Stability of decentralised control mechanisms
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Stability of decentralised control mechanisms
Laurent Massoulié
Thomson Research, Paris
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Congestion control
Point-to-point data flows Data rates regulated by TCP at end-points Multipath versions: “Overlay” TCP
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Peer-to-peer-broadcasting
Pplive, Sopcast,… Hosts exchange data with “overlay” neighbors Aim: real-time playback at all hosts
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Outline
Proportional fairness for congestion controlNew characterisation Implications on stability and insensitivity
“Random useful” packet forwarding for p2p broadcastingOptimality propertiesOpen questions
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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
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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
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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.
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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
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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)
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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
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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]
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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
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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)
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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)(:
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Reversible allocations
Markov process reversible iff for some F,
in which case, stationary distribution:
ss eNFNF exp
sssNNF
ZN log)(exp
1
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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
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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
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Based on joint work with
Andy Twigg, Christos Gkantsidis &
Pablo Rodriguez
P2P broadcasting
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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
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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, …
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Outline Rate-optimal scheme for
edge-capacitated networks
Node-capacitated networks
Application: video streaming
Summary
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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
+
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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
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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
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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
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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
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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)(
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Outline Rate-optimal scheme for
edge-capacitated networks
Node-capacitated networks
Application: video streaming
Summary
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Node-capacitated case
P2P networks constrained by node upload capacity: Cable, ADSL
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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
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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
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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?
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Outline Optimal & decentralized packet forwarding
in edge-capacitated networks
Node-capacitated networks
Application: video streaming
Summary
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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?
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Grid networks 40x40 grid Add shortcut
edges with Pr=0.01
Place source in centre of grid
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Grid networks
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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.
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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)
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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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