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On traffic prediction for resource allocation: A Chebyshev bound based allocation scheme
Computer Communications 31 (2008) 3741–3751
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Computer Communications journal homepage: www.elsevier.com/locate/comcom
On traffic prediction for resource allocation: A Chebyshev bound based allocation scheme R.G. Garroppo, S. Giordano *, M. Pagano, G. Procissi Department of Information Engineering, University of Pisa, Via Caruso 16, I-56122 Pisa, Italy
a r t i c l e
i n f o
Article history: Available online 20 June 2008 Keywords: FIR filters Linear prediction Chebyshev bound Resource allocation
a b s t r a c t The paper presents a predictive approach to network resource allocation techniques. The rationale of this work is to use measurements to estimate future traffic behavior by prediction, and to use such an estimation to define the amount of future network resources that will be required by the considered traffic. In this framework, the paper presents the analysis and performance evaluation of classical and chaotic techniques for network traffic prediction. The performance parameters considered in the analysis are: the accuracy of predictors in capturing the actual behavior of traffic; the computational complexity for a realistic integration of such predictors into experimental testbeds; and the responsiveness with respect to traffic pattern variations. The analysis results show that the classical normalized linear mean square predictor achieves a satisfactory trade-off among the above mentioned metrics as it presents a medium level of complexity while achieving high performance in terms of prediction accuracy and responsiveness to network traffic changes. Then, using the normalized linear mean square predictor, we derive a bandwidth allocation strategy, named statistical delay bound (SDB), which guarantees a probabilistic bound on the delay experienced by packets traversing a network node. The paper presents the performance analysis of SDB showing that, in spite of the simplicity of the adopted predictive algorithm, the proposed measurement based technique allows to fulfill the project requirements and candidates for actual experimentation into prototypal routers which supports QoS mechanisms. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction The evolutionary trend of communication networks in which new services such as Voice over IP, audio/video streaming, videoconferencing and virtual private networks (VPNs) with quality of service are being offered to end users has led to the deployment of network architectures (e.g. DiffServ [1,2]) supporting QoS. Moreover, over the last years, a considerable effort has been dedicated to the optimization of operational IP networks through the MPLS switching paradigm [3] and the so called traffic engineering [4]. The exigence of integration of such approaches arises the necessity of dynamic traffic control mechanisms, such as resource allocation techniques and admission control, to achieve optimal resource utilization and network performance [5]. Model-based approaches have demonstrated their weakness in that the long range dependence (LRD) property of network traffic [6] is not on-line distinguishable from non-stationarity and it does not permit an easy estimation of characteristic parameters (such as the Hurst parameter). This has impaired the development of easy,
* Corresponding author. Tel.: +39 050 2217539. E-mail addresses: [email protected] (R.G. Garroppo), s.giordano@iet. unipi.it (S. Giordano), [email protected] (M. Pagano), [email protected] (G. Procissi). 0140-3664/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2008.05.019
flexible and effective traffic models to be widely used for network dimensioning. In addition to the above mentioned reasons, novel concepts like network self-sizing and reconfigurability have led to consider measurement based approaches as the most promising in order to address (even in approximate ways) common issues in Diffserv aware traffic engineering (DS-TE) [7] context such as constraintbased path computation, LSP dimensioning, scheduler parameters selection and so on. In this scenario, the objective of this paper is to propose a novel measurement based strategy for resource allocation which integrates traffic measures and prediction. To this aim, the comparison of different prediction techniques and the definition of a resource allocation algorithm that uses the information produced by the selected traffic predictor are of paramount relevance. The first part of the paper is devoted to the presentation of several prediction algorithms (classic, chaotic and geometric) and to the investigation of their performance in terms of the above mentioned criteria: accuracy, complexity and responsiveness. The prediction algorithm which will achieve the best trade-off among the three criteria will be integrated into the resource allocation strategy derived in the second part of the paper. As above mentioned, the second part of the article is devoted to the derivation and performance evaluation of the statistical delay
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bound (SDB) resource allocation strategy. SDB aims at computing the service capacity to be allocated a connection in order to guarantee a probabilistic bound on the maximum queueing delay experienced by packets of the connection itself. In summary, the paper is organized as follows. After the presentation of related works on traffic prediction for resource allocation, Section 3 gives the basic notions of time series prediction, while Section 4 reports the performance comparison of three different prediction algorithms. Section 5 describes the SDB allocation strategy whose performance is analyzed in Section 6. Finally, Section 7 concludes the paper with final remarks.
2. Related works Although time series prediction is a classic topic in signal processing, its adoption into networking contexts is becoming quite frequent. Recently, different predictors have been presented and a new active queue management (AQM) scheme exploiting the predictions of future traffic intensity has been proposed in [8]. Focusing on resource allocation algorithms based on traffic prediction, different solutions have been proposed depending on network technology. In particular, in [9] the authors propose an access control protocol for high speed downlink packet access (HSDPA) system based on bandwidth allocation between real time and non-real time traffic. By applying a traffic model to the real time traffic in the current time slot, the system can predict the residual capacity in the next time slot, thus it can optimize the scheduling of non-real time data transmission. It is relevant to note that predictions are based on a model assumption for the considered traffic, and that the time quantum, T p , is of the order of the transmission time interval length (usually tens of ms). In [10], the author proposes a new dynamic satellite bandwidth allocation technique (named predictive resource reservation access, PRRA) which is based on accurate videoconference traffic prediction. The proposed model is then combined with results on data traffic modeling and prediction presented in [11] and is shown to provide very good throughput and delay results. In this case, the time quantum T p is of the order of a round trip time for a GEO satellite system, i.e. around 500 ms. In both these works, the predictions are based on traffic models; it is well known that the accuracy of model-based approaches depend on the reliability of the assumed source models. Often, to simplify the model of the system, Markov chains are used to model source traffic; this assumption ignores the LRD nature of traffic. In our previous work [12], the LRD nature of traffic has motivated the investigation of chaotic prediction to develop bandwidth allocation strategies. Differently from these works, in [13] the dynamic bandwidth allocation strategy designed to support variable bit rate (VBR) video traffic does not assume any model for the traffic source. The strategy predicts the bandwidth requirements for future frames by using adaptive linear prediction that minimizes the mean square error. The adaptive technique does not require any prior knowledge of the traffic statistics nor assume stationarity. In this case, the prediction based bandwidth allocation algorithm has been defined for the renegotiated constant bit rate (RCBR) network service model of ATM. Two traffic predictors on frequency domain have been compared in [14], in terms of complexity and ability to guarantee the project parameter of the resource allocation algorithm. The authors assume the server utilization as the project parameter (hence we refer to that approach as constant server utilization algorithm). The focus of [14] is on VBR video transport over ATM network. As opposed to the previous mentioned works, our goal is to define a bandwidth allocation algorithm based on traffic prediction,
without any assumptions on the statistical features of traffic. Furthermore, our proposal uses the prediction algorithm in the time domain and the allocation strategy has been designed by assuming as project constraint the maximum queueing delay. Moreover, the system model considered in our study is general and does not refer to any specific technology. It can be adapted for resource allocation in cellular, satellite or wired networks. Finally, the prediction algorithm as well as the bandwidth allocation scheme have been selected taken into account the actual implementation of the proposed solutions. The purpose of predictors is to capture the traffic intensity and accurately estimate the future arrivals over prediction intervals [15]. Also, they should exhibit high responsiveness with respect to sudden traffic changes. The huge effort towards actual experimentation of novel network architectures through the set up of field-trials [16,17] adds further critical constraints: ease of implementation and low computational complexity. The proposed solutions have been selected by considering these aspects. 3. Basics of traffic prediction This section briefly presents some of the basic concepts of time series prediction. Since the topic is well known and widely investigated in signal processing, in the following a very concise description will be given and more details are left to the bibliography. Time series prediction can be addressed through statistical or geometric techniques. This paper investigates both classes of algorithms, with special focus on linear and chaotic prediction for the first class, and on polynomial extrapolators for the class of geometric algorithms. None of the presented algorithms, though, require the knowledge of a specific model of the underlying time series to be predicted. Statistical predictors only require the underlying stochastic sequence to be wide sense stationary, while geometric predictors are derived by simple algebraic reasoning. The general problem of prediction can be stated as follows: given a set of observations of a stochastic process xðnÞ, give an estimation ^ xðn þ kÞ of the value xðn þ kÞ that the process x will assume k steps ahead. In other words, given a vector of p observations, x ¼ ½xðnÞ; xðn 1Þ; . . . ; xðn p þ 1Þ, the predicted value ^ x is obtained by
^x ¼ ðxÞ
ð1Þ
where the function is called predictor. The topic of next paragraphs will be the derivation of the analytic expression of for the different classes of predictors investigated. 3.1. Linear prediction Linear prediction occurs whenever the function ðxÞ is linear. In other words, the problem is to determine the impulse response hðnÞ of the linear filter h such that
^xðn þ kÞ ¼ xðnÞ hðnÞ ¼
p1 X
hðiÞxðn iÞ
ð2Þ
i¼0
The filters coefficients can be determined according to arbitrary optimality criteria. One of the most famous and widely adopted prediction algorithm is the so-called linear minimum mean square error (LMMSE) predictor in which, the values hðnÞ are derived by minimizing the mean square error of prediction:
E½e2 ðnÞ ¼ E½ðxðn þ kÞ ^xðn þ kÞÞ2
ð3Þ
In that sense, LMMSE is an optimal algorithm in that, within the class of linear filters, it minimizes the mean square error of predic-
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tion. A thorough description of LMMSE is beyond the objective of this paper and can be found in [18,19]. We only mention here that the derivation of the LMMSE filter requires the knowledge of at least p values of the autocorrelation function of the stochastic process x and inverting a p p matrix. These facts make LMMSE very hard to be implemented on real running equipments. The following paragraphs present, instead, two adaptive versions of LMMSE that will be used in practice in the paper, denoted as least mean square (LMS) predictor and normalized least mean square (NLMS) predictor, respectively.
