A Large Family of Multi-path Dual Congestion Control Algorithms

# A Large Family of Multi-path Dual Congestion Control Algorithms

###### Abstract

The goal of traffic management is efficiently utilizing network resources via adapting of source sending rates and routes selection. Traditionally, this problem is formulated into a utilization maximization problem. The single-path routing scheme fails to react to instantaneous network congestion. Multi-path routing schemes thus have been proposed aiming at improving network efficiency. Unfortunately, the natural optimization problem to consider is concave but not strictly concave. It thus brings a huge challenge to design stable multi-path congestion control algorithms.

In this paper, we propose a generalized multi-path utility maximization model to consider the problem of routes selection and flow control, and derive a family of multi-path dual congestion control algorithms. We show that the proposed algorithms are stable in the absence of delays. We also derive decentralized and scalable sufficient conditions for a particular scheme when propagation delays exist in networks. Simulations are implemented using both Matlab and NS2, on which evaluation of the proposed multi-path dual algorithms is exerted. The comparison results, between the proposed algorithms and the other two existing algorithms, show that the proposed multi-path dual algorithms with appropriate parameter settings can achieve a stable aggregated throughput while maintaining fairness among the involved users.

Dynamic routing, flow control, stability, scalable TCP.

## I Introduction

The Transmission Control Protocol and the Internet Protocol, known as TCP/IP, are widespread for guiding traffic flows in the Internet. In a packet-switch network, a route is computed and selected to send packets from a source user to a destination user, and the sending rate is determined by TCP. Traditionally, a single-path routing scheme is deployed, where the shortest path is chosen by IP routing in terms of hop count or distance, and the flow rate is varied according to congestion level along that path. Ideally, both routes and flow rates should be guided to guarantee the efficiency and fairness in link bandwidth utilization. There thus has long been a desire to direct routes selection and rates variation according to congestion level. However, studies, e.g. [14], have shown that making paths selection consistent with congestion level may result in network occlusions and routing instability. Despite that IP routing is highly scalable, the static or single-path routing scheme fails to react to instantaneous network congestion.

Motivated by the applications in ad-hoc networks and overlay TCP, recently there have been more interests in multi-path routing scheme [21] [22]. In that scheme, packets belonging to the same source-destination pair are transmitted along several routes between them instead of a single path. Notice that these routes might not be disjoint. In order to take advantage of multi-path routing scheme, network users prefer to select the best path among the available routes in terms of high throughput or low latency. However, Wang et al. [15] has shown the instability arose from such interaction between network users and providers, causing barriers to the deployment of this dynamic routing scheme in packet-based networks.

Many researchers devote to find a protocol that can be implemented in a decentralized way by source and routers, and controls the system to a stable equilibrium point which satisfies some basic requirements: high utilization of resources, small queues, and a degree of control over resource allocation. All of these are required to be scalable, i.e., hold for an arbitrary network, with possibly high capacity and delay.

A major difficult for the multi-path congestion control is that the natural optimization problem to consider is concave but not strictly concave. It makes that there are possible existence of multiple equilibriums. Thus, researchers resort to the duality of the primal problem. One attractive consequence of the dual algorithm is that they naturally have equilibrium points which make full use of the limited bandwidth available, while still achieving a notion of fairness between users. Voice [20] is the first considering the stability of multi-path dual algorithm. The method that extended a single-path result to the multi-path case was used in [18] for a primal congestion control algorithm. However, the fairness among different users is not well considered in [20].

In this paper, we propose a generalized multi-path utility maximization model, which is strictly concave and ensure equilibria satisfying desirable static properties. Then we derive a family of multi-path dual congestion control algorithms. We show that the proposed algorithms are stable in the absence of delays, based on which we derive decentralized and scalable sufficient conditions for a particular scheme when propagation delays exist in the networks.

The main contributions of our work can be stated as follows:

1)We propose a generalized multi-path utility maximization model, which can reduce to specific models with different parameter settings. A family of multi-path dual congestion control algorithms derived from the above model can both fully utilize resources under limitation and achieve stability in the presence of propagation delays in network, while maintaining fairness among different users.

