Information Technology Reference
In-Depth Information
In the next chapter we develop fault-tolerant algorithms for the concurrent-push architecture
so that even server failures can be sustained without disruption to ongoing video streams.
Appendices
A.1 Proof of Theorem 10.1
Assume server zero starts the first service round at time t 0 . Since the server clocks are not
precisely synchronized, we let d i be the clock difference between server i and server zero.
Hence d 0 =
0 and max
{ |
d i
d j ||∀
i
,
j
} = τ
and server i will start service round j at time
( t 0 +
jT F ). Let t new be the time a new-session request arrive at the servers. Then the request
will arrive at server i during round
d i +
v i :
t new
( t 0 +
d i )
v i =
(10.45)
T F
and the first video block will be retrieved at round (
2).
Hence the transmission jitter between server i and server k for stripe j can be expressed as:
v i +
1) and transmitted at round (
v i +
δ i , k , j =
( t 0 +
d i +
(
v i +
2
+
j ) T F )
( t 0 +
d k +
(
v k +
2
+
j ) T F )
(10.46)
Substituting equation (10.45) into (10.46) we have
d i +
t new
T F
d k +
t new
T F
( t 0 +
d i )
( t 0 +
d k )
δ i , k , j =
(10.47)
T F
T F
t new
( t 0 + d i )
T F
Without loss of generality, we can assume d i
d k and let H
=
. Then we have
d k +
H
T F
( d i
d k )
δ i , k , j
=
( d i +
H
T F )
+
(10.48)
T F
T F
H
( d i
d k )
=
( d i
d k )
+
H
+
T F
Noting that
x
+
y
x
+
y
we have
T F
( d i
d k )
δ i , k , j
( d i
d k )
+
H
H
(10.49)
T F
T F ( d i
d k )
=
( d i
d k )
T F
Finally, making use of the result that
x
/
y
y
( x
y ) we can then obtain
T F ( d i
d k )
δ i , k , j
=
( d i
d k )
(10.50)
T F
( d i
d k )
(( d i
d k )
T F )
=
T F
and the result follows.
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