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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

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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

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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)

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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

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LTDAssumptions:• 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

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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

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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

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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

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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

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Rounding Algorithm

• Idea: round the bij in LTD-LP to the closet integer

• Rounding algorithm1. Arrange the values of the bij obtained in an optimal

solution of the LTD-LP in decreasing order

2. Starting at the top of the list, set each bij = 1 if the degree constraints would not be violated. Otherwise, set the bij = 0.

3. Stop when all the degree constraints are satisfied or the bijs are exhausted