2)We implement both rate-based and window-based simulations respectively using Matlab and NS2, respectively. To validate the efficiency of the proposed algorithms, a comparison is made between the proposed algorithm and the ones in [18] and [20] under NS2. Results show that the proposed multi-path dual algorithms outperform the later ones in optimal and stable aggregated throughput under an appropriate value of the average window size.

The remainder of this paper is organized as follows. Related work is briefly reviewed in Section II. In Section III, we present the proposed multi-path utility maximization model. Both stability in the absence of delays and in the presence of delays are exhibited in Section IV and Section V, respectively. Following that is the simulation results in Section VI. This paper is finally concluded in Section VII.

## Ii Related Work

In recent years theoreticians have developed a framework that allows a congestion control algorithm such as Jacobson’s TCP to be interpreted as a distributed mechanism solving a global optimization problem: for reviews see [4] and [6]. The framework is based on fluid-flow models, and the form of the optimization problem makes explicit the equilibrium resource allocation policy of the algorithm, which can often be restated in terms of a fairness criterion. And the dynamics of the fluid-flow models allow the machinery of control theory to be used to study stability, and to develop rate control algorithms that scale to arbitrary capacities. The equilibrium and dynamic properties for the related congestion control algorithm based on this framework are summarized in Table I.

These algorithms can be classified into two major groups, i.e., primal algorithms and dual algorithms. In general, the equilibrium point of the algorithm solves the primal (or original) problem, an approximation problem or the relaxed problem (where the capacity constrain is replaced by penalties) respectively [4].

For the single-path case, Vinnicombe [10] derived decentralized and scalable stability conditions for a fluid approximation of a class of Internet-like communications networks operating a modified form of TCP-like congestion control. Dual algorithms are classed two groups, i.e. delay-based and fair dual algorithm [12]. The delay-based dual algorithms allowed a natural interpretation of the link price as either a real or virtual queueing delay [4], [7], [9], [11]. It was, however, difficult to reconcile fairness with stability. Kelly [12] design a class of fair dual algorithm, which can achieve weighted -fairness, and have straightforward delay and stochastic stability properties .

In multi-path case, there are possible existence of multiple equilibriums because that the natural optimization problem to consider is concave but not strictly concave. Meanwhile, when one attempts to use a duality approach, the dual problem may not be differentiable at every point [1]. To circumvent these difficulties, Lin et al. [17] used ideas from proximal point algorithms. Han et al. [19] modified the utility function to ensure a unique equilibrium point and generalized the algorithm for the case of single-path [10] to multi-path. Kelly at al. [18] improved on the results obtained by Han et al. [19], and present an algorithm with a sufficient condition for local stability that is decentralized in the stronger sense that the gain parameter for each route is restricted by the round-trip time of that route. However, the majority of above research focuses on extensions of the primal algorithms proposed by Kelly et al. [4], a class of single-path primal congestion controls. The primal algorithms exhibit a trade-off between rate of convergence and bandwidth utilization at equilibrium since that the desired equilibrium point only solves the relaxed problem.

## Iii A Generalized Multi-path Utility Maximization Model

First we will give the network model and propose a generalized multi-path utility maximization model. Then approximation error and dual problem of the generalized model will be presented.

### Iii-a Network Model

We suppose that the network comprises an interconnection of a set of sources , with a set of resources . Each source identifies a unique source-destination pair. Associated with each source is a collection of routes, each route being a set of resources. If a source transmits along a route , then we write . For a route , we let be the (unique) source such that . We let denote the set of all routes. In the following, we use notations and to denote the cardinalities of sets and respectively.

In our model, a route has associated with it a flow rate , which represents a dynamic fluid approximation to the rate at which the source is sending packets along route at time .

For each route and resource , let denote the propagation delay from to , i.e. the length of time it takes for a packet to travel from source to source along route . Let denote the propagation delay from to , i.e. time it takes for congestion control feedback to reach from resource along route . In the protocols to be considered, a packet must reach its destination before an acknowledgement containing congestion feedback is returned to its source. Further, we assume queueing delays are negligible. Thus for all , , the round trip time for route .