Notice that, at time n, the values xðn þ kÞ, hence eðnÞ, are not known. Therefore, the value eðn kÞ is used, instead, and the one-step NLMS predictor update equation becomes:
3.1.1. Least mean square predictor The LMS algorithm is an adaptive approach which does not require prior knowledge of the autocorrelation structure of the stochastic sequence [18,19]. Thus, it can be used as an on line technique for predicting bandwidth requirements. The algorithm scheme is shown in Fig. 1. The filter coefficients are time varying and are tuned on the basis of the feedback information carried by the error eðnÞ. In the following, we denote the vector of filter coefficients at time n with hn . The values of h adapt dynamically in order to decrease the mean square error. Notice that eðnÞ is defined as in the previous section and xn ¼ ½xðnÞ; xðn 1Þ; . . . ; xðn p þ 1Þ. Indeed, the LMS algorithm is the adaptive version of the optimal LMMSE predictor. The LMS algorithm operates as follows:
Geometric predictors are based on the approximation of the behavior of the time series xðnÞ through elementary functions. These techniques do not involve stochastic analysis of time series while they only involve pure geometric considerations. The basic idea is to determine the ðp 1Þ-order polynomial interpolator of the set of p observations xðnÞ; xðn 1Þ; . . . ; xðn p þ 1Þ available. Such a polynomial is of the form:
(1) initialize the coefficients h0 (2) for each new data, update the filter hðnÞ according to the recursive equation
hnþ1 ¼ hn þ leðnÞxn
ð4Þ
where l is a constant called the step size. If xðnÞ is stationary, the vector of coefficients h converges in the mean to the optimal solution of LMMSE predictor [18,19]. The speed of convergence of the algorithm depends on the value of l. Large values of l result in a faster convergence of the algorithm and in a quicker response to signal changes. However, small values of l result in slower convergence and less fluctuations once the convergence is attained. The selection of l thus, should be made by trade-off between the above phenomena. 3.1.2. Normalized least mean square predictor The normalized LMS algorithm is a modification to the LMS algorithm in which the update equation of the filter’s coefficients is
hnþ1 ¼ hn þ l
eðnÞxn
ð5Þ
kxn k2
where kxn k2 ¼ xn xtn . The main advantage of using NLMS is that it is less sensitive to the step size l. According to [19], NLMS converges in the mean to LMMSE predictor as long as 0 < l < 2.
hnþ1 ¼ hn þ l
ð6Þ
The computational complexity of a p-order NLMS algorithm is that of 2p þ 1 multiplications and p þ 1 additions. 3.2. Geometric prediction
Pp1 ðzÞ ¼ ap1 zp1 þ ap2 zp2 þ þ a1 z þ a0
ð7Þ
and the coefficients an are determined by the condition Pp1 ðkÞ ¼ xðkÞ; n p þ 1 6 k 6 n, that is by solving the linear system (with Vandermonde matrix)
0
1 B1 B B B .. @.
n
.. .
10
np1
n1 .. .
p1
ðn 1Þ .. .
1 n p þ 1 ðn p þ 1Þp1
a0 CB a CB 1 CB . CB . A@ . ap1
1
0
C B C B C¼B C B A @
1
xðnÞ xðn 1Þ .. .
C C C C A
xðn p þ 1Þ
A simple example of this predictor class is the first order predictor, which can be obtained assuming p ¼ 2. From the above relation, this assumption implies that the prediction algorithm considers the observations xðnÞ and xðn 1Þ. In particular, the predicted value, ^xðn þ 1Þ, is obtained by considering the slope of the straight line passing through the two points xðnÞ and xðn 1Þ, i.e.
^xðn þ 1Þ ¼ 2xðnÞ xðn 1Þ: 3.3. Chaotic prediction This section presents the application of chaos theory to time series prediction. For the sake of conciseness, details on the theory of chaos are omitted here; we refer the interested reader to [20] for an excellent overview on the topic. Hence, in this section we will recall only the description of a particular type of chaotic predictor, the so-called radial basis function predictor (RBFP), which has been studied and deeply analyzed in [12]. To make predictions about chaotic time series, the first step is to consider the time series as obtained by sampling a continuoustime scalar variable xðtÞ representing the time evolution of system whose underlying dynamic is that of a strange attractor lying on a D-dimensional invariant manifold. Assume then a sequence of observation xðnÞ ¼ xðnT p Þ; the sequence is referred to as chaotic time series and the sampling period T p is called delay time. The second step is to embed the sequence in a m-dimensional state space. The value m is called embedding dimension. If the strange attractor lies on a D-dimensional invariant manifold, then, necessarily, m P D. On the other hand, Takens theorem assures that m 6 2D þ 1 [21]. The state vectors will be of the form:
0 B B xn ¼ B B @ Fig. 1. LMS algorithm scheme.
eðn 1Þxn1 kx2n1 k
xðnT p Þ
1t
xðnT p T p Þ .. .
C C C C A
xðnT p ðm 1ÞT p Þ
ð8Þ
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The third step is to select a proper interpolation function fN such that
xðn þ 1Þ ¼ fN ðxn Þ:
0
/11 B . B .. B B @ /k1 1
3.3.1. Approximation techniques In the previous paragraph we mentioned interpolation functions, namely functions fN such that
xðn þ 1Þ ¼ fN ðxn Þ;
16n6N1
This, of course, widens the number of techniques that could be used. Typically, they belongs to two different categories: global techniques and local techniques. In the former, the approximant (or interpolant) is applied to the whole series of vectors that can be constructed through the sequence. In the latter case, instead, the prediction is made on the basis of the states in the past that are near to the current one. The rationale of this technique is simple and intuitive: look for pieces of the trajectory in the past that resemble the current one and infer the future behavior according to the way the system evolved in that analogous past condition. Incidentally, this method looks well tailored to the case of network traffic which has widely proven to be selfsimilar. The selection of the number k of ‘‘neighbors” in the past is another critical issue. Typically, and we choose this strategy, k ¼ m þ 1 is assumed [22,23]. Global techniques are computationally too expensive and are not considered in our study. We use the local approach and we select the radial basis functions as approximant functions. 3.3.2. Radial basis function predictor (RBFP) According to the previous discussion, given the set of k neighbors xnj ; j ¼ 1; 2; . . . ; k, of xn we choose a prediction scheme of the form:
^xðn þ 1Þ ¼
k X
kj /ðkxn xnj kÞ þ l
ð9Þ
j¼1
Notice that, in its general expression, the constant l is replaced by a polynomial term (usually not included). The functions / : Rþ ! R, defined as
/ðrÞ ¼ ðr2 þ c2 Þb
b > 1; b–0
ð10Þ
are called radial basis functions. In our analysis we always used b ¼ 0:5. In other words, the prediction (9) can be interpreted by a weighted sum of terms in which the contribution of each neighbor depends inversely on their distance to the current state (closer states give a bigger contribution). The values kj ; j ¼ 1; 2; . . . ; k are determined through the knowledge of the past evolution of the state as
^xðni þ 1Þ ¼
k X
/kk
1
10 1 0 1 xðn1 þ 1Þ 1 k1 CB C B C .. ... CB ... C B C . CB C ¼ B C CB C B C 1 A@ kk A @ xðnk þ 1Þ A 0
l
ð12Þ
0
where
/ij ¼ /ðkxni xnj kÞ:
16n6N1
In many practical contexts, including teletraffic studies, this constraint may be significantly relaxed by allowing fN to be an approximant, that is
xðn þ 1Þ f N ðxn Þ;
/1k .. .. . .
kj /ðkxni xnj kÞ þ l
ð11Þ
4. Prediction algorithms comparison The primary goal of this research is to develop a dynamic and flexible prediction based resource allocation strategy. Thus, at this stage, the prediction algorithms presented in the previous sections will be compared in order to select the one that best fits into the overall research target. In order to define the performance parameters to be considered in the comparison, we first establish the scenario where these predictors will be used (Fig. 2). The model of the resource is that of a single server queue with infinite buffer and deterministic (variable) service rate that will be dynamically adjusted on the basis of the underlying traffic prediction algorithm. A measurement system registers the amount of traffic incoming to the queue over the time intervals ½ðn 1ÞT p ; nT p Þ and produces the time series xðnÞ ¼ xðn T p Þ. The time series xðnÞ is then processed by the prediction block, which returns the predicted value ^ xðn þ 1Þ of traffic expected in the next time interval ½nT p ; ðn þ 1ÞT p Þ. Finally, the allocation block computes the queueing service rate cðn þ 1Þ to be selected in the time interval ½nT p ; ðn þ 1ÞT p Þ, on the basis of the information obtained by predictions. 4.1. Performance parameters The scenario in which the prediction block will be integrated determines the critical parameters to be considered in the comparison of predictors. Firstly, we can notice that, in a time T c T p , the system should ^ predict X½ðn þ 1ÞT p and calculate c½ðn þ 1ÞT p . The parameter T c must be negligible with respect to T p , since the network resources calculated in T c are valid for a time T p . Hence, the prediction algorithm must exhibit low computational complexity. Secondly, the resource allocated depends on the traffic predic^ tion X½ðn þ 1ÞT p . Hence, it is very important to assure a high accuracy of the predictor. A third requirement for the prediction algorithm can be deduced by reminding that the system works ‘‘on line” and that traf-
Input Traffic
Output
Measurement System
j¼1
together with k X
kj ¼ 0
j¼1
Using a matrix notation, the above system of k þ 1 equations is equivalent to the following system:
Prediction System
Resource Allocator
Fig. 2. Prediction-based resources allocation system.
