Local stability with feedback delay

4.4 Stability analysis with the discrete-time model

4.4.1 Local stability with feedback delay

A network of FAST TCP sources is modelled by equations (4.3), (4.4), and (4.7). This generalizes the model in [69] by including feedback delay. When local stability is studied, we ignore all un-congested links (links where prices are zero in equilibrium) and assume that equality always holds in (4.7).

The main result of this section provides a sufficient condition for local asymptotic sta- bility in general networks with common feedback delay.

Theorem 4.4. FAST TCP is locally stable for arbitrary networks if γ ∈ (0,1]and if all sources have the same round-trip feedback delayτi =τ for alli.

The stability condition in the theorem does not depend on the value of the feedback delay, but only on the heterogeneity among them. In particular, when all feedback delays are ignored,τ_i = 0for alli, then FAST TCP is locally asymptotically. This generalizes the stability result in [69].

Corollary 4.1. FAST TCP is locally asymptotically stable in the absence of feedback delay for general networks with anyγ ∈[0,1).

The rest of this subsection is devoted to the proof of Theorem 4.4.

We applyZ-transform to the linearized system and use the generalized Nyquist criterion to derive a sufficient stability condition. Define the forward and backwardZ-transformed routing matricesR_f(z)andR_b(z)as

(Rf(z))li :=





z^−τlif ifR_li = 1

0 ifR_li = 0 and (Rb(z))li :=





z^−τlib ifR_li = 1 0 ifR_li = 0 . The relationτ_li^f +τ_li^b =τ_igives

R_b(z) =R_f(z⁻¹)·diag(z^−τi). (4.19)

DenoteW(z),Q(z), andP(z)as the correspondingZ-transforms ofδw(t),δq(t), andδp(t) for the linearized system. Let q and w be the end-to-end queueing delay and congestion

window at equilibrium. Linearizing (4.7) yields X

i

R_li Ã

δw_i(t−τ_li^f)

d_i+q_i −w_iδq_i(t−τ_li^f) (d_i+q_i)²

!

= 0,

where the equality is used in (4.7). The correspondingZ-transform in matrix form is Rf(z)D⁻¹MW(z)−Rf(z)BQ(z) = 0, (4.20)

where the diagonal matricesB,D, andM are B :=diag

µ w_i (d_i+q_i)²

¶

, M :=diag µ d_i

d_i+q_i

¶

, andD:=diag(d_i).

SinceR_f(z)is generally not a square matrix, we cannot cancel it in (4.20).

Equation (4.3) is already linear, and the correspondingZ-transform in matrix form is

Q(z) =R_b(z)^TP(z). (4.21)

By combining (4.20) and (4.21), we obtain



 I −R^T_b(z) Rf(z)B 0







 Q(z) P(z)



=



 0 Rf(z)D⁻¹M



W(z).

Solving this equation with block matrix inverse gives the transfer function from W(z)to Q(z)

Q(z)

W(z) =R^T_b(z)(R_f(z)BR^T_b(z))⁻¹R_f(z)D⁻¹M.

TheZ-transform of the linearized, congestion window update algorithm is zW(z) =γ(MW(z)−DBQ(z)) + (1−γ)W(z).

By combining the above equations, we get the open-loop transfer functionL(z)fromW(z)

toW(z)as L(z) = −¡

γ¡

M −DBR^T_b(z)(R_f(z)BR^T_b(z))⁻¹R_f(z)D⁻¹M¢

+ (1−γ)I¢ z⁻¹.

A sufficient condition for local stability can be developed based on the generalized Nyquist criterion [23, 31]. Since the open-loop system is stable, if we can show that the eigenvalue loci of L(e^jw) does not enclose −1 for ω ∈ [0,2π), the closed-loop system is stable.

Therefore, if the spectral radius ofL(e^jw)is strictly less than 1 forω ∈[0,2π), the system will be stable.

When z = e^jw, the spectral radii ofL(z) and−zL(z)are the same. Hence, we only need to study the spectral radius of

J(z) : = γ(M−DBR^T_b(z)¡

Rf(z)BR^T_b(z)¢₋₁

Rf(z)D⁻¹M + (1−γ)I.

Clearly, the eigenvalues ofJ(z)are dependent onγ. For any givenz =e^jω, let the eigen- values of J(z) be denoted by λi(γ), i = 1. . . N, as functions of γ ∈ (0,1]. It is clear that

|λ_i(γ)| = |γλ_i(1) + (1−γ)| ≤γ|λ_i(1)|+ (1−γ).

Hence if ρ(J(z)) < 1 for any z = e^jω for γ = 1, it will also hold for all γ ∈ (0,1].

Therefore, it suffices to study the stability condition forγ = 1.

Letµ_i be theith diagonal entry of matrixM withµ_i =d_i/(d_i +q_i). Denoteµ_max :=

max_iµ_i. Since the end-to-end queueing delayq_i cannot be zero at equilibrium (otherwise the rate will be infinitely large), we have q_i > 0 and µ_max < 1. The following lemma characterizes the eigenvalues ofJ(z)withγ = 1.

