EL736 Communications Networks II:
Design and Algorithms
Class4: Network Design Modeling (II)
Yong Liu
10/03/2007
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Outline
 Routing Restriction
 Non-linear Link Dimensioning, Cost and
Delay Functions
 Budget Constraint, Incremental NDP
 Extensions
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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.
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Path Diversity
 “never put all eggs in one basket”
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Lower Bounds on Non-Zero Flows
 the flow volume on a path greater than B if any.
 implicitly limit number of paths
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Limited Demand Split
 only split among k paths
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Node-Link Formulation
 Single Path
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Node-Link Formulation
 equally split among k link-disjoint paths
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Integral Flows
 allocate demand volumes in demand modules
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Nonlinear Link Cost
 Linear Link Cost
 link capacity = link rate
 linear cost: $/bps
 Nonlinear Link Cost
 modular link capacities
 different link modules
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Dimensioning with Modular Links
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Dimensioning with Multiple Modules
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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?
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Minimal Delay Routing
 link delay, network delay, avg. user delay
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Piecewise Linear Approximation of
Convex Function
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Piecewise Linear Approximation of
Convex Function
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From CXP to LP
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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?
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Piecewise Linear Approximation of
Concave Function
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Concave Link Dimensioning
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Budget Constraint
 given budget constraint, maximize the realized
ratio for all demands.
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Incremental NDPs
 design from scratch vs. improve existing
network; sub-optimal solution
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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
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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
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