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fic characteristics can vary in time very quickly. Hence, the prediction algorithm should be able to readily react to sudden variation of the traffic features. In other words, the predictor should by highly responsive. In summary, the performance parameters considered in the comparison will be (1) the complexity, (2) the accuracy, (3) the responsiveness.
4.2. Metrics for the quality of prediction In order to quantify the accuracy of a predictor fN , we introduce a general and unified definition of prediction error r2 ðfN Þ, as
lim
M!1
X 1 1 NþM1 k^xðn þ 1Þ fN ðxn Þk2 M V n¼N
ð13Þ
with the normalizing factor:
M M 1 X 1 X V ¼ lim xðmÞ xðmÞ lim M!1 M M!1 M m¼1 m¼1
ð14Þ
Values of r2 ðfN Þ close to 0 indicate good performance of the predictor. Performance gets worse as r2 ðfN Þ tends to 1. Indeed, r2 ðfN Þ close to zero implies predicted values nearly identical to the actual ones [22]. This condition is excessively restrictive. It is often enough to require a reasonably low value of r2 ðfN Þ and a good match of the statistics of actual and predicted data, such as auto-covariance function, distribution, variance, and so forth.
sary in a PC to obtain a single predicted value. The values obtained with the three considered predictors are summarized in Table 1. We can clearly observe that RBFP exhibits a quite higher complexity with respect to both first order and NLMS predictor. Indeed, the number of clock ticks necessary to obtain a single predicted value when considering the RBFP is double with respect to the NMLS and 3 order of magnitude higher than the first order predictor case. To quantify the time necessary to obtain a single predicted value, we can refer to a PC equipped with a CPU at 1 GHz. Considering this equipment, the computation time of the different algorithms is summarized in Table 2. As we can observe, the RBFP can require a computation time higher than 1 ms to calculate a single predicted value; this value can be inadequate when T p is in the order of tens of ms and for network links with high bandwidth. 4.3.2. Accuracy The accuracy has been evaluated by considering the prediction error defined in Section 4.2. The results are summarized in Table 3, for different sampling time, T p . The table points out the NLMS predictor as the one that achieves the best performance, while the first order predictor, facing a good computational speed, returns the largest value of prediction error. RBFP achieves high accuracy but the selection of its parameters is a long and off-line procedure that strongly depends on the characteristics of the input traffic. Furthermore, as highlighted by the previous paragraph, the RBFP presents a high complexity. For these reasons the best choice for a prediction technique to be implemented in an on-line environment, seems to be the NLMS predictor, that offers a good trade-off between accuracy and computational speed. Finally, we can note that the accuracy decreases as the sampling time, T p , increases.
4.3. Prediction performance analysis In this section, we present numerical results of the comparison among some predictors described in the previous sections. The final result of this section will be the selection of the prediction algorithm which best combines the characteristics of accuracy, low complexity and responsiveness, and that will be adopted in the resource allocation scheme depicted in Fig. 2. The study is carried out by using the traffic data acquired during an emulation of an actual service. In particular, we consider a time series named ‘‘Conf Cell” representing the traffic offered by a videoconference session corresponding to three people in front of a 3-CCD camera. Each sample of the time series represents the number of ATM cells transmitted during a frame period, equal to 1/25 s (40 ms). The available number of samples is equal to 48,496, with mean value and standard deviation equal to 174.064 and 99.21 cells/frame, respectively. In the comparison, we have considered three different types of prediction algorithms: (1) a NLMS of order 20, (2) the first order predictor (denoted in the tables as F.O.P.), (3) the radial basis function predictor.
4.3.3. Responsiveness The last analysis is related to the ability of the prediction algorithm in capturing the variation of traffic characteristics. In this study, we have artificially varied the mean of the measured traffic adding a constant term in the time series xðnÞ, starting from a fixed value of n. The behavior obtained with the first order, the NLMS and the RBFP predictors are shown in Figs. 3–5. From the figures, we can note the high prediction error of the first order prediction at the time 5000 T p , due to high variation of the slope presented by the measured traffic pattern. Indeed, this can be easily explained by observing that the first order predictor predicts the value at time ðn þ 1ÞT p by simply considering the slope of the traffic pattern observed at time nT p . On the contrary, at the time instant of sudden variation, NMLS and RBFP produce an almost negligible error, as can be observed in Figs. 4 and 5. 4.3.4. Final remarks The above presented discussion proves the similar performance of NLMS and RBFP predictors in that they prove to be significantly
Table 2 Computation time to obtain a single predicted value
4.3.1. Complexity A first analysis is devoted to evaluate the complexity of the compared prediction algorithms. As a performance measure for this analysis, we have considered the number of clock ticks neces-
NMLS
F.O.P.
RBFP
660 ls
0.2 ls
1.26 ms
Table 3 Prediction error for different sampling time T p Table 1 Clock ticks necessary to obtain a single predicted value NMLS
F.O.P.
RBFP
0:66 106
213
1:26 106
T p (ms)
NMLS
F.O.P.
RBFP
40 80 120
0.0379 0.0734 0.1125
0.0483 0.0903 0.1493
0.0413 0.0687 0.0985
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650 Actual Data F.O.P. Predicted Data
600
Number of ATM Cells
550 500 450 400 350 300 250 200 4980
4990
5000
5010
5020
5030
5040
Sample, n Fig. 3. Responsiveness of F.O.P. Predictor for T p ¼ 40 ms.
550 Actual Data NLMS Predicted Data
Number of ATM Cells
500
450
400
350
300
250 4980
4990
5000
5010
5020
5030
5040
Sample, n Fig. 4. Responsiveness of NLMS for T p ¼ 40 ms.
accurate as well as enough responsive to react against sudden variation of traffic data. Also, their complexities have the same order of magnitude (though the one achieved by RBFP is twice as large as the one from NLMS); however, RBFP requires a preliminary phase of parameters tuning which may prevent from its use in automatic manner without any need for supervision. For all these reasons, NLMS will be adopted in the prediction-based resource allocation scheme that will be described in the next sections of the paper.
5. Resource allocation This section presents the novel Statistical delay bound resource allocation algorithm based on linear prediction of traffic as well as the basic definition of the CSU algorithm [14] that will be
used in the following section to assess the performance results of SDB. 5.1. Statistical delay bound (SDB) resource allocation algorithm According to the overall system presented in Section 4 and Fig. 2, in the following we derive the statistical delay bound (SDB) resource allocation algorithm which takes advantage of NLMS predictions to dynamically adjust the system service rate in order to guarantee statistically bounded packet delay. In other words, by denoting the overall packet queueing delay with D, and the target delay with d, the service rate cðn þ 1Þ will be calculated in order to meet
PrfD > dg < p where p is an arbitrarily small probability.
ð15Þ
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500 Actual Data RBFP Predicted Data
Number of ATM Cells
450
400
350
300
250
200 4980
4990
5000
5010
5020
5030
5040
Sample, n Fig. 5. Responsiveness of RBFP for T p ¼ 40 ms.
Unfortunately, the statistics of D are strongly dependent on the statistical properties of the input traffic. As already stated in the introduction, the approach presented here is based on measurements and does not assume any a-priori knowledge of input traffic models; therefore, in the following, we address a worst case approach by focusing on the maximum delay D experienced by packets. The condition
PrfD > dg < p
cðn þ 1Þ ¼
^xðn þ 1Þ d
ð17Þ
would guarantee a maximum packet delay of d. Unfortunately, the prediction is, in general, affected by errors; hence, the actual worst case delay experienced by packets is
D¼
ð16Þ
xðn þ 1Þ xðn þ 1Þ ¼ d cðn þ 1Þ ^xðn þ 1Þ
ð18Þ
By denoting the prediction error with ðnÞ, the actual traffic volume can be written as
assures that relation (15) is fulfilled. To show how the service rate is computed, let us begin by assuming the queue empty at the end of the nth time interval and by observing that, if the prediction ^ xðn þ 1Þ were exact, a service rate given by
xðn þ 1Þ ¼ ^xðn þ 1Þ þ ðnÞ
ð19Þ
and Eq. (18) becomes
0.0002 Prediction error pdf 0.00018 0.00016 0.00014
pdf
0.00012 0.0001 8e-05 6e-05 4e-05 2e-05 0 -20000
-15000
-10000
-5000
0
5000
10000
Bytes/Time unit Fig. 6. Histogram of NLMS prediction error.
