mFAST: A Multipath Congestion Control Protocol for High Bandwidth-Delay Connection

Abstract

Today, the smart devices are usually equipped with more than one network interfaces. A multipath congestion control protocol that exploits different paths to transmit data will improve the throughput and high-availability. Many multipath congestion control protocols have been developed in the literature. However, most of them are loss-based algorithms, hence they do not well utilize the bandwidth in high bandwidth-delay product (BDP) connections due to the conservative congestion control. From the single-path Fast TCP, we develop a multipath congestion control protocol, called mFast, for high BDP connections. MFast uses queueing delay to measure the congestion as Fast TCP does. Our framework is based on a network utility maximization model for multipath flows. The features of mFast such as load-balancing, TCP friendliness, and throughput improvement are verified via analysis and extensive simulations.

Appendix A

1.1 Proof of theorem 2

In this proof, we indicate a subflow by indexing $r \in \mathcal F$ for convenience of representation.

Let ∂w_r(t) = w_r(t) − w_r. Linearize at the equilibrium point of the dynamic system (12)–(13) yields

$$ \partial \dot{w}_{r} = J \partial w_{r}(t), $$

(22)

where J is the Jacobian matrix. In general, if the eigenvalues of the Jacobian matrix have negative real parts, then the dynamic system is locally stable at the the equilibrium point.

Let’s denote the following matrices $A=\text {diag}\left (\frac {1}{d_{i} + q_{i}}\right )_{i \in \mathcal F}$, $B=\text {diag}(x_{i})_{i \in \mathcal F} = \text {diag}\left (\frac {w_{i}}{d_{i}+q_{i}}\right )_{i \in \mathcal F}$, and $C=\text {diag}\left (\frac {w_{i}}{(d_{i}+q_{i})^{2}}\right )_{i \in \mathcal F}$. We have C = AB. Let’s denote H be the Hessian matrix of U(x). H is a negative definite matrix since U(x) is strictly concave.

At the equilibrium point, we have

$$ \frac{\partial \dot{w}_{i}}{\partial w_{k}} = \gamma x_{i} \left( \sum\limits_{j \in \mathcal F} H_{ij}\frac{\partial x_{j}}{\partial w_{k}} - \frac{\partial q_{i}}{\partial w_{k}}\right), \forall i, k \in \mathcal F, $$

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in which

$$\begin{array}{@{}rcl@{}} \frac{\partial x_{i}}{\partial w_{i}} & =& \frac{1}{d_{i} + q_{i}} - \frac{w_{i}}{(d_{i} + q_{i})^{2}} \frac{\partial q_{i}}{\partial w_{i}}, \forall i \in \mathcal F, \\ \frac{\partial x_{i}}{\partial w_{k}} & =& -\frac{w_{i}}{(d_{i} + q_{i})^{2}} \frac{\partial q_{i}}{\partial w_{k}} \forall i, k \in \mathcal F, i \neq k. \end{array} $$

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Hence,

$$ J = \gamma B\left( H\left( A-B\frac{\partial \boldsymbol q}{\partial \boldsymbol w}\right) - \frac{\partial \boldsymbol{q}}{\partial \boldsymbol{w}}\right). $$

(25)

By a similar transformation as in [7], we have

$$ \frac{\partial \boldsymbol q}{\partial \boldsymbol w} = R^{T}(RCR^{T})^{-1}RA. $$

(26)

Therefore,

$$ J = \gamma B(H-(HC+I)R^{T}(RCR^{T})^{-1}R)A. $$

(27)

Formula (27) implies

$$\begin{array}{@{}rcl@{}} && \frac{1}{\gamma}B^{-1}JA^{-1}CR^{T} \\ && \qquad = HCR^{T}-(HC+I)R^{T}(RCR^{T})^{-1}RCR^{T} \\ && \qquad = HCR^{T}-(HC+I)R^{T} = -R^{T} \end{array} $$

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Since C = AB, we obtain

$$ (J+\gamma I)BR^{T} = 0. $$

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If the rank of R equals to the number of subflow (full-column rank), then the matrix BR^T has |F| independent vectors. They are also the eigenvectors of J + γI. Hence, all the eigenvalues of J equals to − γ, which implies that the dynamic system is locally asymptotically stable.

1.2 Proof of theorem 3

In the proof, we will use the following lemmas:

Lemma 1

[19, p.53] X and X are two square matrices. Eigenvalues of YX are same as XY.

Lemma 2

[19, p.465] X is a positive definite matrix, Y is a symmetric matrix. XY is a diagonalizable matrix whose eigenvalues are real. Moreover, the matrix XY has the same number of positive, negative, and zero eigenvalues as Y.

Let’s consider the matrix

$$ J_{1} = (H-(HC+I)R^{T}(RCR^{T})^{-1}R)C. $$

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From Lemma 1, γJ₁ and J have the same set of eigenvalues (C = AB). Therefore, we need to prove the real parts of all the eigenvalues of J₁ are negative (0 < γ < 1), or in other words, given J₁ − λI singular, we need to prove λ has a negative real part.

The Woodbury’s formula is stated as follows: if X, Y, X^{− 1} + V Y ^{− 1}U are non-singular, then

$$ (X+UYV)^{-1} = X^{-1} - X^{-1}U(Y^{-1}+VX^{-1}U)^{-1}VX^{-1}. $$

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Let’s denote X = HC − λI, Y = (RCR^T)^{− 1}, U = −(HC + I)R^T, and V = RC. We have J₁ − λI = X + UY V. Hence, X or Y ^{− 1} + V X^{− 1}U must be singular, otherwise, X + UY V is non-singular.

• 1) If X = HC − λI is a singular matrix, then λ is one of the eigenvalues of HC. Lemma 2 implies that λ is real and negative.

