A Concurrent-Push Parallel Server Architecture - Scalable Continuous Media Streaming Systems - page 179

Now for the L.H.S., noting that N S T avg = T F (cf. equations (10.1) and (10.4)) we then have

f +

max { F ( i ) }= ( i + 1) T F + t 0 + max { δ } (10.13)

Using the upper-bound for δ from Theorem 10.1 we obtain

f + max { F ( i ) }= ( i + 1) T F + t 0 + T F + (10.14)

f + = ( i + 2) T F + t 0 +

Similarly, the R.H.S. is

min { P ( i ) }= iN S T avg + min { F ( y − 1) }+ T E (10.15)

f − +

= iT F + yT F + t 0 + T E

Merging equations (10.14) and (10.15), we then have

f + < f − +

( i + 2) T F + t 0 + ( i + y ) T F + t 0 + T E (10.16)

Rearranging, we can then obtain y :

f + − f − −

T E

y > 2 + (10.17)

T F

Knowing the number of groups required, we can then obtain Y from

2 N S

f + − f − −

T E

Y = + (10.18)

T F

10.3.4 Buffer Needed to Prevent Overflow
On the other hand, to guarantee that the client buffer will not be overwhelmed by incoming video
data, we need to ensure that the i th video block group starts playback before the ( i + l − 2)th

video block group is completely received, where l L C / N S . This is because the client buffers

are organized as a circular buffer, and we must have at least one group of N S free buffers
available for video blocks arriving simultaneously from N S servers. Therefore, we need to
ensure that the earliest filling time for group ( i

=

+ l − 2) must be larger than the latest playback

time for group i :

min { F ( i + l − 2) } > max { P ( i ) } (10.19)

Using derivations similar to the previous section, we can obtain the number of buffers needed
to prevent buffer overflow as:

2 N s

f + − f − +

T L

= +

Z (10.20)

T F