CN103095602A - Network congestion control method based on quasi-Newton algorithm - Google Patents

Abstract

The invention relates to a network congestion control method and belongs to the technical field of computer communication. The network congestion control method mainly comprises control methods of a user terminal s and a link l, wherein the control methods are based on quasi-Newton algorithm. The user terminal s includes the following steps: (1) updating time t, the user terminal s receives responded transmission delay to substitute a prior transmission delay; (2) updating time t every time, the user terminal s decides a new rate according to a current path and uses the new rate to transmit under next update; (3) updating time t, the user terminal s transmits a current source rate. The link l control method includes the following steps: (1) updating time t, the link l receives all the transmission rates transmitted from the user terminal through the link l, and the link l uses the currently-received rate to substitute the prior rate; (2) updating time t every time, the link l calculates gl (t) and adjusting the transmission delay through the following formula: y1(t+1)=[y1(t)-muHk1(t) gl(t)]+gl(t), wherein y1 represents gradient direction, mu represents constant coefficient>=0, and Hkl(t) is a relevant matrix; (3) updating time t, the link l exchanges the current transmission delay to the user terminal.

Description

A kind of method for controlling network congestion based on intending Newton's algorithm

Technical field

The present invention relates to method for controlling network congestion, belong to the computer communication technology field.

Background technology

In recent years, along with the extensive use of network and the fast development of information technology, network size, user and application sharply increase, and network is just experiencing increasing packet loss and other mis-behave problem, and wherein network congestion problem is particularly serious.The basic reason that network congestion produces is that the user offers the load of network greater than Internet resources and network throughput, a little less than the immediate cause of congested generation had memory space inadequate, bandwidth capacity deficiency, processor disposal ability, it showed as degradation under Packet Delay increase, losing probability increase, upper layer application systematic function.The hot issue that research has become current network research is controlled in network congestion.The important directions that research is controlled in network congestion is to set up to adapt with network for the characteristics of network to comprise the Mathematical Modeling of various network parameters, and Mathematical Modeling is found the solution and according to the result of finding the solution, the network of reality carried out congestion control.A kind of good Mathematical Modeling can not only well be described the network congestion problem, and can bring convenience for improvement and the design of the algorithm relevant to congestion control.Therefore, method for controlling network congestion a difficult point be improvement and the design of the algorithm relevant to congestion control, and the improvement of the algorithm relevant to congestion control and design are depended on and the adapt foundation of the Mathematical Modeling that comprises various network parameters of network to a great extent.

To this, Kelly has proposed Network Optimization Model, and Low has done further research on this basis, has proposed network utility and has maximized model.Its basic thought is that in network, each source user has separately utility function, how to find the solution the peaked problem of whole network system utility function under the limited condition of the network bandwidth.At present, this model has been widely used in the research fields such as network rate allocation algorithm, internet congestion control protocol, network cross-layer optimizing, and becomes current study hotspot.Control the algorithm of model for network congestion, now proposed the methods such as gradient project algorithms, sub-gradient algorithm and quasi-Newton method both at home and abroad, document has proposed the gradient project algorithms based on end of link; For the not twice differentiable situation of utility function, existing document has proposed sub-gradient algorithm, and has analyzed its convergence; Compare gradient projection method based on the Newton-Like algorithm and have the advantages such as fast convergence rate and Quadratic Termination, also have document to propose the Newton-Like algorithm.These congestion avoidance algorithms between convergence rate, user fairness and the throughput of system aspect also have some problems.

Summary of the invention

Below provide the general introduction of simplification, for aspects more described herein provide basic understanding.This general introduction is not the exhaustive overview of theme required for protection, the scope of not attempting to identify the key/indispensable element of theme required for protection yet or describing theme required for protection.Its unique purpose is to provide in simplified form some concept, as the preamble of more specifically describing that provides after a while.

The object of the present invention is to provide a kind of can guarantee the fairness between the user can improve simultaneously again throughput of system based on the method for controlling network congestion of intending Newton's algorithm.

