Davide Cherubini - PhD Thesis - UniCA Eprints

6.4 LP modelsEquation 6.32 is the objective function to be minimized, while the left handside (LHS) of capacity constraints in equation 6.33, specifies the total amount oftraffic traversing link (i, j) and it is composed by the traffic routed by IS-IS (firstsum) and by the LSP-carried traffic. Obviously, for each link (i, j) this value hasto be smaller than variable umax times the link capacity c ij .Equation 6.34 determines the amount of traffic carried by MPLS and representsa sort of flow conservation defined for each commodity instead of node.It is important to pay attention to constraint 6.36 because it states thatvariable is f is a real variable making the model entirely linear and, as shown inthe previous sections and subsections, a (pure) linear program is “easy” to solve.In fact, changing the definition of variable is f , for example considering a “binary”variable {0, d f } by specifying that every commodity f is carried only byIS-IS or only by MPLS, the problem becomes a Mixed Integer Program (MIP) becauseit integrates linear constraints and linear objective function with an integerconstraint. MIPs are known to be harder to solve with respect to LPs.The solution of GRP gives the optimal distribution of LSPs that minimizesthe maximum bandwidth occupation and this is also an important result becauseit represents a lower bound during the optimization process of IS-IS metrics [43],in normal working conditions.Survivability ConstraintsThe survivability constraints introduced in the model are formalized as follows:⎛ ⎞∑is f · x f,lij + ∑p h + x l ⎝ ∑ij p h⎠ ≤ u max · c ij ∀(i, j), l ∈ Af∈Fh:(i,j)∈P(h)h:l∈P(h)(6.37)Where the constant x l ij specifies the share of MPLS traffic, flowing along linkl that, in case of failure of such link, is rerouted by IS-IS along link (i, j), whilex f,lijl fails.is the share of flow f carried by IS-IS and traversing link (i, j), when the linkNote that, if x f,lij is less than x l ij the constraint specified by equation 6.33 isredundant.54