CSCE619-600 notes

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 
• stay in neighborhood
• 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|