Along with the fast development of Internet, congestion control is one of most critical factor of guaranteeing the Internet robustness.For the congestion problems in current network, the present invention proposes based on the network congestion of optimum theory and control the fairness model.Variable-metric method combines the advantage of gradient method, Newton method and keeps away and abandon their shortcomings separately, only need to calculate the single order partial derivative, need not to calculate second-order partial differential coefficient and inverse matrix thereof, the initial point of target function is selected all without being strict with, fast convergence rate, the present invention will intend Newton's algorithm and is applied to the new model that proposes and carried out the convergence analysis.Simulation result shows that algorithm that the present invention carries has good convergence, has guaranteed the fairness between the user.

Consider S the network user share a network that comprises the L communication link (S={1 wherein, 2 ..., s}, L={1,2 ..., l}).Suppose c _lBe the finite bandwidth of link l, and the transmission rate of hypothesis user s is x _sAnd satisfy m _s≤ x _s≤ M _s, m wherein _sAnd M _sBe respectively minimum value and the maximum of user's transmission rate, define a utility function U _s(x _s) be user's transmission rate x _sFunction, the performance of expression user side s.Now to utility function U _s(x _s) do following hypothesis:

(1) for x _s∈ I _s, I wherein _s=[m _s, M _s], utility function U _s(x _s) be that strict recessed secondary can little increasing function on the domain of definition, and feasible in order to guarantee, and the l ∈ L for all satisfies condition

C wherein _lBe link l total capacity.

(2) at I _sOn, U _s(x _s) curvature range be from 0 to I _s, to all x _s∈ I _s, have $-U_s^{\′\′}\left(x_s\right)\≥1/{\stackrel{\‾}{a}}_s>0,$

For greater than zero constant.

The target that model is controlled in existing network congestion based on optimum theory is to choose suitable transmission rate x=(x _s, s ∈ S), make all user performances reach maximum.In order further to keep the fairness of network, the present invention is further improved above-mentioned model, to the utility function U of each user side _s(x _s) introducing weights ω _s, guarantee the different priority of user, thereby not only guaranteed the fairness of whole network but also guaranteed the maximization of user performance.Concrete model is as follows:

$\mathrm{Max}\underset{s}{\mathrm{\Σ}}{\mathrm{\ω}}_s{U}_s\left(x_s\right)$

$\underset{s}{\mathrm{\Σ}}{x}_s\≤c_l,l=\mathrm{1,2},...,L---\left(1\right)$

m _s≤x _s≤M _s

Known by hypothesis, target function is continuous strictly concave function, and feasible set is for compacting, so there is unique optimal solution in above-mentioned optimization problem.But the constraints from following formula can find out, finding the solution the problems referred to above needs all users to know each other information, and obviously, this is impracticable in real network.We wish that each user side can be found the solution separately with each link under the condition of not losing the network global optimality, namely strive for that each user the situation lower network overall performance that reaches satisfied reaches optimum, thereby the key of solution is to find the solution its dual problem.

Method for controlling network congestion based on intending Newton's algorithm of the present invention comprises following user side s based on the plan Newton's algorithm, the concrete control method of link l

User side s control method is as follows

The first step: update time t, user side s receives the propagation delay time that feeds back, the propagation delay time before replacing with it;

The below further derives the dual form of model proposed by the invention.

Particular content is: at first by the definition Lagrangian:

$L(x,y)=\underset{s}{\mathrm{\Σ}}{\mathrm{\ω}}_s{U}_s\left(x_s\right)-\underset{l}{\mathrm{\Σ}}{y}_l(\underset{s}{\mathrm{\Σ}}{x}_s-c_l)$

$=\underset{s}{\mathrm{\Σ}}({\mathrm{\ω}}_s{U}_s\left(x_s\right)-x_s\underset{l}{\mathrm{\Σ}}{y}_l)+\underset{l}{\mathrm{\Σ}}{y}_l{c}_l$

Wherein, y _l(y _l〉=0) be the upper corresponding Lagrange multiplier of link l, and the throughput of carving link l for time t.

