Logical Topology Design

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. 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 • Rounding algorithm • 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