15000
20000
25000
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D¼ 1þ
ðnÞ d ^xðn þ 1Þ
ð20Þ
Whenever ðnÞ > 0, the worst case delay D is bigger than the target delay d, and the difference can be very large depending on the prediction error. Furthermore, let us make the reasonable assumption that the prediction error ðnÞ has a symmetric zero-mean distribution. Such an assumption is quite logical and is verified in experiments (see Fig. 6). In this hypothesis, half of the time the worst case delay would be larger than the target value d if the service rate is computed as in (17). To sort out this problem, let us correct (17) by adding at the numerator a positive constant n:
cðn þ 1Þ ¼
^xðn þ 1Þ þ n d
ð21Þ
This allows to account for positive value of the prediction error ðnÞ (negative values of ðnÞ would give D < d even in the ideal case (17)). The new worst case delay experienced by packets becomes
D¼
^xðn þ 1Þ þ ðnÞ d ^xðn þ 1Þ þ n
ð22Þ
for which, now, D 6 d if and only if ðnÞ 6 n. In other words, the original problem involving packet delay has been translated by contraction into the following problem involving the prediction error ðnÞ:
PrfD P dtg ¼ PrfðnÞ P ng
ð23Þ
To estimate (23), we use the well known Chebyshev inequality which assures that any random variable X with mean value lX and finite variance r2X satisfies
PrfjX lX j P krX g 6
1 2
k
ð24Þ
By (24) and because of the assumed symmetry of the probability density function of ðnÞ, we have then
PrfðnÞ P kr g 6
1 2
2k
ð25Þ
and, by selecting n ¼ kr
PrfD P dg 6
1 2k
2
¼p
AT p ¼ cðn þ 1Þ T p
AT p ¼ cðn þ 1Þ T p ¼ xðn þ 1Þ
(1) receives as inputs parameter the target delay d and the violation probability p as well asqthe xðn þ 1Þ; ffiffiffiffi predicted value ^ 1 ; (2) computes the parameter k ¼ 2p ^ ðnÞ; (3) estimates r ^ ; (4) computes the parameter n ¼ kr ^ ; (5) calculates the service rate cðn þ 1Þ ¼ xðnþ1Þþn d (6) selects the queueing service rate accordingly. Let us give a brief discussion on the value of the sampling time T p to be used. One more time, let us suppose the prediction algorithm to be exact, that is ^ xðn þ 1Þ ¼ xðn þ 1Þ with service rate
Tp P xðn þ 1Þ d
ð29Þ
and, thus, the constraint
Tp P d
ð30Þ
must be satisfied. 5.2. Constant server utilization (CSU) algorithm The performance of the SDB algorithm will be compared to that proposed in [14], where resources are still allocated on the basis of traffic prediction although considering the server utilization as the project parameter. The allocation algorithm is very simple and the service rate is calculated by keeping constant the server utilization q, defined as the ratio between the average incoming traffic rate and the service rate. In other words, by defining the estimated arrival rate as
^rðn þ 1Þ ¼
^xðn þ 1Þ Tp
ð31Þ
the service rate is computed in order to maintain the ratio
^rðn þ 1Þ ¼q cðn þ 1Þ
ð32Þ
constant. 6. Performance evaluation The performance of the above proposed allocation strategies has been assessed by discrete event simulations driven by the Table 4 Aggregation time T p ¼ 100 ms d (s)
p
EðDÞ (s)
EðqÞ
PrfD P dg
0.025
0.1 0.05 0.01
0.0022 0.0013 0.0004
0.178 0.152 0.095
0.015 0.006 0.00012
0.05
0.1 0.05 0.01
0.0342 0.0165 0.0024
0.355 0.304 0.191
0.12 0.06 0.007
0.075
0.1 0.05 0.01
0.193 0.098 0.012
0.522 0.452 0.287
0.265 0.0175 0.033
ð27Þ
In summary, at any time interval n T p , the algorithm acts as follows:
ð28Þ
which should be greater than xðn þ 1Þ to leave the queue empty at the beginning of the next time slot. This condition gives
ð26Þ
The last equation involves the evaluation of r , for which we use the classical estimator of the sample variance taken over samples of length N, as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u 1 X r^ ðnÞ ¼ t 2 ðn iÞ N 1 i¼1
cðn þ 1Þ ¼ xðn þ 1Þ=d. The total amount of traffic that can be processed by the queue in the time interval T p is then
Table 5 Aggregation time T p ¼ 500 ms d (s)
p
EðDÞ (s)
EðqÞ
PrfD P dg
0.125
0.1 0.05 0.01
0.0023 0.0014 0.0006
0.176 0.155 0.103
0.0004 0.0004 0.0003
0.25
0.1 0.05 0.01
0.033 0.018 0.0028
0.351 0.309 0.206
0.035 0.017 0.0005
0.375
0.1 0.05 0.01
0.228 0.11 0.016
0.522 0.461 0.308
0.132 0.075 0.008
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aggregate LAN traffic measured at Bellcore in 1989 [24]. The prediction algorithm was an NLMS of order 20 (as in Section 4.3) such a choice proved to be a reasonable trade-off between computational complexity and prediction accuracy. Moreover, several values of the time quantum T p have been considered. In the experiments, the target delay bound d and the violation probability p are the input of the system. Incoming traffic is measured over time interval T p of 100 ms, 500 ms and 1 s, respectively. Tables 4–6 show the results obtained for the above listed values of T p by reporting the average measured delay, the mean server utilization and the measured violation probability against several values of target delay bound and violation probability. Table 4 shows that the project requirements are reasonably met for a target delay smaller than 12 T p . Smaller target delay results in average experienced delay and actual violation probability significantly lower than the project parameters. Moreover, the server utilization drops to very low levels. This is due to the very strict delay constraint which requires a large amount of resource to be allocated which, in turn, produces network under-utilization. The server utilization increases for target delay values larger than 12 T p ; unfortunately, in this case, project requirements are not met any longer. In this case, due to prediction error, the hypothesis that at the end of any time interval nT p the queue is empty is not always verified, and the buffer backlog builds-up determining higher packet propagation delays.
Tables 5 and 6 present similar results, while it can be noticed that project requirements are met even for target delays larger that 1 T . The higher delay constraint, indeed, makes possible queueing 2 p backlog phenomena less critical with nearly no impact on the overall performance. For the sake of completeness, Fig. 7 shows the delay complementary probability in the case of SDB running with T p ¼ 500 ms and project constraints d ¼ 250 ms and p ¼ 5%. Analogous experiments have been carried out for the CSU algorithm. As above mentioned, the scope of the CSU strategy is to take advantage of traffic prediction to achieve high network utilization. To assess the performance, the algorithm has been tested with value of T p ¼ 100 ms, T p ¼ 500 ms and T p ¼ 1 s and for different values of target utilization. Since results are similar among each other, for the sake of conciseness we report here the case of T p ¼ 1 s only. Table 7 shows the nominal and the actual server utilization, together with the average packet propagation delay and its 1% quantile. The results show a fair agreement between nominal and actual server utilization. Notice that high values of network load are achieved at the cost of extremely large values of the 1% quantile of the queueing delay. This confirms the rationale behind the algorithm proposed in this paper in that, even in CSU, reasonable values of queueing delay are obtained at (low) utilization values. 6.1. Final remarks
Table 6 Aggregation time T p ¼ 1 s d (s)
p
EðDÞ (s)
EðqÞ
PrfD P dg
0.25
0.1 0.05 0.01
0.003 0.0022 0.0012
0.179 0.159 0.109
0.00126 0.0012 0.0009
0.5
0.1 0.05 0.01
0.0353 0.0191 0.0043
0.356 0.316 0.217
0.017 0.0065 0.0013
0.75
0.1 0.05 0.01
0.257 0.13 0.02
0.530 0.473 0.324
0.088 0.048 0.006
The performance analysis of the proposed algorithm shows that when T p is in the range [0.1 s, 1 s], and the target maximum queueing delay is less than T p , the target performance is satisfied. As mentioned in Section 2, such a range is typical of application targeted for some cellular systems (where the prediction horizon is of the order of a few tens of ms) or dynamics bandwidth allocation in GEO satellite networks (where a prediction horizon of about 500 ms is a reasonable value). In wired networks, although in some cases the short time scale of predictions does not realistically enable end-to-end re-computation of resources (such as LSP remodulation in DiffServ over MPLS [17]), one may consider that on a 1 Gbps link, 100 ms are equivalent to about 8000–1500 bytes long packets. Thus, on a local basis, the prediction horizon looks
1 Delay = 250 ms, p=0.05
Complementary Probability
0.1
0.01
0.001
1e-04
1e-05
1e-06 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Packet Delay Fig. 7. Complementary probability distribution of the packet propagation delay with parameter T p ¼ 500 ms, d ¼ 250 ms, p ¼ 5%.