Because

$\max \underset{s}{\mathrm{\Σ}}({\mathrm{\ω}}_s{U}_s\left(x_s\right)-x_s\underset{l}{\mathrm{\Σ}}{y}_l)$

$=\underset{s}{\mathrm{\Σ}}\max ({\mathrm{\ω}}_s{U}_s\left(x_s\right)-x_s\underset{l}{\mathrm{\Σ}}{y}_l)$

Therefore the dual function of former problem is:

$D\left(y\right)=\max L(x,y)=\underset{s}{\mathrm{\Σ}}{A}_s\left(y^s\right)+\underset{l}{\mathrm{\Σ}}{y}_l{c}_l$

s.t.m _s≤x _s≤M _s

A wherein _s(y ^s)=max ω _sU _s(x _s)-x _sy ^s(2)

$y^s=\underset{l}{\mathrm{\Σ}}{y}_l$

Therefore the dual form of former problem is

$D:\underset{y\≥0}{\min }D\left(y\right)---\left(3\right)$

The second step of user side control method: in each update time of t, the path estimation value that user side s is current according to it

Select a new speed

Then with this speed rates until next update;

Particular content is: because Lagrangian is decomposable, so the target function D (y) of dual problem can be broken down into S independently subproblem find the solution respectively.Again because the constraints in former problem is linear, so the antithesis ditch is zero, and necessarily has dual optimal solution x _s(y ^*).And x _s(y ^*) but through type (2) calculates, the Kuhn-Tucker theorem according to classical in Optimum Theory is got by single order necessity condition:

$x_s\left(y\right)={\left[{\left({\mathrm{\ω}}_s{U_s}^{\′}\left(y\right)\right)}^{-1}\right]}_{m_s}^{M_s}$

Wherein, ${\left[{\left({\mathrm{\ω}}_s{U_s}^{\′}\left(y\right)\right)}^{-1}\right]}_{m_s}^{M_s}=\min \left\{\max \right\{{\left({\mathrm{\ω}}_s{U_s}^{\′}\left(y\right)\right)}^{-1},\}{m}_s,M_s\},$ (ω _sU _s') ^-1Expression ω _sU _s' contrary, wherein to put forward implication with the front identical for each letter.

The user side control method the 3rd the step: update time t, user side s passes on present rate;

Link l control method is as follows

For the derivation algorithm of link l, existing document has proposed the correlation techniques such as Projected Gradient, sub-gradient algorithm and quasi-Newton method.From the overall situation, intending Newton's algorithm is to solve the most effective class algorithm in unconstrained optimization method, and it has convergence rate faster, and this algorithm has Quadratic Termination.Therefore, the present invention will solve the new model that proposes with intending Newton's algorithm.Intending Newton's algorithm is to be based upon on the basis of Newton method, to use matrix H _kThe second dervative that replaces iteration function, and needn't calculate the Hessian matrix, when making H _kRegularly positive, the direction that algorithm produces is descent direction, and algorithm is as follows:.

The first step of chainlink control method: update time t, link l obtains the speed that user side transmits, the speed before replacing with the transmission rate that receives recently;

The second step of chainlink control method: at each update time of t, link l calculates sub-gradient direction g _l(t) and adjust its propagation delay time by following formula:

y _l(t+1)＝[y _l(t)-μH _kl(t)g _l(t)] ⁺

g _l(t) represent sub-gradient direction, μ is the constant coefficient greater than zero, H _kl(t) be corresponding matrix.