Lemma 4.2. Whenz =e^jω withω ∈ [0,2π)andγ = 1, the eigenvalues ofJ(z)have the following properties:

1. There areLzero eigenvalues with the corresponding eigenvectors as the columns of matrixM⁻¹DBR^T_b(z).

2. The nonzero eigenvalues have moduli less than1ifτ_max−τ_min <1/4, whereτ_max= max_iτ_i andτ_min = min_iτ_i .

Proof: Atγ = 1, the matrixJ(z)is

M −DBR_b^T(z)(R_f(z)BR^T_b(z))⁻¹R_f(z)D⁻¹M.

It is easy to check that

J(z)M⁻¹DBR^T_b(z) = DBR^T_b(z)−DBR_b^T(z) = 0.

SinceM⁻¹DBR^T_b(z)has full column rank, it consists ofLlinearly independent eigenvec- tors ofJ(z)with corresponding eigenvalue 0. This proves the first assertion.

For the second assertion, suppose thatλ is an eigenvalue ofJ(z)for a givenz. Define matrixAas

A: = J(z)−λI = (M −λI)−DBR^T_b(z)(R_f(z)BR^T_b(z))⁻¹R_f(z)D⁻¹M,

which is singular by definition. Based on the matrix inversion formula (see, e.g., [62]) (J + EHS)⁻¹ =J⁻¹−J⁻¹E(H⁻¹+SJ⁻¹E)⁻¹SJ⁻¹,

ifJ +EHS is singular, then eitherJ orH⁻¹+SJ⁻¹Eis singular. We can let

J :=M −λI, E :=−DBR^T_b(z), H := (R_f(z)BR^T_b(z))⁻¹, andS :=R_f(z)D⁻¹M.

SinceA = J +EHS is singular, eitherJ = M −λI orH⁻¹ +SJ⁻¹E is singular. The second term can be reformulated intoR_f(z)(B−M(M −λI)⁻¹B)R^T_b(z).

Case 1: M −λIis singular. SinceM is diagonal, then 0< λ= di

d_i+q_i =µi ≤µmax <1.

Case 2: R_f(z)(B−M(M −λI)⁻¹B)R^T_b(z)is singular.

It is clear that

B−M(M −λI)⁻¹B =diag¡

1−µ_i(µ_i−λ)⁻¹β_i¢

=−λdiag µ βi

µ_i−λ

¶ ,

where β_i is the ith diagonal entry of matrix B. Hence, λ = 0 is always an eigenvalue, which is claimed before. Ifλis nonzero, it has to be true that

det µ

R_f(z)diag µ β_i

µ_i−λ

¶ R^T_b(z)

¶

= 0. (4.22)

Whenz =e^jω, we havez⁻¹ =z. Hence, equation (4.19) can be rewritten as R^T_b(z) =diag(z^−τi)R^T_f(z) =diag(z^−τi)R^∗_f(z).

Substituting the above equation into (4.22) withz =e^jω yields det

µ

R_f(z)diag

µe^−jωτiβi

µ_i−λ

¶ R^∗_f(z)

¶

= 0. (4.23)

Therefore, the following formula is also zero

e^{−j(ωτmax+ψ)}det µ

Rf(z)diag

µe^j(θi+ψ)β_i µ_i−λ

¶ R^∗_f(z)

¶

= 0.

whereθ_i = (τ_max−τ_i)ω, andψ can be any value. Whenτ_max−τ_min <1/4, we have 0≤θ_i = (τ_max−τ_i)ω≤π/2.

Suppose that there is a solution such that |λ| ≥ 1. Based on Lemma 4.3, which will be presented later, there exists a ψ s.t. Im(diag¡

e^j(θi+ψ)β_i/(µ_i−λ))¢

is a positive diagonal matrix. Therefore the imaginary part of matrix R_f(z)diag¡

e^j(θi+ψ)β_i/(µ_i−λ))¢ R^∗_f(z) is positive definite, and the real part is symmetric. From Lemma 4.4 below, it has to be nonsingular. This contradicts the equation

det µ

R_f(z)diag

µe^j(θi+ψ)β_i µ_i−λ

¶ R^∗_f(z)

¶

= 0.

Hence, we have|λ|<1.

The proof of Theorem 4.4 will be complete after the next two lemmas.

Lemma 4.3. Suppose that0< µ_i <1and0≤θ_i < π/2. If|λ| ≥1, there exists aψsuch that

Im

µe^j(θi+ψ)β_i µ_i−λ

¶

>0 for i= 1. . . N.

Proof: See Appendix 4.6.3.

Lemma 4.4. If the real part of a complex matrix is symmetric, and the imaginary part is positive definite, then the matrix is nonsingular.

Proof: See Appendix 4.6.4.