Lecture 21

1. Lecture 21 Mathematical Models Used To Model Telecommunication Design Problems

2. Robust Designs for WDM Routing and Provisioning Jeff Kennington, Karen Lewis, Eli Olinick Southern Methodist University Augustyn Ortynski, Gheorghe Spiride Nortel Networks

3. Objective of the work • Develop a robust design procedure for WDM routing and provisioning problems. • These problems come in three varieties based upon the protection requirements • no protection, • 1+1 protection • shared protection • So far we have studied the “no protection” case

4. The problem • Given • The network topology • An estimate of the traffic demands, and routing assumptions • Equipment capacity, modularity, and unit cost assumptions • Determine • Working and protection channel routing • Required number of network elements at nodes and on links. • Several versions of this problem • Depending on protection requirements

5. The goal • Design for given point forecast • However, • Traffic growth is difficult to predict • Uncertain point forecasts to start with • Therefore, • An optimal design for an erroneous forecast may prove to be inferior. • The goal is to develop a network design that will be robust over a variety of demand forecasts.

6. The proposed approach • Consider a set of scenarios, each with a given probability of occurrence • A fixed budget to cover cap expenses • Create a network design that minimizes the regret over the range of scenarios, while the total equipment cost is below the budget • The regret associated with a design penalizes non-robust designs

7. Equipment modeling – sample network link Note: the cost of WDM couplers is included in the LTE/R cost

8. Equipment modeling • Nodal equipment • LTEs have a given modularity • Line equipment • Regenerators have given modularity • Optical amplifiers have a larger modularity

9. Other assumptions • Demand is expressed in DS3 • Line capacity is OC192 • Routing candidate paths are computed and fed into the model • In this analysis we consider the first k-shortest paths as candidates for each demand • A given maximum number of candidate routings is considered for each demand

10. Modeling uncertainty

11. Solution approaches • Robust optimization • Design a network that minimizes regret • Other approaches from the literature • Stochastic Programming • Minimize overall cost (equip. + penalty) • Worst-Case • Minimize the maximum cost • Mean-Value • Compute expected value of demand and use the basic design approach

12. What is regret? Time 0 – Build Network Time t later – Demand is known Case 1: Under Provision (can not meet demand for some (o,d) pairs) Case 2: Over Provision (there is excess capacity) Regret is a piece-wise linear approximation to a quadratic

13. Regret example 2.51E+08 2.18E+08 2.01E+08 1.51E+08 Regret 1.23E+08 1.01E+08 5.40E+07 5.10E+07 1.40E+07 1.00E+06 0 2000 4000 6000 8000 10000 12000 14000 16000 Positive underprovisioning

14. Regret example

15. Basic design model Minimize cx (equip. cost) Subject to Ax = b (structural const) Bx = r (demand const) 0<x<u (bounds) xj integer for some j (integrality) Integer Linear Program

16. Decision variables

17. Constant definitions

18. Routing for scenario s

19. Robust model

20. Robust model (cont.)

21. Mean-Value model

22. Stochastic Programming model

23. Worst Case model

24. Source Total Nodes 67 Total Links 107 Total Demand Pairs 200 Number of Paths/Demand 4 Total Demand Scenarios 5 Source Total Nodes 18 Total Links 35 Total Demand Pairs 100 Number of Paths/Demand 4 Total Demand Scenarios 5 Test problems overview • Regional US network – DA problem • European multinational network – KL problem

25. Scenario Prob. LTEs Rs As CPU Seconds Equipment Cost 1 0.15 24,996 3962 563 0.5 1,848,000,000 2 0.20 39,456 6502 864 0.5 2,925,000,000 3 0.30 51,882 8074 1101 0.5 3,791,000,000 4 0.20 65,086 10,122 1355 0.6 4,742,000,000 5 0.15 76,848 12,447 1584 0.5 5,630,000,000 Expected — 51,749 8,208 1096 — 3,792,000,000 Value DA – method comparison

26. DA – results

27. Budget Provisioning Sce nario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Totals under LTE 11 381 2908 7838 16,536 27,674 under R 0 0 91 504 2076 2671 under A 1 4 49 126 254 434 5,630,000,000 over LTE 38,137 24,048 14,149 5875 2811 85,020 over R 6851 4311 2830 1195 442 15,629 over A 863 565 373 196 95 2092 under LTE 253 566 3969 13,832 25,322 43,942 DA – under/over-provisioning under R 119 101 724 2080 4419 7443 under A 12 17 124 307 535 995 3,787,000,000 over LTE 27,482 13,334 4312 971 699 46,798 over R 4205 1647 698 6 20 6576 over A 513 217 87 16 15 848 under LTE 2731 14,180 26,145 39,157 50,922 133,135 under R 720 3082 4552 6555 8880 23,789 under A 82 341 572 822 1051 2868 1,848,000,000 over LTE 3663 653 191 0 3 4510 over R 352 147 45 0 0 517 over A 52 10 4 0 0 66

28. KL – individual scenarios

29. CPU Unrouted Scaled Budget Method LTEs Rs As Equip. Cost Seconds Demand Regret Mean Value 25,124 15,350 1221 3,094,700,000 1.1 15.4% 1.41 Stoch. Prog. 20,264 14,168 996 2,644,620,000 0.6 20.8% 1.94 4,554,610 ,000 Worst Case 17,977 11,812 872 2,279,830,000 1.1 27.8% 4.05 Robust Opt. 27,520 21,348 1614 3,890,840,000 1.0 5.6% 1.00 Mean Value 23,978 15,382 1198 3,028,460,000 0.5 15.4% 1.11 Stoch. Prog. KL – method comparison 20,264 14,168 996 2,644,620,000 0.2 20.8% 1.52 3,032,69 0,000 Worst Case 17,977 11,812 872 2,279,830,000 0.4 27.9% 3.20 Robust Opt. 23,967 15,548 1181 3,032,690,000 200.0 13.1% 1.00 No Feasible Mean Value — — — ? 100% — Solution Stoch. Prog. 12,154 7456 666 1,537,150,000 2.7 42.7% 1.19 1,539,360,000 Wors t Case 12,782 7222 645 1,539,360,000 1.9 44.4% 1.71 Robust Opt. 13,562 7172 575 1,539,360,000 5.6 43.3% 1.00

30. Budget Provisioning Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Totals under LTE 965 1484 3993 7311 12,263 26,016 under R 294 722 1793 3500 6676 12,985 under A 22 48 112 259 442 883 4,554,610,000 over LTE 15,718 11,511 7494 5536 4060 44,319 over R 14,367 10,379 7358 5652 4418 42,174 ove r A 998 704 548 418 296 2964 under LTE 102 295 2302 6207 12,602 21,508 under R 52 263 1932 3963 8592 14,802 under A 14 34 139 296 615 1098 3,032,690,000 over LTE 11,151 6619 2099 729 696 21,294 over R 8369 4164 1741 359 578 15,211 over A 565 265 150 30 44 1054 under LTE 1372 5473 10,518 15,733 22,161 55,257 under R 918 5272 8641 12,024 16,434 43,289 under A 120 440 603 880 1185 3228 1,539,360,00 over LTE 2166 1542 60 0 0 3768 over R 815 753 30 0 KL – under/over-provisioning 0 1598 over A 57 57 0 0 0 114