Stability of decentralised control mechanisms

Stability of decentralised control mechanisms

Laurent Massoulié

Thomson Research, Paris

Congestion control

Point-to-point data flows Data rates regulated by TCP at end-points Multipath versions: “Overlay” TCP

Peer-to-peer-broadcasting

Pplive, Sopcast,… Hosts exchange data with “overlay” neighbors Aim: real-time playback at all hosts

Outline

Proportional fairness for congestion controlNew characterisation Implications on stability and insensitivity

“Random useful” packet forwarding for p2p broadcastingOptimality propertiesOpen questions

Network Bandwidth Allocation problem Flows of distinct types, sS Ns such flows Which rate s to type s flows? Vector (s )sS : must lie in set C C: captures physical network constraints

Convex Non-increasing

Network capacity set C,single path flows Fixed routes & link capacity constraints

(i)C 0+1 ≤ C1, 0+2 ≤ C2

Polyhedral, convex non-increasing capacity set C

N0 C1

N2

C2

N1

Network capacity set C, multi-path flows

Type s flows can use network paths from set P(s) Bandwidth: s= pP (s)p Network capacity (path var.): {p} C’

path variables: ab + cba + bac ≤ c,…

)( ,

sPp psps CC

ca

bc

2c

a

bcCapacity set C (class variables):

b + c ≤ 2c, a + c ≤ 2c, b + a ≤ 2c.

Utility-maximising allocations

Maximise sS Ns Us(s/Ns) over C Distributed control mechanisms (single- and multi-path) known

A special case: Us(x)= ws [x1--1]/(1-) if 1, ws log(x) if =1 (w,)-fair allocations

In terms of Kuhn-Tucker multipliers:

TCP square root formula:

“TCP-fairness” corresponds to =2, ws=1/RTT2s

a

sll

as

s

s pwN

/1/1

)(

11

spTN ss

s

The dynamic set-up

Type s file transfers: start at instants of Poisson process, rate s

File sizes: Exponential distribution (s)[or general i.i.d.]

Markov process: Ns++ at rate s, Ns-- at rate s s where s: result of congestion control

(time scale separation assumption)

Objectives of congestion control

Maximise schedulable region, defined as

R = Set of vectors of loads s=s/s such that Markov process ergodic

Make performance insensitive to assumption of exponential service times

Previous results

Optimal schedulable region R=int(C)

Exponential service times Max-min fairness [Konstantopoulos et al. 99] (w,a)-fairness [Bonald-M. 01] General utility-maximisation schemes [Ye 03]

General i.i.d. service times Balanced fairness [Bonald-Proutiere 02-04]

exactly insensitive; no known distributed control to achieve it Max-min fairness [Bramson 05]

Proportional fairness

Definition:

Alternative characterisation:

where J: Fenchel-Legendre transform of (log of) capacity set C:

sss xN logargmax: C x

sss yNuJ

Ce :y ysup:)(

NN

JN

ss exp

Main application

Theorem

Proportional Fairness achieves maximal schedulable region R=int(C) for arbitrary phase-type service time distributions

(more generally, for original dynamics augmented by Markovian user routing)

Proof insights

PF “almost” reversible:

Suggests proof outline: The “right” Lyapunov function is given by

Apply suitable Lyapunov function criteria for ergodicity (Foster, Rybko-Stolyar, Dai, Robert)

)()(expexp ss

s eNJNJNN

JN

s

ssNNJNL log)(:

Reversible allocations

Markov process reversible iff for some F,

in which case, stationary distribution:

ss eNFNF exp

sssNNF

ZN log)(exp

1

Reversible allocations (ctd)

“Rate function” of equilibrium distribution

decreases along “fluid dynamics” of system

(by decrease of Kullback-Leibler divergence between current and stationary distributions)

s

ssNNFNL log)(:

NNdt

dssss

Congestion control – summary

Characterisation of proportional fairnessYields new stability resultExplains previously observed reversibility on

particular topologies (hypergrids)could yield finer results, e.g.

characterisation of rate function at equilibrium

Based on joint work with

Andy Twigg, Christos Gkantsidis &

Pablo Rodriguez

P2P broadcasting

Broadcast problem

Transmit data from source to all nodes Unstructured (overlay) network Nodes have no global knowledge

Models many p2p applications Content distribution Video-on-Demand Live video streaming

Broadcast problem Goal: Efficient decentralized schemes

Metrics: broadcast rate & playback delay

Constraints: Edge capacities (well studied, centralized)

[distributed] Node capacities (less explored)

Models different nodes in P2P networks: ADSL, cable, …

Outline Rate-optimal scheme for

edge-capacitated networks

Node-capacitated networks

Application: video streaming

Summary

Edge-capacitated case: background

λ* = min number of edges to disconnect some node from s Can be achieved by packing edge-disjoint spanning trees

[Edmonds,Lovasz, Gabow,…] centralized algorithms

broadcast rate, λ* = min [ mincut(s,i): iV ][Edmonds, 1972]

1

1

1

a

s

b

c

1

1 1

a

s

b

c

a

s

b

c

+

Challenges Aim for decentralised schemes

No explicit tree construction simplifies management with node churn

Manage tension between timeliness and diversity in-order delivery from s to a & b reduces potential

rate from 2 to 1.

