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Logical Topology Design. 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

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## Logical Topology Design

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**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**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**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)**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**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**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**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**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**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**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

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