A Staggered-Push Parallel Server Architecture - Scalable Continuous Media Streaming Systems - page 225

To guarantee video playback continuity, we must ensure that all video blocks arrive before
their respective playback deadlines. Therefore, we need to ensure that for all video blocks, the
latest arrival time must be smaller than the earliest playback time:

max { f ( i ) } < min { p ( i ) }∀ i ≥ 0 (12.23)

Using the bounds from equation (12.21) and (12.22), we can rewrite equation (12.23) as

f + + ( i + 1) T F + t 0 + d mod ( i , N S ) < iT avg + f ( Y − 1) + T E (12.24)

or

f + + f − + ( i + 1) T F + t 0 + d mod ( i , N S ) < iT avg + YT F + t 0 + d mod ( Y − 1 , N S ) + T E

(12.25)

From equations (12.1) and (12.16) we know that T avg = T F , rearranging and solving for Y ,

we then obtain

f + −

f − −

T E + ( d mod ( i , N S ) − d mod ( Y − 1 , N S ) )

Y > 1 + (12.26)

T F

Since max {| d i − d j ||∀ i , j }= τ , the worst case is

f + −

f − −

T E + τ

Y > 1 + (12.27)

T F

which is the number of buffers that must be prefilled before beginning video playback.
Similarly, to guarantee that the client buffer will not be overwhelmed by incoming video
data, we need to ensure that the i th video block starts playback before the ( i

2)th video
block is completely received. This is because the client buffers are organized as a circular
buffer. Therefore, we need to ensure that

+ L C −

min { f ( i + L C − 2) } > max { p ( i ) }∀ i ≥ ( L C − Z ) (12.28)

Again using the bounds from equations (12.21) and (12.22), we can rewrite equation (12.28)

as

f − + ( i + L C − 1) T F + t 0 + d mod ( i + L C − 2 , N S ) > iT avg + f ( L C − Z − 1) + T L (12.29)

or

f − + f +

( i + L C − 1) T F + t 0 + d mod ( i + L C − 2 , N S ) > iT avg + ( L C − Z ) T F + t 0 +

+ d mod ( L C − Z − 1 , N S ) + T L (12.30)

Similarly, rearranging and solving for Z we obtain

f + −

f − + T L + ( d mod ( L C − Z − 1 , N S ) − d mod ( i + L C − 2 , N S ) )

Z > 1 + (12.31)

T F