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 16 th 2010 CDC 2010, Atlanta, USA C. Briat [KTH / ] 1/16
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 C. Briat [KTH / ] 2/16
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 C. Briat [KTH / ] 3/16
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 C. Briat [KTH / ] 4/16
Introduction Problem Network Model Stability Analysis Conclusion and Future Works Buffer model (infinite capacity) 1 τ ( t ) ˙ = c [ φ ( t ) − r ( φ ( t ) , τ ( t ))] � c if C ( t ) r ( φ ( t ) , τ ( t )) = φ ( t ) otherwise C ( t ) = [ φ ( t ) > c ] or [ τ ( t ) > 0] τ ( t ) ˙ [ − 1 , + ∞ ) ∈ ◮ Hybrid linear model with linear constraints ◮ Flow integrator ◮ Aggregated flows ◮ Queue saturation → 3rd mode C. Briat [KTH / ] 5/16
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 t s as a reference ◮ Forward operator f ( t ) = t + τ ( t ) . ◮ Reception time t r = f ( t s ) ◮ Intuitive, easy for modeling Reception time t r as a reference ◮ Backward operator g = f − 1 exists iff φ > 0 ◮ Sending time t s = g ( t r ) ◮ Less intuitive but better for analysis C. Briat [KTH / ] 6/16
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 r i (if possible) ? C. Briat [KTH / ] 7/16
Introduction Problem Network Model Stability Analysis Conclusion and Future Works Extended buffer model (2) Output flows - Closed form expression �� � 1 � τ ( t ) ˙ = φ i ( t ) − r i ( φ t , τ t ) c i i φ i ( g ( t )) c if C ( g ( t )) r i ( φ t , τ t ) = � j φ j ( g ( t )) φ i ( t ) otherwise ◮ g ( t ) = t − τ ( g ( t )) ◮ Delayed flow proportion ◮ Can be extended to more complex buffers, e.g. multiple output capacities C. Briat [KTH / ] 8/16
Introduction Problem Network Model Stability Analysis Conclusion and Future Works Forward vs. Backward protocol model Forward protocol model - t s is the reference z ( t s + RTT t s ) ˙ = P ( z ( t s + RTT t s ) , τ ( t s + T f ) , T ) w ( t s + RTT t s ) = h ( z ( t s + RTT t s )) RTT t s = T f + τ ( t s + T f ) + T b ◮ Not very easy to work with Backward protocol model - t r is the reference z ( t r ) ˙ = P ( z ( t r ) , τ ( g ( t r − T b )) , T ) w ( t r ) = h ( z ( t r )) τ ( g ( t r − T b )) = τ ( t r − T b − τ ( g ( t r − T b ))) ◮ Standard form for dynamical systems ◮ Implicit state-dependent delay ! C. Briat [KTH / ] 9/16
Introduction Problem Network Model Stability Analysis Conclusion and Future Works Complete network model Single-buffer/Single-user 1 τ ( t ) ˙ = c [ φ ( t ) − r ( φ ( t ) , τ ( t ))] z ( t ) ˙ = P ( z ( t ) , τ ( g ( t − T b )) , T f , T b ) w ( t ) = h ( z ( t )) φ ( t ) = Φ( w ( t ) , τ ( t ) , T b , T f ) ◮ Functional Φ converts windows sizes into flows [Jacobsson] φ ( t ) = w ( t − T f ) φ ( t ) = w ( t − T f ) T + τ ( t ) , T + τ ( t ) + ˙ w ( t − T f ) C. Briat [KTH / ] 10/16
Introduction Problem Network Model Stability Analysis Conclusion and Future Works Delays properties Propagation delays T f , T b ◮ Constant delays ◮ Bounded Queuing delay τ ( g ( t − T b )) - Single buffer case ◮ Bounded in practice ◮ Derivative belongs to ( −∞ , 1) : c � 1 − if C ( g ( t )) D + [ τ ( g )]( t ) = φ ( g ( t )) 0 otherwise ◮ Well-posedness problems do not occur ( ˙ τ ( t ) < 1 ) ◮ Many results on time-delay systems can be applied (Lyapunov-Krasovskii Theory) C. Briat [KTH / ] 11/16
Introduction Problem Network Model Stability Analysis Conclusion and Future Works Queuing delay τ ( g ( t − T b )) - Multiple buffer case ◮ Two delays: τ 1 ( g 1 ( g 2 ( t ))) and τ 2 ( g 2 ( t )) ◮ D + [ τ 1 ( g 1 ( g 2 ))]( t ) given by ( φ 1 ( g 1 ( g 2 )) + φ 2 ( g 1 ( g 2 )) − c 1 ) c 2 if C 1 ( g 1 ( g 2 )) and C 2 ( g 2 ) ( φ 1 ( g 1 ( g 2 ) + φ 2 ( g 1 ( g 2 ))) φ 3 ( g 2 ) + φ 1 ( g 1 ( g 2 )) c 1 c 1 1 − if C 1 ( g 1 ( g 2 )) and not C 2 ( g 2 ) φ 1 ( g 1 ( g 2 )) + φ 2 ( g 1 ( g 2 )) 0 otherwise ◮ May exceed one when c 2 > c 1 . C. Briat [KTH / ] 12/16
Introduction Problem Network Model Stability Analysis Conclusion and Future Works FAST-TCP FAST-TCP protocol � τ ( g ( t − T b )) � w ( t ) = γ ˙ − T + τ ( g ( t − T b )) 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): α τ ∗ = w ∗ = α (1 + T/τ ∗ ) , φ ∗ = α/τ ∗ c (1 − δ ∗ ) , ◮ Equilibrium flow independent of the propagation delay (also in the multiple users case) C. Briat [KTH / ] 13/16
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 1 τ ∗ < T, τ ∗ ( T − 1) + T 2 > 0 , γ < τ ∗ ( T − 1) + T 2 (quasipolynomials) C. Briat [KTH / ] 14/16
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 C. Briat [KTH / ] 15/16
Introduction Problem Network Model Stability Analysis Conclusion and Future Works Thank you for your attention C. Briat [KTH / ] 16/16
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