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ROBUST CONGESTION CONTROL IN HIGH SPEED COMMUNICATION NETWORKS: A MODEL PREDICTIVE CONTROL APPROACH
m
Copyright 02002 IFAC
ELSEVIER
15th Triennial World Congress, Barcelona, Spain
PUBLICATIONS
www. elsevier.com/locate/ifac
ROBUST CONGESTION CONTROL IN HIGH SPEED COMMUNICATION NETWORKS: A MODEL PREDICTIVE CONTROL APPROACH Hua 0.Wang *,**,I Yongru Gu * Huajing Fang **
*Laboratoryfor Intelligent and Nonlinear Control (LINC) Department of Electrical and Computer Engineering Duke University, Durham, NC 2 7708-0291 USA ** Centerfor Nonlinear and Complex Systems Department of Control Science and Engineering Huazhong Universi@of Science and Technology Wuhan 430074, China
Abstract: In this paper, the design of explicit rate-based congestion control in high speed communication networks is considered. At a bottleneck node, there are multiple best-effort sources competing with other high priority cross traffic sources. The goal of congestion control is to achieve high link utilization, low packet loss, low delay, and fairness among the best-effort sources. In this paper, the high priority traffic is described by an autoregressive integrated moving average (ARIMA) process. To deal with the propagation delays associated with the best-effort sources, model predictive control, particularly, generalized predictive control, techniques are proposed to solve the congestion problem here. It is demonstrated that the proposed controller performs well and is robust to delay uncertainties. In addition, in a multiple-nodes configuration, the controller provides max-min fairness. Copyright 02002 IFAC Keywords: Communication control applications, Model-based control, Predictive control, Communication networks, Time delay, Robustness.
status in the networks. This is referred to as rate-based congestion control. The goal of congestion control is to achieve high link utilization, low packet loss, low delay and delay variation. This can be accomplished by designing a controller to maintain the best-effort traffic buffer at the bottleneck at a target length.
1. INTRODUCTION Traffic in high speed communication networks can be broadly classified as high priority cross traffic and best-effort service traffic. For example, in the context of Asynchronous Transfer Mode (ATM) networks, the best-effort traffic corresponds to Available Bit Rate (ABR) traffic while the high priority cross traffic corresponds to Constant Bit Rate (CBR) and Variable Bit Rate (VBR) traffic. Generally, as indicated, the high priority cross traffic is of high priority and its flow cannot be regulated by the networks. Thus, the service rate for the best-effort traffic at the node is the total link bandwidth deducted by the rate of the high priority cross traffic. In addition, the networks can exercise control over the best-effort traffic by assigning rates to its sources based on congestion
Early rate-based congestion control work involves binary and proportional feedback schemes (Fendick, et. al., 1992; Ramakrishan and Jain, 1990 ; Roberts, 1994; Yin and Hluchyj, 1994). One drawback of these schemes is that they often encounter problems of stability, oscillatory dynamics and require large amount of buffer to avoid packet loss. Later, as of a consequence, people consider explicit rate schemes such as those in (Benmohamed and Meerkov, 1993; Charny, et. al., 1995; Kolarov and Ramamurthy, 1999; Lengliz and Kamoun, 2000). As the product of propagation delay and bandwidth of the networks continues to increase, people began to consider the problem of so called “action delay” in their congestion controller deign. Altman Imer and their colleagues (Altman, et.
This research was supported in part by the Lord Foundation of North Carolina, the U S . Army Research Office Grant DAAD19-0001-0504, the Ministly of Education of China, and the Hubei Natural Science Foundation.
103
al., 1999; Imer, et. al., 2001) use Linear-QuadraticGaussian (LOG) team techniaues to solve the action delay problem. Mascolo et al. 11996,2000) have used Smith principle to compensate propagation delays in their schemes for ATM networks and TCP/IP networks. Ozbay (1998) Wang (2000) have proposed to use robust H” controllers. Various congestion control mechanisms proposed for high speed communication networks can be classified into two categories: pro-active control and reactive control. The above control schemes fall into the reactive mechanism. They may be too slow to be effective in high speed communication networks. As indicated, pro-active congestion control techniques aims at prevent congestion by taking appropriate actions before it actually occurs. In this paper we use the pro-active control approach, particularly, model-based predictive control (MPC), to study congestion control in high speed communication networks. In our MPC-based congestion control scheme, a model including time delays is employed to predict the buffer level and the best-effort traffic rates are adjusted according to the prediction results. MPC has been found to be a quite robust type of control in industry application. It can handle time delays explicitly. In addition, having noticed the high priority cross traffic, especially, at local communication networks, is time-varying and non-stationery, we model the service rate (or the available bandwidth) for best-effort traffic as an autoregressive integrated moving average (ARIMA) process, while most of the above papers assume that it is constant, slowly-varying, or stationery. By modeling the service rate as an ARIMA process, our controller can handle both bursty cross traffic and non-stationery cross traffic. In addition, since in our scheme the bottleneck node will issue the same rate command to the best-effort sources, the basic fairness can automatically be provided without tuning any parameters.
