congestion control scheme in atm networks based on fast tracing-queue

Journal of Systems Engineering and Electronics, Vol. 18, No. 1, 2007, pp. 101 -110

Congestion control scheme in ATM networks based on fast tracing- queue* Liu Zhixin & Guan Xinping

Inst. of Electrical Engineering, Yanshan Univ., Qinhuangdao 066004, P. R. China (Received September 29,2005)

Abstract: One of the more challenging and unresolved issues in ATM networks is the congestion control of available bit rate (ABR). The dynamic controller is designed based on the control theory and the feedback mechanism of explicit rates. With the given method of a chosen parameter, it can guarantee the stability of the controller and closed loop system with propagation delay and bandwidth oscillation. It needs less parameters(only one) to be designed. The queue length can converge to the given value in the least steps. The fairness of different connections is considered further. The simulations show better performance and good quality of service(QoS) is achieved.

Keywords: ATM networks, Congestion control, Control theory, Tracing-queue, QoS.

1. Introduction

Asynchronous transfer mode(ATM) transfer principle has been recommended by the International Telecom- munication Union(1TU) as the transfer vehicle for broad band networks. Many data applications are highly bursty and have no way of predicting data traffic requirements in advance, but have well-defined packet loss requirement and can tolerate time-varying and unpredictable packet delays. More importantly, they are able to modify their data transfer rates according to network loading. Thus the notion of elastic traffic services was introduced, by which the data transfer rates are adjusted at the source, depending on the available bandwidth in the network. A representative example of this kind of service is the available bit rate (ABR) service in ATM networks. Users of this service dynamically share the available bandwidth in an equitable fashion. They will attempt to minimize the cell loss at the existence of delay and utilize the available bandwidth fully. An end-to-end rate-based feedback control scheme has been specified by the ATM Forum"I, which is a rate-based closed-loop control. This scheme uses feedback information from the network to control the rate at which each source can transmit cells in the network. The feedback information is carried by special cells called resource management cells(RM cells). There are three mechanisms for a switch to write the congestion status onto RM cells: explicit forward congestion indication (EFCI) marking, relative rate(RR)

marking and explicit rate(ER) marking. At least one of these have to be implemented on a switch for the rate-based flow control. Among these methods, ER scheme is the most famous and practical"', where the switch may use the ER field of forward andor backward RM cells to indicate the rate it can support.

Some algorithms have been proposed for the computation of the ER. Two famous algorithms are enhanced proportional rate control algorithm(EPRCA) and explicit rate identification congestion avoidance algorithm(ERICAA). In EPRCA, the value of ER is calculated at the switch according to the current queue length and arrival cell rate. ERICAA is based on measurement. In this scheme, bandwidth and network states are needed and a load factor is adopted to indicate the load of the current network. So some addi- tional memory for storage and computation is necessary. The main drawback of these control schemes is that they do not have a formal theory to support them and the improved results are verified through simulations. Besides, with the mechanism of measurement, the noise in estimation cannot be avoided. Assistant steps may slow the response and worsen the QoS.

As those methods based on practice and simulation cannot provide an ideal analysis, such as stability, and closed loop performance, a number of algorithms based on control theory have been proposed recently. %o proportional differential(PD) controllers are proposed in Refs.[2, 31, respectively. The stability and robustness of a closed-loop system are considered. In Refs.[4, 51, proportional integral(P1) controllers are

This project was supported partly by the Outstanding Youth Scientific Foundation of China(60525303), the National Natural Science. Foundation of China ~60404022.606040121 and the Natural Science Foundation of Hebei hovince of China~0050003901.

