1. Logical Topology Design

2. Logical Topology vs. Physical Topology
Optical layer provides lightpaths between pairs of client layer equipment (SONET TMs, IP routers, ATM switches)
The lightpaths and the client layer network nodes form a logical topology
The OXCs and optical fibers form a physical topology

3. Logical Topology Design
Lightpath can eliminate electronic processing at intermediate nodes in the client layer => save client layer switch ports/electronic processing
Cost: more wavelength required at the optical layer
Ideally: use a fully-connected logical topology, i.e., setup a lightpath between every pair of source-destination nodes
Not possible for larger networks due to limit on # wavelengths per fiber
4 node ring example

4. Logical Topology Design
Design logical topology based on given traffic patterns and the physical topology
Traffic routed over logical topology
Traffic may travel more than one logical hops
A logical topology can be reconfigured by changing the set of lightpaths
Adaptability (when traffic patterns change)
Self-healing capability (when physical topology changes due to network component failures)
Upgradability (when physical topology changes due to addition or upgrading of network components)

5. A Logical Topology Design Problem (LDT)
Given:
Physical topology
Packet arrival rates for every source-destination pair
Objective:
Compute a logical topology with minimal congestion (congestion is the maximum traffic routed over a logical link)
Why minimize congestion?
Low congestion leads to low packet queuing delay
LT can accommodate the maximum traffic scale-up
Note: need solve the packets routing problem together with LDT

6. LTD
Assumptions:
No limit on the number of wavelengths in the optical layer
All lightpaths are bidirectional: if we set up a lightpath from node i to node j, we also set up a lightpath from node j to node i
Each IP router has at most Δ input ports and Δ output ports
constrains cost of IP routers and number of lightpaths
Traffic between the same pair of nodes can be split over different paths

7. Mathematical Formulation
See handout for problem formulation
The objective functions and the constraints are linear functions of the variables
Linear program (LP): all variables are real
Integer linear program (ILP): all variables must take integer values
Mixed integer linear program (MILP): some variables must take integer values
There are efficient algorithms for solving LPs
ILPs and MILPs are NP-hard

8. A Heuristic for LTD-MILP
Use LP-relaxation and rounding
Terms used in mathematical programming
Feasible solution: any set of values of the variables that satisfy all the constraints
Optimal solution: a feasible solution that optimizes the objective function
Value: value of the objective function achieved by any optimal solution

9. A Heuristic for LTD-MILP
LP-relaxation: if we replace the constraints bij  {0,1} by 0  bij  1, LTD-MILP reduces to LDT-LP
The value of the LTD-LP is a lower bound on the value of the LTD-MILP
The bound is called the LP-relaxation bound
Routing-LP: the values of the bij are fixed at 0 or 1 such that the degree constraints are satisfied
The problem is to route the packets over the logical topology to minimize the congestion
The value of routing-LP is an upper bound on the value of LTD-MILP

10. A Heuristic for LTD-MILP
Solve LTD-LP
Fix the values of bij in LTD-LP to 0 or 1 using the rounding algorithm
Solve the routing-LP

11. Rounding Algorithm Idea: round the bij in LTD-LP to the closet integer
Arrange the values of the bij obtained in an optimal solution of the LTD-LP in decreasing order
Starting at the top of the list, set each bij = 1 if the degree constraints would not be violated. Otherwise, set the bij = 0.
Stop when all the degree constraints are satisfied or the bijs are exhausted