The notation denotes that if and if . We abuse notations to use to denote and to use to denote for any and such that .

### Iii-B A Generalized Multi-path Utility Maximization Model

A utility function is associated to each source , which is an increasing, strictly concave and continuously differentiable function of over the range . And if . As an example, suppose that

 Us(ys)=⎧⎨⎩wsy1−αs1−α,α≠1wslogys,α=1 (2.1)

for , so that the resource shares obtained by different sources are weighted -fair [5]. When , the cases and correspond respectively to an allocation which achieves maximum throughput, is proportionally fair or is max-min fair [5]. TCP fairness, in the case where each source has just a single route, corresponds to the choice with the reciprocal of the square of the (single) round trip time for source [6]. Define the demand function , a continuous, strictly decreasing function. The demand functions derived from the class of utility functions defined in (2.1) is

 Ds(λs)=(wsλs)1/α. (2.2)

For the convenience of making analysis, first we introduce routing matrix to succinctly express the relationships between routes and resources. Let if , so that resource lies on route , and set otherwise. This defines a 0-1 matrix . The aggregate rate for sources is . Since we wish the total network utility to be high, it is desirable for a congestion control algorithm to asymptotically solve the classical Kelly formulation:

 maximizex≥0∑s∈SUs(∑r∈sxr)subject toAx≤c, (2.3)

where with being the capacity of resource . Note, even if is strictly concave, the whole objective function is not, due to the linear relationship in .

A generalized model for the multi-path utility maximization problem (2.3) is to

 maximizex≥0∑s∈SUs(uqs)subject to us≤γ∑r∈sx1qr+(1−γ)y1qs∑r∈sxr=ys, s∈SAx≤c, (2.4)

where and . Given , (2.4) reduces to the one proposed by Voice in [20] with and to (2.3) with . The motivation for such formulation is that (2.4) can reduce to the classical Kelly formulation (2.3) and the one in [20] under different parameter settings. The advantages of formulation (2.4) are two folds, we can not only provide a direct insight into the reason for the stability of the dual congestion control with respective to previous work in [4] and [7], but also avoid choosing sufficient large parameter to approximate (2.3) with respective to the work in [20], which implies large risk of numerical instability.

To ensure that the objective function is strictly concave, we make Assumption H: For each is a strictly decreasing function of . This is true for the weighted fairness utility function (2.1) if an appropriate is chosen, such as , where and .

### Iii-C Approximation Error

To solve the non-strict concave of the objective function in (2.3) with , only the fraction of the power of aggregate rate is substituted by in problem (2.4). We can bound how far the solution to (2.4) is from maximizing aggregate user utility.

###### Lemma 1 (Approximation error)

Let be any optimal solution to (2.3), and be the optimal solution to (2.4). We have

 ∑s∈SUs(∑r∈sx′r)≥∑s∈SUs(∑r∈sxr) (2.5)

and

 ∑s∈SUs(eγ∑r∈sxr)≥∑s∈SUs(∑r∈sx′r), (2.6)

where error factor and denotes the number of route serving for source .

Proof: It’s obvious that is feasible for (2.3). So the inequality (2.5) is followed by the optimality of to (2.3).

For and , we have with the famous Hölder inequality [3]. Combining it with the facts that and increasing, we have

 eγ∑r∈sxr≥uqs. (2.7)

Now let . It’s obvious that is feasible for (2.4). By the optimality of to (2.4), we have

 Us(uqs)≥Us(u′sq). (2.8)

Since is a subadditive function and is increasing, we have

 u′sq≥∑r∈sx′r (2.9)

for . Combing the inequalities (2.7), (2.8), (2.9) and the fact that is increasing, we get the desired result (2.6).

Remark 1: It can be verified that . The error factor is increasing with . And the facts that and hold. So for given . The lemma 1 in [20] is a special case of Lemma 1 here with .

