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(12.5)
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and Nfull (n, m) = NS m N (n m , N S m, 1). (12.7)
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12.4 Traf c Overlapping
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If server clock jitter is greater than zero, then transmissions from two or more servers destined to the same client will overlap and multiply the transmission rate in the overlapping interval (Figure 12.7). This could cause congestion at the network and the client, resulting in the packet being dropped. To avoid traf c overlapping, we can sacri ce some server and network bandwidth, and transmit video data at a rate higher than RV , say RORT (Figure 12.8). We call this scheme overrate transmission (ORT) for obvious reasons. The transmission window will then be reduced to a time interval of Tw = Q RORT
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(12.8)
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We can guarantee that there will be no transmission overlapping by ensuring that Tw + < TF (12.9)
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Rearranging, we can then obtain the minimum transmission rate needed to avoid traf c overlapping: RORT > Q RV Q RV (12.10)
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Figure 12.7 Traf c overlapping due to server clock jitter
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Figure 12.8 Preventing traf c overlap by over-rate transmission
A Staggered-Push Parallel Server Architecture
Since the transmission rate must be positive and less than in nity, we have the condition that < Q = TF RV (12.11)
In other words, the server clock jitter must be smaller than a micro-round. Note that under this condition, traf c overlapping involves at most two servers and the data rate is doubled to 2RV in the overlapping region. As RORT in equation (12.10) can become very large when the denominator becomes small, the useful operating range for over-rate transmission is actually limited by: RORT < 2RV (12.12)
Substituting equation (12.10) into equation (12.12) and rearranging we can then determine the maximum clock jitter for which ORT is applicable: < Q = 0.5TF 2RV (12.13)
Therefore, ORT can prevent traf c overlapping if clock jitter is less than half of a micro-round. With ORT, the maximum network bandwidth needed at each server will be increased to CORT = RORT = Q RV Q RV (12.14)
12.5 Buffer Management
In this section, we present buffer management algorithms for the server and the client, and derive the respective buffer requirements. For simplicity, we ignore network delay and delay jitter. However, the effect of network delay and delay jitter can be incorporated in the same way as clock jitter and the same derivations are still valid.
12.5.1 Server Buffer Requirement
There are N S micro-rounds in a macro-round, therefore the duration of a macro-round, denoted by TR , is given by TR = NS Q RV (12.15)
As buffers are released after each micro-round, this scheduler requires only 2 Q buffers for each server, regardless of the number of servers and clients in the system. Therefore, existing servers do not need any upgrade when one scales up a system by adding more servers.
12.5.2 Client Buffer Requirement
Many studies on VoD system have assumed that video data are consumed periodically by the video decoder. However, as previously discussed in Section 10.3.2, hardware and software
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video decoders consume xed-size data blocks only quasi-periodically. Given the average video data rate, RV , and block size, Q, the average time for a video decoder to consume a single block is Tavg = Q RV (12.16)
To quantify the randomness of video block consumption time, we employ the consumption model proposed in Section 10.3.2, reproduced below for sake of completeness. De nition 12.1. Let T i be the time the video decoder starts decoding the ith video block, then the decoding-time deviation of video block i is de ned as TDV (i) = Ti i Tavg T0 (12.17)
and decoding is late if TDV (i) > 0 and early if TDV (i) < 20. The maximum lag in decoding, denoted by TL , and the maximum advance in decoding, denoted by TE , are de ned as follows: TL = max{TDV (i)| i 0} TE = min{TDV (i)| i 0} (12.18) (12.19)
The bounds TL and TE are implementation-dependent and can be obtained empirically. Knowing these two bounds, the playback instant for video block i, denoted by p(i), is then bounded by max{(T0 + i Tavg + TE ), 0} p(i) (T0 + i Tavg + TL ) (12.20)
Buffers are used at the client to absorb these variations to prevent buffer under ow (which leads to playback hiccups) and buffer over ow (which leads to packet dropping). Let L C = (Y + Z ) be the number of buffers (each of Q bytes) available at the client, organized as a circular buffer. The client pre lls the rst Y buffers before starting playback to prevent buffer under ow, and reserves the last Z buffers for incoming data to prevent buffer over ow. We rst determine the lower bound for Y . Let t0 be the time (with respect to the admission scheduler s clock) when the rst block of a video session begins transmission. Let di be the clock jitter between the admission scheduler and server i. Without loss of generality, we can assume that the video title is striped with block zero at server zero. Then the time for block i to be completely received by the client, denoted by f (i), is bounded by ((i + 1)TF + t0 + f + d mod (i,N S ) ) f (i) ((i + 1)TF + t0 + f + + d mod (i,N S ) ) (12.21) where f + and f are used to model the maximum transmission time deviation due to randomness in the system, including transmission rate deviation, CPU scheduling, bus contention, etc. Since the client begins video playback after lling the rst Y buffers, the playback time for video block 0 is simply equal to f (Y 1). Setting T0 = f (Y 1) in equation (12.20) then the playback time for video block i is bounded by ( f (Y 1) + i Tavg + TE ) p(i) ( f (Y 1) + i Tavg + TL ) (12.22)