2. NETWORK MODEL AND PROBLEM FORMULATION
time unit corresponds to the interval over which the rate available to the best-effort sources is determined. The best-effort sources are controllable sources and emit at rates specified by a central controller residing at the bottleneck node. Associated with each source is a source-dependent round-trip delay or “action delay”, which consists of so called “downstream delay” and “upstream delay” (Altman, 1999). Assume that the number of the best-effort sources is M and the round-trip delay for source i is di time units where di is a positive integer. Without loss of any generality, assume that dm’s are ordered such that
0 5 d 5 d , 5 . . .IdM 5 d where d and d correspond to the minimum and maximum round-trip delays. Consider the example in Figure 1, let q(k) denote the buffer length at time instant k, then the dynamics of the buffer is described by:
where ri(k) denote the total number of packets from best-effort source i that arrive in the buffer in the interval [k, k + l ) , and c(k) denotes the number of packets that depart from this buffer in the same time interval. Note that ri(k)represents source i’s rate. c(k) represents the service rate for the best-effort traffic. The controller, residing at the bottleneck node, makes control decision at each time step. The congestion control problem is to determine a rate command ui(k) to relay to source i, i = 1, . . . , M , at time k to achieve some overall performance objective. However, source i will not immediately change its rate to the new rate ui(k).It takes time for the rate command to reach the source and subsequently for the traffic with the new rate to affect the state of the node. So the buffer’s input rate from source i is equal to the rate command di time units ago, i.e., ri(k)= u i ( k - d J , i = 1,. . . ,M. Therefore, we can obtain the buffer dynamics as follows: M
@
<
i................................................................................................ Buffer BottleneckNode
:
@a
HS I mgh Pnonty Source 1 BS I Best-effort Source I
Fig. 1. Model at a Bottleneck Node In this paper, we will consider congestion control design for a single bottleneck node, which is shared by multiple best-effort traffic sources and other high priority cross-traffic sources, shown in Figure 1. The mathematical model we will adopt to describe the buffer’s behavior is a discrete-time model, where a
Our objectives are to achieve high link utilization, low packet loss, low delay, etc. These objectives can be accomplished by regulating the buffer length at the bottleneck node around a desired level. Tracking such a nominal buffer length is desirable to avoid loss due to overflow and waste of link capacity due to underflow. In addition, we are interested in a fair share of the available bandwidth among the best-effort sources at the bottleneck node. So in our design, we let u1 ( k ) = u,(k) = . . . = uM(k)and as a result, we have: M
Then, the buffer dynamics can be characterized by:
104
whereA(zP1)= 1 -zP1 andB(zP1)= &z-(dj-d).
.
The high-priority cross traffic represents, for example, CBWVBR in ATM network, and it determines the service rate that the best-effort traffic experiences at the bottleneck node. For convenience, instead of specifying the cross traffic distribution, we specify the “service process,” which is the difference between the rate of cross traffic and the link bandwidth at the bottleneck node. We assume that the service process is represented by an ARIMA process:
Objective Function J
Fig. 2. Overall control Structure
c n[q(i+klk) + c h,[Au(i+ N2
~ ( z -) ( 1 - z- ) ( -c - c(k)) = P ( Z ) e ( k ) (5
+
+
-qd12
J=E{
i=N1
NU
+ +
k - 1)12} where C(zP1) = 1 clzP1 c2zP2 . . . C ~ ~ Z - ~ C , (8) i= 1 P(zP1)= 1+plzP1+p2zP2+.. .+ p , , ~ - ~ p . The driving term { e ( k ) } k r lis a zero-mean i.i.d Gaussian sewhere n’s and 4 ’ s are positive constants, q(i klk) is the prediction of the buffer length at time i + k, quence with variance 0,”. e,, i = 1, . . . , nc andp,, i = and qd is the target buffer length. Nl and N, are the 1, . . . , nd are the parameters characterizing the prominimum and maximum cost horizons and Nu is the cess, which can be estimated based on maximum likelihood criterion. -c is the mean service rate. -c is not control horizon. In the buffer model (7), there is a time delay d. The buffer length will not be affected required to be known as we can see in the derivation by current explicit rate issued by the node until d of our controller later. Note that (1 - z-’) -c = 0, then 1 time units later. Therefore there is no reason for we have: Nl to be less than d 1. In our design, we will let Nl = d + 1, N, = N + d and Nu to be the same as the maximum cost horizon. The first term in the cost function represents the penalty for deviating from the Substitute equation (6) into equation (4), we have: desired buffer length qd. The second term represents a penalty for the control effort. It can help the closedloop system to be stable. A(z-l)q(k+ 1) =B(z-l)z-du(k)
+
+
+
Multiply A = 1 -zP1 to the both sides of equation (7), we can get A”(z-l)q(k+ 1) =B(z-l)z-dAu(k)
In the following, based on model (7), we develop an algorithm to calculate the explicit rate u(k) to avoid congestion in high speed communication networks.
+
where
3. MODEL-BASED PREDICTIVE CONGESTION CONTROL
a(,-’)
= (1 - zP1)A( z
(9)
C(z-1)
1.
