derived from network topology. Constraint (3) ensure
that link flows do not violate link capacity and (4) says
that li nk flo ws are nonne gative . The PRI MAL_I solution
specifies the , and so we have the optimal path with
arbitrary splitting for all sessions that minimize MLU.
Our objective is to find a practical method suited for IP
netwo rks that fo rces the t raffic to go through a set of op-
timal paths. To achieve this goal we should find a set of
new link metrics such that all paths which are specified
by PRIMALI problem can also be obtained by the short-
est path algor ithm in regard t o the new metrics. It means
that if
0
k
ij
X>
, then l ink should be selected by ses-
sion k according to the shortest path algorithm. Here we
assume that the shortest path algorithm is OSPF that
supports Equal-Cost Multi-Path (ECMP).
The load balancing methods introduced in [6,7] are
based on primal optimization problem. In this paper we
consider the dual optimization problem (DUAL_II)
which is obtained with respect to Lagrange Multipliers.
In other word we aim to distribute traffic by deter mining
the links weight. And It will be shown that t he Lagra nge
Multipliers comparable with constraint (3) can be inter-
preted as OSPF link metrics that satisfy the load balanc-
ing objective. Link metric in OSPF protocol must be an
inte ger be tween 1 and 655 35 but we will sho w in secti on
IV tha t the La gr an ge Multip lie rs that are o btains fro m the
solution of DUAL_II problem and comparable with con-
straint (3), do not satisfy this range in general. So the
following definition gives us the choice of an alternative
weight set.
Definition 1: Two weights set
{ }
1
L
ii
Ww
=
=
and
{ }
1
L
ii
Ww
=
′′
=
are equivalent with respect to a given graph
(,)
GNA with
L
links, if and only if the shortest paths
between any arbitrary nodes in
G
are the same consi-
dering any one of these two weight sets.
3. The Dual Problem
Before Before Defining the Lagrange multiplayer
{ }
1
N
ii
p=
comparable with constraint (2) and
{ }
,(,)
ij ij A
w
compara-
ble with constr a int (3), the Lagrange polynomial is:
0
:( ,):(.)
(,)
(,,,)
k
ij
X
kk
i kijji
kK iNj i jAjjiA
k
ijij ij
ij AkK
LXMLUP W
MLUp DXX
w XC
∈∈∈ ∈
∈∈


=+ −+



+−



∑∑∑ ∑
∑∑
(5)
For more details can be referred to chapter 5 of [9].To
achieve the dual problem the following equation should
be satisfied for each feasible X and
MLU
.
(,,, )MLULXMLUP W
(6)
To satisfy (6) we must have:
0
ij
w
. Now the func-
tion
(,)
gW P is defined as bellow:
,
(, )min(,, ,)
X MLU
gWPLXMLUP W=
(7)
So we have:
,
(, )
(, )
(, )min
()
(1)
kk
ii
X MLU
k
ij jiij
k K ijA
ij ij
ij A
gW PpD
Xp pw
MLUw C
∈∈
=
+ −+
+−
∑∑
∑∑
(8)
Because
ij
w
and k
ij
X are positive values, we have:
(,)
1
(,) 1
kk
iiijijij ij
k Ki N
ijijij ij
ijA
pDppwandC w
gW PppworC w
∈∈
−≤ =
=−∞− ≥≠
∑∑ ∑
(9)
The dual function is defined to maximize
(,)gW P
when all
ij
w
are positive values. Equation (10) shows
the DUAL_I problem that is the dual function of
PRIMAL_I.
