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1
An Arc-Path Model for OSPF Weight Setting Problem
Dr.Jeffery Kennington
Anusha Madhavan
2
Agenda
Introduction
Node-Arc Model
Arc-Path Model
Empirical Analysis and Comparison
Conclusion
3
Introduction
OSPF – Open Shortest Path First Interior Gateway Protocol
Routing Information in an autonomous system
Link State based Algorithm The state of the interface or link is used to decide the path on
which the information is routed Multiple links with same state is possible. Demand to a
destination can be routed on multiple paths.
4
Routing using OSPF
Routers maintain database with link state information, weights computed using link state, IP address etc.
This database in each router is updated by transmitting Link State Advertisements throughout the autonomous system
A shortest path tree is constructed by each router with itself as the root node and based on weights in the database.
5
Illustration of OSPF
Router 2
Router 3
Router 4 Router 5
Router 1
Router 6
aTraffic from a to z =1200
z
[1, 600]
[1, 600]
[1, 300] [1, 300]
[2,300]
OC-48,
[3,600]
[Weight, Flow]
OC-12, 622 Mbps
[2,300]
2488 Mbps
6
Disadvantages
Lack of prior knowledge of point to point demands may result in congestion as seen in link OC-12
Updating the weights based on the link state information
7
Node-Arc Weight Setting Problem The weight-setting problem as defined by Pioro and
Medhi is as follows: Given: the network topology, the link capacities, and a set of
point-to-point demands Find: Weights and flows for each link Constraints: the demands must be satisfied, the capacities
cannot be violated, and at each node the total entering flow to a given destination is split equally among all out-going links that lie on the shortest paths to that destination.
8
Illustration of Node-Arc Model
1
2
3
4
5
6
7
8
Routing 30 units from Node 2 to Node 7
Routing 10 units from Node 4 to 7
15
15
5+7.5
5+7.5
15 15
5+7.5
+5+7.5
30 10
1
1
1
1
1
1
1
i j fij
wij
lij =0
cij=50
9
Illustration of Node-Arc Model
1
2
3
4
5
6
7
8
Routing 50 units from Node 2 to 7
50
25
25
25
25
50
51
1
1
1
1
2,
1
i j fij
Wij, cij
lij =0
cij=50
10
Node-Arc Model Formulation
Definition of sets, parameters and variables denotes the set of nodes (routers) denotes the set of links (unordered pairs of nodes) denotes the set of arcs denotes the demand volume to be routed from origin to
destination denotes the set of pairs such that > 0 is the sum of demand volumes (i.e.)
Ddo
odhH),(
NLE
odh od
D ),( do odhH
Definition of sets, parameters and variables denotes the set of nodes (routers) denotes the set of links (unordered pairs of nodes) denotes the set of arcs denotes the demand volume to be routed from origin to
destination denotes the set of pairs such that > 0 is the sum of demand volumes (i.e.)
N
11
Node-Arc Model Formulation
Set denotes the capacity of arc The requirement for commodity at node , denoted by
is defined below:
DdodD ),(:
Dd dig
dig
DddihDds
sd
,,),(
,idh DdDdi ,),(
,0 otherwise
Eji ),(ijci
12
Node-Arc Model Formulation
denotes the weight on arc denotes the flow on arc with destination denotes the distance from node to node on the
shortest path to node denotes the common value of the flow assigned to arcs
originating at and contained in shortest paths from to
The binary decision variable = 1 if arc belongs to the shortest path to node ; else 0
Eji ),(dijx Dd ijw
idr i dDd
idyi i
Dd diju ),( ji
d
13
Illustration of Node-Arc Model
5.12
15
47
27
y
y
i j fij
wij, cij
lij =0
cij=50
5.12
15745
723
x
x
Routing 30 units from Node 2 to Node 7
Routing 10 units from Node 4 to 7
1
2
3
4
5
6
7
8
15
15
5+7.5
5+7.5
15 15
5+7.5
+5+7.5
30 10
1
1
1
1
1
1
1
2
3
47
27
r
r
0
1
1
768
745
723
u
u
u
14
Formulation of Node-Arc Problem
Objective: minimize
Constraints: The first set of constraints ensures that demand at node is
satisfied and ensures the conservation of flow at each node
The arc capacity constraints are
Eji
ijw),(
dk
Ejk Eki
dik
dkj gxx
),( ),(DdNk ,
ijDd
dij cx
Eji ),(
15
Formulation of Node-Arc Problem (contd.)
The third set of constraints ensures that flows on shortest paths for each pair are equal.
The fourth set of constraints prevents flow on any arc that is not in a shortest path for demand node .
The next set of constraints ensures that the lengths of shortest paths for an pair are equal.
Huxy dij
dijid )1(0 EjiDd ),(,
),( do
dHux d
ijdij EjiDd ),(,
),( do
Hurwr dijidijjd )1(0 EjiDd ),(,
idijjddij rwru 1 EjiDd ),(,
16
Formulation of Node-Arc Problem (contd.)