102 Liu Zhixin & Guan Xinping

designed to quicken the response of the sources and force the queue length to the desired point. Reference[6] proposes a dual proportional-pus-differential controller(DPDC). The wide range feedback delay is considered in this literature. In a similar way, Reference[7] introduces a detecting/relieving congestion mechanism based on probability, and designs the linear PID controller to update the probability depending on the network load status, so the self-oscillation caused by the nonlinear control law is reduced, and the link utilization increased. In Ref.[8], a continuous time queuing model is studied and stability conditions for various controllers are obtained. However, performance analysis is done only under the assumptions of no delays and the system being continuously observed, which seems unrealistic in the ABR context. In Refs.[9, 101, to deal with the round trip delay, the Smith principle is used to design the ER controller. In Refs.[ 11, 121, the robust control theory and digital filer method are employed, based on the linear model, to deal with the congestion in networks, respectively. In Ref.[13], a feedback control scheme, based on adaptive and inverse adaptive control theory is presented. But the algorithms are complex. In Ref. [14], the authors propose a one-layer neural network adaptive control methodology to prevent congestion in high-speed ATM networks. The buffer dynamics at the switch are modeled as nonlinear discrete-time system, and an "(neural network)-based predictive controller is designed to predict the explicit values of the transmission rates of the source so as to prevent congestion. But it ignores the transfer delay in the model also. In Ref.[15], the Routh-Hurwitz stability criterion is used to quantify the relationship between the rate of adaptation of the flow rate of the data sources and the time delay, which is required for stable operation of the flow control process. However, it does not analyze the response from the oscillation of available bandwidth to the queue length. All these schemes try to find a tradeoff between the simplicity and the effectiveness. In this article, a method for the design of a rate-base controller is developed, which has the following merits:

(1) Operating over a wide range of round trip time;

(2) Guaranteeing the stability of closed loop

system and controller; (3) Tracing the desired queue length in the buffer

in the least steps; (4) Needing less parameters in ER controller(on1y

one to be chosen); ( 5 ) Achieving madmin fairness for the connec-

tions, which are in the framework of closed loop based explicit rate(ER) mechanism.

2. Network system model

In this section an explicit rate control mechanism based on a simplified "fluid is described and motivated. A generic switch as a reference model is considered here@], which is shown in Fig.1. The switch has K input/output links, each input link is connected with the bus in the switch and output link is associated with a buffer and a logical queue designated for this link. The transmission of ATM services is based on the virtual connection. In the networks, the source sends an RM cell to every N , data cells, along the logical connection. Among all links the connection passing by, the one, which restrains the rate to the minimum value, is called the bottleneck link. It is assumed that there is only one bottleneck link in a logical connection. Without loss of generality, the assumption is made that there are K VCs bottlenecked at the considered node as shown in Fig. 1.

Switch

bandwidth

Fig.1 Model of the switch

One associated with each output linkj~=1,2;-., K ) is an ER controller, which computes the ER value according to the current state of the networks periodically. The period is T s. With the basic mechanism, successive RM cells of the connection,

Congestion control scheme in ATM networks based on fast tracingqueue 103

which is bottlenecked here, will be tamped with a new ER value during a T period. The ER will reach source i after delay zy (i=1,2,-,K), it is called backward propagation delay. z y is the propagation delay between the bottleneck node and the source i along the backward path. The source receipt of a backward RM cell adjusts its sending rate based on source end algorithm at time t- r y . The new rate rj(t) is always bound by the MCR(minimum cell rate) and PCR(peak cell rate). This is to say ri(t) must satisfy the following constraints.

MCR x < M C R ri ( t ) = PCR x > PCR (1) I ri ( t ) else

The effect of the source adaptation becomes apparent at the bottleneck node after another delay zf , where $is the propagation delay between the source i and the congested buffer in the considered node. The sum of the propagation delay in the path between the source and the corresponding buffer j , is denoted by r3 = z + z j” . For simplicity, it is denoted as zi . In a general case, the propagation delay is dominant compared to other delays, Let time be slotted and each slot be the sample period T. The control law is computed during T, and is executed at time t(n-1), t(n), t(n+l), - * - . Let q(n) be the number of cells at time n, that is, queue length. The size of the buffer being finite, it is taken to be B. The simplified propagation model is described in Fig.2.