11

1

1

a

s

b

1

2

1

a

b

c

Random Useful packet forwarding

Let P(u) = packets received by u

for each edge (u,v)send a random packet from P(u) \ P(v)

New packets injected at rate λ

λ

a

s

b

c

Assumptions: G: arbitrary edge-capacitated graph Min(mincut(G)): λ*

Poisson packet arrivals at source at rate λ Pkt transfer time along edge (u,v): Exponential

random variable with mean 1/c(u,v)

TheoremWith RU packet forwarding, Nb of pkts present at source not yet broadcast:A stable, ergodic process.

RU packet forwarding: Main result

a

s

b

s,a

s

s,b

s,a,b

c

s,a,c s,b,c

s,a,b,c

s,a,c

Correct description of state space: Number of packets XA present exactly at nodes u A, for any set of nodes A(plus state of packets in flight on edges)

Optimality of RU – proof

Optimality proof

s,a

s

s,b

s,a,bs,a,c s,b,c

s,a,b,c

s,a,c

Identify fluid dynamics:

λ ??

λ

Random Block Choice

These capture the original system’s dynamics after some space/time rescaling;

• Prove that solution of fluid dynamics converges to zero when λ < λ*by exhibiting suitable Lyapunov function:

VAxxL AA : sup)(

Outline Rate-optimal scheme for

edge-capacitated networks

Node-capacitated networks

Application: video streaming

Summary

Node-capacitated case

P2P networks constrained by node upload capacity: Cable, ADSL

Node-capacitated case

P2P networks constrained by node upload capacity: Cable, ADSL

How to allocate upload capacity to neighbours? By Edmonds thm, optimum can be achieved by

assigning node capacities to edges and packing spanning trees

a

s

b

c

4

2

2

a

s

b

c

2 a

s

b

c

a

s

b

c

Most-deprived neighbour selection

for each node u choose a neighbour v maximizing |P(u)\P(v)| If u=source, and has fresh pkt, send random

fresh pkt to v Otherwise send random pkt from P(u)\P(v) to v

Distributed: uses only local information Can estimate |P(u) \ P(v)| efficiently

Optimality properties Let λ* be the optimal rate that can be achieved by

a feasible allocation of edge capacities {c* ij}.

Theorem: For the complete graph and injection rate λ < λ* , system ergodic under fresh/RU pkt forwarding to most deprived neighbour.

More general networks?

Outline Optimal & decentralized packet forwarding

in edge-capacitated networks

Node-capacitated networks

Application: video streaming

Summary

Video streaming Model

Assume feasible injection rate λ Source begins sending at time 0 At time D, users start playing back at rate λ

Packets not yet received are skipped p = fraction of skipped packets

How much delay to achieve target p?

Grid networks 40x40 grid Add shortcut

edges with Pr=0.01

Place source in centre of grid

Grid networks

Delay/loss trade-off for RU policy

Expected fraction of skipped packets is (1-1/k)D ~ e-D/k

s

v

network

A toy model: Let k=expected Nb of packets s has and v doesn’t Approximate the network by the following:

Source begins with k packets 1..k Source receives new packets at rate λ Source gives randomly useful packets to v at rate λ

k reflects connectivity between s and v

Fraction of skipped packets decreases exponentially with delay D Can be used to determine suitable playback delay at receiver v.

Simulation

0.155000

0.251000

0.4384

0.2128

Fraction of nodes

Uplink capacity

Random graph (n=500,p=0.05)

Distribution of node capacities as observed in Gnutella [Bharambe et al]

Optimal rate, λ* ≤ 1180

Delay < 1000 inter-pkt send times (<1min)

Conclusions Edge-capacitated networks

Random Useful pkt forwarding achieves optimal broadcast rate

Future: Understand topology impact on delays Extend to dynamic networks

Node-capacitated networks“Most deprived” neighbour selection appears to

perform well Proven rate-optimal for complete graphs Future: optimal for other networks?

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