From the above equation, we know that C(zP1) can be absorbed into A”(zP1)and B(zP1).For simplicity, in the following we will only consider the case where C(zP1)is equal to 1. Then the ABR buffer dynamics becomes:
As mentioned early, we need to design a controller to regulate the buffer length at the bottleneck node around a desired level. In the design of such a controller, an important issue is how to deal with time delays. MPC is widely used in industry for its capability of handling time delays. This motivates us to use model-based predictive control to deal with the congestion control problem here. Particularly, we will use the technique of generalized predictive control (GPC) developed in (Clarke, et. al., 1987).
A”(z-l)q(k+ 1) =B(z-l)z-dAu(k) +P(z- ) e(k)
(10)
This is a so-called CARMA (controlled autoregressive moving average) model (Clarke, et. al., 1987).
Our overall congestion control structure is shown in Figure 2. First, based on model (7), the future buffer lengths for a determined prediction horizon N are predicted. Then, a set of future best-effort source rates is calculated by optimizing a determined criterion in order to keep the buffer length as close as possible to the desired buffer length. The criterion we adopt here is of the form of a quadratic function, which is:
The objective of predictive control is to compute future best-effort source rate sequence u ( k ) , u(k+ l ) , . . . in such a way that the future buffer length q(k+ j ) is driven close to qd. This is accomplished by minimizing J . In order to optimize the cost function the optimal prediction of q(k+ j ) for j 2 d 1 and j 5 N d need to be obtained. From (10), we know that the buffer level at time k + j is:
+
105
+
where G,(zP1) has the form of x { i i g j , i z - i , and G,(zP1)and H,(zp1) can be obtained by solving the following Diophantine equation:
Equation (1 1) can be rewritten as:
B ( z - ' ) E , + ~ ( z - ~=) G,(z-')P(z-') + z - j H ( z P 1 )(19) So we have:
where E,(zp1)a n d 5 (.-I) can be obtained by solving the following Diophantine equation:
P(z-')
= E , ( z - ' ) ~ ( z - ' )+zPiF,(zp1)
(13)
In equation (12), e ( k - 1) can be computed in terms of data available at time k, i.e., e ( k - 1) is given from (10) as
Therefore, for j
=
1,. . . ,N, we can get
q = G A u + H ( z - ' ) A u ( k - 1) + F ( Y 1 ) q ( k ) (21) where
Substitute (14) into equation (12), we obtain:
As the degree of polynomial E,(zp1) is j - 1 the noise terms in equation (15) are all in the future. The best prediction of q ( k + j ) is therefore (Astrom and Wittenmark, 1990):
4(k+jlk) =
B(z-I)E, (z-1) A u ( k + j - d - 1) P(z-1) The last two terms in equation (21) only depend on the past. If we group them into f , we can obtain:
we can obtain the j
+ d ahead optimal prediction as:
@(k+j+dIk)=
B(z-l)Ej+d(z-l) A u ( k + j - 1) P(z-1)
q=GAu+f Substitute (22) into cost function (S), we have J = ( G A u + f - q d ) T r ( G A U + f - qd)
+AuTAAu In the cost function (S), we need to obtain the future rate change sequence A u ( k ) , A u ( k + l ) , . . ., A u ( k + N - l ) , so we need to extract future rate change A u ( k + j ) for j 2 0 from equation (17). To do this, we can rewrite parts, i.e.,
B(2-')Ejfd(2-1) P(2-1)
where
qd
=
[qd qd ... qdl
r=diag[x
A u ( k + j - 1) into two
fi
T
...,rNl
A = diag[h, h2 . . . , A N ] Equation (23) can be written as: J
106
=
-1A u T W A u 2
+ b T A u+ f o
where
1500,
I
,
,
j
j
!
!
:
.
,
,
,
I _Th;Y.nhOr.nr....Tr-. '
(r
"."...". ,
..... 60001
..
..
!
..
I
,. I
.. .
I
,..
,..
,..
.. .
I
.. .
I
.
.
I
.
.. .
,
I
I
The minimum of J can be now found by making the gradient of J equal to zero, which leads to:
AU = -Wplb
(25)
;mot
The first element of AU together with u ( k - 1) will be used by the node to generate current explicit rate command u ( k ) , which will be sent to the sources. Other elements of AU are rejected. The procedure is repeated at the next time instant.
~-.,,,,,,,,,,,,,~ .
.. . j
.
.. .
.. .
j
j
.
.
.
High Pnonty Traffic, Buffer Length and ABR Sources' Rate
One source of error is the assumption that the delay di, i = 1, .... M , is an integer multiple of the unit time. In addition, there is variability in the delays due to queuing.
4.1 Simulation Setup
In the following, we will test the performance of robustness of our controller against action delay uncertainty by considering B(zpl) uncertainty in equation (27). In the modeling we let B(zpl) = 1 + z p l +zp2 while the real B(zpl) may be 1 + 2zp1, 1 + 2zp2, and 3, respectively. A representative robustness result is illustrated in Figure 4 where the real model for B(zpl) is 1 + 2zp1 while in the modeling we let B(zpl) = 1 + zpl +zp2. Similar results hold for the other two cases. Our controller performs very well in all cases.
