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Nonlinear state-dependent delay modeling and stability analysis of - - PowerPoint PPT Presentation

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works Nonlinear state-dependent delay modeling and stability analysis of Internet congestion control Corentin Briat H. Hjalmarsson, K.H. Johansson, U. T. Jnsson,

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

Nonlinear state-dependent delay modeling and stability analysis of Internet congestion control

Corentin Briat

  • H. Hjalmarsson, K.H. Johansson, U. T. Jönsson, G. Karlsson, H. Sandberg

KTH, Stockholm, Sweden December 16th 2010 CDC 2010, Atlanta, USA

  • C. Briat [KTH /

] 1/16

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

The congestion problem in networks

Network Elements

◮ Buffers/Servers/Routers ◮ Media ◮ Users

Congestion problem - QoS deterioration

◮ Data loss ◮ Too large delay

Need for protocols for congestion control

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] 2/16

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

Problem definition

Modeling problem

◮ Precisely represent each element (user, medium, buffer) ◮ Provide precise adapters/interfaces

Modular construction of networks models Congestion control problem

◮ Define efficient protocols achieving performance specifications ◮ Fairness ◮ Efficiency ◮ Cross-traffic adaptation ◮ Analyze network stability and dynamic performance ◮ Local vs. global stability ◮ Static and dynamic performance ◮ Delays effects on stability

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

How to model networks ?

Packet level

◮ Asynchronous discrete-time large-scale (hybrid) system ◮ Perfect for simulation (NS-2) ◮ Few tools available for analysis

Flow level

◮ Valid when packets size small w.r.t. transfer speed (e.g. kbit vs Mbit/s) ◮ Fluid-flow models, continuous-time ◮ Difficult to transpose packets level effects to a flow level ◮ Good models for representation and analysis

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] 4/16

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

Buffer model (infinite capacity)

◮ Hybrid linear model with linear constraints ◮ Flow integrator ◮ Aggregated flows ◮ Queue saturation → 3rd mode

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] 5/16

˙ τ(t) = 1 c [φ(t) − r(φ(t), τ(t))] r(φ(t), τ(t)) = c if C(t) φ(t)

  • therwise

C(t) = [φ(t) > c] or [τ(t) > 0] ˙ τ(t) ∈ [−1, +∞)

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

Queuing delay map

The time is not universal

◮ Temporal order of reaction ◮ Different models for different reference times

Sending time ts as a reference

◮ Forward operator f(t) = t + τ(t). ◮ Reception time tr = f(ts) ◮ Intuitive, easy for modeling

Reception time tr as a reference

◮ Backward operator g = f−1 exists iff φ > 0 ◮ Sending time ts = g(tr) ◮ Less intuitive but better for analysis

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] 6/16

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

Extended buffer model (1)

Problem of flow separation

◮ Crucial for interconnections description ◮ Flows are aggregated in the previous model ◮ How to split up r into a sum of atomic ri (if possible) ?

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] 7/16

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

Extended buffer model (2)

Output flows - Closed form expression ˙ τ(t) = 1 c

  • i

φi(t) −

  • i

ri(φt, τt)

  • ri(φt, τt)

=    φi(g(t))c

  • j φj(g(t))

if C(g(t)) φi(t)

  • therwise

◮ g(t) = t − τ(g(t)) ◮ Delayed flow proportion ◮ Can be extended to more complex buffers, e.g. multiple output capacities

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

Forward vs. Backward protocol model

Forward protocol model - ts is the reference ˙ z(ts + RTTts) = P(z(ts + RTTts), τ(ts + Tf), T) w(ts + RTTts) = h(z(ts + RTTts)) RTTts = Tf + τ(ts + Tf) + Tb

◮ Not very easy to work with

Backward protocol model - tr is the reference

◮ Standard form for dynamical systems ◮ Implicit state-dependent delay !

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] 9/16

˙ z(tr) = P(z(tr), τ(g(tr − Tb)), T) w(tr) = h(z(tr)) τ(g(tr − Tb)) = τ(tr − Tb − τ(g(tr − Tb)))

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

Complete network model

Single-buffer/Single-user ˙ τ(t) = 1 c [φ(t) − r(φ(t), τ(t))] ˙ z(t) = P(z(t), τ(g(t − Tb)), Tf, Tb) w(t) = h(z(t)) φ(t) = Φ(w(t), τ(t), Tb, Tf)

◮ Functional Φ converts windows sizes into flows [Jacobsson]

φ(t) = w(t − Tf) T + τ(t) , φ(t) = w(t − Tf) T + τ(t) + ˙ w(t − Tf)

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] 10/16

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

Delays properties

Propagation delays Tf, Tb

◮ Constant delays ◮ Bounded

Queuing delay τ(g(t − Tb)) - Single buffer case

◮ Bounded in practice ◮ Derivative belongs to (−∞, 1):

D+[τ(g)](t) =

  • 1 −

c φ(g(t)) if C(g(t))

  • therwise

◮ Well-posedness problems do not occur ( ˙

τ(t) < 1)

◮ Many results on time-delay systems can be applied (Lyapunov-Krasovskii Theory)

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] 11/16

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

Queuing delay τ(g(t − Tb)) - Multiple buffer case

◮ Two delays: τ1(g1(g2(t))) and τ2(g2(t)) ◮ D+[τ1(g1(g2))](t) given by

         (φ1(g1(g2)) + φ2(g1(g2)) − c1)c2 (φ1(g1(g2) + φ2(g1(g2)))φ3(g2) + φ1(g1(g2))c1 if C1(g1(g2)) and C2(g2) 1 − c1 φ1(g1(g2)) + φ2(g1(g2)) if C1(g1(g2)) and not C2(g2)

  • therwise

◮ May exceed one when c2 > c1.

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

FAST-TCP

FAST-TCP protocol ˙ w(t) = γ

  • −

τ(g(t − Tb)) T + τ(g(t − Tb)) w(t) + α

  • ◮ γ tunes the bandwidth of the protocol (control sense)

◮ α rules out the bandwidth-delay product (communication network sense) ◮ Network equilibrium point (buffer+user):

τ ∗ = α c(1 − δ∗) , w∗ = α(1 + T/τ ∗), φ∗ = α/τ ∗

◮ Equilibrium flow independent of the propagation delay (also in the multiple users

case)

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

Stability Analysis

The single-user/single-buffer network is

◮ globally exponentially stable without delays (Lyapunov theory) ◮ locally exponentially stable independently of the delay when

τ ∗ > T where τ ∗ = α/c (small gain)

◮ locally delay-dependent exponentially stable when

τ ∗ < T, τ ∗(T − 1) + T 2 ≤ 0 (quasipolynomials)

◮ locally delay-dependent exponentially stable when

τ ∗ < T, τ ∗(T − 1) + T 2 > 0, γ < 1 τ ∗(T − 1) + T 2 (quasipolynomials)

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

Conclusion and Future Works

Conclusion

◮ Accurate models for FIFO buffers ◮ theoretical delay modeling and analysis ◮ Local stability analysis based on time-delay systems theory

Future works

◮ More general buffer models (priorities, multiple output links/capacities) ◮ Nonlinear stability analysis ◮ Study of limit cycles in more complex topologies

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] 15/16

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Introduction Problem Network Model Stability Analysis Conclusion and Future Works

Thank you for your attention

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