Congestion Control
- Fundamental
- Design controllers for each r(n) so that R(n) converges
- Each flow can only rely on its loal state
- Controller model
- Differential equation: dy(t)/dt = F(y(t), f(t), t)
- Recurrences: y(n+1) = y(n) + F(y(n), f(n), n)
- Stationarity
- Continuous
- dx(t)/dt = F(x(t))
- stationary point: F(x*) = 0
- Discrete:
- x(n+1) = F(x(n))
- fixed point: F(x*) = x*
- Corollary
- if a system does not have stationary points, it either ocscillates or diverges
- Stability
- Whether the system returns back to the stationary state after a small disturbance or not
- Lyapunov stability theory
- Lyapunov-stable
- quasi-asymptotically stable
- converge to stationary point
- asympototically stable
- both Lyapunov and quasi-asympototially stable
- Stability-analysis
- Solving differential equations
- Graphical analysis
- Lemma
- If the system converges to a stationary point monotonically, it is asymptotically stable
- For linear systems, local stability implies global stability
- Comparison
- Discrete systems are usually more strict in their requirements for stability because continuous systems make infinitely small steps and avoid drastic jumps of discrete systems
- Linear Stability
- Model
- A system of N equations: dx(t)/dt = Ax(t) + b
- A is non singular i.e. det(A) ≠ 0, there is no linear dependency between the rows.
- y(t) = x(t) + A^-1b
- dy(t)/dt = Ay(t)
- Eigenvalue and eigenvector
- Av = λv, λ is eigenvalue, v is eigenvector, Mi is set of linearly independent engienvectors corresponding to λi
- det(A - λI) = 0
- P(λ) = det(A-λI) = (λ-λ1)^m1...(λ-λk)^mk, sum(mi) = N
- M = (M1,...,Mk) = (v1,...,vl)
- if A has N different eigenvalues, l = N
- Determine the stability
- Continuous system
- Solution: $z(n) = \sum_{i=1}^{N}{c_iv_ie^{\lambda_it}}$
- r = max Re(λi)
- Discrete system
- Solution: $z(n) = \sum_{i=1}^{N}{e_iv_i\lambda_i^n}$
- spectral radius max|λi|