Concrete steps are as follows:

If D (y) is continuously differentiable, target function is asked local derviation:

$g_l\left(t\right)=\frac{\∂D}{{\∂y}_l}\left(y\right)=c_l-\underset{s}{\mathrm{\Σ}}{x}_s\left(y\right)$

In real network, utility function is general function, does not usually possess two subdifferentiabilities, at this moment, and g _l(t) be taken as sub-gradient direction.At each update time of t, link l calculates g _l(t), and by following formula adjust its unit propagation delay time:

y _l(t+1)＝[y _l(t)-μH _kl(t)gl(t)] ⁺

Wherein, H _kl(t) with

The pass be:

${\left[{\▿}^{2}D\left(y\left(t\right)\right)\right]}_{\mathrm{ll}}=-\underset{s\∈S\left(l\right)}{\mathrm{\Σ}}\frac{{\∂x}_s}{{\∂y}_l}\left(y\left(t\right)\right)$

$=-\underset{s\∈S\left(l\right)}{\mathrm{\Σ}}{x}^{\′}\left(y_l^s\left(t\right)\right)$

$\≈-\underset{s\∈S\left(l\right)}{\mathrm{\Σ}}\frac{x_s\left(t\right)-x_s(t-1)}{y^s\left(t\right)-y^s(t-1)}$

$\≈-\underset{s\∈S\left(l\right)}{\mathrm{\Σ}}\frac{x_s\left(t\right)-x_s(t-1)}{y_l\left(t\right)-y_l(t-1)}$

$\frac{{\hat{x}}^l\left(t\right)-{\hat{x}}^l(t-1)}{y_l\left(t\right)-y_l(t-1)}$

Wherein,

Total source speed, y _l(t) be the throughput that time t carves link l.

The 3rd step of chainlink control method: update time t, link l is to the current propagation delay time of user side exchange.

Compared with the prior art, the present invention has the following advantages:

1. the present invention is based on optimum theory and proposed more to be applicable to actual network congestion control fairness new model, added weights in model, guarantee the fairness of system;

2. found the solution new model with the plan Newton's algorithm, and respectively can be little with regard to the utility function secondary and two kinds of situation analysis of non-differentiability this convergence.Can find out that by analyzing us the fairness of algorithm is good.

3. simulation result shows, algorithm that the present invention carries can converge to optimal value faster, has good convergence, and has guaranteed the fairness of system.

The present invention proposes based on the network congestion of optimum theory and control the fairness model, it is closer to the real network model, the end subscriber shared network bandwidth resources in network, and guarantee the fairness of network with different weights.

For addressing relevant purpose before reaching, unite the following description and drawings herein some illustrative aspects of theme required for protection is described.But several in the variety of way of each principle that can use theme required for protection only indicated in these aspects, and all aspects and equivalence techniques scheme thereof attempted to comprise in this claimed theme.When considered in conjunction with the accompanying drawings,

Other advantage and novel feature will become apparent from the following specifically describes.

Description of drawings

Fig. 1: 3 users' network topology structure schematic diagram;

Fig. 2: system's convergence graph picture of 3 users;

Fig. 3: do not add temporary 3 users' system optimal speed;

Fig. 4: add temporary 3 users' system optimal speed;

Fig. 5: 9 users' network topology structure schematic diagram;

Fig. 6: system's convergence graph picture of 9 users;

Fig. 7: do not add temporary 9 users' system optimal speed;

Fig. 8: add temporary 9 users' system optimal speed;

Embodiment

Referring to accompanying drawing, provide the specific embodiment of the present invention, be used for formation of the present invention is further illustrated.

Embodiment 1

The method for controlling network congestion based on the plan Newton's algorithm of the present embodiment comprises following user side s based on the plan Newton's algorithm, the concrete control method of link l

User side s control method comprises the following steps

(1) update time t, user side s receives the propagation delay time that feeds back, the propagation delay time before replacing with it;

(2) in each update time of t, the path estimation value that user side s is current according to it

Select a new speed

Then with this speed rates until next update;

(3) update time t, user side s transmits present rate;

The detailed process of system model is:

Utility function U to each user side _s(x _s) introducing weights ω _sThereby, not only having guaranteed the fairness of whole network but also guaranteed the maximization of user side s performance, concrete model is as follows:

$\mathrm{Max}\underset{s}{\mathrm{\Σ}}{\mathrm{\ω}}_s{U}_s\left(x_s\right)$

$\underset{s}{\mathrm{\Σ}}{x}_s\≤c_l,l=\mathrm{1,2},...,L---\left(1\right).$

m _s≤x _s≤M _s

The dual form of said system model:

Now define Lagrangian:

$L(x,y)=\underset{s}{\mathrm{\Σ}}{\mathrm{\ω}}_s{U}_s\left(x_s\right)-\underset{l}{\mathrm{\Σ}}{y}_l(\underset{s}{\mathrm{\Σ}}{x}_s-c_l)$

$=\underset{s}{\mathrm{\Σ}}({\mathrm{\ω}}_s{U}_s\left(x_s\right)-x_s\underset{l}{\mathrm{\Σ}}{y}_l)+\underset{l}{\mathrm{\Σ}}{y}_l{c}_l$

Wherein, y _l(y _l〉=0) be the upper corresponding Lagrange multiplier of link l.

Because

$\max \underset{s}{\mathrm{\Σ}}({\mathrm{\ω}}_s{U}_s\left(x_s\right)-x_s\underset{l}{\mathrm{\Σ}}{y}_l)$

$=\underset{s}{\mathrm{\Σ}}\max ({\mathrm{\ω}}_s{U}_s\left(x_s\right)-x_s\underset{l}{\mathrm{\Σ}}{y}_l)$

Therefore dual function is:

$D\left(y\right)=\max L(x,y)=\underset{s}{\mathrm{\Σ}}{A}_s\left(y^s\right)+\underset{l}{\mathrm{\Σ}}{y}_l{c}_l$

s.t.m _s≤x _s≤M _s

Wherein

A _s(y ^s)＝maxω _sU _s(x _s)-x _sy ^s （2）

$y^s=\underset{l}{\mathrm{\Σ}}{y}_l$

Therefore dual form is

$D:\underset{y\≥0}{\min }D\left(y\right)---\left(3\right).$

Link l control method comprises the following steps

(1) update time t, link l receives from the transmission rate of all user sides by link l transmission, the speed before replacing with recently received speed;

(2) at each update time of t, link l calculates g _l(t) and adjust its propagation delay time by following formula:

y _l(t+1)＝[y _l(t)-μH _kl(t)g _l(t)] ⁺

g _l(t) represent sub-gradient direction, μ is the constant coefficient greater than zero, H _kl(t) be corresponding matrix.

(3) update time t, link l is to the current propagation delay time of user side exchange.

Provide the present invention and put forward the convergence theorem.

Theorem: suppose B ₀A preliminary positive definite matrix, y ₀It is an initial point.Following hypothesis is arranged:

(1) D (y): R ⁿ→ R ¹Continuously differentiable

(2) Ω={ y ∈ R ⁿ| D (y)≤D (y ₀) increase progressively, and have constant m and M, satisfy

m‖z‖ ²≤z ^TG(y)z≤M‖z‖ ²。

{ x so _sConverge to y ^*, { x wherein _sIt is the optimal solution that BFGS produces.

Proof:

$m\left(t\right)=\frac{q{\left(t\right)}^{T}p\left(t\right)}{p{\left(t\right)}^{T}p\left(t\right)},$ $M\left(t\right)=\frac{q{\left(t\right)}^{T}q\left(t\right)}{q{\left(t\right)}^{T}p\left(t\right)}---\left(5\right)$

Can obtain:

m(t)≥m,M(t)≤M (6)

According to (4) formula, take to follow the tracks of operation,

$\mathrm{Trace}\left(B(t+1)\right)$

$=\mathrm{Trace}\left(B\left(t\right)\right)-\frac{{\left|\right|B\left(t\right)p\left(t\right)\left|\right|}^2}{p{\left(t\right)}^{T}B\left(k\right)p\left(t\right)}+\frac{{\left|\right|q\left(t\right)\left|\right|}^2}{q{\left(t\right)}^{T}p\left(t\right)}---\left(7\right)$