The boundary conditions are:
wij 1 Eji ),(
dijx
0 EjiDd ),(,
idr DdNi ,
idy DdNi ,
diju }1,0{ EjiDd ),(,
0
0
17
Arc-Path Weight Setting Problem
The objective is to split the flow equally on limited paths of an demand pair
By limiting the candidate paths, all possible flow combinations can be enumerated
A unique pattern number is assigned to a possible flow distribution in the paths
The selection of a pattern suggests if there is a single or multiple shortest paths for a pair
The weights on the arcs can be computed based on the pattern selection subject to capacity constraints
),( do
),( do
18
Illustration of Arc-Path Model
1
2
3
4
5
6
7
8
Routing 30 units from Node 2 to Node 7
Routing 10 units from Node 4 to 7
19
Flow Distribution Pattern for 2 (o,d) Pairs
p 1 p 2 p 3 p 4 p 5
v1 30 0 0
v2 0 30 0
v3 0 0 30
v4 15 15 0
v5 0 15 15
v6 15 0 15
v7 10 10 10
v8 10 0
v9 0 10
v10 5 5
Demand (2,7) Demand (4,7)
Pattern
Path1p - 2->3->5->7
2p - 2->4->5->7
3p - 2->4->6->7
4p - 4>5->7
5p - 4->6->7
Patterns v7 and v10 are selected
Candidate Paths
20
Illustration of Arc-Path Model
1
2
3
4
5
6
7
8
Routing 30 units from Node 2 to Node 7
Candidate Paths – 2->3->5->7; 2->4->5->7;2->4->6->7
Routing 10 units from Node 4 to 7
Candidate Paths – 4->5->7; 4->6->7
10
20
10
10
10 10 +10
10
30 10
1
1
1
1
1
1
1
i j fij
wij
lij =0
cij=50
+5
+5
+5
+5
21
Arc-Path Model Formulation
Definition of sets, parameters and variables denotes the set of paths for demand pair P = denotes the denote the set of paths computed
using the least hops criterion. denotes the arcs in path P denotes the paths that include arc denotes the pattern numbers for each pair
Assumptions The demand values for all are equal The number of candidate paths for each demand pair is 3 (i.e.)
odP Ddo ),(
Ddo
),( odP
pJ p
ijQ Eji ),(odR ),( do
Ddo ),(
3odP Ddo ),(
22
Arc-Path Model Formulation (contd.)
Definition of sets, parameters and variables Set denote the set of patterns that are associated with a
single path for demand pair Set denote the set of patterns that are associated with two
paths for demand pair Set denote the set of patterns that are associated with three
paths for demand pair Let be the path associated with pattern and let set
Let and be the paths associated with each and let
1odC
Ddo ),(2odC
Ddo ),(3odC
Ddo ),(
p 1odCv
pTodv 1
2odCv ap bp
baodv ppT ,2
23
Arc-Path Model Formulation (contd.)
Let and be the paths associated with each and let
ba pp , cp 3odCv
cbaodv pppT ,,3 p 1 p 2 p 3 p 4 p 5
v 1 30 0 0
v 2 0 30 0
v 3 0 0 30
v 4 15 15 0
v 5 0 15 15
v 6 15 0 15
v 7 10 10 10
v 8 10 0
v 9 0 10
v 10 5 5
Demand (2,7) Demand (4,7)
Pattern
Path
321
3277
327
542
4710247
312
27322
27212
27654227
51
4741
4798147
31
2721
271127321
127
,,
,
,,,,,
,
,,
7
10
654
98
321
pppTvC
ppTvC
ppTppTppTvvvC
pTpTvvC
pTpTpTvvvC
v
v
vvv
vv
vvv
24
Arc-Path Model Formulation (contd.)
Let be the set of patterns The flow in path P , pattern is stored in matrix Let Let be the flow of path P Let be the length of path P The binary variable is 1 if pattern is selected; and 0,
otherwise
DV 7,,2,1 p vpb
0,: vpp bVvvVpf p
pl p
vq Vv
25
Formulation of Arc-Path Problem
Objective: minimize
Constraints: The first set of constraints ensures that that only one
pattern is selected for each pair.
The arc capacity constraints are
Eji
ijw),(
),( do
1 odRv
vq Ddo ),(
ijvQp Vv
vp cqbij p
Eji ),(
26
Formulation of Arc-Path Problem (Contd.)
The third set of constraints calculates the length of each path
P
The fourth set of constraints guarantee that the weights on arcs
The following sets of constraints ensure that if a pattern with flow on a single path is chosen then the length of that path is shortest.
pJji
ij lwp
),(
p
),( ji and are equal ),( ij
jiij ww Lji ),(
111 \,,,),()1()1( odvododvodvtp TPtTpCvDdoqMll
27
Formulation of Arc-Path Problem (Contd.)
The following sets of constraints ensure that if a pattern with flow on two paths is chosen then the lengths of those two paths are shortest
The last sets of constraints ensure that the lengths of multiple paths with flow are equal
222 \,,,),()1()1( odvododvodvtp TPtTpCvDdoqMll
tpTtTpCvDdoqMll odvodvodvvtp ,,,,),()1( 222
tpTtTpCvDdoqMll odvodvodvvtp ,,,,),()1( 333
28
Formulation of Arc-Path Problem (Contd.)
The boundary conditions are given below:
P
1620 ijw and integer Eji ),(
0,0 pp lf p
}1,0{vq Vv
29
Empirical Analysis
Comparison between node-arc and arc-path models
The test cases were generated from 6 different networks
The demand value was fixed at 10 units
The capacity on the arcs were generated randomly in the range [50,100]
Each test case had a maximum of 2 hours to compute weights on the arcs
30
Summary and Conclusions
OSPF algorithm involves developing a shortest path tree by each router with itself as the root node
Data packets to any other router are directed along this shortest path tree.
In this approach the point-to-point demands are disregarded
A node-arc based integer programming model to determine the optimal weights for a given problem instance was presented
The node-arc model balances flow by splitting the incoming flow at a node equally among the outgoing arcs.
31
Summary and Conclusions
The node-arc model is very difficult to solve An improved alternative model using an arc-path
approach was presented The arc-path model splits the flow at origin node equally
among all the outgoing paths. The arc-path approach allows the user to restrict the
number of candidate paths Restricting the solution space allows much larger
problems to be solved using the arc-path approach
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