1 EL736 Communications Networks II: Design and Algorithms Class4: Network Design Modeling (II) Yong Liu 10/03/2007

2 Outline  Routing Restriction  Non-linear Link Dimensioning, Cost and Delay Functions  Budget Constraint, Incremental NDP  Extensions

3 Introducing Routing Restriction  enforce the resulting routes w./w.o. certain properties  path diversity vs. limited split  equal splitting vs. arbitrary splitting  modular flows vs. unmodular flows  extend the basic formulation by introducing additional routing constraints.

4 Path Diversity  “never put all eggs in one basket”

5 Lower Bounds on Non-Zero Flows  the flow volume on a path greater than B if any.  implicitly limit number of paths

6 Limited Demand Split  only split among k paths

7 Node-Link Formulation  Single Path

8 Node-Link Formulation  equally split among k link-disjoint paths

9 Integral Flows  allocate demand volumes in demand modules

10 Nonlinear Link Cost  Linear Link Cost  link capacity = link rate  linear cost: $/bps  Nonlinear Link Cost  modular link capacities  different link modules

11 Dimensioning with Modular Links

12 Dimensioning with Multiple Modules

13 Convex Cost Functions  Convex Function   non-negative second order derivative  local minimum-> global minimum  good approx. for link delay   split demand if possible  how to split?

14 Minimal Delay Routing  link delay, network delay, avg. user delay

15 Piecewise Linear Approximation of Convex Function

16 Piecewise Linear Approximation of Convex Function

17 From CXP to LP

18 Concave Link Dimensioning Functions  Concave Function   non-positive second derivative, unique maximum  Erlang B-Loss Formula (extend to real domain)  Implications  merge resources if possible  conflict?

19 Piecewise Linear Approximation of Concave Function

20 Concave Link Dimensioning

21 Budget Constraint  given budget constraint, maximize the realized ratio for all demands.

22 Incremental NDPs  design from scratch vs. improve existing network; sub-optimal solution

23 Extensions: nodes  constraints on nodes  node cost: input/output ports, link termination, switching fabric, installation, …  reliability: node disjoint  virtual graph  two copies for a node: receiving/sending  directed link from receiving copy to sending copy  incorporating node constraints  node cost represented by link cost on its virtual link  node-disjoint in real graph link-disjoint in virtual graph

24 Extensions: nodes  link-path formulation  load on a node:  reliability against node failures: no node carries more than certain share for a demand  link-path formulation  node-link formulation