stability of decentralised control mechanisms
DESCRIPTION
Stability of decentralised control mechanisms. Laurent Massouli é 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,… - PowerPoint PPT PresentationTRANSCRIPT
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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