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Optimal Multi-Path Routing and Bandwidth Allocation under Utility Max-Min Fairness

Jerry Chou and Bill LinUniversity of California, San Diego

IEEE IWQoS 2009Charleston, South CarolinaJuly 13-15, 2009

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Outline

• Problem

• Approach

• Application to optical circuit provisioning

• Summary

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Basic Max-Min Fair Allocation Problem

• Motivation: Bandwidth allocation is a common problem in several network applications

• Example:

A D

10 10

B1010

C

C1: AD C2: BD C3: CD

C1 C2 C3

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Maxincrease

SaturatedflowsFully allocated link

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Utility Max-Min Fairness

100/)( 21 rr 100/)12()( 2

2 rrr 100/)403()(3 rrC1: AD C2: BD C3: CD

0 BW 10

utility

1

0

utility

1

0

utility1

00 BW 10 0 BW 10

Path of C1 Allocation Utilities

ABD (5, 5, 10) (0.25, 0.85, 0.70)

Utility functions capture differences in benefitsfor different commodities

A D

10 10

B1010

C

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Utility Max-Min Fairness

100/)( 21 rr 100/)12()( 2

2 rrr 100/)403()(3 rrC1: AD C2: BD C3: CD

0 BW 10

utility

1

0

utility

1

0

utility1

00 BW 10 0 BW 10

A D

10 10

B1010

C

Path of C1 Allocation Utilities

ABD (5, 5, 10) (0.25, 0.85, 0.70)

ABD (6.8, 3.2, 10) (0.47, 0.47, 0.70)

Utility functions capture differences in benefitsfor different commodities

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Utility Max-Min Fairness

100/)( 21 rr 100/)12()( 2

2 rrr 100/)403()(3 rrC1: AD C2: BD C3: CD

0 BW 10

utility

1

0

utility

1

0

utility1

00 BW 10 0 BW 10

A D

10 10

B1010

C

Path of C1 Allocation Utilities

ABD (5, 5, 10) (0.25, 0. 85, 0.70)

ABD (6.8, 3.2, 10) (0.47, 0.47, 0.70)

Multi-path (8, 4, 8) (0.64, 0.64, 0.64)

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Freedom of choosing multi-path routing achieveshigher min utility and more fair allocation

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Prior Work

• Utility max-min fair allocation only considered fixed (single-path) routing

• Optimal multi-path routing only considered weighted max-min and max-min fairness

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Why is the Problem Difficult?

• Why is optimal multi-path routing and allocation under utility max-min fairness difficult?

→ Unlike conventional fixed (single) path max-min fair allocation problems1. Cannot assume a commodity is saturated just

because a link that it occupies in the current routing is full

2. Once a commodity is saturated, cannot assume its routing is fixed in subsequent iterations

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Example

• At iteration i, suppose we route both flows AD and AE with 5 units of demand

AD:5

AE:5

If routing is fixed after iteration, AD would be at most 5

A D

10/10 5/10

B

0/10

CE

5/5

0/10

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Example

• At iteration i+1, suppose we want to route AD with 10 units of demand

AD:10

AE:5

Route of AD must change to increase

A D

5/10 0/10

B

10/10

CE

5/5

10/10

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Outline

• Problem

• Approach– OPT_MP_UMMF– ε-OPT_MP_UMMF

• Application to optical circuit provisioning

• Summary

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OPT_MP_UMMF

• Step 1: Find maximum common utility that can be achieved by all unsaturated commodities

• Step 2: Identify newly saturated commodities

• Step 3: Assign the utility and allocation for each newly saturated commodity

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Key Differences

• A commodity is truly saturated only if its utility cannot be increased by any feasible routing – Requires testing each commodity for saturation

separately

• To guarantee optimality, fix the utility, not the routing after each iteration

Fix utility,not routing

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Comments• Although OPT_MP_UMMF achieves optimal

solution, both Steps 1 & 2 require solving non-linear optimization problems

Step 1 Step 2

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ε-OPT_MP_UMMF

• Instead of solving a non-linear optimization problem, find maximum common utility by means of binary search

• Test if a common utility has feasible multi-path routing by solving a Maximum Concurrent Flow (MCF) problem

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Maximum Concurrent Flow (MCF)

• Given network graph with link capacities and a traffic demand matrix T, find multi-path routing that can satisfy largest common multiple of T

• If < 1, means demand matrix cannot be satisfied

• If > 1, means bandwidth allocation can handle more traffic than specified demand matrix

• MCF well-studied with fast solvers

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Find Maximum Utility• Determine demand matrix by utility functions• Find feasible routing by querying MCF solver

– If <1, decrease utility, otherwise increase utility

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2010 30 40 50

4060

80

100

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2010 30 40 50

4060

80

100

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2010 30 40 50

4060

80

100

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2010 30 40 50

4060

80

100

BW BW BW BW

Utility(%

)

Utility(%

)

Utility(%

)

Utility(%

)

C = 100Max utility Traffic (T)

1 (50,50,50,50) 0.5

.0.6±ε (10,40,10,40) 1

0.5 (10,30,10,40) 1.25

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Outline

• Problem

• Approach

• Application to optical circuit provisioning

• Summary

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Optical Circuit Provisioning Application• Provision optical circuits for Ingress-Egress (IE) pairs

to carry aggregate traffic between them

• Goal is to maximize likelihood of having sufficient circuit capacity to carry traffic

WDM links

Optical circuit-switched long-haul backbone cloud

Boundaryrouters

Opticalcircuit switches

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Optical Circuit Provisioning (cont’d)• Utility curves are Cumulative Distribution Functions

(CDFs) of “Historical Traffic Measurements”

• Maximizing likelihood of sufficient capacity by maximizing utility functions

• Route traffic over provisioned circuits by default

• Adaptively re-route excess traffic over circuits with spare capacity

• Details can be found in– Jerry Chou, Bill Lin, “Coarse Circuit Switching by Default,

Re-Routing over Circuits for Adaptation”, Journal of Optical Networking, vol. 8, no. 1, Jan 2009

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Experimental Setup

• Abilene network– Public academic network– 11 nodes, 14 links (10 Gb/s)

• Historical traffic measurements– 03/01/4 – 04/21/04

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Example

Seattle

Sunnyvale

Indianapolis

Denver

Los Angeles Kansas City

Chicago New York

Washington

Atlanta

Houston

SeattleNY:90% time ≤ 6Gb/s50% time ≤ 4Gb/sAllocate: 6Gb/s

SunnyvaleHouston:90% time ≤ 6Gb/s80% time ≤ 4Gb/sAllocate: 4Gb/s

Seattle NY has 90% acceptance probabilitySunnyvale Houston has 80% acceptance probability

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Comparison of Allocation Algorithms• WMMF: Single-path weighted max-min fair

allocation– Use historical averages as weights– Only consider OSPF path

• UMMF: Single-path utility max-min fair allocation– Only consider OSPF path

• MP_UMMF: Multi-path utility max-min fair allocation– Computed by our algorithm

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Individual Utility Comparison• Reduce link capacity to 1 Gb/s• MP_UMMF has higher utility for most flows

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Minimum Utility Comparison • MP_UMMF has greater minimum utility

improvement under more congested network

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Excess Demand Comparison

• Simulate traffic from 4/22/04-4/26/04• MP_UMMF has much less excess demand

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Summary of Contributions

• Defined multi-path utility max-min fair bandwidth allocation problem

• Provided algorithms to achieve provably optimal bandwidth allocation

• Demonstrated application to optical circuit provisioning

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Thank You