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ISP and the P2P application. In other words, P2P applications and ISPs can
solve the subproblem (13) and (12) independently at first and then transfer the
optimal solution t k and s e to iTracker. In the next moment, iTracker will update
p e by solving (11).
With respect to v ʲ ( s e ) = log( s e ), we can express the explicit solution of (12)
and the engineering significance of its dual variable as follows:
Theorem 2. When β =1 , the subproblem (12) has the explicit optimization
s e = 1
1
c e −b e
,
p e
,
p e
1
c e −b e
c e
b e , 0
p e <
.
t e }
Theorem 3. Suppose that
{
is the solution of (9) and
{
p e }
is the solution
of the dual problem (11) .Then
e
E ,andwhen s e = c e
b e ,
1
c e
p e
[0 ,
);
b e
when s e <c e
b e ,
1
c e k t e − b e
p e =
.
As we see from theorem 3, to any link e , the larger its free link capacity is,
the lower its link price ( p e ); the smaller its free link capacity is, the higher its
link price ( p e ). This makes the ISPs to control trac on the non-most-congested
links more eciently when dealing with networks with bottleneck links.
3 Simulation Methodology
We have built a discrete-event package for simulation. We have followed the
method de-scribed in [12] and performed the simulations by implementing the
native BitTorrent protocol. We have also calculated the trac on every link in
order to estimate the link utilization. Our simulation includes keeping statistics
of the trac on bottleneck links, interdomain links, and intradomain links in
P2P, P4P, and Improved-P4P . Further, we have studied the benefits of P2P
applications in P4P and Improved-P4P by varying the swarm size.
3.1 Assumptions
- We have ignored the propagation delay because the propagation delay relates
only to small control packets. We believe that this simplification has very
little impact on the conclusion because of the following:
1. The downloading time depends on the transmission time of the packets.
2. In practice, the pipelined processing mechanism of BitTorrent reduces
most of the propagation delay of the packets.
- We have followed the method proposed in [13] and assumed that all TCP
sessions share the link's capacity equally in the stable state.
- After finishing downloading, all peers leave the network immediately.
 
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