5.9 Appendix ??“ Derivation of (9)
We want to derive the result for:
1
1
(1 )
N
j N
j
m jp p Np ??’
=
= ?‹… ?‹… ??’ + ?‹… ??‘ . (35)
with ( ) i v p e L = p and max 1 i N = ?¤ + p .
Consider:
( )( )
( )
1
1
1
1
1
N N
j N
j
p p
p p p p
p
??’
=
??’
= + + + =
??’ ??‘ ??¦ . (36)
Taking the derivative of both sides with respect to p, we have:
( )
( )
( )
( )
( )
( )
( )
( )
1
1
2
1
1 1 1
1 1 1 1
N N N N N
j
j
p p p pN p p p d j p
dp p p p p
??’
??’
=
??® ??? ??’ ??’ ?‹… ??’ ??? ??? ?‹… = = ??’ + ??? ??? ??’ ??’ ??’ ??’ ??° ??»
??‘ . (37)
or:
( )
( ) ( )
1
2
1
1
1 1
N N N
j
j
p N p j p
p p
??’
=
??’ ?‹… ?‹… = ??’
??’ ??’ ??‘ . (38)
Hence:
( ) ( )
( )
1
1
1
1
1
N N
j N
j
p
j p p N p
p
??’
=
??’
?‹… ??’ + ?‹… =
??’ ??‘ . (39)
Cross-layer Optimized Video Streaming over Wireless Multi-hop Mesh Networks
127
5.10 References
[1] K. Holt, ???Wireless LAN: past, present, and future,??? Proc. IEEE Design, Automation,
and Test in Europe, DATE, vol. 3, pp. 92-93, 2005.
[2] W. Wei, and A. Zakhor, ???Multipath unicast and multicast video communication over
wireless ad hoc networks,??? Proc.
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