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where (8a) is the original feasible condition, (8b) and (8c) are the dual feasible
conditions, and (8d) is the complementarity condition. Further, because of the
optimization of (4)
(
b
e
+
k
t
e
)
/c
e
,
α
=max
e∈E
where
t
e
is the solution of (3). Based on (8d), we know that all
p
e
swhose
corresponding links do not achieve the maximum utilization equal to 0, and
p
e
s
whose corresponding links achieve the maximum are equal to or greater than
0. Moreover, by (8b), we know that there exists at least one
p
e
that doesn't
equal to 0.
As we see in the above theorems, all link prices of links that do not achieve
the most congested state are equal to 0, i.e., for a flow, the link price of each
non-most-congested link is equal to that of another such link. This property of
the multiplier makes MLU invalid when dealing with networks with bottleneck
links.
In Figure 2, suppose that the capacity of each link is 1, the tra
c demand
between node 1 and node 3 is 1, and the tra
c demand between node 3 and
node 4 is 0.9. If we set MLU as the optimization objective of the ISP, the trac
on link (1, 3) will be 0.9 and the trac on link (1, 2) will be 0.1. Now, there are
two bottleneck links in the network, i.e. link (1, 3) and link (3, 4), which lead
to the situation that of the two links between node 1 and node 3, one is very
congested and the other is free.
Fig. 2.
An Example of Network with Bottleneck Links
2.3 Improved Cooperation Algorithm
We introduce a
link utility function
as the ISP optimization objective and verify
that the new objective can better utilize the network resources by carrying out
a theoretical analysis and experimental simulations. In this section, we propose
a cooperative algorithm of ISP and P2P with a
link utility function
as the ISP
optimization objective, and analyze some issues with this objective.
We follow the methods introduced in congestion control[11], and make the
link utility function
the ISP optimization objective. Consider that
maximize
{s
e
},t
k
∈T
k
,∀k
e
v
ʲ
(
s
e
)
(9a)
t
e
−
subject to
s
e
≤
c
e
−
b
e
,
∀
e
∈
E,
(9b)
k