’ bi

Fig.2 Simplified propagation model with single bottleneck link

Then the dynamics of the queue in the buffer is q(n+l)=Sat , [q(n)+R(n) T-C(n) r] (2)

where 0, x < o

Sats{x}= B, x > B 1 x, else

R(n) is the whole sending rate of VC’s at time n, C(n) is the available bandwidth for the ABR service, B is the buffer size. Let S be the set of all VCs that are

routed through the buffer j . As transmission rate is always determined by the smallest ER reported by the switch along the connection path, that is, bottleneck node, there will be some connections bottlenecked elsewhere along the path. Let those connections be in the set B. Let A=S-B, the rest are bottlenecked in bufferj. So R(n) is defined as the following

(3)

where r(n) is the common rate computed at the bufferj by the ER controller at time n, and fi = int( zi IT). Fraction of the time delay .Zi is neglected, since time is assumed to be slotted. Let d be defined as the largest round-trip delay measured in time T, that is, d= max ti . Then R(n) can be written as

R ( n ) = C r ( n -f;) + uj ( n ) is A

& A

d

R ( n ) = cZi ( n ) r ( n - i) + Uj ( n ) (4) i=l

where, &(n) is the number of bottlenecked VCs at the portj, whose round-trip delay is i time slots. The open loop system can be

4 ( n + 1) = Sat, ( 4 ( n ) +

[ i i i ( n ) r ( n - i ) + U j ( n ) .T - c ( n > .T> (5)

i=l 1 r(n) is the control law and it will be written in the ER field in the backward RM cell during the next period. For the rest of the connections, which are bottlenecked elsewhere, the ER field is unchanged.

3. Design of the controller

3.1 Fast queue-tracing controller

Around the network equilibrium point, the saturation has little effect on the performance of the closed loop system“]. SO the saturation nonlinearities of (5 ) are removed, when the system stability is analysed, and rewritten as

4 ( n + l ) = 4 ( 4 +

(6) cZi ( n ) r ( n - i ) + U j ( n ) - C ( n ) i=l

The single bottlenecked node network is described as a linear dynamic control system with time-delay in Fig.3. Note that the system is described in 2-transformation here.

In Fig.3, Q d is the desired queue length, which is determined according to buffer size. Here it is chosen as Q d =0.5B. R(n), Q(n), E(n) are control law, queue


length of the buffer and the error between Qd and Q(n), respectively. G,(z) is the Z-transformation of the controller to be designed. G,(z) is the Z-transformation of the plant. According to Fig.3 and Eq.(6), G,(z) is given by

Fig.3 Block diagram of the control system

1 G,(z)=-

1 - z-I (7)

Let K be the total number of VCs bottlenecked here,

so 1 =K. Propagation delay is the most important

element that worsens the performance of the network. To guarantee the robustness of the algorithm, consider the worst situation, that is, all connections have the maximum delay. This is to say

i=l

i=l i=l

the available bandwidth is a disturbance in the designed flow rate control system'". Then, -C(n)+U(n) can be regarded as the disturbance for the control system, when the stability of closed loop system is analysed. It is assumed that this term is equal to zero here. The response of this term is analyzed in Section 3.2. Let G(z) be Z-transformation of the global plant. On the basis of the above assumptions, it can be described as the following

- -d L G(z)=-

1 - z-1

Let

(9)

They are the Z-transformations of closed loop system and error system, respectively. Then Eq. (1 1) holds.

(11) Solving Eq. (lo), this is obtained

@( z ) + ae ( z ) = 1

Let E(n)=Qd - Q(n), the task is to design controller G,(z) to make the output Q(n) trace the desired queue length Qd, that is, E(n) tends to zero with the increase of n. From Eq. (lo), it follows:

According to the final value theory, it is known that, (Z) = Qd ( Z ) @e (2) (13)

e(m)=lim(l- z-')E(z)=(l- Z-')-@~(Z) Qd (14) z+l 1 - 2-1

It is necessary that Ge(z) contains the factor (1-z-') to guarantee e(m)=O. So we let

(15) From Eq. (1 3), QJz) is a given term, the less the terms Qe(z ) contains, the faster E(n) tends to zero, that is, output, Q(n), traces the desired queue length. If F(z) is chosen as 1, Ge ( z ) contains the least terms, then

Qe(z ) = (1 - z - ' )F ( z )

that can make Q(n) trace Qd at the fastest speed. "-1 4. , the controller However, for G,(z) =

G(z)(l- 2-l)

contains the factor zd-', which can not be realized. To eliminate the unstable zeros (including the term of z-' for the realization) and poles, @(z) and Qe(z) are chosen in a proper fashion, respectively, under the condition of Eq. (1 1). Here is chosen

(17) @(z) = z -dM (z)

Qe(z) = ( l - z V ( z ) where

p , q are the highest powers of M(z),F(z) respectively, and d+p=q+ 1.

