1. Optimal Multi-Path Routing and Bandwidth Allocation under Utility Max-Min Fairness
Jerry Chou and Bill Lin
University of California, San Diego
IEEE IWQoS 2009
Charleston, South Carolina
July 13-15, 2009 1

2. Outline Problem Approach Application to optical circuit provisioning
Summary

3. Basic Max-Min Fair Allocation Problem
Motivation: Bandwidth allocation is a common problem in several network applications
Example:
C1: AD
C2: BD
C3: CD
Saturated
flows
Fully allocated link B C1 C2 C3 10 10 A D 10 5
Max
increase 10 C

4. Utility Max-Min Fairness
C1: AD
C2: BD
C3: CD 1 1 1
utility
utility
utility BW BW BW B
Path of C1
Allocation
Utilities
ABD
(5, 5, 10)
(0.25, 0.85, 0.70) 10 10 A D 10 10 C
Utility functions capture differences in benefits
for different commodities

5. Utility Max-Min Fairness
C1: AD
C2: BD
C3: CD 1 1 1
utility
utility
utility BW BW BW B
Path of C1
Allocation
Utilities
ABD
(5, 5, 10)
(0.25, 0.85, 0.70)
(6.8, 3.2, 10)
(0.47, 0.47, 0.70) 10 10 A D 10 10 C
Utility functions capture differences in benefits
for different commodities

6. Utility Max-Min Fairness
C1: AD
C2: BD
C3: CD 1 1 1
utility
utility
utility BW BW BW B
Path of C1
Allocation
Utilities
ABD
(5, 5, 10)
(0.25, 0. 85, 0.70)
(6.8, 3.2, 10)
(0.47, 0.47, 0.70)
Multi-path
(8, 4, 8)
(0.64, 0.64, 0.64) 10 10 A 6 D 10 10 2 C
Freedom of choosing multi-path routing achieves
higher min utility and more fair allocation

7. 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

8. 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 problems
Cannot assume a commodity is saturated just because a link that it occupies in the current routing is full
Once a commodity is saturated, cannot assume its routing is fixed in subsequent iterations

9. If routing is fixed after iteration, AD would be at most 5
Example
At iteration i, suppose we route both flows AD and AE with 5 units of demand
If routing is fixed after iteration, AD would be at most 5 B
0/10
0/10 A D
AD:5
5/10
10/10 E
AE:5 C
5/5

10. Route of AD must change to increase
Example
At iteration i+1, suppose we want to route AD with 10 units of demand
Route of AD must change to increase B
10/10
10/10 A D
AD:10
0/10
5/10 E
AE:5 C
5/5

11. Outline Problem Approach Application to optical circuit provisioning
OPT_MP_UMMF
ε-OPT_MP_UMMF
Application to optical circuit provisioning
Summary

12. 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

13. 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

14. Comments
Although OPT_MP_UMMF achieves optimal solution, both Steps 1 & 2 require solving non-linear optimization problems
Step 1
Step 2

15. 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

16. 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 l of T
If l < 1, means demand matrix cannot be satisfied
If l > 1, means bandwidth allocation can handle more traffic than specified demand matrix
MCF well-studied with fast solvers

17. Find Maximum Utility Determine demand matrix by utility functions
Find feasible routing by querying MCF solver
If l<1, decrease utility, otherwise increase utility
100
100
100
100
Utility(%)
Utility(%)
Utility(%)
Utility(%) 80 80 80 80 60 60 60 60 40 40 40 40 20 20 20 20 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 BW BW BW BW
Max utility
Traffic (T) l 1
(50,50,50,50)
0.5
C = 100
0.5
(10,30,10,40)
1.25 .
0.6±ε
(10,40,10,40) 1

18. Outline Problem Approach Application to optical circuit provisioning
Summary

19. 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
Boundary
routers
Optical
circuit switches

20. 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

21. Experimental Setup Abilene network Historical traffic measurements
Public academic network
11 nodes, 14 links (10 Gb/s)
Historical traffic measurements
03/01/4 – 04/21/04

22. Example Seattle  NY has 90% acceptance probability
90% time ≤ 6Gb/s
50% time ≤ 4Gb/s
Allocate: 6Gb/s
Seattle
Sunnyvale
Indianapolis
Denver
Los Angeles
Kansas City
Chicago
New York
Washington
Atlanta
Houston
SunnyvaleHouston:
90% time ≤ 6Gb/s
80% time ≤ 4Gb/s
Allocate: 4Gb/s
Seattle  NY has 90% acceptance probability
Sunnyvale  Houston has 80% acceptance probability

23. 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
MP_UMMF: Multi-path utility max-min fair allocation
Computed by our algorithm

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

25. Minimum Utility Comparison
MP_UMMF has greater minimum utility improvement under more congested network

26. Excess Demand Comparison
Simulate traffic from 4/22/04-4/26/04
MP_UMMF has much less excess demand

27. 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

28. Thank You