Optimal NetworkFlow Allocation

EE 384YAlmir Mutapcic and Primoz Skraba27/05/2004

Problem StatementOptimal network flow allocation

Find flow allocation which minimizes certain performance criterion

Lowest average delay through the networkMinimize maximum link utilizationFair bandwidth allocation and QoS agreements

Trade-off between optimality and simplicityDevise practical schemes with low computational complexity and guaranteed performance bounds

Motivation

Internet backbone and PoPs are over-engineeredOvercome link failuresUnderutilized (multiple routes exist)

Current ProtocolsTypically find shortest path(s)Do not directly minimize delay through PoPs

BackgroundIS-IS & OSPF

Link-state routing protocolsLimited load balancingManually tuned to a few routes (traffic engineering)

MPLSRe-labels packets in the internal network

Previous workResource Allocation (minimize max link utilization)Routing Heuristics

Informal FormulationOptimal network flow allocation

Decide how to distribute packets from a particular flow across the network links (x variables)Satisfy conservation laws

Definition of a flowAggregate flow to each destination (sink) nodeEvery other node can be a source to the sink nodeSource-sink vector

Mathematical Formulation

Convex cost function for each link iTotal link i traffic (x is flow’s traffic)Node-link incidence matrix – A Link capacity vector – C

Piece-wise Linear ApproximationGoal: Convert problem into LP

Approximate convex function by K (PWL) segmentsEpigraph minimization (p variables)

Cost Function

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

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Minimize delay over all links

Delay for one linkM/M/1 queue delay

A convex functionProblem

Complex algorithms Slow convergence

PWL ApproximationHow to approximate?

UniformMSEMin-Max

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

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Uniform s plit

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

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10Min-Max s plit

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

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10MS E s plit

PWL Optimization AlgorithmCentralized “one-shot” algorithm (K > 100)

Computational intensiveVery accurate results for underutilized networks

Centralized iterative algorithm (K < 10)Solve LP for the given K (start)Identify link segment k = 1,…,K* that contains traffic flowSplit marked link segment into K more segmentsUpdate LP constraints (slopes and intercepts)Repeat until stopping criteria satisfied

Algorithm SimulationExperimental Setup

MATLAB linprog()Sprint IP backbone network topologyTraffic matrix

Uniform trafficSparsity pattern

ResultsUniform traffic, unit capacities, (K = 2, 3, 5)

1 2 3 4 5 6 7 860

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74objective value

1 2 3 4 5 6 7 810

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10-1

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optimality gap

Results

1 2 3 4 5 6 7 80

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250total cumulative iterations

Iterations (how many?)Stopping criteria

Feasability10E-6

ConvergenceAlways findsfeasible solution(if one exists)

More Results

1 2 3 4 5 6 7 810-4

10-3

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100optimality gap

Gaussian traffic withsparsity pattern, unit C

1 2 3 4 5 6 7 828.7

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29.6objective value

1 2 3 4 5 6 7 80

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200total cumulative iterations

Even More Results

1 2 3 4 5 6 710

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optimality gap

Heavy Gaussian traffic with sparsity, random C

1 2 3 4 5 6 760

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85objective value

1 2 3 4 5 6 70

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200total cumulative iterations

Traffic Distribution

-3 -2 -1 0 1 2 3-3

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Total flow allocation

Computational ComplexityLP interior-point algorithms

number of arithmetic operationsnumber of iterations

M is number of variables + inequality constraintsFor our problem:

M = LF + LK + LFor i iterations = iL(K+F+i/2+3/2)

Storage ComplexityMemory storage requirements

F – FlowsL – LinksN – NodesK – number of segments

(KLFN + N + KLF)α ≈ KLF(N + 1)αith iteration: (K + i)LF(N + 1)α

Distributed algorithmCentralized implementationEasily distributed (especially PWL approximation)Dual methods

Subgradient ascentLagrangian relaxation

Path augmentation approach

Protocol ImplementationRouting protocols

MPLS-like labelingAt most M flowsM – Number of edge routers or PoPsDEST determines pi

DEST IP PACKETp1

p2

p3

Edge RoutersFull look-up

DEST – ID of edge router where packet leaves PoPIdentifies flows within PoP

Congestion ControlAll congestion control can be done at the edgesDetect when traffic not admissible – not feasible

Drop packets at edgeEstimate flows – recalculation at substantial changeShould not occur often – large aggregation of flows

ConclusionMost links underutilized (IP backbone)But still cannot guarantee performance (e.g. delay)Optimal network flow allocation could helpWe suggest a practical algorithm

Converges to near optimal solutionsFew iterationsStandard LPCan make it distributed

Special thanks to Yashar Ganjali for all his help!!

Questions?

PWL Approximation (1)Convert PWL problem into LP

Optimal Network Flow AllocationProblem StatementMotivationBackgroundInformal FormulationMathematical FormulationPiece-wise Linear ApproximationCost FunctionPWL ApproximationPWL Optimization AlgorithmAlgorithm SimulationResultsResultsMore ResultsEven More ResultsTraffic DistributionComputational ComplexityStorage ComplexityDistributed algorithmProtocol ImplementationEdge RoutersConclusionPWL Approximation (1)