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Table 7 Aggregation time T p ¼ 1 s
q
EðqÞ
EðDÞ (s)
1%-quantile (s)
0.95 0.8 0.5 0.3
0.903 0.789 0.517 0.316
11.46 3.387 0.26 0.028
43 21 3.5 0.6
sufficient to change scheduling configurations to either allow an extra budget of bandwidth to high priority traffic that is foreseen to increase its bitrate or to grant extra resources to low priority traffic (such as TCP) upon a short time decreasing activity of valuable traffic is predicted. Furthermore, in other contexts, even short prediction times may be of interest, in particular when dealing with multimedia traffic. This is typically the case of a sudden increase of the number of VoIP calls or audio/video streaming flows on a backbone link as a consequence of emergencies or occasional flash crouds events (such as free tickets to the first caller, sport events like penalty kicks on a soccer game, etc.). In all those cases, even a time scale of a second might be critical in that network nodes build-up a considerable backlog that has to be served in the following seconds. The availability of an effective prediction technique may allow to catch-up the growth of traffic volumes by adaptively remodulating resources; on the contrary, the amount of voice and video packets stuck in the buffers will suffer from intolerable delays (if not discarded at all), and this phenomenon may persist for a significant amount of time thereafter. 7. Conclusion The paper addresses the design of two modules of a resource allocation system based on traffic prediction. In particular, the paper compares different prediction algorithms to be used in the system, and proposes a resource allocation algorithm, named SDB, aimed at guaranteeing a probabilistic bound on the maximum packet queueing delay. Three main prediction techniques have been analyzed in the paper: the NLMS predictor, the first order predictor and the radial basis function predictor. Performance of the three algorithms has been compared in terms of their accuracy in capturing actual traffic behavior, their implementing complexity and their responsiveness. The results show the high complexity of the RBFP, while a very low computation time is required by the first order predictor. On the contrary, RBFP presents higher accuracy and responsiveness with respect to the first order predictor. Differently from these two prediction schemes, the NLMS permits to achieve an adequate trade-off among complexity and accuracy–responsiveness. Indeed, the analysis has shown that the NLMS has a medium complexity, but permits to achieve the same performance of the RBFP in terms of accuracy and responsiveness. Upon the above results, the paper presents the novel SDB resource allocation strategy based on linear traffic prediction (implemented via the NLMS algorithm) and on the estimation of queueing delay through the Chebishev bound. The scope of the algorithm is to determine the minimum service capacity to be allocated in order to guarantee a probabilistic bound on the maximum delay experienced by packets traversing the resource. The results show that the project constraints are met as long as the values of delay and aggregation time satisfy the underlying analytic hypotheses. Such a technique candidates for managing quantitative QoS requirements in that it guarantees limited packet delays with assigned violation probability.
To properly assess the performance of the SDB algorithm, a second strategy (CSU) aiming at maintaining a given level of server utilization is considered. Such a requirement is well met in our experiments and, thanks to predictions, high values of server utilization are achieved even in the presence of VBR traffic. Unfortunately, this is obtained at the cost of very large queueing delays, which can be decreased by paying off in terms of a significant reduction of the network utilization. Indeed, at low traffic loads, results obtained under CSU and under the SDB technique are consistent, and prove the effectiveness of the latter in providing statistical guarantees on the packet delay introduced by network resources. Possible drawbacks of the SDB algorithm can be summarized into two points. The first one comes from the use of the Chebyshev bound that, in some cases, may be pretty loose and may results in bandwidth overprovisioning. The second drawback is the one already pointed out in Section 6, as we noticed that for target delays larger than half of the sampling time T p , the project requirement might not be met; indeed, in these cases, the hypothesis of empty queue at the end of any time interval may not be verified, and the buffer backlog builds-up determining higher packet propagation delays. However, this problem reduces as T p increases since, in this case, queueing backlog phenomena become less critical. Acknowledgment This work has been sponsored by the Italian MIUR research program MIMOSA. References [1] S. Blake, et al., An Architecture for Differentiated Services, IETF RFC, 2475, December 1998. [2] B.E. Carpenter, K. Nichols, Differentiated services in the Internet, Proceedings of the IEEE 90 (9) (2002). [3] MPLS Resource Center, http://www.mplsrc.com/. [4] D. Awduche, et al., Overview and Principles of Internet Traffic Engineering, IETF RFC 3272, 2002. [5] H.L. Lu, I. Faynberg, An architectural framework for support of quality of service in packet networks, IEEE Communications Magazine (2003). [6] K. Park, W. Willinger (Eds.), Self-Similar Network Traffic and Performance Evaluation, John Wiley and Sons, 2000. [7] F. Le Faucheur (Ed.), Protocol Extensions for Support of Diffserv-Aware MPLS Traffic Engineering, IETF RFC 4124, June 2005. [8] S.S. Oruganti, M. Devetsikiotis, Analyzing robust active queue management schemes: a comparative study of predictors and controllers, in: Proceedings of ICC 2003, Anchorage, Alaska, May 2003. [9] Y.H. Chan, T. Randhawa, S. Hardy, Traffic prediction based access control using different video traffic models in 3G CDMA high speed data networks, in: IWCMC ’06: Proceedings of the 2006 International Conference on Wireless Communications and Mobile Computing, Vancouver, British Columbia, Canada, 2006, pp. 227–232. [10] P. Koutsakis, On providing dynamic resource allocation based on multimedia traffic prediction in satellite systems, Computer Communications 30 (2) (2007) 404–415. [11] F. Chiti, R. Fantacci, F. Marangoni, Advanced dynamic resource allocation schemes for satellite systems, in: Proceedings of IEEE ICC 2005, vol. 3, Seoul, Korea, pp. 1469–1472. [12] R.G. Garroppo, S. Giordano, S. Lucetti, G. Procissi, Chaotic prediction and application to resource allocation strategies, in: Proceedings of the IEEE ICC 2004, 2004. [13] A.M. Adas, Using adaptive linear prediction to support real-time vbr video under rcbr network service model, in: Proceedings of the IEEE/ ACM Transaction on Networking, vol. 6, no. 5, October 1998, pp. 635– 644. [14] S. Chong, S.-Q. Li, J. Gosh, Predictive dynamic bandwidth allocation for efficient transport of real-time VBR video over ATM, IEEE Journal of Selected Area in Communications 13(1) (1995). [15] A. Sang, S. Li, A predictability analysis of network traffic, in: Proceedings of INFOCOM, 2000, pp. 342–351. [16] G.Carrozzo, N. Ciulli, S. Giordano, Multi access inter domain architecture for QoS provisioning in MPLS/DiffServ networks, in: TANGO Project Symposium, January 2004. [17] R.G. Garroppo, S. Giordano, F. Mustacchio, F. Oppedisano, G. Procissi, A flexible measurement system for supporting traffic engineering in DiffServ/MPLS networks, in: Workshop on Traffic Engineering, Protection and Restoration for NGI, Lund, Sweden, May 27–28, 2004.
R.G. Garroppo et al. / Computer Communications 31 (2008) 3741–3751 [18] M. Hayes, Statistical Digital Signal Processing and Modeling, Wiley & Sons, 1996. [19] S. Haykin, Adaptive Filter Theory, Prentice-Hall, 1991. [20] T.S. Parker, L.O. Chua, Chaos: a tutorial for engineers, Proceedings of the IEEE 75 (1987) 982–1008. [21] F. Takens, Detecting strange attractors in turbulence, in: D. Rand, L. Young (Eds.), Dynamical Systems and Turbulence of Lecture Notes in Mathematics, vol. 898, Springer, 1981, pp. 366–381.
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[22] M. Casdagli, Nonlinear prediction of chaotic time series, Physica D 35 (1989) 335–356. [23] J. Farmer, J. Sidrowich, Predicting chaotic time series, Physical Review Letter 59 (1987) 45–848. [24] W.E. Leland, M.S. Taqqu, W. Willinger, D.V. Wilson, On the self-similar nature of Ethernet traffic (extended version), IEEE/ACM Transactions on Networking (TON) 2 (1) (1994) 1–15.