Consider Figure 1, we assume it is an ATM network, and the best-effort sources are 3 ABR sources. The high priority sources are either CBR or VBR sources. The link capacity is 155 Mbps. The unit time is 3 ms, then the link capacity is equivalent to 1097 cells per unit time. In addition, we assume the desired ABR buffer length is 3000. We model the CBR and VBR sources' traffic collectively as a high priority traffic. The data of this high priority traffic comes from a mixture of the video traces from MPEG-1 video traces (1997), a CBR source with a rate of 141.5 cells per unit time, and an Om'Off VBR source with a mean rate of 141.5 cells per unit time. The mean rate of the high priority traffic is 456 cells per unit time. It is shown in Figure 3. Then the mean available bandwidth for the three ABR sources is 641 cells per unit time. We assume the action delays for sources 1,2, and 3 are 2, 3, and 4 unit times, respectively. Thus the ABR buffer model at the bottleneck is:
+
:
Fig. 3.
4. SIMULATION STUDIES
q(k+ 1) = q(k) u ( k - 2)
.
5000
,.
,.
.
.
j.
j.
,.
,
,
.,
,.
,.
,.
4000 I
8 3000 I
: 2000 2,000~
..
..
ABRmean queue length 2978
! l .. .
.. ..
' ! ; - I .. ..
.. ..
..
+ u ( k - 3)
+u(k-4)
-
~ ( k ) (26)
i.e.,
Fig. 4. Robustness Simulation: Real B ( z - ' )
whereA(zpl) = 1 -zpl, andB(zpl) = 1 +zp1+zp2.
4.3 Multiple Sources Multiple Nodes Configuration .for Max-min Fairness
If the controller perform well, then the buffer length will be regulated around 3000 and the mean rate will be 213.7 cells per unit time. Figure 3 shows the simulation results. We can see the buffer length is controlled around 3000 as expected. In addition, the mean ABR sources' rate is 213, which is very near the expected value 2 13.7 cells per unit time.
=
1f 2 z - l
In real networks, there may be multiple bottlenecks. In such cases, one important performance index is max-min fairness. In this section, we want to check the performance of max-min fairness of our controller under a two-node network configuration shown in Figure 5. High priority traffics 1 and 2 are of the same kind as the one in the previous section. The mean rate for high priority traffic 1 is 420 cells per unit time, and for high priority traffic 2 it is 228 cells per unit time. Thus the mean service rates at nodes 1 and 2 are 677 and 869 cells per unit time. At each node, there is a congestion controller running. We assume the delays
4.2 Robustness In addition, we consider the robustness of our controller against modeling errors of the action delays.
107
I
,
-0 -0
sw2
Fig. 5. Max-min Fairness Simulation Setup Figure 6 and 7 show the max-min fairness simulation results. The buffers 1 and 2 are regulated around 3000 as expected. Sources 1, 2, and 3 emit at a rate very near to their max-min fairness share of 225.7 cells per unit time. Source 4 and 5's rate is also close to their ideal value of 321.5 cells per unit time. Therefore, our design can achieve max-min fairness.
800,
,
,
,
,
,
,
,
,
I
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"
7""
a""
6""
8""
4"""
49""
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48""
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T,me 13ms
Fig. 6. Max-min Fairness at Node
........j . . .
1
. . j ........
..................
...
i
Fig. 7. Max-min Fairness at Node 2
5. CONCLI JSION
In this paper, a preventive congestion controller has been proposed using a model predictive control scheme for high speed communication networks. It is demonstrated that the controller performs very well under various criteria such as max-min fairness and high link utilization and is robust to time delay uncertainties. Model predictive control appears to be a viable approach to develop congestion control algorithms for high speed communication networks.
