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Unformatted text preview: smission rate is rg
In channel is in bad state, the transmission rate is rb
Let a r.v. R be the transmission rate. Let the channel be in bad state
with probability q , then
pR (r ) = q,
1 − q, if r = rb ;
if r = rg . We want to transmit a ﬁle over this wireless link. Let a r.v. X be the
length of a ﬁle, measured in KB . X follows a Geometric distribution
with parameter p :
1
pX (x ) = (1 − p )x −1 p , x = 1, 2 . . . ; E [X ] = .
p
Question: What is the PMF of the transmission time? Is the
question welldeﬁned?
M. Chen (IE@CUHK) ENGG2430C lecture 8 5 / 26 Transmission Time of a File over a Wireless Link * Additional assumption: X and R are independent
Let a r.v. T = X be the transmission time; compute its PMF and
R
E [T ].
Its PMF is given by (divide and conquer)
pT (t ) = pX R (rb t rb )pR (rb ) + pX R (rg t rg )pR (rg )
Compute the expected transmission time:
E [T ] = E (T R = rb )pR (rb ) + E (T R = rg )pR (rg )
E [X ]
E [X ]
=
·q +
· (1 − q )
rb
rg
1 q 1−q
=
+
p rb
rg
How will E [T ] change if X and R are correlated? (The same
approach still works; only diﬀers in details.)
M. Chen (IE@CUHK) ENGG2430C lecture 8 6 / 26 Transmission Time of a File over a Wireless Network
* (From radio.weblogs.com) We want to transmit the ﬁle to a friend across a wireless network
Transmission rates of links, denoted by Ri , i = 1, 2, . . . , are i.i.d. with
the same PMF as R
The number of links to traverse, denoted by N , is determined by
routing and is uniformly distributed in {n1 , n1 + 1, . . . , n2 }
Question: What is the expected endtoend transmission time? Assume
that Ri (i = 1, 2, . . . ), X , and N are independent.
M. Chen (IE@CUHK) ENGG2430...
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This document was uploaded on 03/31/2014.
 Spring '14

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