Fault-Tolerant Algorithms for the Concurrent-Push Architecture - Scalable Continuous Media Streaming Systems - page 213

or

T E
(11.57)
Substituting the lower bound of equation (11.52) into equation (11.57) and noting T F =
( N S −

( i

D F ≤ i ( N S − f + +

+ 1) T F + t 0 + τ + K ) T a v g + min { F F ( y − 1) } +

K ) T a v g we get

( i

D F

≤ i ( N S −

K ) T a v g + y ( N S − f + +

+

1)( N S − K ) T a v g + t 0 + τ + K ) T a v g + t 0

D F + T E f − + + (11.58)

Rearranging, we can obtain y from

1

f + − f − −

+ τ +

T E

y Before = (11.59)

( N S − K ) T a v g

Similarly, if the failure occurs after playback has begun, then the continuity condition
becomes

min P After ( i )

max { F ( i ) } ≤ (11.60)

or

( i

D F ≤ i ( N S − T E

(11.61)

f + + + 1) T F + t 0 + τ + K ) T a v g + min { F N ( y − 1) } +

Solving, we can obtain y from

1

f + − f − −

+ τ +

T E + D F

y After = (11.62)

( N S − K ) T a v g

Similarly for Z , we also need to consider the two cases. First, for the case where failure
occurs before playback begins, we have

min { F ( i + l − 2) }≥ max { P Before ( i ) } (11.63)

or

( i

f − > i ( N S − T L

(11.64)

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

Substituting the upper bound of equation (11.52) into equation (11.64), 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

D F + T L

f + + + τ + (11.65)

= −

Rearranging we can obtain z ( l y )as

1

f + − f − +

+ τ +

T L + D F

z Before = (11.66)

( N S − K ) T a v g