### Iii-D Dual Problem of the Generalized Model

Given vectors and , let . The Lagrangian of (2.4) is . For any , we

 maximizeUs(uqs)−∑r∈s(λr−νs)xr−νsyssubject tous≤γ∑r∈sx1qr+(1−γ)y1qsxs≥0, (2.10)

where is the rate vector for source . We denote the optimal objective function value of (2.10) by . It can be verified that is finite only if for all and . Otherwise, . In the following derivation, we assume for all and .

The Karush-Kuhn-Tucker condition [1] for (2.10) is

 qU′s(uqs)uq−1s−ηs=0 (2.11a) γ(ηs/q)x−1pr−(λr−νs)=0,r∈s (2.12a) (1−γ)(ηs/q)y−1ps−νs=0 (2.13a) us=γ∑r∈sx1qr+(1−γ)y1qs. (2.14a)

Let , i.e., . By (2.11a), we have

 ηs/q=U′s(¯ys)¯y1ps. (2.15)

By (2.12a) and (2.13a), we get

 x1pr=γηs/qλr−νs, y1ps=(1−γ)ηs/qνs. (2.16)

Substituting (2.16) into (2.14a), we obtain

 ¯yp−1ps=(ηs/q)p−1(γp∑r∈s(λr−νs)1−p+(1−γ)pν1−ps).

Plugging (2.15) into this equation, we get

 U′s(¯ys)=(γp∥\boldmathλs−νs∥1−p1−p+(1−γ)pν1−ps)11−p.

Then, by the definition of the demand function (2.2), we have

 ¯ys=Ds((γp∥\boldmathλs−νs∥1−p1−p+(1−γ)pν1−ps)11−p). (2.17)

By (2.15), we obtain . And by (2.16), we have that and , where and is defined in (2.17).

Finally, the Lagrangian dual problem of (2.4) is

 minimize\boldmathμ≥0∑s∈SWs(% \boldmathλs,νs)+cT\boldmathμ. (2.18)

The control laws that we will propose can be viewed as decentralized dual algorithms to solve problem (2.18) and the primal problem (2.4) simultaneously.

## Iv A Large Family of Multi-path Dual Congestion Control Algorithms

In this section, we will derive a large family of multi-path dual congestion control algorithms from the generalized multi-path utility maximization model. The stability with the absence of delay will be stated later.

### Iv-a Multi-path Dual Congestion Control Algorithms

In Section III, a generalized utility model have been formulated, which can reduce to specific models with different parameters. We are now in a position to state a family of multi-path dual algorithm, which can be described as follows.

For all resources

 ddtμj(t)=κj(μj(t))(zj(t)−cj)+μj(t) (3.1)

for some positive function , and for all sources ,

 ddtνs(t)=κs(νs(t))(ys(t)−∑r∈sxr(t)) (3.2)

for some positive function , where

 zj(t)=∑j∈rxr(t),xr(t)=¯ys(r)(t)(γU′s(r)(¯ys(r)(t))λr(t)−νs(r)(t))p, (3.3)
 ys(t)=¯ys(t)((1−γ)U′s(¯ys(t))νs(t))p,λr(t)=∑j∈rμj(t), (3.4)

and

 ¯ys=Ds((γp∥\boldmathλs(t)−νs(t)∥1−p1−p+(1−γ)pνs(t)1−p)11−p). (3.5)
###### Theorem 1

Let be an equilibrium point of the system (3.1)-(3.2), and let be defined through (3.3)-(3.5). Then solves the dual problem (2.18). And is unique and solves the primal problem (2.4), where denotes .

###### Lemma 2

Let and . Then the objective of (2.18) is differentiable with derivative and where and is defined by (2.17) .

Proof: The objective function of (2.4) is strictly concave; hence, the objective function of (2.18) is convex and differentiable with and , where and solve problem (2.10) [1]. By the definition of the system (3.1)-(3.5), we get the desired results.

### Iv-B Global Stability

Assume that the matrix has full row rank and is strictly decreasing. This condition and Assumption H are sufficient to deduce that the system (3.1)-(3.5) has a unique equilibrium point. The first is the general assumption to make sure the unique equilibrium point when analyzing the dual congestion control algorithms, such as [9], [12] and [20]. Note, the assumption is needed only for the links that would be a bottleneck; so our assumption is quite generic. Following is the first main result of this paper with a strict proof, where the proof is completely different from the heuristic one given in [20].