Altman, E., T. BaSar, and R. Srikant (1999). Congestion control as a stochastic control problem with action delays. Automatica, vol. 35, no. 12, pp. 19371950. Wstrom, K.J. and B. Wittenmark (1990). Computercontrolled systems: Theoq and design. Second Edition, Prentice Hall, Englewood Cliffs, NJ. Benmohamed, L. and S. M. Meerkov (1993). Feedback control of congestion in packet switching networks: the case of a single congested node. IEEE/ACM Trans. on Networking, vol. 1, pp. 693707. Charny, A,, D. D. Clark and R. Jain (1995). Congestion control with explicit rate indication. Proc. of IEEE ICC'95, Seattle, WA, vol. 3, pp. 1954-1963. Clarke, D.W., C. Mohtadi and P. S. Tuffs (1987). generalized predictive control - Part I. The basic algorithm. Automatica, vol. 23, no. 2, pp. 137-148. Fendick, K.W., M. A. Rodrigues and A. Weiss (1992). Analysis of a rate-based feedback control strategy for long haul data transport. Performance Evaluation. 16,6744. Imer, O.C., S. Compans, T. BaSar, and R. Srikant (2001). Available bit rate congestion control in ATM networks. IEEE Control Systems Magazine, vol. 21, no. 1, pp. 38-56. Kolarov, A. and G. Ramamurthy (1999). A controltheoretic approach to the design of an explicit rate controller for ABR service.IEEE/ACM Trans. Networking, vol. 7, no. 5, pp. 741-753. Lengliz, I. and F. Kamoun (2000). A rate-based flow control method for ABR service in ATM networks. ComputerNetworks. vol. 34, no. 1, pp. 129-138. Mascolo, S., D. Cavendish and M. Gerla (1996). ATM rate based congestion control using a Smith predictor: an EPRCA implementation. Proc. IEEE INFOCOM '96, vol. 2, pp. 569 -576, San Francisco, CA. Mascolo, S. (2000) Smith's principle for congestion control in high speed data networks. IEEE Trans. on Automatic Control, vol. 42, no. 2, pp. 358-364. Ozbay, H., S. Kalyanaraman and A. Iftar (1998). On rate-based congestion control in high speed networks: Design of an H" based flow controller for single bottleneck. Proc. of 1998 American Control Conference, pp. 2376-2380, Philadelphia, PA. Ramakrishnan, K. and R. Jain (1990). A binary feedback scheme for congestion avoidance in computer networks with a connectionless network layer. ACM Transactions on Computer Systems. vol. 8, no. 2, pp. 158-181. Roberts, L. (1994). Enhanced PRCA (Proportional rat-control algorithm). Available at http://ww. atmforum. org. Wang, L.Y. and L. Schwiebert (2000). Robust control and rate coordination for efficiency and fairness in ABR traffic with explicit rate marking. Proc. of the American Control Confirence, Chicago, IL, pp. 1975-1979. Yin, N. and M. G. Hluchyj (1994). On closed-loop rate control for ATM cell relay networks. Proceedings sf infocorn 94. Los Alamitos, CA, USA, vol. 1, pp. 99-108. MPEG-1 video traces (1997). Available at http://nero.informatik.uni-wuerzburg.de/MPEG/.
108
Copyright 02002 IFAC
ELSEVIER
15th Triennial World Congress, Barcelona, Spain
PUBLICATIONS
www. elsevier.com/locate/ifac
ROBUST CONGESTION CONTROL IN HIGH SPEED COMMUNICATION NETWORKS: A MODEL PREDICTIVE CONTROL APPROACH Hua 0.Wang *,**,I Yongru Gu * Huajing Fang **
*Laboratoryfor Intelligent and Nonlinear Control (LINC) Department of Electrical and Computer Engineering Duke University, Durham, NC 2 7708-0291 USA ** Centerfor Nonlinear and Complex Systems Department of Control Science and Engineering Huazhong Universi@of Science and Technology Wuhan 430074, China
Abstract: In this paper, the design of explicit rate-based congestion control in high speed communication networks is considered. At a bottleneck node, there are multiple best-effort sources competing with other high priority cross traffic sources. The goal of congestion control is to achieve high link utilization, low packet loss, low delay, and fairness among the best-effort sources. In this paper, the high priority traffic is described by an autoregressive integrated moving average (ARIMA) process. To deal with the propagation delays associated with the best-effort sources, model predictive control, particularly, generalized predictive control, techniques are proposed to solve the congestion problem here. It is demonstrated that the proposed controller performs well and is robust to delay uncertainties. In addition, in a multiple-nodes configuration, the controller provides max-min fairness. Copyright 02002 IFAC Keywords: Communication control applications, Model-based control, Predictive control, Communication networks, Time delay, Robustness.
status in the networks. This is referred to as rate-based congestion control. The goal of congestion control is to achieve high link utilization, low packet loss, low delay and delay variation. This can be accomplished by designing a controller to maintain the best-effort traffic buffer at the bottleneck at a target length.
1. INTRODUCTION Traffic in high speed communication networks can be broadly classified as high priority cross traffic and best-effort service traffic. For example, in the context of Asynchronous Transfer Mode (ATM) networks, the best-effort traffic corresponds to Available Bit Rate (ABR) traffic while the high priority cross traffic corresponds to Constant Bit Rate (CBR) and Variable Bit Rate (VBR) traffic. Generally, as indicated, the high priority cross traffic is of high priority and its flow cannot be regulated by the networks. Thus, the service rate for the best-effort traffic at the node is the total link bandwidth deducted by the rate of the high priority cross traffic. In addition, the networks can exercise control over the best-effort traffic by assigning rates to its sources based on congestion
Early rate-based congestion control work involves binary and proportional feedback schemes (Fendick, et. al., 1992; Ramakrishan and Jain, 1990 ; Roberts, 1994; Yin and Hluchyj, 1994). One drawback of these schemes is that they often encounter problems of stability, oscillatory dynamics and require large amount of buffer to avoid packet loss. Later, as of a consequence, people consider explicit rate schemes such as those in (Benmohamed and Meerkov, 1993; Charny, et. al., 1995; Kolarov and Ramamurthy, 1999; Lengliz and Kamoun, 2000). As the product of propagation delay and bandwidth of the networks continues to increase, people began to consider the problem of so called “action delay” in their congestion controller deign. Altman Imer and their colleagues (Altman, et.
