1. Wavelength-Routed Optical Networks: Linear Formulation, Resource Budgeting Tradeoffs, and a Reconfiguration Study Dhritiman Banergee and Biswanath Mukherjee, Member, IEEE Reporter:Pey_Jiuan He 2006-02-22

2. 2 Outline Problem Statement Linear Formulation Conclusion

3. 3 Problem Statement 2 1 3 0 4 PT Traffic matrix T1

4. 4 Linear Formulation Given: Number of nodes in the network = N. Maximum number of wavelengths per fiber = W Physical topology P mn denotes the number of fibers interconnecting node m and node n Fiber length matrix, fiber distance d mn from node m to node n, d mn is expressed as a propagation delay. d mn = d nm, and d mn =∞ if P mn =0 Shortest path delay matrix D, D sd denotes the delay over the shortest path between nodes s and d.

5. 5 Given: Lightpath length bound α,1 ≦ α<∞, bounds the delay over a lightpath between two nodes i and j, with respect to the shortest path delay D ij between them, the maximum permissible propagation delay over the lightpath between the two nodes i and j is α* D ij Number of transmitters at node i =T i ( ≧ 1) Numberof receivers at node i = R i ( ≧ 1)

6. 6 Given: Traffic matrix Λ sd denotes the average rate of traffic flow (packets/s) from node s to node d, with Λ ss = 0 for s,d = 1,2,3…N. Capacity of each channel = C (bits/s) Maximum loading per channel =β, 0< β<1, β restricts the queueing delay on a lightpath from getting unbounded by avoiding excessive link congestion.

7. 7 Variables: Virtual topology: The variable V ij denotes the number of lightpaths from node i to node j in the virtual topology. Traffic routing: The variable denotes the traffic flowing from node s to node d and employing V ij as an intermediate virtual link. Physical topology route: The variable denotes the number of lightpaths between nodes i and j being routed though fiber link mn.

8. 8 Objective: Optimality Criterion The objective function minimizes the average packet hop distance in the network. origin:

9. 9 Constraints: On virtual topology connection matrix V ij :

10. 10 Constraints: On physical route variables :

11. 11 Constraints: On virtual topology traffic variables β : loading per channel C: channel capacity

12. 12 Optional constraints: Physical topology as a subset of the virtual topology: Bound lightpath length: d mn : fiber distance from node m and n α: lightpath length bound D ij : delay over the shortest path

13. 13 Simplifying Assumptions Relax Wavelength-continuity constraints :  Total number of lightpaths routed through a fiber is less than or equal to W  mean that a lightpath from node i to node j is assigned the wavelength k (where k=1,2,…,W) nonlinear

14. 14 Simplifying Assumptions Queueing delays are also intentionally ignored, the exact optimization function for delay minimization is as follow: nonlinear

15. 15 Simplifying Assumptions The number of variables and equations in the formulation are reduced.  Tracking a limited number of alternate shortest paths (eg. K=2), such that the selected routes are within a constant factor of the shortest-path distance between the given s-d pair.

16. 16 Simplifying Assumptions The current formulation allows bifurcated routing of packet traffic. To specify nonbifurcated routing of traffic, we use new variables as (21) which are only allowed to take binary values.

17. 17 Simplifying Assumptions The objective function becomes

18. 18 Conclusion Translating above formulation into CPLEX model, and get the path set-up table to form the virtual topology