• 2) If HC − λI is non-singular, then X^{− 1} + V Y ^{− 1}U must be a singular matrix.

  $$\begin{array}{@{}rcl@{}} && X^{-1}+VY^{-1}U \\ && \qquad = RCR^{T} - RC(HC-\lambda I)^{-1}(HC+I)R^{T} \\ && \qquad = R(C - C(HC-\lambda I)^{-1}(HC+I))R^{T} \\ && \qquad = R(C - CX^{-1}(X +(\lambda + 1)I)R^{T}) \\ && \qquad = (\lambda + 1)R(CX^{-1})R^{T} \\ && \qquad = (\lambda + 1)R(H - \lambda C^{-1})^{-1}R^{T}. \end{array} $$

If λ = − 1, then λ is real and negative. Otherwise, R(H − λC^{− 1})^{− 1}R^T must be singular.

H − λC^{− 1} is a block diagonal matrix in which each block corresponds to a flow. We analyze a specific block associated with multipath flow s. Assuming that s has N subflows with subflow rates x_s,1,…,x_s,N. $x_{s} = {\sum }_{n = 1}^{N} x_{s,n}$. Let’s denote $\theta _{s,k} = \frac {x_{s,k}}{x_{s}}$ and $g_{s,k} = \frac {(1-\epsilon )\alpha _{s}\theta ^{2}_{s,k}}{\epsilon \alpha _{s} + \lambda w_{s,k}}$, for all k ∈ s.

$$\begin{array}{@{}rcl@{}} &&H_{s} - \lambda C_{s}^{-1} = \\ &&\left[\begin{array}{cccc} \!-\frac{(1-\epsilon)\alpha_{s}}{{x_{s}^{2}}} -\frac{\epsilon\alpha_{s}}{x_{s,1}^{2}} & -\frac{(1-\epsilon)\alpha_{s}}{{x_{s}^{2}}} & {\ldots} & -\frac{(1-\epsilon)\alpha_{s}}{{x_{s}^{2}}} \\ -\frac{(1-\epsilon)\alpha_{s}}{{x_{s}^{2}}} & \!\!-\frac{(1-\epsilon)\alpha_{s}}{{x_{s}^{2}}} -\frac{\epsilon\alpha_{s}}{x_{s,2}^{2}} & {\ldots} & -\frac{(1-\epsilon)\alpha_{s}}{{x_{s}^{2}}} \\ {\ldots} &{\ldots} &{\ldots} &{\ldots} \\ -\frac{(1-\epsilon)\alpha_{s}}{{x_{s}^{2}}} & -\frac{(1-\epsilon)\alpha_{s}}{{x_{s}^{2}}} & {\ldots} & \!-\frac{(1-\epsilon)\alpha_{s}}{{x_{s}^{2}}} - \frac{\epsilon\alpha_{s}}{x_{s,N}^{2}} \end{array}\right] \\ &&= -\frac{(1-\epsilon)\alpha_{s}}{{x^{2}_{s}}} \left( \text{diag}\left( \frac{1}{g_{s,k}}\right)_{k \in s} + \mathbf{1}\mathbf{1}^{T}\right) \\ &&= -\frac{(1-\epsilon)\alpha_{s}}{{x^{2}_{s}}} (G_{s} + \mathbf{1}\mathbf{1}^{T}), \end{array} $$

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where 1 is an all-ones column vector, and $G_{s} = \text {diag}\left (\frac {1}{g_{s,k}}\right )_{k \in s}$.

From Sherman-Morrison’s formula,

$$\begin{array}{@{}rcl@{}} (H_{s} - \lambda C_{s}^{-1})^{-1} & \! = & -\frac{{x^{2}_{s}}}{(1 - \epsilon)\alpha_{s}}\left( G_{s} + \mathbf{1}\mathbf{1}^{T}\right)^{-1} \\ &\! = & -\frac{{x^{2}_{s}}}{(1 - \epsilon)\alpha_{s}}\left( G_{s}^{-1} - \frac{G_{s}^{-1}\mathbf{1}\mathbf{1}^{T} G_{s}^{-1}}{1 + \mathbf{1}^{T} G_{s}^{-1}\mathbf{1}}\right). \end{array} $$

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In the special case in which there is a multipath flow f₁ and a single-path flow f₂ on a bottleneck link (Fig. 2), the routing matrix R becomes an all-ones row vector and R(H − λC^{− 1})^{− 1}R^T is a scalar number. Moreover, x₁ = Nx_1,k and w_1,k = w_1,1 for all k ∈ f₁. We obtain

$$\begin{array}{@{}rcl@{}} && R(H - \lambda C^{-1})^{-1}R^{T} = \\ && \quad - \frac{{x^{2}_{1}}}{(1-\epsilon)\alpha_{1}}\frac{{\sum}_{k \in f_{1}} g_{1,k}}{1 + {\sum}_{k \in f_{1}} g_{1,k}} - \frac{{x^{2}_{2}}}{(1-\epsilon)\alpha_{2}}\frac{g_{2}}{1 + g_{2}}, \end{array} $$

where $g_{2} = \frac {(1-\epsilon )\alpha _{2}}{\epsilon \alpha _{2} + \lambda w_{2}}$. Solving the equation R(H − λC^{− 1})^{− 1}R^T = 0 for λ yields

$$ \lambda = -\frac{ \frac{(1+(N-1)\epsilon) \alpha_{1}}{{x_{1}^{2}}} + \frac{\alpha_{2}}{{x_{2}^{2}}}}{N\frac{w_{1,1}}{{x_{1}^{2}}} + \frac{w_{2}}{{x_{2}^{2}}}}. $$

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We have λ < 0, hence, the dynamic system of one-bottleneck link is locally stable at the equilibrium point.