The closed loop system is stable and the output can track the desired queue length, if @ ( z ) , Qe(z ) ,G,(z) are chosen as the above fashions and equation (20) is satisfied.

Theorem 1

P

C r n i = l (20) i=O

Proof As p is a positive integer, closed loop Z-transformation @ ( z ) has a finite number of terms

Congestion control scheme in ATM networks based on fast tracing-queue 105

in its power-series expansion inverse powers of z. Thus the output signal will go to a finite value in a finite number of sample periods. So the closed loop system is stable. According to the final value theory, it holds that

Qd (% + -k ' * * mp) with the condition of Eq.(20), the following is obtained

Q<-> = Qd

Substituting Eqs. (17)-(19) to (l l) , this follows, ~ - ( m , z - ~ + Y Z - ~ - ' +m,z-d-2+. . .+

+ mpz-d-P ) = -d-p+l m,-,z 1 + v; - 1)z-l + <f, - f,)z" + . . . +(fq - f )z-q - fqZ-q-l

(21)

q-1

The coefficients of the corresponding term should be same on both sides of the equation. One has

fi -1=o

f d - 1 - f d - 2 = O

f d - fd -1 = -% (22) f d + l - f d =-q

- f q =-mp

Theorem 2 The controller is stable and realiza- ble, if all the roots of Eq. (23) are located in the unit circle of the Z plant, where the coefficients satisfy the constraint of Eq.(22).

the poles of Gc(z) are the roots of F(z). Multiply the factor zq on both sides of (19), to get (23). According to the stability theory of discrete system, it is known that Theorem 2 holds.

Next how to determine the coefficients of the controller is discussed.

The velocity of tracing depends on the number of the terms in the given closed loop transfer function. To quicken the response, M(z) is chosen as a one-order polynomial.

M ( z ) = m, + qz-' (24)

The highest power of F(z) can be determined q=d+p- 1 =d.

F ( z ) = 1 + fiz-' + f2Z-* + - -. + f d Y d Theorem 3 The controller is stable and reali-

zable, the output can track desired queue length in the least steps, if the roots of Eq. (25) are located in the unit circle.

From Eqs.(22) and (24), it is easy to obtain the result of Eq.(25).

For simplicity, the round trip time is assumed to be not larger than thrice the times of computation

period. It is reasonable, as B - z, is chosen,where zmax

can be estimated in advance. So the maximum delay is obtained in Eq.(8), d is 3. The velocity of tracing depends on the number of the terms in the given closed loop Z-transformations @(z) = z - ~ M ( z ) . For d

is a constant here, it depends on M(z). Now M(z) is chosen as a one-order polynomial. The highest power of F(z) can be determined q=d+p- 1=3.

Theorem 4 The controller is stable and reali- zable, the output can track desired queue length in the least steps, if (26) is satisfied.

F ( z ) = 1 + z-'+ z-' + . * * + z-~'' + m,z-d (25)

1 3

F ( z ) = 1 + z-' + z-2 + qy3 o<m, <1

(26)

The detailed proof for Theorem 4 can be seen in Appendix A. The relationship between m,ml is shown in Fig. 4. And the location of eigenvalues of F(z) are shown in Fig. 5.

" O 4 Irn 4

Fig.4 The relationship Fig.5 The location of of m o m eigenvalue of F(z)

1 3

Remark The assumption, B- t,,, , may not

be available all the time, because the propagation delay is variable and the period T is fixed. Though z,, can be estimated in advance and then T is chosen, satisfying the assumption, the larger the T is, the worse the performance may be. So a list of parameters is made to deal with the varied propagation delay. If the delay has large oscillation, d and ml can


be adjusted to guarantee the stability of both the controller and closed loop system. Similarly, the following conclusion can be obtained.

The controller is stable with d=4,5,6,***, if ml is in the interval (0,l).