max .(10)
k
kk
t
kK
pD
(11)
kk
ijij
ppw−≤
(, )
1 (12)
ij ij
ij A
Cw
=
Maximum Load Balancing with Optimized Link Metrics
Copyright © 2012 SciRes. JSEA
16
0 (13)
k
k
s
p=
0 (14)
ij
w
As the primal and dual problems are linear, strong
duality holds and according to complementary slackness
in KKT theorem if
ˆ
k
ij
X
is optimal solution of PRIM-
AL_I and
{
}
ˆˆ ,
k
ij ij
wp
is the optimal solution of DUAL_I
we have:
ˆ
ˆˆˆ
.() 0
kk k
ij ijij
Xp pw
−+ =
(15)
Equation (15) indicates that if session k passes link
(, )ij
then
kk
j iij
ppw−=
. According to theorem 1 in [8]
if
{ }
(,)
ˆ
ij ij A
w
is used as a link metric in a shortest path
algorithm, all non empt y links (
0
k
ij
X>
) will be included
among the selected paths by the shortest path algorithm
procedure.
4. Practical Requirements
{ }
(,)
ij ij A
w
that is calculated from the DUAL_I are equal
to or greater than zero. But as we mentioned before
OSPF link metrics cannot be zero. We show that there
exists a weight set equivalent to
{ }
(,)
ij ij A
w
that can be
obtained using the new optimization problem.
Proposition 1: consider
with weight set
{ }
(,)
ij ij A
w
and some scalars
{ }
1
N
ii
δ
=
corresponding to
each link and node respectively. If we change the link
weight to
ijijji
ww
δδ
= +−
, then the weight set
{ }
(,)
ij ij A
w
and
{ }
(,)
ij ij A
w
are equivalent weight sets
with respect to
[8]. To achieve non-zero
weight set we changed the weights of the links according
to algorithm 1.
Algorithm 1:
Step 1: For each session
kK
assign the scalar set
{ }
1
N
k
ii
δ
=
as follows:
If there exists at least one directed path to node i from
source node of session k (k
s), the n
k
i
δ
is equal to th e
length of the longest hop-count non-loopy path from
k
sto i.
Else
k
i
δ
is equal to zero.
Step 2: Assign the
max
k
i
k
δ
as the final scalar
i
δ
Step 3:
If the
0
ji
δδ
−≥
then
ijijji
ww
δδ
= +−
Else
1
ij ij
ww= +
.
So according to Proposition 1 and algorithm 1, there
exists an equivalent weight set to
{ }
(,)
ij ij A
w that all of
them are greater or equal to one. Thus we can assume
.
ij ij
ij
Cw H=
.
Theorem 1: consider two optimization p roblems called
DUAL_I and DUAL_II.
DUAL_I:
(,)
max .
1
0
0
k
k
kk
t
kK
kk
ijij
ij ij
ij A
k
s
ij
pD
ppw
Cw
p
w
−≤
=
=
DUAL_II:
(,)
max .
0
1
k
k
kk
t
kK
kk
ij ij
ij ij
ij A
k
s
ij
pD
ppw
CwH
p
w
′′ ′
−≤
=
=
If
{ }
(,)
ˆ
ij ij A
w
is an optimal solution of DUAL_I and
{ }
(,)
ˆ
ij ij A
w
is an optimal solution of DUAL_II, the sets
{ }
(,)
ˆ
ij ij A
w
and
{ }
(,)
ˆij ij A
w
are equivalent weight sets
with respect to
(,)GNA
.
Proof: Consider the optimization problem PRIM-
AL_II.
PRIMAL_II:
(,)
max. .
1
0
0
k
k
kk
t
kK
kk
ijij
ij ij
ij A
k
s
ij
HpD
ppw
Cw
p
w
−≤
=
=
It is clear that the ij
X
s which minimizes the objec-
tive function of the problem PRIMALL_II are the same
as the ones which cause the problem PRIMALL_I to be
optimized.
The dual of PRIMAL_II is the DUAL_TEMP.
DUAL_TEMP:
(, )
max .
