Combining Multipath Routing and Combining Multipath Routing and Congestion Control for RobustnessCongestion Control for Robustness

Peter KeyPeter Key

Motivation

• Performance of Internet /overlays unpredictable – and hard to manage

• Multiple ownership / policies (eg BGP) can exacerbate performance problems

• … but diversity increasing– Multihoming– Mesh

• So why not harness the diversity?

Outline

• Motivation• Framework• Resource Pooling

– 2 resources– the function– Cutset dimensioning

• Multipath – Coordinated control– Fluid dynamics

• Route choices & architectures• Concluding remarks

Framework

• Network: capacited graph – G=[X,J]

• Edges have capacity Cj

• Routes s S , sets of edges

• Demands type r, associated source –destination, can use a set of routes

• Link-route incidence matrix A,

route flow incidence matrix B

Framework

• File Transfers– Arrival rate – Mean file size

– Nr in progress

• Streaming – Arrival rate

– Mean holding time

– Mr in progress

• Rate allocation : – xr to both types (fair share): exact rate depends on utility

function

rload r

rr

1r rF

r1

r

lrlr

rrr CxMN :

)(

Resource sharing

Fixed Routing

Dynamic Routing

Fixed Proportions

FMean transf er time

C-

Capacity C

2

F

2p

1-p

1 FTransf er time

2 C-

p 1-pTransf er time F

C-2 C-2 (1 )p p

2 max( ,1 )Stable C p p

F

– Assume performance measure

– non-decreasing in 1st arg., non-increasing in second

– Dimensioning means

– Eg

Performance : the function

( , )C

( , ) some measurable C D D

( , ) =(- ,0)C C , D1( , ) ( )C C

( , )C

Take a cutset C of the Graph G

Under resource pooling, necessary performance conditions are

Becomes interesting when related to sufficiency

Cutset Dimensioning

,j r

j

r jjC

C

D

C

C

Node cutsets:

(Keslassy et al)

• Symmetric case; – Valiant load balancing, – Dynamic routing

Mesh Network Example

6

1

5 4

2

3

and ij j ij j

r r

2 per link suffi cient traffi c matrices

rN

1r

N 2r

N

Multipath Routing: Utility functions

• Utility function associated with type r flow

• increasing, strictly concave etc– Eg TCP,

– Putting w=k/(RTT)2 implies familiar

( )rU x

( )U x w x

1 1 TCP f air:

( )r

r

xT p r

Cost functions

• Now require “cost” convex, and

• True for packet marking etc • with “prices” pj,:

• is prob of drop/marking at j when load is yj,

• Eg small buffer model

( , )j C

0

( , ) /jy

j j j j jy C p z C dz

( ; ) ( ; )L z C Lz LC

j j jp y C

min 1,j j

j j

by yC Cjp

Multipath

Maximise

;r r sr sr j r j s sr sr jr s j r s

N U B x N A B x C

over

• Coordinated

• Single utility function across possible routes flow can choose– Single dependence on RTT

0srx

Fluid Dynamics

• Scale arrival rates and capacities by large number L and take limits

• Gives limiting ODE (FLLN)

1

, 1

rrrr ML

mNL

n

( ) ( ) ( ) ( );

( ) ' ( )r r r r r

r r r r

n t n t x n t mt C

m t m t

Limit Theorems

• Theorem:– Under multipath routing, there is a unique

invariant point– System in Lyapunov stable (under mild

conditions on )– Allocation is only non-zero to routes s for

which “prices” on route are equal– When no streaming, offered load is split

optimally, independent of utility functions

ˆ ˆˆ,r r r r r rm n x

j

Remarks

• Prices on route s are

• Unless “prices” are equal on different routes, only one route is used

• Coordinated multipath chooses load fractions to minimise total “cost”– if no streaming traffic present, fractions

independent of utility functions

ˆ' ( ; )j s j j jj

A y C

rs

Remarks

• Coordinated multipath chooses load fractions to minimise total “cost”rs

,

ˆ ;rrj j s sr sr sr j

j s r

A B x C

Route Choices

• How to search for low cost paths?– Use 2 per nominal route, (eg “direct” +1)– Periodically add new route at random– Probe to chose which route to drop

• Cf “Sticky Random” DAR– “Power of 2”, Mitzenmacher

• Theorem: Under random path resampling, mulitpath routing will find an optimal feasible load split, if one exists

Architecture

• Need path diversity– Dual homing– Multiple addresses (eg IPv6)

• For overlays, wireless, or the Internet?

• Need coordinated congestion control, uncoordinated, parallel, inefficient (see Laurent’s talk …)– at transport or application layer

Summary: Multipath routing/multi-access

• Source /edge routing

• Halve delay (processor sharing)

• Resilience • Simpler dimensioning (cutsets)

C

C

C

C

22 vs 2p p

1 1

vs 2 2 ) ( )C C

• Robust routing provides robustness to– Traffic variations /uncertainty– Routing / BGP / Network operators

• Need to combine multipath routing with congestion control

• Challenges: – Time-scales for route adaptation– Removing RTT bias of TCP?

Summary

References

• Fluid models of integrated traffic and multipath routing, Peter Key & Laurent Massoulié, QUESTA, June 2006

• Network Programming methods for loss networks, Gibbens and Kelly, JSAC 1995

• Stability of end-to-end algorithms for joint routing and rate control, Kelly and Voice, CCR, 2005

• Dynamic Alternative Routing, Gibbens, Kelly and Key, ITC, 1989.