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Fault-Tolerant Algorithms for the Concurrent-Push Architecture
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or (i + 1) TF + t0 + + f + + D F i (N S K ) Tavg + min {FF (y 1)} + TE (11.57) Substituting the lower bound of equation (11.52) into equation (11.57) and noting TF = (N S K )Tavg we get (i + 1) (N S K ) Tavg + t0 + + f + + D F i (N S K ) Tavg + y (N S K ) Tavg + t0 + f + D F + TE Rearranging, we can obtain y from yBefore = 1 + + f + f TE (N S K ) Tavg (11.59) (11.58)
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Similarly, if the failure occurs after playback has begun, then the continuity condition becomes max {F(i)} min PAfter (i) or (i + 1) TF + t0 + + f + + D F i (N S K ) Tavg + min {FN (y 1)} + TE (11.61) Solving, we can obtain y from yAfter = 1 + + f + f TE + D F (N S K ) Tavg (11.62) (11.60)
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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{PBefore (i)} or (i + l 1) (N S K ) Tavg + t0 + f > i (N S K ) Tavg + max {FF (y 1)} + TL (11.64) Substituting the upper bound of equation (11.52) into equation (11.64), we have (i + l 1) (N S K ) Tavg + t0 + f > i N S Tavg + y (N S K ) Tavg + t0 + + f + + D F + TL Rearranging we can obtain z = (l y) as z Before = 1 + + f + f + TL + D F (N S K ) Tavg (11.66) (11.65) (11.63)
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For the second case, playback is not delayed by the failure. Hence we have min{F(i + l 2)} max{PAfter (i)} or (i + l 1) (N S K ) Tavg + t0 + f > i (N S K ) Tavg + max {FN (y 1)} + TL (11.68) Substituting the upper bound of equation (11.51) into equation (11.68), we have (i + l 1) (N S K ) Tavg + t0 + f > i N S Tavg + y (N S K ) Tavg + t0 + + f + + TL Rearranging we can obtain z = (l y) as z After = 1 + + f + f + TL (N S K ) Tavg (11.70) (11.69) (11.67)
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Assuming failure occurs during transmission of group j, then the lling time for group i of a video stream started at time t0 is bounded by (i + 1) Tavg + t0 + f FN (i) (i + 1) Tavg + t0 + f + + , 0 i < j (11.71) i j (11.72)
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(i + 1) Tavg + t0 + f + D F FF (i) (i + 1) Tavg + t0 + f + + + D F ,
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Merging equation (11.71) and equation (11.72) gives the universal bounds for F(i): (i + 1) Tavg + t0 + f F(i) (i + 1) Tavg + t0 + f + + + D F , i Similarly, the playback schedule is bounded by i Tavg + FF (Y 1) + TE PBefore (i) i Tavg + FF (Y 1) + TL for the case where failure occurs before playback begins, and i Tavg + FN (Y 1) + TE PAfter (i) i Tavg + FN (Y 1) + TL (11.75) (11.74) (11.73)
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for the case where failure occurs after playback has begun. Invoking the continuity condition, we can obtain the corresponding bounds for Y as follows: YBefore = YAfter = f + f TE + Tavg +1 +1 (11.76) (11.77)
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