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Computer Communications journal homepage: www.elsevier.com/locate/comcom
On traffic prediction for resource allocation: A Chebyshev bound based allocation scheme R.G. Garroppo, S. Giordano *, M. Pagano, G. Procissi Department of Information Engineering, University of Pisa, Via Caruso 16, I-56122 Pisa, Italy
a r t i c l e
i n f o
Article history: Available online 20 June 2008 Keywords: FIR filters Linear prediction Chebyshev bound Resource allocation
a b s t r a c t The paper presents a predictive approach to network resource allocation techniques. The rationale of this work is to use measurements to estimate future traffic behavior by prediction, and to use such an estimation to define the amount of future network resources that will be required by the considered traffic. In this framework, the paper presents the analysis and performance evaluation of classical and chaotic techniques for network traffic prediction. The performance parameters considered in the analysis are: the accuracy of predictors in capturing the actual behavior of traffic; the computational complexity for a realistic integration of such predictors into experimental testbeds; and the responsiveness with respect to traffic pattern variations. The analysis results show that the classical normalized linear mean square predictor achieves a satisfactory trade-off among the above mentioned metrics as it presents a medium level of complexity while achieving high performance in terms of prediction accuracy and responsiveness to network traffic changes. Then, using the normalized linear mean square predictor, we derive a bandwidth allocation strategy, named statistical delay bound (SDB), which guarantees a probabilistic bound on the delay experienced by packets traversing a network node. The paper presents the performance analysis of SDB showing that, in spite of the simplicity of the adopted predictive algorithm, the proposed measurement based technique allows to fulfill the project requirements and candidates for actual experimentation into prototypal routers which supports QoS mechanisms. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction The evolutionary trend of communication networks in which new services such as Voice over IP, audio/video streaming, videoconferencing and virtual private networks (VPNs) with quality of service are being offered to end users has led to the deployment of network architectures (e.g. DiffServ [1,2]) supporting QoS. Moreover, over the last years, a considerable effort has been dedicated to the optimization of operational IP networks through the MPLS switching paradigm [3] and the so called traffic engineering [4]. The exigence of integration of such approaches arises the necessity of dynamic traffic control mechanisms, such as resource allocation techniques and admission control, to achieve optimal resource utilization and network performance [5]. Model-based approaches have demonstrated their weakness in that the long range dependence (LRD) property of network traffic [6] is not on-line distinguishable from non-stationarity and it does not permit an easy estimation of characteristic parameters (such as the Hurst parameter). This has impaired the development of easy,
* Corresponding author. Tel.: +39 050 2217539. E-mail addresses: [email protected] (R.G. Garroppo), s.giordano@iet. unipi.it (S. Giordano), [email protected] (M. Pagano), [email protected] (G. Procissi). 0140-3664/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2008.05.019
flexible and effective traffic models to be widely used for network dimensioning. In addition to the above mentioned reasons, novel concepts like network self-sizing and reconfigurability have led to consider measurement based approaches as the most promising in order to address (even in approximate ways) common issues in Diffserv aware traffic engineering (DS-TE) [7] context such as constraintbased path computation, LSP dimensioning, scheduler parameters selection and so on. In this scenario, the objective of this paper is to propose a novel measurement based strategy for resource allocation which integrates traffic measures and prediction. To this aim, the comparison of different prediction techniques and the definition of a resource allocation algorithm that uses the information produced by the selected traffic predictor are of paramount relevance. The first part of the paper is devoted to the presentation of several prediction algorithms (classic, chaotic and geometric) and to the investigation of their performance in terms of the above mentioned criteria: accuracy, complexity and responsiveness. The prediction algorithm which will achieve the best trade-off among the three criteria will be integrated into the resource allocation strategy derived in the second part of the paper. As above mentioned, the second part of the article is devoted to the derivation and performance evaluation of the statistical delay
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bound (SDB) resource allocation strategy. SDB aims at computing the service capacity to be allocated a connection in order to guarantee a probabilistic bound on the maximum queueing delay experienced by packets of the connection itself. In summary, the paper is organized as follows. After the presentation of related works on traffic prediction for resource allocation, Section 3 gives the basic notions of time series prediction, while Section 4 reports the performance comparison of three different prediction algorithms. Section 5 describes the SDB allocation strategy whose performance is analyzed in Section 6. Finally, Section 7 concludes the paper with final remarks.
2. Related works Although time series prediction is a classic topic in signal processing, its adoption into networking contexts is becoming quite frequent. Recently, different predictors have been presented and a new active queue management (AQM) scheme exploiting the predictions of future traffic intensity has been proposed in [8]. Focusing on resource allocation algorithms based on traffic prediction, different solutions have been proposed depending on network technology. In particular, in [9] the authors propose an access control protocol for high speed downlink packet access (HSDPA) system based on bandwidth allocation between real time and non-real time traffic. By applying a traffic model to the real time traffic in the current time slot, the system can predict the residual capacity in the next time slot, thus it can optimize the scheduling of non-real time data transmission. It is relevant to note that predictions are based on a model assumption for the considered traffic, and that the time quantum, T p , is of the order of the transmission time interval length (usually tens of ms). In [10], the author proposes a new dynamic satellite bandwidth allocation technique (named predictive resource reservation access, PRRA) which is based on accurate videoconference traffic prediction. The proposed model is then combined with results on data traffic modeling and prediction presented in [11] and is shown to provide very good throughput and delay results. In this case, the time quantum T p is of the order of a round trip time for a GEO satellite system, i.e. around 500 ms. In both these works, the predictions are based on traffic models; it is well known that the accuracy of model-based approaches depend on the reliability of the assumed source models. Often, to simplify the model of the system, Markov chains are used to model source traffic; this assumption ignores the LRD nature of traffic. In our previous work [12], the LRD nature of traffic has motivated the investigation of chaotic prediction to develop bandwidth allocation strategies. Differently from these works, in [13] the dynamic bandwidth allocation strategy designed to support variable bit rate (VBR) video traffic does not assume any model for the traffic source. The strategy predicts the bandwidth requirements for future frames by using adaptive linear prediction that minimizes the mean square error. The adaptive technique does not require any prior knowledge of the traffic statistics nor assume stationarity. In this case, the prediction based bandwidth allocation algorithm has been defined for the renegotiated constant bit rate (RCBR) network service model of ATM. Two traffic predictors on frequency domain have been compared in [14], in terms of complexity and ability to guarantee the project parameter of the resource allocation algorithm. The authors assume the server utilization as the project parameter (hence we refer to that approach as constant server utilization algorithm). The focus of [14] is on VBR video transport over ATM network. As opposed to the previous mentioned works, our goal is to define a bandwidth allocation algorithm based on traffic prediction,
without any assumptions on the statistical features of traffic. Furthermore, our proposal uses the prediction algorithm in the time domain and the allocation strategy has been designed by assuming as project constraint the maximum queueing delay. Moreover, the system model considered in our study is general and does not refer to any specific technology. It can be adapted for resource allocation in cellular, satellite or wired networks. Finally, the prediction algorithm as well as the bandwidth allocation scheme have been selected taken into account the actual implementation of the proposed solutions. The purpose of predictors is to capture the traffic intensity and accurately estimate the future arrivals over prediction intervals [15]. Also, they should exhibit high responsiveness with respect to sudden traffic changes. The huge effort towards actual experimentation of novel network architectures through the set up of field-trials [16,17] adds further critical constraints: ease of implementation and low computational complexity. The proposed solutions have been selected by considering these aspects. 3. Basics of traffic prediction This section briefly presents some of the basic concepts of time series prediction. Since the topic is well known and widely investigated in signal processing, in the following a very concise description will be given and more details are left to the bibliography. Time series prediction can be addressed through statistical or geometric techniques. This paper investigates both classes of algorithms, with special focus on linear and chaotic prediction for the first class, and on polynomial extrapolators for the class of geometric algorithms. None of the presented algorithms, though, require the knowledge of a specific model of the underlying time series to be predicted. Statistical predictors only require the underlying stochastic sequence to be wide sense stationary, while geometric predictors are derived by simple algebraic reasoning. The general problem of prediction can be stated as follows: given a set of observations of a stochastic process xðnÞ, give an estimation ^ xðn þ kÞ of the value xðn þ kÞ that the process x will assume k steps ahead. In other words, given a vector of p observations, x ¼ ½xðnÞ; xðn 1Þ; . . . ; xðn p þ 1Þ, the predicted value ^ x is obtained by
^x ¼ ðxÞ
ð1Þ
where the function is called predictor. The topic of next paragraphs will be the derivation of the analytic expression of for the different classes of predictors investigated. 3.1. Linear prediction Linear prediction occurs whenever the function ðxÞ is linear. In other words, the problem is to determine the impulse response hðnÞ of the linear filter h such that
^xðn þ kÞ ¼ xðnÞ hðnÞ ¼
p1 X
hðiÞxðn iÞ
ð2Þ
i¼0
The filters coefficients can be determined according to arbitrary optimality criteria. One of the most famous and widely adopted prediction algorithm is the so-called linear minimum mean square error (LMMSE) predictor in which, the values hðnÞ are derived by minimizing the mean square error of prediction:
E½e2 ðnÞ ¼ E½ðxðn þ kÞ ^xðn þ kÞÞ2
ð3Þ
In that sense, LMMSE is an optimal algorithm in that, within the class of linear filters, it minimizes the mean square error of predic-
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tion. A thorough description of LMMSE is beyond the objective of this paper and can be found in [18,19]. We only mention here that the derivation of the LMMSE filter requires the knowledge of at least p values of the autocorrelation function of the stochastic process x and inverting a p p matrix. These facts make LMMSE very hard to be implemented on real running equipments. The following paragraphs present, instead, two adaptive versions of LMMSE that will be used in practice in the paper, denoted as least mean square (LMS) predictor and normalized least mean square (NLMS) predictor, respectively.