The locations of roots of controller are given in Fig.6. It can be seen that the larger the d is, the more unstable the controller will be. The reason is that the roots of controller are close to unit circle inch by inch. The simulations also verify the remark.

Therefore, the controller is

d=5 &6

Fig.6 The location of roots of Controller witb different d

The control law is 1 d -1

r(n) = C m , e ( n - i) - C r ( n - j ) - m,r(n - d ) (28) i=O j=1

and error signal is

where h(n ) is described in the next part.

3.2 The feed-forward controller

To reduce the output signal caused by disturbance D(n), the feed-forward controller is adopted as shown in Fig. 7.

The Z-transformation, from D(n) to Q(n), is denoted as Q&). From Fig.7 we can get

4 n ) = Qd + - Q(n> (29)

Q D ( z ) = { - D ( z ) + [-Q,(z) + D(z)F~(z)IG,(z)z-~ lG0(z)

I U I

I J

Fig.7 The block graph with feed-forward structure

If we choose F&) as the following

It is clear that Eq. (32) holds.

Namely, the oscillation of disturbance has no effect on output. We rewrite Eq.(31) as

Qdn)=O (32)

(33) [I + z-l + 2-* + * + z-d+l + q z - d ]

m,yd + qyd- l

and get

(34)

The time domain relation between D(n) and b (n) can be described as

fi(n)(m,z-d + q z - d - l ) =

D(n)(l+z-l +2-2 + * * - + . z - ~ - ' + q ~ - ~ )

m0&n - d ) + F?@(n - d -1) =

D(n) + D(n - 1) + D(n - 2) + ..- + q D ( n - d ) (35)

If d and T are small, and the oscillation of D(n) is not too violent, D(n)+D(n-l)+*-+D(n-d) can be used to substitute the term, D(n+d)+D(n+d-l)+D(n+d-2)+--- +Wn) .Then

.. 1

m0 D(n) =-[D(n) + D(n - 1) + D(n - 2) + . . a +

q ~ ( n - d ) - q h ( n - 111 (36) It needs to be pointed out that the error maybe exists. So the response caused by D(n) is not zero in fact. The simulations also verify it. If the parameters are chosen appropriately, the effect on output caused by disturbance can be reduced effectively. Otherwise the exact predictor is needed to predict the available bandwidth ford steps.

3.3 The fairness of VCs

MAX-MIN fairness is a frequently adopted fairness ~riterion"~'. MAX-MIN fairness can be defined as follows.

(1) Every bandwidth request is satisfied or if any requests are not satisfied, bandwidth not used by satisfied requests must be equally allocated to unsatisfied requests. The bandwidth allocated for each unsatisfied request is no less than bandwidth allocated for a satisfied request.

(2) Condition( 1) is applied to each link.

Congestion control scheme in ATM networks based on fast tracing-queue 107

If a rate-based flow control method follows MAX-MIN fairness criterion, it can be said that it is a rate-based control method with MAX-MIN fairness. As mentioned above, a VC from source to destination usually passes several links. The link with the smallest bandwidth allocated for the VC is called a bottleneck link. For a VC with a bottleneck link, its bandwidth in other links is limited by the bandwidth allocated in the bottleneck link. So the VC has an equal amount of bandwidth allocation in all links.

This proposed method is based on the framework of ER feedback mechanism. The RM cell is sent from the source to every Nrm cell with the initial ER value. RM cell may pass several links and switches. When it passes by a switch, the associated controller computes the available resource for each source. The update of ER stored in RM can be divided into three phases.

Phase 1 Collect the local useful information in the switch and received RM cell, such as available bandwidth, queue length, the arrival rate, the number of VCs, the ER value of the arrived RM cell which is modified in upstream switch, and so on.

Phase 2 Compute the ideal control law r(n) according to the structure shown in Fig.7 and other related parameters. Here the following method is adopted, to determine the rate of connections, which are not bottlenecked in the local switch, and achieve the parameter q ( n ) in (5) . If the former sending rate, which is calculated by the controller in the last period, is bigger than the ER of received RM cell, it is considered that they are bottlenecked elsewhere. It is assumed they are in the set B, and &(n)= CCR,..

j r B

The flow chart of this process is described in Appendix B. This is to say the phase is the process of looking for the bottleneck node. So the assumption is reasonable that there is only one bottleneck link for each VC. Then the control law(advising sending rate for source end) is achieved according to the above formulations. The flow chart of control law computation is described in Fig. 8.