This research was supported in part by the Lord Foundation of North Carolina, the U S . Army Research Office Grant DAAD19-0001-0504, the Ministly of Education of China, and the Hubei Natural Science Foundation.
103
al., 1999; Imer, et. al., 2001) use Linear-QuadraticGaussian (LOG) team techniaues to solve the action delay problem. Mascolo et al. 11996,2000) have used Smith principle to compensate propagation delays in their schemes for ATM networks and TCP/IP networks. Ozbay (1998) Wang (2000) have proposed to use robust H” controllers. Various congestion control mechanisms proposed for high speed communication networks can be classified into two categories: pro-active control and reactive control. The above control schemes fall into the reactive mechanism. They may be too slow to be effective in high speed communication networks. As indicated, pro-active congestion control techniques aims at prevent congestion by taking appropriate actions before it actually occurs. In this paper we use the pro-active control approach, particularly, model-based predictive control (MPC), to study congestion control in high speed communication networks. In our MPC-based congestion control scheme, a model including time delays is employed to predict the buffer level and the best-effort traffic rates are adjusted according to the prediction results. MPC has been found to be a quite robust type of control in industry application. It can handle time delays explicitly. In addition, having noticed the high priority cross traffic, especially, at local communication networks, is time-varying and non-stationery, we model the service rate (or the available bandwidth) for best-effort traffic as an autoregressive integrated moving average (ARIMA) process, while most of the above papers assume that it is constant, slowly-varying, or stationery. By modeling the service rate as an ARIMA process, our controller can handle both bursty cross traffic and non-stationery cross traffic. In addition, since in our scheme the bottleneck node will issue the same rate command to the best-effort sources, the basic fairness can automatically be provided without tuning any parameters.
2. NETWORK MODEL AND PROBLEM FORMULATION
time unit corresponds to the interval over which the rate available to the best-effort sources is determined. The best-effort sources are controllable sources and emit at rates specified by a central controller residing at the bottleneck node. Associated with each source is a source-dependent round-trip delay or “action delay”, which consists of so called “downstream delay” and “upstream delay” (Altman, 1999). Assume that the number of the best-effort sources is M and the round-trip delay for source i is di time units where di is a positive integer. Without loss of any generality, assume that dm’s are ordered such that
0 5 d 5 d , 5 . . .IdM 5 d where d and d correspond to the minimum and maximum round-trip delays. Consider the example in Figure 1, let q(k) denote the buffer length at time instant k, then the dynamics of the buffer is described by:
where ri(k) denote the total number of packets from best-effort source i that arrive in the buffer in the interval [k, k + l ) , and c(k) denotes the number of packets that depart from this buffer in the same time interval. Note that ri(k)represents source i’s rate. c(k) represents the service rate for the best-effort traffic. The controller, residing at the bottleneck node, makes control decision at each time step. The congestion control problem is to determine a rate command ui(k) to relay to source i, i = 1, . . . , M , at time k to achieve some overall performance objective. However, source i will not immediately change its rate to the new rate ui(k).It takes time for the rate command to reach the source and subsequently for the traffic with the new rate to affect the state of the node. So the buffer’s input rate from source i is equal to the rate command di time units ago, i.e., ri(k)= u i ( k - d J , i = 1,. . . ,M. Therefore, we can obtain the buffer dynamics as follows: M
@
<
i................................................................................................ Buffer BottleneckNode
:
@a
HS I mgh Pnonty Source 1 BS I Best-effort Source I
Fig. 1. Model at a Bottleneck Node In this paper, we will consider congestion control design for a single bottleneck node, which is shared by multiple best-effort traffic sources and other high priority cross-traffic sources, shown in Figure 1. The mathematical model we will adopt to describe the buffer’s behavior is a discrete-time model, where a
Our objectives are to achieve high link utilization, low packet loss, low delay, etc. These objectives can be accomplished by regulating the buffer length at the bottleneck node around a desired level. Tracking such a nominal buffer length is desirable to avoid loss due to overflow and waste of link capacity due to underflow. In addition, we are interested in a fair share of the available bandwidth among the best-effort sources at the bottleneck node. So in our design, we let u1 ( k ) = u,(k) = . . . = uM(k)and as a result, we have: M
Then, the buffer dynamics can be characterized by:
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whereA(zP1)= 1 -zP1 andB(zP1)= &z-(dj-d).
.
