Download Clique principle Application to flow aggregation Two flows routed on

Clique principle
Application to flow aggregation
Two flows routed on a 2x5 grid network. These flows can either be routed using a shortest path
algorithm or aggregated to use a minimal number of nodes. To reduce the size of the conflict graph,
only interferences between used links will be considered to build the conflict graph and derive
cliques. The utilization rate of cliques is the sum of utilization rate of links composing the clique.
Unused links will have a utilization rate of 0% and won't impact the utilization rate of cliques
they're in.
Without aggregation (shortest path routing) :
Involved links : (1 → 2), (2 → 3), (3 → 4), (4 → 5), (10 → 9), (9 → 8), (8 → 7), (7 → 6)
Conflict graph associated to involved links :
1→2
2→3
7→6
8→7
3→4
9→8
4→5
10→9
Cliques :
• C1 : (1 → 2), (2 → 3), (3 → 4), (4 → 5), (8 → 7)
• C2 : (7 → 6), (8 → 7), (9 → 8), (10 → 9), (3 → 4)
• C3 : (3 → 4), (4 → 5), (8 → 7), (9 → 8), (10 → 9)
• C4 : (2 → 3), (3 → 4), (4 → 5), (8 → 7), (9 → 8)
• C5 : (1 → 2), (2 → 3), (3 → 4), (8 → 7), (7 → 6)
Considering that links have a capacity of 100 units and that flows f1 and f2 have a demand of 20,
the utilization rate of cliques would be :
• C1 : 20/100 + 20/100 + 20/100 + 20/100 + 20/100 = 100%
• C2 : 20/100 + 20/100 + 20/100 + 20/100 + 20/100 = 100%
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C3 : 20/100 + 20/100 + 20/100 + 20/100 + 20/100 = 100%
C4 : 20/100 + 20/100 + 20/100 + 20/100 + 20/100 = 100%
C5 : 20/100 + 20/100 + 20/100 + 20/100 + 20/100 = 100%
With aggregation :
Involved links : (1 → 2), (2 → 3), (3 → 4), (4 → 5), (10 → 5), (5 → 4), (4 → 3), (3 → 1), (2 → 1),
(1 → 6)
Conflicts graph associated to involved links :
1→2
2→3
2→1
1→6
3→4
3→2
4→5
10→5
4→3
5→4
Cliques :
• C1 : (1 → 2), (2 → 3), (3 → 4), (4 → 5), (3 → 2), (4 → 3), (2 → 1)
• C2 : (2 → 1), (3 → 2), (4 → 3), (5 → 4), (2 → 3), (3 → 4), (4 → 5)
• C3 : (1 → 6), (2 → 1), (3 → 2), (4 → 3), (1 → 2), (2 → 3), (3 → 4)
• C4 : (10 → 5), (3 → 4), (4 → 5), (3 → 2), (4 → 3), (5 → 4)
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Considering that links have a capacity of 100 units and that flows f1 and f2 have a demands of 20,
the utilization rate of cliques would be :
• C1 : 20/100 + 20/100 + 20/100 + 20/100 + 20/100 + 20/100 + 20/100 = 140%
• C2 : 20/100 + 20/100 + 20/100 + 20/100 + 20/100 + 20/100 + 20/100 = 140%
• C3 : 20/100 + 20/100 + 20/100 + 20/100 + 20/100 + 20/100 + 20/100 = 140%
• C4 : 20/100 + 20/100 + 20/100 + 20/100 + 20/100 + 20/100 = 120%
With a QoS requirement of 20 for each flows, the utilization rate of cliques overpass 100% and we
overload the network. Lowering down the demands of flows to 14,28 would reduce the utilization
rates of cliques become :
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C1 : 14,28/100 + 14,28/100 + 14,28/100 + 14,28/100 + 14,28/100 + 14,28/100 +14,28/100 =
99,96 (~100 %)
C2 : 14,28/100 + 14,28/100 + 14,28/100 + 14,28/100 + 14,28/100 + 14,28/100 +14,28/100 =
99,96 (~100 %)
C3 : 14,28/100 + 14,28/100 + 14,28/100 + 14,28/100 + 14,28/100 + 14,28/100 +14,28/100 =
99,96 (~100 %)
C4 : 14,28/100 + 14,28/100 + 14,28/100 + 14,28/100 + 14,28/100 +14,28/100 = 85,68%
Giving more resources to one these two flows would increase the utilization rates of C1, C2 and C3
over 100%. The limit of the network is reached when aggregation is used.