###### Theorem 2

Given the system defined by (3.1)-(3.5). Then the unique equilibrium point is globally asymptotically stable.

The proof of Theorem 2 can be found in Appendix.

## V Delay Stability

As transmission delay universally exists in network environment, in this section we will present a particular scheme, where the proposed algorithms can achieve stability in the presence of propagation delays.

### V-a Choice of Scheme

When we include propagation delays, we get the following algorithms, for which we can provide scalable, decentralized stability conditions.

For all resources ,

 ddtμj(t)=κjμj(t)p(∑j∈rxr(t−Trj)−cj)+μj(t) (4.1)

for some positive constant , and for all sources ,

 ddtνs(t)=κsνs(t)p(ys(t)−∑r∈sxr(t−Tr)), (4.2)
 ddt¯ys(t)=qρsp(γ∑r:r∈sxr(t−Tr)1q (4.3) +(1−γ)ys(t)1q−¯ys(t)1q)

for some positive constants and , where

 xr(t)=¯ys(r)(t)(γU′s(r)(¯ys(r)(t))λr(t)−νs(r)(t))p, (4.4)
 ys(t)=¯ys(t)((1−γ)U′s(¯ys(t))νs(t))p, (4.5)

and

 λr(t)=∑j∈rμj(t−Tjr). (4.6)

Compared with (3.1)-(3.5), we set and as the dynamic gain factor for resource and source respectively. In addition, we relax the algebraic equation (3.5) to differential equation (4.3) for the existence of delay.

We first give some properties of the equilibrium point, which are useful in the proof of the main result.

###### Lemma 3

We have , where .

Proof: We have for we suppose Assumption H holds. It can be checked that

 bs=(q−1)uq−2sU′s(¯ys)+qu2(q−1)sU′′s(¯ys)=−qu2(q−1)sU′s(¯ys)as.

Combining it with nonnegative by concave with , we get the desired result.

###### Lemma 4

Let be an equilibrium point of system (4.1)-(4.3) and defined by (4.4)-(4.6). Then for each we have that

 xr¯y1ps(r)U′s(r)=γx1qrλr−νs(r) (4.7)

and

 1+νs(r)λr−νs(r)+∑j∈rμjλr−νs(r)≤2γ. (4.8)

Proof: Firstly, we have by (4.4). Then power the bothside of this equation with , we get

 x1pr=¯y1ps(r)γU′s(r)(¯ys(r))λr−νs(r). (4.9)

Multiply (4.9) with and combine the fact , we have the equation (4.7).

Similarly with (4.9), we have

 y1ps(r)=¯y1ps(r)(1−γ)U′s(r)(¯ys(r))νs(r) (4.10)

from (4.5). Combine (4.9) with (4.10), we get

 νs(r)λr−νs(r)=(1γ−1)(xrys(r))1p.

Combining it with , we have . Then

 1+νs(r)λr−νs(r)+∑j:j∈rμjλr−νs(r)=1+νs(r)λr−νs(r)+λrλr−νs(r)=2+2νs(r)λr−νs(r)≤2γ.

### V-B Main Result

We now turn to our main concern, the local stability of the system (4.1)-(4.6).

Define a link to be almost saturated if at which both and condition

 ∑r:j∈rxr=cj,j∈J (4.11)

holds. We thus rule out the possible degeneracy that both terms in the product (4.1) might vanish.

###### Theorem 3

Let be an equilibrium point of system (4.1)-(4.6) with no almost saturated links, and suppose that for all ,

 κj∑j∈rxrTr<γπ4, (4.12)

and for all ,

 κs∑r∈sxrTr<γπ4 (4.13)

and

 ρsas∑r∈sx1qrTr<π4. (4.14)

Then there exists a neighborhood of such that for any initial trajectory with lying within the neighborhood , converge as to the solution to the optimization problem (2.18) and converge as to the solution to the optimization problem (2.4), where .