The high-priority cross traffic represents, for example, CBWVBR in ATM network, and it determines the service rate that the best-effort traffic experiences at the bottleneck node. For convenience, instead of specifying the cross traffic distribution, we specify the “service process,” which is the difference between the rate of cross traffic and the link bandwidth at the bottleneck node. We assume that the service process is represented by an ARIMA process:
Objective Function J
Fig. 2. Overall control Structure
c n[q(i+klk) + c h,[Au(i+ N2
~ ( z -) ( 1 - z- ) ( -c - c(k)) = P ( Z ) e ( k ) (5
+
+
-qd12
J=E{
i=N1
NU
+ +
k - 1)12} where C(zP1) = 1 clzP1 c2zP2 . . . C ~ ~ Z - ~ C , (8) i= 1 P(zP1)= 1+plzP1+p2zP2+.. .+ p , , ~ - ~ p . The driving term { e ( k ) } k r lis a zero-mean i.i.d Gaussian sewhere n’s and 4 ’ s are positive constants, q(i klk) is the prediction of the buffer length at time i + k, quence with variance 0,”. e,, i = 1, . . . , nc andp,, i = and qd is the target buffer length. Nl and N, are the 1, . . . , nd are the parameters characterizing the prominimum and maximum cost horizons and Nu is the cess, which can be estimated based on maximum likelihood criterion. -c is the mean service rate. -c is not control horizon. In the buffer model (7), there is a time delay d. The buffer length will not be affected required to be known as we can see in the derivation by current explicit rate issued by the node until d of our controller later. Note that (1 - z-’) -c = 0, then 1 time units later. Therefore there is no reason for we have: Nl to be less than d 1. In our design, we will let Nl = d + 1, N, = N + d and Nu to be the same as the maximum cost horizon. The first term in the cost function represents the penalty for deviating from the Substitute equation (6) into equation (4), we have: desired buffer length qd. The second term represents a penalty for the control effort. It can help the closedloop system to be stable. A(z-l)q(k+ 1) =B(z-l)z-du(k)
+
+
+
Multiply A = 1 -zP1 to the both sides of equation (7), we can get A”(z-l)q(k+ 1) =B(z-l)z-dAu(k)
In the following, based on model (7), we develop an algorithm to calculate the explicit rate u(k) to avoid congestion in high speed communication networks.
+
where
3. MODEL-BASED PREDICTIVE CONGESTION CONTROL
a(,-’)
= (1 - zP1)A( z
(9)
C(z-1)
1.
From the above equation, we know that C(zP1) can be absorbed into A”(zP1)and B(zP1).For simplicity, in the following we will only consider the case where C(zP1)is equal to 1. Then the ABR buffer dynamics becomes:
As mentioned early, we need to design a controller to regulate the buffer length at the bottleneck node around a desired level. In the design of such a controller, an important issue is how to deal with time delays. MPC is widely used in industry for its capability of handling time delays. This motivates us to use model-based predictive control to deal with the congestion control problem here. Particularly, we will use the technique of generalized predictive control (GPC) developed in (Clarke, et. al., 1987).
A”(z-l)q(k+ 1) =B(z-l)z-dAu(k) +P(z- ) e(k)
(10)
This is a so-called CARMA (controlled autoregressive moving average) model (Clarke, et. al., 1987).
Our overall congestion control structure is shown in Figure 2. First, based on model (7), the future buffer lengths for a determined prediction horizon N are predicted. Then, a set of future best-effort source rates is calculated by optimizing a determined criterion in order to keep the buffer length as close as possible to the desired buffer length. The criterion we adopt here is of the form of a quadratic function, which is:
The objective of predictive control is to compute future best-effort source rate sequence u ( k ) , u(k+ l ) , . . . in such a way that the future buffer length q(k+ j ) is driven close to qd. This is accomplished by minimizing J . In order to optimize the cost function the optimal prediction of q(k+ j ) for j 2 d 1 and j 5 N d need to be obtained. From (10), we know that the buffer level at time k + j is:
+
105
+
where G,(zP1) has the form of x { i i g j , i z - i , and G,(zP1)and H,(zp1) can be obtained by solving the following Diophantine equation:
Equation (1 1) can be rewritten as:
B ( z - ' ) E , + ~ ( z - ~=) G,(z-')P(z-') + z - j H ( z P 1 )(19) So we have:
where E,(zp1)a n d 5 (.-I) can be obtained by solving the following Diophantine equation:
P(z-')
= E , ( z - ' ) ~ ( z - ' )+zPiF,(zp1)
(13)
In equation (12), e ( k - 1) can be computed in terms of data available at time k, i.e., e ( k - 1) is given from (10) as
Therefore, for j
=
1,. . . ,N, we can get
q = G A u + H ( z - ' ) A u ( k - 1) + F ( Y 1 ) q ( k ) (21) where
Substitute (14) into equation (12), we obtain:
As the degree of polynomial E,(zp1) is j - 1 the noise terms in equation (15) are all in the future. The best prediction of q ( k + j ) is therefore (Astrom and Wittenmark, 1990):
4(k+jlk) =
B(z-I)E, (z-1) A u ( k + j - d - 1) P(z-1) The last two terms in equation (21) only depend on the past. If we group them into f , we can obtain:
we can obtain the j
+ d ahead optimal prediction as:
@(k+j+dIk)=
B(z-l)Ej+d(z-l) A u ( k + j - 1) P(z-1)
q=GAu+f Substitute (22) into cost function (S), we have J = ( G A u + f - q d ) T r ( G A U + f - qd)
+AuTAAu In the cost function (S), we need to obtain the future rate change sequence A u ( k ) , A u ( k + l ) , . . ., A u ( k + N - l ) , so we need to extract future rate change A u ( k + j ) for j 2 0 from equation (17). To do this, we can rewrite parts, i.e.,
B(2-')Ejfd(2-1) P(2-1)
where
qd
=
[qd qd ... qdl
r=diag[x
A u ( k + j - 1) into two
fi
T
...,rNl
A = diag[h, h2 . . . , A N ] Equation (23) can be written as: J
106
=
-1A u T W A u 2
+ b T A u+ f o
where
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j
j
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:
.
