A Concurrent-Push Parallel Server Architecture - Scalable Continuous Media Streaming Systems - page 178

(b) The maximum lag in decoding, T L , and maximum advance in decoding, T E , are defined
as:

T L = max { T DV ( i ) |∀ i ≥ 0 } (10.6)

T E = min { T DV ( i ) |∀ i ≥ 0 } (10.7)

(c) The peak-to-peak decoding-time deviation is defined as

T DV = T L − T E (10.8)

Assume the bounds T L and T E are known, the time between the consumption of two video
blocks i and j ( j > i ) will be bounded by

max { (( j − i ) T avg − T DV ) , 0 }≤ t ≤ (( j − i ) T avg + T DV ) (10.9)

We use buffers at the client to absorb these variations to prevent buffer underflow and buffer
overflow during playback. Let there be L C =

Z ) buffers (each Q bytes) at the client,
organized as a circular buffer. The client starts video playback once the first Y buffers are
completely filled with video data. We prefill buffers before playback to avoid buffer underflow,
and reserve the last Z buffers for incoming data to avoid buffer overflow.

( Y +

10.3.3 Buffer Needed to Prevent Underflow
Since all N S servers transmit data to a client concurrently, the client will be receiving N S
video blocks simultaneously. Hence Y must be multiples of N S . We let y

Y / N S and consider
groups of N S buffers in the follow derivations (i.e., group zero consists of blocks 0 to N S −

=

1,

group one consists of blocks N S to 2 N S − 1, and so on.).

Among the N S servers, let the earliest transmission for the first round start at time t 0 , then
the last transmission for the first round must start at time t 0 + δ . Therefore, the time for video

block group i to be completely filled, denoted by F ( i ), is bounded by

f − ) f + ) (( i + 1) T F + t 0 + ≤ F ( i ) ≤ (( i + 1) T F + t 0 + δ + (10.10)

where f + ( f + ≥ 0) and f − ( f − ≤

0) are used to model the maximum transmission time devi-
ation due to randomness in the system, including transmission rate deviation, CPU scheduling,
bus contention, etc.
Since the client starts playing the video after filling the first y groups of buffers, the playback
time for video block group 0 is simply F ( y

− 1). From Section 10.3.2, setting T 0 = F ( y − 1)

then the playback time for video block group i , denoted by P ( i ), is bounded by

{ iN s T avg + F ( y − 1) + T E }≤ P ( i ) ≤{ iN s T avg + F ( y − 1) + T L } (10.11)

To guarantee video playback continuity, we must ensure that a video block group arrives
before playback deadline. In the worst-case scenario, the latest filling time must be smaller
than the earliest playback time, i.e.

max { F ( i ) } < min { P ( i ) } (10.12)