The proof of Theorem 3 can be found in Appendix.

### V-C Result for α-fair Utility Function

Corresponding to weighted -fair utility function, defined by equation (2.1), the defined in Lemma 3 becomes . Then the local stability condition (4.14) reduces to

 ρs(αp−1)∑r∈sx1qrTr¯ys

Since and , the conditions (4.12), (4.13) and

 ρs(αp−1)∑r∈sxrTrys

are sufficient to derive the local stability for this special case.

These conditions are attractive because they are local and decentralized. Let the maximize available rate for source be . They lead to a highly scalable parameter choice scheme: each source and link chooses their gain parameters to be and for some , where is the average round trip time of packets transmitted by source and is the average round trip time of packets passing through resource . As a desirable feature, the gain parameters in the proposed algorithms can be derived from local information only. Independence of state information in networks leads conditions for delay stability to be scalable and decentralized.

Remark 2: If , the conditions (4.12) and (4.15) reduce to the sufficient ones determined by Voice [20] except they use an estimation of the average round trip time of packets transmitted by source . The delay stability condition for single-path fair dual algorithm given in [12] has minor difference from these condition, which was derived by linearizing the system about the flow rate .

## Vi Simulation Experiments

We will further investigate the proposed algorithm by simulation experiments, which mainly focus on two aspects:

a) To evaluate the performance of the proposed algorithm in terms of stability and convergence rate. We implement our algorithm in Matlab and explore the influence of parameters on these two features.

b) To study the performance of the proposed algorithm in network environment. We use NS2 to implement the proposed algorithm in a window-based network environment and make performance comparison with the other existing two algorithms.

In the sequel, we take the utility with the form of (2.1) with and for all .

### Vi-a Matlab Simulation

To really achieve the best tradeoff between stability and convergence one has to carefully select a few parameters. To make our work really usefully for network designers and administers, we provide some discussion and rules of thumb to select the values. We use Matlab to implement the proposed algorithm and investigate how parameters affect the algorithm’s performance. The network topology used is Abilene backbone network, shown in Fig.1 (a), which have fourteen 100-Mb/s links with 2-ms delay. The discussion explain the intuitive meaning of changing a parameter in each direction.

First, we select a set of with fixed values, and vary the gains to explore the trade-off between the convergence rate and stability; Then, we vary the values of parameter within the interval , and study how to tune the gains to make the system achieve the optimal performance.

Stability and convergence rate. First, we set and observe how the gains and to trade off between the convergence rate and the stability.

For simplicity, we first fix link gain to be , and study the effect of source gain on system performance. Here we run the simulation with three different values , and respectively. In each simulation we set step to 5ms and run the simulation for 50s. The results is shown in Fig.2 (a), where axis represents the aggregate throughput of four routes. All simulations with three different parameters will achieve global stability roughly at 5s. We notice that, however, when , the coverage is slightly slow. And when , there exists big oscillation before stability reaches. It seems that is the best choice among these three values of .

And then we fix source gain to be to explore the influence of link gain on system performance. We run simulations with three different values , and respectively. Each simulation runs 50s with step of 5ms. The simulation results are illustrated in Fig.2 (b), where the trends of the aggregate throughput are similar to those in Fig.2 (a). Simulation with a larger value of link gain converges faster and meanwhile experiences more severe oscillation. We notice that, with and , the algorithm can achieve a better trade-off between the stability and convergence rate.

Approximation Parameters. In the proposed algorithms, there exist two parameters, and which control the error factor in Lemma 1. As also exists in the algorithm proposed by Voice [20], here we only consider the influence of on system performance.

As shown in Fig.2 (b), simulation with link gain converges fast, we thus set and in the following simulations. We run simulations with varying from 0 to 1, and explore how source gain should react to different values of in order to achieve the optimal performance. Each simulation runs with step = 5ms and lasts for 50s.