Notice that, at time n, the values xðn þ kÞ, hence eðnÞ, are not known. Therefore, the value eðn kÞ is used, instead, and the one-step NLMS predictor update equation becomes:
3.1.1. Least mean square predictor The LMS algorithm is an adaptive approach which does not require prior knowledge of the autocorrelation structure of the stochastic sequence [18,19]. Thus, it can be used as an on line technique for predicting bandwidth requirements. The algorithm scheme is shown in Fig. 1. The filter coefficients are time varying and are tuned on the basis of the feedback information carried by the error eðnÞ. In the following, we denote the vector of filter coefficients at time n with hn . The values of h adapt dynamically in order to decrease the mean square error. Notice that eðnÞ is defined as in the previous section and xn ¼ ½xðnÞ; xðn 1Þ; . . . ; xðn p þ 1Þ. Indeed, the LMS algorithm is the adaptive version of the optimal LMMSE predictor. The LMS algorithm operates as follows:
Geometric predictors are based on the approximation of the behavior of the time series xðnÞ through elementary functions. These techniques do not involve stochastic analysis of time series while they only involve pure geometric considerations. The basic idea is to determine the ðp 1Þ-order polynomial interpolator of the set of p observations xðnÞ; xðn 1Þ; . . . ; xðn p þ 1Þ available. Such a polynomial is of the form:
(1) initialize the coefficients h0 (2) for each new data, update the filter hðnÞ according to the recursive equation
hnþ1 ¼ hn þ leðnÞxn
ð4Þ
where l is a constant called the step size. If xðnÞ is stationary, the vector of coefficients h converges in the mean to the optimal solution of LMMSE predictor [18,19]. The speed of convergence of the algorithm depends on the value of l. Large values of l result in a faster convergence of the algorithm and in a quicker response to signal changes. However, small values of l result in slower convergence and less fluctuations once the convergence is attained. The selection of l thus, should be made by trade-off between the above phenomena. 3.1.2. Normalized least mean square predictor The normalized LMS algorithm is a modification to the LMS algorithm in which the update equation of the filter’s coefficients is
hnþ1 ¼ hn þ l
eðnÞxn
ð5Þ
kxn k2
where kxn k2 ¼ xn xtn . The main advantage of using NLMS is that it is less sensitive to the step size l. According to [19], NLMS converges in the mean to LMMSE predictor as long as 0 < l < 2.
hnþ1 ¼ hn þ l
ð6Þ
The computational complexity of a p-order NLMS algorithm is that of 2p þ 1 multiplications and p þ 1 additions. 3.2. Geometric prediction
Pp1 ðzÞ ¼ ap1 zp1 þ ap2 zp2 þ þ a1 z þ a0
ð7Þ
and the coefficients an are determined by the condition Pp1 ðkÞ ¼ xðkÞ; n p þ 1 6 k 6 n, that is by solving the linear system (with Vandermonde matrix)
0
1 B1 B B B .. @.
n
.. .
10
np1
n1 .. .
p1
ðn 1Þ .. .
1 n p þ 1 ðn p þ 1Þp1
a0 CB a CB 1 CB . CB . A@ . ap1
1
0
C B C B C¼B C B A @
1
xðnÞ xðn 1Þ .. .
C C C C A
xðn p þ 1Þ
A simple example of this predictor class is the first order predictor, which can be obtained assuming p ¼ 2. From the above relation, this assumption implies that the prediction algorithm considers the observations xðnÞ and xðn 1Þ. In particular, the predicted value, ^xðn þ 1Þ, is obtained by considering the slope of the straight line passing through the two points xðnÞ and xðn 1Þ, i.e.
^xðn þ 1Þ ¼ 2xðnÞ xðn 1Þ: 3.3. Chaotic prediction This section presents the application of chaos theory to time series prediction. For the sake of conciseness, details on the theory of chaos are omitted here; we refer the interested reader to [20] for an excellent overview on the topic. Hence, in this section we will recall only the description of a particular type of chaotic predictor, the so-called radial basis function predictor (RBFP), which has been studied and deeply analyzed in [12]. To make predictions about chaotic time series, the first step is to consider the time series as obtained by sampling a continuoustime scalar variable xðtÞ representing the time evolution of system whose underlying dynamic is that of a strange attractor lying on a D-dimensional invariant manifold. Assume then a sequence of observation xðnÞ ¼ xðnT p Þ; the sequence is referred to as chaotic time series and the sampling period T p is called delay time. The second step is to embed the sequence in a m-dimensional state space. The value m is called embedding dimension. If the strange attractor lies on a D-dimensional invariant manifold, then, necessarily, m P D. On the other hand, Takens theorem assures that m 6 2D þ 1 [21]. The state vectors will be of the form:
0 B B xn ¼ B B @ Fig. 1. LMS algorithm scheme.
eðn 1Þxn1 kx2n1 k
xðnT p Þ
1t
xðnT p T p Þ .. .
C C C C A
xðnT p ðm 1ÞT p Þ
ð8Þ
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The third step is to select a proper interpolation function fN such that
xðn þ 1Þ ¼ fN ðxn Þ:
0
/11 B . B .. B B @ /k1 1
3.3.1. Approximation techniques In the previous paragraph we mentioned interpolation functions, namely functions fN such that
xðn þ 1Þ ¼ fN ðxn Þ;
16n6N1
This, of course, widens the number of techniques that could be used. Typically, they belongs to two different categories: global techniques and local techniques. In the former, the approximant (or interpolant) is applied to the whole series of vectors that can be constructed through the sequence. In the latter case, instead, the prediction is made on the basis of the states in the past that are near to the current one. The rationale of this technique is simple and intuitive: look for pieces of the trajectory in the past that resemble the current one and infer the future behavior according to the way the system evolved in that analogous past condition. Incidentally, this method looks well tailored to the case of network traffic which has widely proven to be selfsimilar. The selection of the number k of ‘‘neighbors” in the past is another critical issue. Typically, and we choose this strategy, k ¼ m þ 1 is assumed [22,23]. Global techniques are computationally too expensive and are not considered in our study. We use the local approach and we select the radial basis functions as approximant functions. 3.3.2. Radial basis function predictor (RBFP) According to the previous discussion, given the set of k neighbors xnj ; j ¼ 1; 2; . . . ; k, of xn we choose a prediction scheme of the form:
^xðn þ 1Þ ¼
k X
kj /ðkxn xnj kÞ þ l
ð9Þ
j¼1
Notice that, in its general expression, the constant l is replaced by a polynomial term (usually not included). The functions / : Rþ ! R, defined as
/ðrÞ ¼ ðr2 þ c2 Þb
b > 1; b–0
ð10Þ
are called radial basis functions. In our analysis we always used b ¼ 0:5. In other words, the prediction (9) can be interpreted by a weighted sum of terms in which the contribution of each neighbor depends inversely on their distance to the current state (closer states give a bigger contribution). The values kj ; j ¼ 1; 2; . . . ; k are determined through the knowledge of the past evolution of the state as
^xðni þ 1Þ ¼
k X
/kk
1
10 1 0 1 xðn1 þ 1Þ 1 k1 CB C B C .. ... CB ... C B C . CB C ¼ B C CB C B C 1 A@ kk A @ xðnk þ 1Þ A 0
l
ð12Þ
0
where
/ij ¼ /ðkxni xnj kÞ:
16n6N1
In many practical contexts, including teletraffic studies, this constraint may be significantly relaxed by allowing fN to be an approximant, that is
xðn þ 1Þ f N ðxn Þ;
/1k .. .. . .
kj /ðkxni xnj kÞ þ l
ð11Þ
4. Prediction algorithms comparison The primary goal of this research is to develop a dynamic and flexible prediction based resource allocation strategy. Thus, at this stage, the prediction algorithms presented in the previous sections will be compared in order to select the one that best fits into the overall research target. In order to define the performance parameters to be considered in the comparison, we first establish the scenario where these predictors will be used (Fig. 2). The model of the resource is that of a single server queue with infinite buffer and deterministic (variable) service rate that will be dynamically adjusted on the basis of the underlying traffic prediction algorithm. A measurement system registers the amount of traffic incoming to the queue over the time intervals ½ðn 1ÞT p ; nT p Þ and produces the time series xðnÞ ¼ xðn T p Þ. The time series xðnÞ is then processed by the prediction block, which returns the predicted value ^ xðn þ 1Þ of traffic expected in the next time interval ½nT p ; ðn þ 1ÞT p Þ. Finally, the allocation block computes the queueing service rate cðn þ 1Þ to be selected in the time interval ½nT p ; ðn þ 1ÞT p Þ, on the basis of the information obtained by predictions. 4.1. Performance parameters The scenario in which the prediction block will be integrated determines the critical parameters to be considered in the comparison of predictors. Firstly, we can notice that, in a time T c T p , the system should ^ predict X½ðn þ 1ÞT p and calculate c½ðn þ 1ÞT p . The parameter T c must be negligible with respect to T p , since the network resources calculated in T c are valid for a time T p . Hence, the prediction algorithm must exhibit low computational complexity. Secondly, the resource allocated depends on the traffic predic^ tion X½ðn þ 1ÞT p . Hence, it is very important to assure a high accuracy of the predictor. A third requirement for the prediction algorithm can be deduced by reminding that the system works ‘‘on line” and that traf-
Input Traffic
Output
Measurement System
j¼1
together with k X
kj ¼ 0
j¼1
Using a matrix notation, the above system of k þ 1 equations is equivalent to the following system:
Prediction System
Resource Allocator
Fig. 2. Prediction-based resources allocation system.