$\det \left(B(t+1)\right)=\det \left(B\left(t\right)\right)\frac{q{\left(t\right)}^{T}p\left(t\right)}{p{\left(t\right)}^{T}B\left(t\right)p\left(t\right)}---\left(8\right)$

If

$\cos \mathrm{\θ}\left(t\right)=\frac{p{\left(t\right)}^{T}B\left(t\right)p\left(t\right)}{\left|\right|p\left(t\right)\left|\right|\left|\right|B\left(t\right)p\left(t\right)\left|\right|}$ (9)

$s\left(t\right)=\frac{p{\left(t\right)}^{T}B\left(t\right)p\left(t\right)}{p{\left(t\right)}^{T}p\left(t\right)}$

θ (t) represents the angle between p (t) and B (t) p (t), therefore, can release

$\frac{{\left|\right|B\left(t\right)p\left(t\right)\left|\right|}^2}{p{\left(t\right)}^{T}B\left(k\right)p\left(t\right)}$

$=\frac{{\left|\right|B\left(t\right)p\left(t\right)\left|\right|}^2{\left|\right|p\left(t\right)\left|\right|}^2}{{\left(p{\left(t\right)}^{T}B\left(k\right)p\left(t\right)\right)}^2}\frac{p{\left(t\right)}^{T}B\left(k\right)p\left(t\right)}{{\left|\right|p\left(t\right)\left|\right|}^2}---\left(10\right)$

$=\frac{s\left(t\right)}{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="1303131549063.png,1303131549064.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&theta;}\left(t\right)}$

Can be obtained by (6) and (8) formula

$\det \left(B(t+1)\right)$

$=\det \left(B\left(t\right)\right)\frac{q{\left(t\right)}^{T}p\left(t\right)}{p{\left(t\right)}^{T}p\left(t\right)}\frac{p{\left(t\right)}^{T}B\left(t\right)p\left(t\right)}{p{\left(t\right)}^{T}p\left(t\right)}---\left(11\right)$

$=\det \left(B\left(t\right)\right)\frac{m\left(t\right)}{s\left(t\right)}$

If

ψ(B)＝Trace(B)-ln(det(B)) (12)

According to (7)-(11), can obtain

$\mathrm{\&psi;}\left(B(t+1)\right)$

$=\mathrm{Trace}\left(B(t+1)\right)-\ln \left(\det \left(B(t+1)\right)\right)$

$=\mathrm{Trace}\left(B\left(t\right)\right)+M\left(t\right)-\frac{s\left(t\right)}{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="1303131549063.png,1303131549064.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&theta;}\left(t\right)}$

$-\ln \left(\det \left(B\left(t\right)\right)\right)-\ln m\left(t\right)+\ln s\left(t\right)$

$=\mathrm{\&psi;}\left(B\left(t\right)\right)+(M\left(t\right)-\ln m\left(t\right)-1)+$

$[1-\frac{s\left(t\right)}{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="1303131549063.png,1303131549064.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&theta;}\left(t\right)}+\ln \frac{s\left(t\right)}{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="1303131549063.png,1303131549064.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&theta;}\left(t\right)}]+\mathrm{<figure-callout\; id="2"\; label="ln\; cos"\; filenames="1303131549063.png,1303131549064.png"\; state="\{\{state\}\}">ln</figure-callout>}{\cos }^{}{}^2\mathrm{\&theta;}\left(t\right)$

According to formula (5), the formula above repeatedly adopting can obtain

$0<\mathrm{\&psi;}\left(B(t+1)\right)\&le;\mathrm{\&psi;}\left(B_0\right)+\mathrm{ct}+\underset{j=1}{\overset{\mathrm{<figure-callout\; id="2"\; label="t\; ln\; cos"\; filenames="1303131549063.png,1303131549064.png"\; state="\{\{state\}\}">t</figure-callout>}}{\mathrm{\&Sigma;}}}\ln {\cos }^{}{}^2\mathrm{\&theta;}\left(j\right)---\left(13\right)$

Wherein, c=M-lnm-1＞0.

