Performance Oriented Virtual Topology Design

The considered performance metric to maximize is the total amount of routed traffic from the given traffic matrix. The total number of lightpaths (|L|) is given. Nodal interface constraints are also considered as in resource oriented optimization. We developed an ILP formulation and also implemented a heuristic for the solution of this problem.

Mixed Integer Linear Programming (MILP) Formulation

As in the previous section, we make the simplifying assumption that traffic be- tween each node pair is splittable so that integer routing variables can be relaxed. The resulting formulation gives us an upper bound on the objective function.

The sets are the same as in the ILP formulation proposed in Section 3.2.1. The decision variables xdh

ij and yij are also the same as in the previous formulation. A

new decision variable, zh

d, which represents the amount of routed traffic belonging

to demand d in hour h, is introduced. The rest of the formulation is as follows: Objective Function: MaximizeX h∈H X d∈D zh d (3.8) Subject to: X j∈N xdh ij − X j∈N xdh ji =         

zdh, if i is the source of demand d

−zdh, if i is the destination of demand d

0, otherwise

X d∈D xdh ij + X d∈D xdh ji ≤ yij · C · Umax for i = 1, ..., |N|; j = 1, ..., |N| (i 6= j); h = 1, ..., |H| (3.10) X j∈N yij ≤ Omax for i = 1, ..., |N|; (i 6= j) (3.11) z_dh ≤ Tdh for d = 1, ..., |D|; h = 1, ..., |H| (3.12) X i∈N X j∈N yij = L (3.13)

The objective function is to maximize the amount of routed traffic from the demand set D. Eq. (3.10),(3.10) and (3.11) are the flow, capacity and optical interface constraints, respectively; similar to the constraints used in the ILP formulation presented in Section 3.2.1. (3.12) implies that the amount of routed traffic belonging to a demand cannot exceed the total traffic offered by that demand. The last type of constraints (3.13) state that the total number of established lightpaths should be equal to the predefined value.

Even this simplifying formulation takes a considerable amount of time (up to several weeks depending on the number of nodes and the number of lightpaths) to solve using the commercial ILP solver CPLEX.We use the solution of this relaxed formulation as an upper bound to evaluate the performance of our proposed heuristic algorithm.

Tabu Search Based Heuristic Algorithm

In this algorithm, the search space consists of moves that satisfy the nodal inter- face constraints. The traffic matrix is input to the algorithm and the objective function is the amount of traffic that can be routed on the current topology over all |H| hours. A move is defined as tearing down an existing lightpath and setting up a new lightpath between two nodes. A valid move is defined as a move that is not in the tabu list and results in a topology that satisfies the interface constraints. The initial solution is generated by a random topology generator algorithm. This algorithm places the lightpaths randomly between the nodes without violating the nodal interface constraints. If the generated topol- ogy is not connected, the algorithm is run from the beginning until a connected topology is generated. The flowchart of the algorithm is given in Figure 3.4.

The Tabu Search algorithm starts from the initial solution. At each iteration, the set of valid moves (M) are calculated. For each move m in M , the resulting objective function, i.e. the amount of traffic that can be routed on the resulting virtual topology, is calculated.

To calculate the objective function, an offline routing algorithm based on the shortest path approach is run with the traffic matrix for each hour. For a single hour h, the traffic flows are sorted according to their traffic rate in a decreasing order and they are served in that order. The link weight (Wi,j) for the lightpath

between nodes i and j is calculated dynamically as 1/Cresi,j where Cresi,j is

the residual capacity of the lightpath, if the LSP is routed on that lightpath. If a lightpath does not have sufficient capacity to carry the current flow, it is assigned a predetermined extremely high cost.

The flows are routed along the minimum cost path, calculated by using the Dijkstra’s algorithm. The total amount of routed traffic is calculated as the sum of the all the routed traffic in each hour. Among the moves in M, the one resulting

in the topology with the highest objective value is chosen. If there are multiple topologies resulting in the highest objective function, a tie-breaker parameter is used to determine which topology to choose. The tie breaker parameter is calculated as:

X

i,j∈N

X

h∈H

Sij· Tdh, d is the demand between nodes i and j (3.14)

In (3.14), Sij denotes the number of hops on the shortest path between i and j

in the resulting topology.

Among the topologies with the same objective function, the one with the low- est tie breaker parameter is chosen. As it can be seen, the tie-breaker parameter favors setting up the lightpaths between the nodes with higher traffic between. Before the chosen move is done, it is added to the tabu list with the current topology. Each entry in the tabu list has four elements; the set up lightpath, the torn down lightpath, current topology and the tenure value. While searching if a possible move is tabu or not, if there is an entry in the tabu list with the first three elements being the same with the current move, the algorithm decides that the move is tabu and excludes it from the set of valid moves. The tenure value is chosen uniformly randomly between an upper and a lower limit. After a move is made, the tenure of each entry in the list is decremented by one. The entries with zero tenure value are removed from the list.

In order to search different regions of the solution space effectively, we in- troduced a diversification step to the algorithm. If there is no iteration in the objective function for a certain number of times (U1) , the algorithm jumps to another element of the solution space randomly. This is achieved by running the random topology generator algorithm. The stopping criterion of the algorithm is the case of no improvements in the best solution for a specific number (U2) of times. If a solution with a higher objective value or an equal objective value and a lower tie breaker value than the best solution found so far cannot be found for a given number of iterations, the algorithm records the best solution and stops.

Table 3.3: The percentages of the routed traffic by the tabu search algorithm and the ILP upper bound

# of lightpaths 13 14 15 16 17 Tabu Search 86.897 93.885 98.816 99.971 100 Upper Bound 93.403 97.886 100 100 100 Optimality Gap (%) 6.966 4.087 1.184 0.029 0

To evaluate the performance of the proposed algorithm, an upper bound on the percentage of routed traffic for a given number of lightpaths, that can be achieved by any algorithm is obtained by solving the MILP formulation intro- duced in Section 3.2.2. The percentages of traffic carried by each algorithm are given in Table 3.3 for different number of lightpaths. These results are obtained using a lightpath capacity C = 3tgen2_{. According to the lower bounds, at least}

15 lightpaths are needed to route all the traffic demands. Proposed TS-MVTD algorithm achieves this using 17 lightpaths, however it can route close to 100% of the traffic demands for also 15 and 16 lightpaths.