An axiomatic fluid-flow model for congestion control analysis - - PowerPoint PPT Presentation
Introduction Modeling networks Building Blocks Examples Conclusion An axiomatic fluid-flow model for congestion control analysis Corentin Briat H. Hjalmarsson, K.H. Johansson, U.T. Jnsson, G. Karlsson, H. Sandberg and E.A. Yavuz KTH,
Introduction Modeling networks Building Blocks Examples Conclusion
Corentin Briat
E.A. Yavuz KTH, Stockholm, Sweden December 13th 2011 CDC 2011, Orlando, USA
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◮ Introduction ◮ Modeling networks ◮ Building blocks ◮ Simulations ◮ Conclusion and Future Works
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Network Elements
◮ Buffers/Servers/Routers (queuing delays, loss) ◮ Transmission channels (delays, limited capacity) ◮ Users (senders, receivers)
Transmission principle
◮ Information split into packets ◮ User A sends packets destined to User B ◮ Packets routed through the network ◮ User B receives packets and acknowledges them (ACK packet) ◮ User A receives ACK packets → transmission successful
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Reason
◮ Network infrastructure ’small’
Consequence: QoS deterioration
◮ Presence of bottlenecks ◮ Large delays ◮ Data loss
Congestion control ?
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Concepts and variables
◮ Flight-size: controlled variable ◮ Congestion window: reference ◮ Congestion measure: measurement
Congestion measures
◮ Data loss: simple but aggressive ◮ Delay: complex but preventive
Objectives
◮ Stability ◮ Performance criteria
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Small star topology
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Internet
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diagram/model, and vice-versa.
compromising existing ones.
Similar properties are desired for congestion control modeling
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The last ingredient. . .
◮ Flight-size: controlled variable ◮ Congestion window: reference ◮ Congestion measure: measurement ◮ Flow: control input
Flow definition
◮ Quantity of information: N [bit] ◮ Flow: φ [bit/s] ◮ Quantity of information having passed through point x between t0 and t:
Nx(t0, t) := t
t0
φ(x, s)ds
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Core idea
◮ Information conservation law (Axiom 1)
Independent models
◮ Transmission channels ◮ Buffers/Queues ◮ Users
Properties of the model
◮ Obtained model is explicit, modular and scalable ◮ Similar to electrical networks
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Observations
◮ If a packet is in transit in a network element, it must have entered it in the past. ◮ We can count the number of packets PE(t) in an element E at time t simply by
counting the entering packets during a certain amount of time. Conservation law statement There exists t0(t) ≤ t such that the following equality holds: PE(t) = t
t0(t)
φ(s)ds.
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Balance equation P ′
E(t)
= φ(t)
− φo(t)
= φ(t) − t0(t)′φ(t0(t)) Output flow model The output flow of element E, φo(t), is given by φo(t) := t0(t)′φ(t0(t)).
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Assumptions
◮ Lossless ◮ Constant propagation delay T > 0
Quantity of information PE(t) = t
t−T
φ(s)ds I/O Relationship and output flow P ′
E(t)
= φ(t) − φ(t − T) φo(t) = φ(t − T)
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˙ τ(t) = 1 c
j
φj(t) − r(t) r(t) = c if C(t)
φj(t)
Buffer model
◮ Flow integrator ◮ Hybrid system
Model inaccuracies
◮ Aggregate output flows r(t) ! ◮ How to connect buffers to other buffers ? ◮ Does this model describe a FIFO queue ?
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Difficulties
◮ Integration destroys the information ◮ Indistinguishability of the output flows due to aggregation
Solution
◮ Apply the conservation law ’around’ the buffer
PE(t) = cτ(t) = t
g(t)
φ(s)ds (1) where g(t) = t − τ(g(t)).
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FIFO buffer model ˙ τ(t) = 1 c
[φj(t) − rj(t)] rj(t) = φj(g(t))c
if C(t) φj(t)
g(t) = t − τ(g(t))
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Description variables
◮ Window size w: number of desired outstanding packets ◮ Congestion measure µ: delays, packet loss, etc ◮ Sending flow φo(t) and ACK flows φ(t)
Continuous-time model of user protocol (Axiom 3) Protocol state : ˙ z(t) = P(z(t), µ(t)) Window size : w(t) = W(z(t), µ(t))
◮ How to relate flows, flight-size and window size ?
φo(t) = U(w(t), φ(t)) ?