Phase 3 Update the value of ER in RM. It is assumed that there are K connections bottlenecked here totally. The rate is satisfied Eq.(l) constraint. For each connection bottlenecked here, the absolutelideal fair rate is

(37)

Since the existence of estimated error during the above phase, the calculated rate may be larger than the ER value stored in the received RM. The bottleneck link does not locate in the considered switch. The ER value cannot be changed. In this phase, when the switch receives the backward RM(BRM) cell, the rule of modifying ER field is

Thus the dynamic fair rate allocated to each VC is obtained.

ER(n) =min{ER,r,,(n)} (38)

Set d by estimation and determine rn,, rn,

4 Get Q(n) by measurement

Calculate D(n) according to Appenddix B

Calculate D(n) according to (36)

Calculate e(n) according to (29)

+ Calculate rfn) according to (28)

Fig.8 The flow chart of control law computation

The source end behavior can refer to Ref. [17], and it is neglected here.

4. Simulations

Figure 9 is adopted as the network topology in the simulations. For simplicity, it is considered the single bottleneck link, that is, there are n(n=5) ABR VCs and a VBR connection that share a link(the VBR services refer to all those traffic whose priority is higher than ABR).

155M VBR

ABR 1

ABR5

SWI

Fig.9 The network topology with VBR tr&c

MCR and PCR values are set at 0 Mb/s and 155 Mbh respectively. The maximum round trip time is 30 ms, and sample period is T=10 ms. The desired queue length is 200 cells and the buffer size of switches is 400 cells.

Scenario 1 The scenario is tested for the non-


oscillatory condition of available bandwidth(i.e., VBR traffk is constant). The output queue length is shown in Fig. 10, when the controller has different parameters. The queue length of the buffer can converge to the desired value quickly. But it has overshot. The main reason is the existence of error between the predictive available bandwidth at the beginning, as shown in (36). Besides, with a different parameter of the controller, rnl , the system has different dynamic performance. If rnl is changed, the roots of F(z) are changed.

Scenario 2 It is assumed that the background traffk is bursty flow, the output is shown in Fig.11. When the VBR is oscillating around 100 Mbs and the magnitude is in the range [-20 Mbs, 20 Mbs] and [-50 Mbs, 50 Mbs], which is denoted as VBRl, VBR2, respectively, the queue length exhibits a wave that is shown in Fig.12. It is caused by the imprecise available bandwidth used in the feed-forward control- er. It is also in the finite range(buffer size). Thus there is nearly no cell lost. The bigger the oscillated range is, the worse the dynamic output performance will be as shown in Fig.12. And there are cells to be dropped when the oscillation is too intense. The quality of service, such as cell loss rate, throughput, propagation delay, may be worse.

0- 00- tis r i s

-m,=0.5; -m,=0.3 -VBRI; -VBM Fig.10 The queue length with Fig.11 The background tra&c

constant available bandwidth of VBR service

In another scenario the performance of our rate control scheme is investigated. Here it is assumed that the VBR services is on-off traffic, the switch time is 20 ms. Then the available bandwidth is equal to the difference between the link capacity and occupancy by VBR, as shown in Fig.13. When rnl=0.5, the dynamic queue length is around 200 cells. It can be seen that the maximum of the queue length is nearly 320 cells. the buffer size of the switch is set at 400 cells as shown in Fig.14, so there are no lost cells. And there are queuing cells waiting for service all the time, there is nearly no waste of bandwidth. With different control parameters the sending rate of source and link

utilization are shown in Fig.15. and Fig.16, respectively.