0
0
k
k
kk
t
kK
kk
ijij
ij ij
ij A
k
s
ij
pD
ppw
CwH
p
w
′′ ′
−≤
=
=
From complementar y slackness theorem
ˆˆˆˆ
.() 0
kk k
ij ijij
Xpp w
′′′
−+ =
(16)
Since the optimal solutions of the PRIMAL_I and
Maximum Load Balancing with Optimized Link Metrics
Copyright © 2012 SciRes. JSEA
17
PRIMAL_II are the same, thus the weight sets
{ }
(,)
ˆ
ij ij A
w
and
{ }
(,)
ˆ
ij ij A
w
are equivalent weight sets.
The weight set
{ }
(,)
ij ij A
wis the feasible set of the prob-
lem DUAL_TEMP and hold (16). Therefore it is the op-
timal solutio n o f t his problem.
So we in this way, we were able to obtain optimal
weights that do not include any link with weight 0 by
limiting the constraint
0
ij
w
to
1
ij
w
. This converts
the problem DUAL_TEMP to DUAL_II. Figure 1 shows
the fl ow c hart of our method.
With ECMP routing a flow arriving at a node is split
evenl y ove r the l ink s on the sho rtest paths fro m this node
to the destination. It should be mentioned that arbitrary
routing is not possible once ECMP in OSPF is used. so in
OSPF environment we can never obtain the optimal
routing but we can get close to it as much as possible.
Objec tive function that is used in [8] is min
. The second term in this ob-
jective function cause to minimizes
(,)
k
ij
ij A
X
in addi-
tion to
MLU
. In this case the weight set resulted from
the dual function is
(where
{ }
(,)
ij ij A
w
are
Lagrange multipliers that correspond to the non-equal
constraint). The routing algorithm that we use in this
paper is OSPF. This protocol splits the traffic equally
among the available shortest paths, so we prefer traffic
splitting as much as possible even if it passes through
longer paths. As the constant
r
in the second term pre-
vents the flow to go through long paths we assume that
0r=
.
Figure 1 . Maximum load balancing flow chart.
5. Simulation Results
In this section we simulate the OSPF protocol with its
default link metrics and with the metrics that are calcu-
lated using the optimizatio n problem.
A. Scenario I : In fir st scenario the simula tion plat for m
is shown in Figur e 2.
All links in this network are DS3 with 44.7 Mbps rate.
We suppose FIFO as a queuing policy. The session is a
VOIP with GSM quality and the average bit rate is 40
Mbps. Node 1 is the source node and node 2 is the desti-
nation.
Table 1 show the solution of primal problem ( )
which indicate t he paths that mini mize maximum utiliza-
tion. Solution of DUAL_I problem is shown in Ta ble 2
and Ta b l e 3 show the solution of DUAL_II problem.
Figure 2. Simulation network to pology.
Table 1. O pt imum f lows.
X12
20
X13 20
X23 0
X24 20
X25 0
X35 20
X54 20
Table 2. The Soluti on of DUAL_ I.
w12 0.0056
w13 0.0017
w23 0
w24 0.0056
w25 0
w35 0.0077
w54 0.0017
Table 3. The Solution of DUAL_II.
W12 2 .0177
W13 1 .3567
W23 1 .0000
W24 1 .9823
W25 1 .0000
W35 1 .2843
w54 0.0017
Maximum Load Balancing with Optimized Link Metrics
Copyright © 2012 SciRes. JSEA
18
Table 1 sho w that t he opti mum pat hs are 1->2->4 and
1->3->5. These paths can obtain in a shortest path algo-
rithm regardin g to the weig hts that show in Table 2. Ta-
ble 2 and Table 3 are the equivalent weight s with respect
to the graph that shows in Figure 2. So we use the sub-
optimal weigh set in Table 3 instead of default OSPF
link metrics. Figure 3 show tha t in rec ent me thod p acket
drop decrease signifi cantly.
In fowlloing scenarioes (senario 2-) the simulation
plat for m is s ho wn in Fig ur e 4. We co mpare MLU, BWE
(Band width Efficiency), Nu mber of Over Utilized Links,
IP Traffic Dropped and IP Traffic Received for all scena-
rios
Figure.3. D ropping traffic in scenario1 for default protocol
and new method.