Got by general inaccuracy line search convergence theorem:

‖g(t)‖cosθ(t)→0

In order to prove

We first prove and have sequence { t _jSatisfied { cos θ _j} 〉=δ＞0.

Suppose cos θ _j→ 0, therefore, there is t ₀＞0, for j＞t ₀Satisfy

lncos ²θ _j＜-2c

According to (13), we can derive,

$0<\mathrm{\&psi;}\left(B_0\right)+\mathrm{ct}+\underset{j=1}{\overset{{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="1303131549062.png,1303131549063.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}{\mathrm{\&Sigma;}}}\mathrm{<figure-callout\; id="2"\; label="ln\; cos"\; filenames="1303131549063.png,1303131549064.png"\; state="\{\{state\}\}">ln</figure-callout>}{\cos }^{}{}^2\mathrm{\&theta;}\left(j\right)+\underset{j={\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="1303131549062.png,1303131549063.png"\; state="\{\{state\}\}">t</figure-callout>}}_0+1}{\overset{t}{\mathrm{\&Sigma;}}}(-2c)$ t＞t ₀

$=\mathrm{\&psi;}\left(B_0\right)+\underset{j=1}{\overset{{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="1303131549062.png,1303131549063.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}{\mathrm{\&Sigma;}}}\mathrm{<figure-callout\; id="2"\; label="ln\; cos"\; filenames="1303131549063.png,1303131549064.png"\; state="\{\{state\}\}">ln</figure-callout>}{\cos }^{}{}^2\mathrm{\&theta;}\left(j\right)+2t_{0}c-\mathrm{tc}$

The first of above-mentioned formula and third part are that positive definite is limited, and second portion is negative definite.The 4th part is negative definite also, and t → ∞ hypothesis is false.Show and have { t _jSatisfied { cos θ _j} 〉=δ＞0.Therefore

$\underset{t\&RightArrow;\&infin;}{\mathrm{li}\min f}\left|\right|g\left(t\right)\left|\right|=0$

Top formula represents { y (t) } → y ^*Theorem is set up.

In the present embodiment, we carry out emulation experiment, and checking the present invention carries the BFGS algorithm complexity, considers respectively 3 users and 9 users' situation.Intending Newton method with projection and gradient in algorithm compares.

Number of users is the situation of 3

We adopt network topology structure shown in Figure 1, and this network has 2 connections and 3 users.

Fig. 2 has provided total effectiveness under algorithm that the present invention puies forward, and as can be seen from the figure, algorithm that the present invention carries reaches convergence fast than gradient algorithm and the more effective algorithm that makes of plan Newton's algorithm.

Fairness is to estimate the important indicator of an algorithm quality, and it has determined under the identical network resources supplIes, the occupy situation of user to Internet resources.Fig. 3,4 have provided respectively source speed in the situation of system's not weighting of weighted sum for three users, as can be seen from the figure, by weighting can fine assurance user fairness.

Number of users is the situation of 9

In order further to verify algorithm complexity of the present invention, we consider larger system, as shown in Figure 5,7 connections and 9 users are arranged.

As can be seen from Figure 6, algorithm that the present invention carries still has good convergence

Fig. 7 and 8 is the optimal rate of system user, as can be seen from Figure 7, exists a speed to be almost 0 user.The speed that can guarantee the user by weighting is comparatively average, as shown in Figure 8, thus the fairness of the system of assurance.

Foregoing comprises some examples of theme required for protection.Certainly, each combination that can conceive of describing each assembly and method for the purpose of describing theme required for protection is impossible, but one skilled in the art will recognize that, many other combination and conversion are possible.Thus, theme required for protection is intended to comprise this type of change, modification and the variant within all spirit and scope that fall into appended claims.In addition, describe in detail or claims in use on the meaning that term " comprises ", this term intention is inclusive as term " comprises ", explains when being used as the transition word as " comprising " in claims.