The simulation results exhibit in Fig. 3. From Fig.3 (a), we notice that parameter will influence the characteristics of both convergence and oscillation. Simulation with a smaller value of experiences greater oscillation before achieving stability. The value of should decrease with the increase of to guarantee the simulation converges. Besides, the convergence throughput, at which the algorithm achieve global stability, is influenced by the choice of . Fig.3 (b) shows that a smaller value of will lead to a smaller gap between the optimal throughput and the one obtained with the proposed algorithm.

### Vi-B NS2 Simulation

The proposed congestion control algorithm is derived from a rate-based model. TCP, however, is a window-based control protocol. Now we use NS2 to implement TCP-Reno for further investigation.

The topology used is shown in Fig.1 (b). Here we consider the case of three source-destination pairs, namely from 1 to 4, 1 to 2 and 1 to 3, and each link-capacity along the route is set to 100Mb/s with a one way propagation delay of 2ms. In simulation, each source-destination pair is attached to multiple routes. For example, pair 1 and 2 has two routes, from 1 to 2 directly and from 1 to 2 via 3.

1)Stability and convergence rate According to the rate-based simulation, we implement the proposed algorithm with the parameters shown in Table II, which guarantee the algorithm achieving its optimal performance. The simulation lasts for 5 seconds and 7 TCP connections start at the same time and during the whole simulation.

The performance of the proposed algorithm in terms of each source-destination’s aggregated throughput and the whole network’s aggregated throughput is shown in Fig.4. As is shown, the aggregated throughput with the proposed algorithm achieves equilibrium at about 1.5s.

2)Performance comparison with different values of Now we vary the values of within the interval [0,1] and achieve the optimal performance. Throughput of the three source-destination pairs is plotted in Fig.5 (a) (e), where is set to 0.2, 0.4, 0.6, 0.8, 1.0 respectively.

All the three source-destination pairs can achieve equilibrium after a period of oscillation, however, with different approximate aggregated throughput when varies from 0.2 to 1.0. According to Fig.6, better consideration of fairness is shown when the equilibrium conditions are achieved if and when , there exists a trade-off among different source-destination pairs in terms of aggregated throughput. Thus, possibly, we can set to different values to meet different demands when facing different situations of the network, meanwhile remaining equilibrium conditions.

Finally, throughput for TCP connections does not make sense. Particularly when experimenting with congestion control. It counts even re-transmits as useful packets. We measure the goodput. Usually goodput can be substantially different than throughput when the congestion control makes the window oscillate. We would like to know, how our schemes with their oscillations cpmpare with the alternatives.

## Vii Conclusion

This paper considers the well known problem of joint multi-path routing and flow control, and brings new insight to congestion control algorithms. Specifically, we propose a generalized multi-path utility maximization model, and then derive a family of muti-path dual congestion control algorithm. Based on the results in this paper, one can understand the unstability of the natural muti-path dual congestion control algorithm which is a special cases in the proposed family. The one proposed in [20], a special case in this family, is at a risk to choose a sufficient large to approximation the solution of the original problem. Simulation results show that the proposed algorithm can achieve a more optimal and stable aggregate throughput with an appropriate value of the average window size than the others multi-path congestion control algorithms.

Future work is mainly focused on enriching simulation experiments of the proposed dual multi-path congestion control algorithms. The excellent performance of the proposed algorithms in real network environment is the ultimate goal we are pursuing.

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## Appendix A Appendix

The proof of Theorem 2 is presented as follows:

Proof: The proof is based on Lasalle’s invariance principle applied to a suitable Lyapunov function. Now we introduce the candidate Lyapunov function For any state vector of system (3.1)-(3.2), it can be seen that is feasible for the dual problem (2.18). For is the unique solution of (2.18), we have .

We now take the derivative of along trajectories of our system:

 ddtV=∑j∈J∂W∂μj˙μj+∑s∈S∂W∂νs˙νs=∑j∈J(cj−zj)˙μj+∑s∈S(∑r∈sxr−ys)˙νs=∑j∈Jvj−∑s∈Sκs(νs)(ys−∑r∈sxr)2,

where we have denoted . Note that the second equality follows from Lemma 2. We will now show that for each . For this, we must apply the dynamic equations (3.1)-(3.2), and distinguish between the two cases:

(a) . Here