Figure 4. Si mulation Network T opology.
In fowlloing scenarioes (senario 2-) the simulation
plat for m is s ho wn in Fig ur e 4. We co mpare MLU, BWE
(Bandwidth Efficiency), Num ber of Over Utilized Links,
IP Traffic Dropped and IP Traffic Received for all scena-
rios
Scenario 2: In this scenario R1 is the traffic source and
R13 is the tra ffic destinat ion. Table 4 and Table 5 show
the Subo ptimal Link Weights that obt in by DUAL_II .
MLU in new method decreases from 91.3 percent to
36.3 percent and BWE increases from 7.6 percent to 10
percent as shown in Table 6.
Table 4. Solution of DUAL_ II f or Senario2.
W1_2 56.84 W3_6 1.000
W2_1
55.56
W6_3
1.000
W1_3
1.000
W3_7
1.000
W3_1
1.000
W7_3
1.000
W1_4
1.000
W4_9
1.000
W4_1
1.000
W9_4
1.000
W2_5
1.000
W4_8
1.000
W5_2
1.000
W8_4
1.000
W2_11
1.000
W5_12
1.000
W11_2
1.000
W12_5
1.000
W2_3
1.000
W6_11
1.000
W3_2
1.000
W11_6
1.000
W3_4
2.000
W6_10
1.000
W4_3
1.000
W10_6
1.000
Table.5. Solution of D UAL_ II, cont for Senario2.
W6_7 55.84 W10_13 1.000
W7_6
1.000
W13_10
1.000
W7_10
1.000
W10_14
1.000
W10_7
1.000
W14_10
1.000
W7_9
1.283
W11_12
1.000
W9_7
1.000
W12_11
1.000
W8_9
1.000
W11_13
1.000
W9_8
1.000
W13_11
1.000
W9_10
1.000
W12_13
1.000
W10_9
1.000
W13_12
1.000
W9_15
2.000
W13_14
1.000
W15_9
1.000
W14_13
1.000
W10_11
1.000
W14_15
1.000
W11_10
1.000
W15_14
1.000
Table 6. MLU and BWE v alue s for Senar io2.
Default
Algorithm Suboptimal
Algorithm
M LU 91.3 36.3
BWE 7.6 10
Number of Over
Utilized Lin k
0
0
Maximum Load Balancing with Optimized Link Metrics
Copyright © 2012 SciRes. JSEA
19
IP Traffic Dropped and IP traffic Received do not
change in this scenario because there is no congestion
Scenario 3: in this scenario we have three source-des-
tination pairs
11
(, )
sd ,
22
(, )sd
,
33
(, )sd
that are origi-
nated from R1 to R13, R5 to R9 and R4 to R2 respec-
tively. MLU in the new method decreases from 137 per-
cent to 91.3. The Number of over-utilized links also de-
creases from eight links to two links. BWE increases
from 26.6 percent to 31.4 percent, Table 7. F igure 5 and
Figure 6 show the comparison of IP Traffic Dropped and
IP Traffic Received
6. Conclusion
In this paper we show that Optimization Theory can help
Internet protocols work better. We use a duality theor y to
find a weight set that improve the routing protocols effi-
ciencies. As a matter of fact ro uting is the most i mportant
aspect of Internet Traffic Engineering. So we focus on
routing protocols and introduce a practical method that
optimizes Link Metrics. Previous optimization methods
suffer from practical issues but our method could be im-
plemented with Routing Protocols that based on
Table 7. MLU and BWE values for Senario 3.
Default
Algorithm Suboptimal
Algorithm
M LU 137 91.3
BWE 29.6 31.4
Number of Over
Utilized Link 8 2
Figure 5. Ssenario2 IP traffic dro pped.
Figure 6 . Ssenario2’s IP traffic received.
shortest paths. Our simulation results show significant
improvement on network e fficiency.
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