Fault-Tolerant Algorithms for the Concurrent-Push Architecture - Scalable Continuous Media Streaming Systems - page 212

or

( i

f + + τ ≤ iT a v g + YT a v g + f − + T E + 1) T a v g + t 0 + t 0 + (11.48)

Rearranging, we can obtain Y from

f + −

f − −

T E + τ

Y = + 1 (11.49)

T a v g

Using similar derivations, we can obtain Z from

f + −

f − + T L + τ

Z = + 1 (11.50)

T a v g

A.3 Derivations of Buffer Requirement for Block Striping under PRT
Assume that a failure occurs during the transmission of group j , then for those groups received
before a failure (i.e., 0 ≤ i < j ), the filling time is bounded by

( i

f − ≤ ≤ ( i f + ,

+ 1) T F + t 0 + F N ( i ) + 1) T F + t 0 + τ + 0 ≤ i < j (11.51)

However, groups transmitted after the failure ( i j ) will be deferred due to server reconfigu-

ration. According to Section 11.3.2, the worst-case delay due to reconfiguration is D F seconds.
Hence, the maximum filling time is bounded by
( i

≥

D F ≤ ≤ ( i D F ,

f − + f + + + 1) T F + t 0 + F F ( i ) + 1) T F + t 0 + τ + i ≥ j

(11.52)

Merging (11.51) and (11.52) gives bounds for the general case:
( i

f − ≤ ≤ ( i D F , f + + + 1) T F + t 0 + F ( i ) + 1) T F + t 0 + τ + ∀ i (11.53)

The bounds for P ( i ) depend on whether a failure occurs before or after playback has begun.
Specifically, if a failure occurs before playback begins, then playback will be delayed up to
D F seconds due to the need to reconfigure the servers to complete the prefill process:
i ( N S −

T L
(11.54)
Otherwise, if the failure occurs after playback has begun, then the playback schedule will not
be affected:
i ( N S −

T E ≤ ≤ i ( N S −

K ) T a v g + F F ( y − 1) + P Before ( i ) K ) T a v g + F F ( y − 1) +

T L
(11.55)
Now if the failure occurs before playback begins, then invoking the continuity condition
gives

T E ≤ ≤ i ( N S − K ) T a v g + F N ( y − 1) + P After ( i ) K ) T a v g + F N ( y − 1) +

min P Before ( i ) max { F ( i ) } ≤ (11.56)