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
Outline
• Problem
• Approach
• Application to optical circuit provisioning
• Summary
2
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
Fully allocated link
A
10
B
10
C
10
10
D
Saturated
flows
C1
C2
C3
5
Max
increase
3
Utility Max-Min Fairness
C1: AD
1 ( r )  r 2 / 100
C2: BD
2 ( r )  ( r 2  12r ) / 100
C3: CD
3 ( r )  (3r  40) / 100
1
1
1
utility
utility
utility
0
0
0
0
BW
A
10
10
0
B
10
C
10
BW
D
10
0
Path of C1 Allocation
ABD
(5, 5, 10)
BW
10
Utilities
(0.25, 0.85, 0.70)
10
Utility functions capture differences in benefits
for different commodities
4
Utility Max-Min Fairness
C1: AD
1 ( r )  r 2 / 100
C2: BD
2 ( r )  ( r 2  12r ) / 100
C3: CD
3 ( r )  (3r  40) / 100
1
1
1
utility
utility
utility
0
0
0
0
BW
A
10
10
0
B
10
C
10
10
BW
D
10
0
Path of C1 Allocation
BW
10
Utilities
ABD
(5, 5, 10)
(0.25, 0.85, 0.70)
ABD
(6.8, 3.2, 10) (0.47, 0.47, 0.70)
Utility functions capture differences in benefits
for different commodities
5
Utility Max-Min Fairness
C1: AD
1 ( r )  r 2 / 100
C2: BD
2 ( r )  ( r 2  12r ) / 100
C3: CD
3 ( r )  (3r  40) / 100
1
1
1
utility
utility
utility
0
0
0
0
BW
10
A 6
10
2
10
0
B
C
10
10
BW
D
10
0
Path of C1 Allocation
BW
10
Utilities
ABD
(5, 5, 10)
ABD
(6.8, 3.2, 10) (0.47, 0.47, 0.70)
Multi-path (8, 4, 8)
(0.25, 0. 85, 0.70)
(0.64, 0.64, 0.64)
Freedom of choosing multi-path routing achieves
higher min utility and more fair allocation
6
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
7
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
1. 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
8
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
C
E
5/5
AE:5
9
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
C
E
5/5
AE:5
10
Outline
• Problem
• Approach
– OPT_MP_UMMF
– ε-OPT_MP_UMMF
• Application to optical circuit provisioning
• Summary
11
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
12
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
13
Comments
• Although OPT_MP_UMMF achieves optimal
solution, both Steps 1 & 2 require solving nonlinear optimization problems
Step 1
Step 2
14
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 multipath routing by solving a Maximum
Concurrent Flow (MCF) problem
15
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
16
Find Maximum Utility
• Determine demand matrix by utility functions
• Find feasible routing by querying MCF solver
– If l<1, decrease utility, otherwise increase utility
10 20 30 40 50
BW
C = 100
10 20 30 40 50
BW
Utility(%)
100
100
80
60
40
20
Utility(%)
Utility(%)
Utility(%)
100
80
60
40
20
80
60
40
20
10 20 30 40 50
BW
100
80
60
40
20
10 20 30 40 50
BW
Max utility
Traffic (T)
1
(50,50,50,50)
0.5
(10,30,10,40)
.
(10,40,10,40)
0.6±ε
l
0.5
1.25
1
17
Outline
• Problem
• Approach
• Application to optical circuit provisioning
• Summary
18
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
Boundary
routers
WDM links
Optical
circuit switches
Optical circuit-switched long-haul backbone cloud
19
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
20
Experimental Setup
• Abilene network
– Public academic network
– 11 nodes, 14 links (10 Gb/s)
• Historical traffic measurements
– 03/01/4 – 04/21/04
21
Example
SeattleNY:
90% time ≤ 6Gb/s
50% time ≤ 4Gb/s
Allocate: 6Gb/s
Seattle
New York
Chicago
Sunnyvale
Los Angeles
Denver
Kansas City
Indianapolis
SunnyvaleHouston:
90% time ≤ 6Gb/s
80% time ≤ 4Gb/s
Allocate: 4Gb/s
Washington
Atlanta
Houston
Seattle  NY has 90% acceptance probability
Sunnyvale  Houston has 80% acceptance probability
22
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
23
Individual Utility Comparison
• Reduce link capacity to 1 Gb/s
• MP_UMMF has higher utility for most flows
24
Minimum Utility Comparison
• MP_UMMF has greater minimum utility
improvement under more congested network
25
Excess Demand Comparison
• Simulate traffic from 4/22/04-4/26/04
• MP_UMMF has much less excess demand
26
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
27
Thank You
28