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I
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I
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I
.
.
I
.
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,
I
I
The minimum of J can be now found by making the gradient of J equal to zero, which leads to:
AU = -Wplb
(25)
;mot
The first element of AU together with u ( k - 1) will be used by the node to generate current explicit rate command u ( k ) , which will be sent to the sources. Other elements of AU are rejected. The procedure is repeated at the next time instant.
~-.,,,,,,,,,,,,,~ .
.. . j
.
.. .
.. .
j
j
.
.
.
High Pnonty Traffic, Buffer Length and ABR Sources' Rate
One source of error is the assumption that the delay di, i = 1, .... M , is an integer multiple of the unit time. In addition, there is variability in the delays due to queuing.
4.1 Simulation Setup
In the following, we will test the performance of robustness of our controller against action delay uncertainty by considering B(zpl) uncertainty in equation (27). In the modeling we let B(zpl) = 1 + z p l +zp2 while the real B(zpl) may be 1 + 2zp1, 1 + 2zp2, and 3, respectively. A representative robustness result is illustrated in Figure 4 where the real model for B(zpl) is 1 + 2zp1 while in the modeling we let B(zpl) = 1 + zpl +zp2. Similar results hold for the other two cases. Our controller performs very well in all cases.
Consider Figure 1, we assume it is an ATM network, and the best-effort sources are 3 ABR sources. The high priority sources are either CBR or VBR sources. The link capacity is 155 Mbps. The unit time is 3 ms, then the link capacity is equivalent to 1097 cells per unit time. In addition, we assume the desired ABR buffer length is 3000. We model the CBR and VBR sources' traffic collectively as a high priority traffic. The data of this high priority traffic comes from a mixture of the video traces from MPEG-1 video traces (1997), a CBR source with a rate of 141.5 cells per unit time, and an Om'Off VBR source with a mean rate of 141.5 cells per unit time. The mean rate of the high priority traffic is 456 cells per unit time. It is shown in Figure 3. Then the mean available bandwidth for the three ABR sources is 641 cells per unit time. We assume the action delays for sources 1,2, and 3 are 2, 3, and 4 unit times, respectively. Thus the ABR buffer model at the bottleneck is:
+
:
Fig. 3.
4. SIMULATION STUDIES
q(k+ 1) = q(k) u ( k - 2)
.
5000
,.
,.
.
.
j.
j.
,.
,
,
.,
,.
,.
,.
4000 I
8 3000 I
: 2000 2,000~
..
..
ABRmean queue length 2978
! l .. .
.. ..
' ! ; - I .. ..
.. ..
..
+ u ( k - 3)
+u(k-4)
-
~ ( k ) (26)
i.e.,
Fig. 4. Robustness Simulation: Real B ( z - ' )
whereA(zpl) = 1 -zpl, andB(zpl) = 1 +zp1+zp2.
4.3 Multiple Sources Multiple Nodes Configuration .for Max-min Fairness
If the controller perform well, then the buffer length will be regulated around 3000 and the mean rate will be 213.7 cells per unit time. Figure 3 shows the simulation results. We can see the buffer length is controlled around 3000 as expected. In addition, the mean ABR sources' rate is 213, which is very near the expected value 2 13.7 cells per unit time.
=
1f 2 z - l
In real networks, there may be multiple bottlenecks. In such cases, one important performance index is max-min fairness. In this section, we want to check the performance of max-min fairness of our controller under a two-node network configuration shown in Figure 5. High priority traffics 1 and 2 are of the same kind as the one in the previous section. The mean rate for high priority traffic 1 is 420 cells per unit time, and for high priority traffic 2 it is 228 cells per unit time. Thus the mean service rates at nodes 1 and 2 are 677 and 869 cells per unit time. At each node, there is a congestion controller running. We assume the delays
4.2 Robustness In addition, we consider the robustness of our controller against modeling errors of the action delays.
107
I
,
-0 -0
sw2
Fig. 5. Max-min Fairness Simulation Setup Figure 6 and 7 show the max-min fairness simulation results. The buffers 1 and 2 are regulated around 3000 as expected. Sources 1, 2, and 3 emit at a rate very near to their max-min fairness share of 225.7 cells per unit time. Source 4 and 5's rate is also close to their ideal value of 321.5 cells per unit time. Therefore, our design can achieve max-min fairness.
800,
,
,
,
,
,
,
,
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7""
a""
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........j . . .
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Fig. 7. Max-min Fairness at Node 2
5. CONCLI JSION
In this paper, a preventive congestion controller has been proposed using a model predictive control scheme for high speed communication networks. It is demonstrated that the controller performs very well under various criteria such as max-min fairness and high link utilization and is robust to time delay uncertainties. Model predictive control appears to be a viable approach to develop congestion control algorithms for high speed communication networks.
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