400r,. . . , , . . , . 81201 ' ' ' ' ' ' ' ' ' I

./I

--Quael; -Queue2 tls

Fig. 12 The queue length with Fig.13 The available bandwidth different VBR services under on-off VBR traffic

350- 101 , . , , 1

u s t is -m,=0.5; -m,=0.3

Fig. 14 Queue length with Fig. 15 Sending rate with ml=0.5 different parameters

At last the queue length is compared with Ref. [7], under the same conditions. Here let ml=0.3. The simulation result is shown in Fig. 17. The magnitude of oscillation is reduced significantly.

tJS tis -m,=0.5; -m,=0.3 -PID control; -New method

Fig.16 Link utilization with Fig.17 The comparison of ifferent parameters queue length

5. Conclusions

A flow control algorithm for ABR services management is presented. It is under the mechanism of explicit rate(ER). The new controller is composed of a queue-tracing controller and a feed-forward controller, which are used to trace the given queue length in the least steps and reduce the output response caused by disturbance, respectively. High bandwidth fairness level is achieved in the presence of bursty VBR traffic is based on the MAX-MIN fairness principle. With a rigorous theoretical analysis, the stable conditions of closed loop system and the controller are given. This mechanism needs less parameters(on1y one parameter to be designed in controller) and can tolerate large delay. From simulations, it is known that the closed

Congestion control scheme in ATM networks based on fast tracing-queue 1 09

-loop system is stable and the output can trace the desired queue length as quickly as possible under the designed controller. The feed-forward controller shows good performance of reducing disturbance. It needs to be pointed out that the more peaceful the background flow is, the better the dynamic performance is. If the uncontrolled service can be predicted exactly, the effect of disturbance can be reduced further. The prediction of available band width for ABR service is the next work to be researched.

Appendix A

Lemma(Jury stability criterion): Consider the characteristic equation P(z) is polynomial of z,

D(z)=Q +a1z+a2z +.--+a,z", a,& The sufficient and necessary condition of all roots of D(z)=O located in the unit circle is that the following n conditions are satisfied.

2

(2) laol<la,l (3) lb&dbn-ll (4) Icol<lc,-2l ( 5 ) Idol>lc,-31-~(n)l qo 1>1 qa I

Where the parameters ak, bk, ..*, q2 are defined as follows

...

Po P3 Po Pz Po P1 qo = / P3 Po I? q1 =I P3 Pl 1 9 qz = I p 3 p z l

The proof of Theorem 4.

controller is

If the parameter ml is in the interval [0,1], it is easy to test that the following formulations hold.

When n is 3, the characteristic equation of

~(z)=rn~+z-'+z-~+z"

( 1 ) D(1)=3+mpO, D(-l)=ml-l<O (2) uo=m1<a3=1

( 3 ) IboH -1 k 1 - qZ Ib21=ll -mlI=l-ml Ibol>lb21

Conditions (1)-(3) hold, Jury stability criterion is satisfied.

Similarly, it can be validated that Jury stability criterion is satisfied with ml E (O,l), when n=4, 5, 6;-. Due to the lack of space, the proofs are neglected here.

Appendix B

We give the flow chart of Phase 2 as shown in Fig. 18. It is assumed that the process will be completed in time slot nT. First, we give definitions of variables in the flow chart.

N: the total number of connections passing by the considered buffer j .

si: the active state of the ith connection, and has the following rule

1

U(n) = CCR, (n) I "; I D(n)= C, - C,, - U(n)

Fig.18 The flow chart of computation of

available bandwidth for ABR

(1 if there are at least N , sells arrival during the time slot nT,(No=l or 2) i Si=

L O otherwise

ZC the number of connections bottlenecked in

S: the set of N connections. B: the set of connections bottlenecked elsewhere. CL: the total link capacity. D: the bandwidth for ABR. CVBR: the bandwidth occupied by high priority

buffer j .

services.

U: the sum bandwidth of connections in set B.


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Liu Zhirdn was born in 1976. He received the B.S. and M.S. degrees both in Yanshan University in 2000 and 2003, respectively. He is now a Ph.D. candidate of Department of Electrical Engineering, Yanshan University. His research interests include congestion control of communication networks and intelligent control. E-mail: lzxauto@ysu. edu. cn

Gum Xinping was born in 1963 and now is a professor and doctoral advisor of Department of Electrical Engineering, Yanshan University. He is also the member of IEEE. His research interests include robust congestion control in communication network, chaos control.

congestion control scheme in atm networks based on fast tracing-queue

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