a v g + y ( N S − + l − 1)( N S − K ) T a v g + t 0 + K ) T a v g + t 0 + τ

f + + T L

+ (11.69)

Rearranging we can obtain z = ( l − y )as

1

f + − f − +

+ τ +

T L

z After = (11.70)

( N S − K ) T a v g

A.4 Derivations of Buffer Requirement for Sub-Schedule Striping
under PRT
Assuming failure occurs during transmission of group j , then the filling time for group i of a
video stream started at time t 0 is bounded by
( i

f − ≤ ≤ ( i f + + τ ,

+ 1) T a v g + t 0 + F N ( i ) + 1) T a v g + t 0 + 0 ≤ i < j

(11.71)

( i

D F ≤ ≤ ( i D F , f − + f + + τ +

j
(11.72)
Merging equation (11.71) and equation (11.72) gives the universal bounds for F ( i ):
( i

+ 1) T a v g + t 0 + F F ( i ) + 1) T a v g + t 0 + i ≥

f − ≤ ≤ ( i D F , ∀

f + + τ + + 1) T a v g + t 0 + F ( i ) + 1) T a v g + t 0 + i (11.73)

Similarly, the playback schedule is bounded by
iT a v g +

T E ≤ ≤ iT a v g + T L F F ( Y − 1) + P Before ( i ) F F ( Y − 1) + (11.74)

for the case where failure occurs before playback begins, and
iT a v g +

T E ≤ ≤ iT a v g + T L

F N ( Y − 1) + P After ( i ) F N ( Y − 1) + (11.75)

for the case where failure occurs after playback has begun.
Invoking the continuity condition, we can obtain the corresponding bounds for Y as follows:

f + −

f − − T E + τ

Y Before = + 1 (11.76)

T a v g

f + −

f − −

T E + τ + D F

Y After = + 1 (11.77)

T a v g