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Protocol Equations ˙ z(t) = P(z(t), µ(t)) w(t) = W(z(t), µ(t)) Sending flow model φo(t) =
wi(t) + φ(t) if Ti(t)
˙ πi(t) = if Ti(t) ˙ wi(t) + φ(t)
Ti(t) = ([πi(t) = 0] ∧ [ ˙ wi(t) + φ(t) ≥ 0])
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2.8 3 3.2 3.4 3.6 3.8 4 4.2 0.01 0.015 0.02 0.025 0.03 0.035 0.04
Time [sec] Queue size [sec]
Proposed model Ratio link model Static link model Joint link model NS−2
Scenario 1
◮ c = 100Mb/s, ρ = 1590 bytes ◮ Propagation delays: T1 = 3.2ms and T2 = 117ms ◮ Initial congestion window sizes: w0
1 = 50 and w0 2 = 550 packets
◮ At 3s, w1 is increased to 150 packets.
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4.8 5 5.2 5.4 5.6 5.8 6 6.2 0.06 0.065 0.07 0.075 0.08 0.085
Time [sec] Queue size [sec]
Proposed model Ratio link model Static link model Joint link model NS−2
Scenario 2
◮ c = 100Mb/s, ρ = 1590 bytes, no cross-traffic ◮ Propagation delays: T1 = 10ms and T2 = 90ms ◮ Initial congestion window sizes: w0
1 = 210 and w0 2 = 750 packets
◮ At 5s, w1 is increased to 300 packets.
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9.5 10 10.5 11 11.5 12 12.5 13 0.07 0.08 0.09
Time [sec] Queue size 1 [sec]
9.5 10 10.5 11 11.5 12 12.5 13 0.045 0.05 0.055 0.06 0.065 0.07
Time [sec] Queue size 2 [sec]
Proposed Model NS−2
Scenario 3
◮ c1 = 72Mb/s, c2 = 180Mb/s, ρ = 1448 bytes, no cross-traffic ◮ Propagation delays: T1 = 120ms, T2 = 80ms and T3 = 40ms ◮ Initial congestion window sizes: w0
1 = 1600, w0 2 = 1200 and w0 3 = 5 packets
◮ At 10s, w2 is increased to 1400 packets.
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9.5 10 10.5 11 11.5 12 12.5 13 0.2 0.205 0.21 0.215 0.22 0.225
Time [sec] Queue 1 [sec]
9.5 10 10.5 11 11.5 12 12.5 13 0.045 0.05 0.055 0.06 0.065 0.07
Time [sec] Queue 2 [sec]
Proposed Model NS−2
Scenario 4
◮ c1 = 72Mb/s, c2 = 180Mb/s, ρ = 1448 bytes, δ1 = c1/2 ◮ Propagation delays: T1 = 120ms, T2 = 80ms and T3 = 40ms ◮ Initial congestion window sizes: w0
1 = 1600, w0 2 = 1200 and w0 3 = 5 packets
◮ At 10s, w2 is increased to 1400 packets.
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4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 0.05 0.1 0.15 0.2
Time [sec] Queue size [sec]
Proposed model Joint & Ratio link model Static link model NS−2 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 −300 −200 −100 100
Time [sec] ACK-buffer π [Pkt]
Scenario 5
◮ c = 12.5Mb/s, ρ = 1040 bytes, no cross-traffic ◮ Propagation delay: T = 150ms ◮ Initial congestion window size: w0 = 500 packets ◮ At 5s, w1 is halved.
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4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 0.05 0.1 0.15 0.2
Time [sec] Queue size [sec]
Proposed model Joint & Ratio link model Static link model NS−2 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 −300 −200 −100 100
Time [sec] ACK-buffer π [Pkt]
Scenario 6
◮ c = 25Mb/s, ρ = 1040 bytes, δ = c/2 ◮ Propagation delay: T = 150ms ◮ Initial congestion window size: w0 = 500 packets ◮ At 5s, w1 is halved.
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◮ Model obtained from few principles: 1 conservation law and 2 basic models ◮ ’Kirchhoff’s laws’ (conservation of flows and quantity of information):
Current [A] Flow [bit/s] Voltage [V] Quantity of information [bit]
◮ Network component models expressed in terms of these variables: ◮ Transmission channel ◮ Queues ◮ Users ◮ Describe quite well the reality for considered topologies ◮ Model is modular, scalable, topologically identical ◮ Suitable for building (graphical) simulators
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Future Works
◮ Global stability analysis of some protocols ◮ Input/ouput approaches ◮ Incorporate packet dropping and queue saturation in the model
Ideas
◮ Design new elements (like tanks) ◮ Approximations for easier analysis ◮ Design of new protocols, e.g. rate-based protocols (get rid of the congestion
window)
◮ Adapt the modeling spirit to other fields where conservation law may be defined
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