Claims (4)

1. the method for controlling network congestion based on the plan Newton's algorithm, is characterized in that, comprises following user side s based on the plan Newton's algorithm, the control method of link l; Wherein

User side s control method comprises the following steps

(1) update time t, user side s receives the propagation delay time that feeds back, the propagation delay time before replacing with it;

(2) in each update time of t, the path estimation value that user side s is current according to it

Determine a new speed

Then with this speed rates until next update;

(3) update time t, user side s transmits current source speed;

Link l control method comprises the following steps

(1) update time t, link l receives all from user side transmission rates by link l transmission, the speed before link l replaces with recently received speed;

(2) at each update time of t, link l calculates g _l(t) and adjust its propagation delay time by following formula:

y _l(t+1)＝[y _l(t)-μH _kl(t)g _l(t)] ⁺

g _l(t) represent sub-gradient direction, μ is the constant coefficient greater than zero, H _kl(t) be corresponding matrix.

(3) update time t, link l is to the current propagation delay time of user side exchange.

2. according to the described method for controlling network congestion based on intending Newton's algorithm of claim 1, it is characterized in that,

Utility function U to each user side _s(x _s) introducing weights ω _s, concrete system model is as follows

$\mathrm{Max}\underset{s}{\mathrm{\&Sigma;}}{\mathrm{\&omega;}}_s{U}_s\left(x_s\right)$

$\underset{s}{\mathrm{\&Sigma;}}{x}_s\&le;c_l,l=\mathrm{1,2},...,L---\left(1\right).$

m _s≤x _s≤M _s

3. according to the described method for controlling network congestion based on intending Newton's algorithm of claim 2, it is characterized in that the dual form of said system model:

The definition Lagrangian:

$L(x,y)=\underset{s}{\mathrm{\&Sigma;}}{\mathrm{\&omega;}}_s{U}_s\left(x_s\right)-\underset{l}{\mathrm{\&Sigma;}}{y}_l(\underset{s}{\mathrm{\&Sigma;}}{x}_s-c_l)$

$=\underset{s}{\mathrm{\&Sigma;}}({\mathrm{\&omega;}}_s{U}_s\left(x_s\right)-x_s\underset{l}{\mathrm{\&Sigma;}}{y}_l)+\underset{l}{\mathrm{\&Sigma;}}{y}_l{c}_l$

Wherein, y _l(y _l〉=0) be the upper corresponding Lagrange multiplier of link l

Because

$\max \underset{s}{\mathrm{\&Sigma;}}({\mathrm{\&omega;}}_s{U}_s\left(x_s\right)-x_s\underset{l}{\mathrm{\&Sigma;}}{y}_l)$

$=\underset{s}{\mathrm{\&Sigma;}}\max ({\mathrm{\&omega;}}_s{U}_s\left(x_s\right)-x_s\underset{l}{\mathrm{\&Sigma;}}{y}_l)$

Therefore dual function is:

$D\left(y\right)=\max L(x,y)=\underset{s}{\mathrm{\&Sigma;}}{A}_s\left(y^s\right)+\underset{l}{\mathrm{\&Sigma;}}{y}_l{c}_l$

s.t.m _s≤x _s≤M _s

Wherein

A _s(y ^s)＝maxω _sU _s(x _s)-x _sy ^s （2）

$y^s=\underset{l}{\mathrm{\&Sigma;}}{y}_l$

Therefore dual form is

$D:\underset{y\&GreaterEqual;0}{\min }D\left(y\right)---\left(3\right).$

4. according to claim 1 or 2 or 3 described method for controlling network congestion based on intending Newton's algorithm, it is characterized in that

Intending Newton's algorithm is to be based upon on the basis of Newton method, to use matrix H _kThe second dervative that replaces iteration function, and needn't calculate the Hessian matrix, when making H _kRegularly positive, the direction that algorithm produces is descent direction.

Images