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fic characteristics can vary in time very quickly. Hence, the prediction algorithm should be able to readily react to sudden variation of the traffic features. In other words, the predictor should by highly responsive. In summary, the performance parameters considered in the comparison will be (1) the complexity, (2) the accuracy, (3) the responsiveness.
4.2. Metrics for the quality of prediction In order to quantify the accuracy of a predictor fN , we introduce a general and unified definition of prediction error r2 ðfN Þ, as
lim
M!1
X 1 1 NþM1 k^xðn þ 1Þ fN ðxn Þk2 M V n¼N
ð13Þ
with the normalizing factor:
M M 1 X 1 X V ¼ lim xðmÞ xðmÞ lim M!1 M M!1 M m¼1 m¼1
ð14Þ
Values of r2 ðfN Þ close to 0 indicate good performance of the predictor. Performance gets worse as r2 ðfN Þ tends to 1. Indeed, r2 ðfN Þ close to zero implies predicted values nearly identical to the actual ones [22]. This condition is excessively restrictive. It is often enough to require a reasonably low value of r2 ðfN Þ and a good match of the statistics of actual and predicted data, such as auto-covariance function, distribution, variance, and so forth.
sary in a PC to obtain a single predicted value. The values obtained with the three considered predictors are summarized in Table 1. We can clearly observe that RBFP exhibits a quite higher complexity with respect to both first order and NLMS predictor. Indeed, the number of clock ticks necessary to obtain a single predicted value when considering the RBFP is double with respect to the NMLS and 3 order of magnitude higher than the first order predictor case. To quantify the time necessary to obtain a single predicted value, we can refer to a PC equipped with a CPU at 1 GHz. Considering this equipment, the computation time of the different algorithms is summarized in Table 2. As we can observe, the RBFP can require a computation time higher than 1 ms to calculate a single predicted value; this value can be inadequate when T p is in the order of tens of ms and for network links with high bandwidth. 4.3.2. Accuracy The accuracy has been evaluated by considering the prediction error defined in Section 4.2. The results are summarized in Table 3, for different sampling time, T p . The table points out the NLMS predictor as the one that achieves the best performance, while the first order predictor, facing a good computational speed, returns the largest value of prediction error. RBFP achieves high accuracy but the selection of its parameters is a long and off-line procedure that strongly depends on the characteristics of the input traffic. Furthermore, as highlighted by the previous paragraph, the RBFP presents a high complexity. For these reasons the best choice for a prediction technique to be implemented in an on-line environment, seems to be the NLMS predictor, that offers a good trade-off between accuracy and computational speed. Finally, we can note that the accuracy decreases as the sampling time, T p , increases.
4.3. Prediction performance analysis In this section, we present numerical results of the comparison among some predictors described in the previous sections. The final result of this section will be the selection of the prediction algorithm which best combines the characteristics of accuracy, low complexity and responsiveness, and that will be adopted in the resource allocation scheme depicted in Fig. 2. The study is carried out by using the traffic data acquired during an emulation of an actual service. In particular, we consider a time series named ‘‘Conf Cell” representing the traffic offered by a videoconference session corresponding to three people in front of a 3-CCD camera. Each sample of the time series represents the number of ATM cells transmitted during a frame period, equal to 1/25 s (40 ms). The available number of samples is equal to 48,496, with mean value and standard deviation equal to 174.064 and 99.21 cells/frame, respectively. In the comparison, we have considered three different types of prediction algorithms: (1) a NLMS of order 20, (2) the first order predictor (denoted in the tables as F.O.P.), (3) the radial basis function predictor.
4.3.3. Responsiveness The last analysis is related to the ability of the prediction algorithm in capturing the variation of traffic characteristics. In this study, we have artificially varied the mean of the measured traffic adding a constant term in the time series xðnÞ, starting from a fixed value of n. The behavior obtained with the first order, the NLMS and the RBFP predictors are shown in Figs. 3–5. From the figures, we can note the high prediction error of the first order prediction at the time 5000 T p , due to high variation of the slope presented by the measured traffic pattern. Indeed, this can be easily explained by observing that the first order predictor predicts the value at time ðn þ 1ÞT p by simply considering the slope of the traffic pattern observed at time nT p . On the contrary, at the time instant of sudden variation, NMLS and RBFP produce an almost negligible error, as can be observed in Figs. 4 and 5. 4.3.4. Final remarks The above presented discussion proves the similar performance of NLMS and RBFP predictors in that they prove to be significantly
Table 2 Computation time to obtain a single predicted value
4.3.1. Complexity A first analysis is devoted to evaluate the complexity of the compared prediction algorithms. As a performance measure for this analysis, we have considered the number of clock ticks neces-
NMLS
F.O.P.
RBFP
660 ls
0.2 ls
1.26 ms
Table 3 Prediction error for different sampling time T p Table 1 Clock ticks necessary to obtain a single predicted value NMLS
F.O.P.
RBFP
0:66 106
213
1:26 106
T p (ms)
NMLS
F.O.P.
RBFP
40 80 120
0.0379 0.0734 0.1125
0.0483 0.0903 0.1493
0.0413 0.0687 0.0985
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650 Actual Data F.O.P. Predicted Data
600
Number of ATM Cells
550 500 450 400 350 300 250 200 4980
4990
5000
5010
5020
5030
5040
Sample, n Fig. 3. Responsiveness of F.O.P. Predictor for T p ¼ 40 ms.
550 Actual Data NLMS Predicted Data
Number of ATM Cells
500
450
400
350
300
250 4980
4990
5000
5010
5020
5030
5040
Sample, n Fig. 4. Responsiveness of NLMS for T p ¼ 40 ms.
accurate as well as enough responsive to react against sudden variation of traffic data. Also, their complexities have the same order of magnitude (though the one achieved by RBFP is twice as large as the one from NLMS); however, RBFP requires a preliminary phase of parameters tuning which may prevent from its use in automatic manner without any need for supervision. For all these reasons, NLMS will be adopted in the prediction-based resource allocation scheme that will be described in the next sections of the paper.
5. Resource allocation This section presents the novel Statistical delay bound resource allocation algorithm based on linear prediction of traffic as well as the basic definition of the CSU algorithm [14] that will be
used in the following section to assess the performance results of SDB. 5.1. Statistical delay bound (SDB) resource allocation algorithm According to the overall system presented in Section 4 and Fig. 2, in the following we derive the statistical delay bound (SDB) resource allocation algorithm which takes advantage of NLMS predictions to dynamically adjust the system service rate in order to guarantee statistically bounded packet delay. In other words, by denoting the overall packet queueing delay with D, and the target delay with d, the service rate cðn þ 1Þ will be calculated in order to meet
PrfD > dg < p where p is an arbitrarily small probability.
ð15Þ
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500 Actual Data RBFP Predicted Data
Number of ATM Cells
450
400
350
300
250
200 4980
4990
5000
5010
5020
5030
5040
Sample, n Fig. 5. Responsiveness of RBFP for T p ¼ 40 ms.
Unfortunately, the statistics of D are strongly dependent on the statistical properties of the input traffic. As already stated in the introduction, the approach presented here is based on measurements and does not assume any a-priori knowledge of input traffic models; therefore, in the following, we address a worst case approach by focusing on the maximum delay D experienced by packets. The condition
PrfD > dg < p
cðn þ 1Þ ¼
^xðn þ 1Þ d
ð17Þ
would guarantee a maximum packet delay of d. Unfortunately, the prediction is, in general, affected by errors; hence, the actual worst case delay experienced by packets is
D¼
ð16Þ
xðn þ 1Þ xðn þ 1Þ ¼ d cðn þ 1Þ ^xðn þ 1Þ
ð18Þ
By denoting the prediction error with ðnÞ, the actual traffic volume can be written as
assures that relation (15) is fulfilled. To show how the service rate is computed, let us begin by assuming the queue empty at the end of the nth time interval and by observing that, if the prediction ^ xðn þ 1Þ were exact, a service rate given by
xðn þ 1Þ ¼ ^xðn þ 1Þ þ ðnÞ
ð19Þ
and Eq. (18) becomes
0.0002 Prediction error pdf 0.00018 0.00016 0.00014
0.00012 0.0001 8e-05 6e-05 4e-05 2e-05 0 -20000
-15000
-10000
-5000
0
5000
10000
Bytes/Time unit Fig. 6. Histogram of NLMS prediction error.
15000
20000
25000
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R.G. Garroppo et al. / Computer Communications 31 (2008) 3741–3751
D¼ 1þ
ðnÞ d ^xðn þ 1Þ
ð20Þ
Whenever ðnÞ > 0, the worst case delay D is bigger than the target delay d, and the difference can be very large depending on the prediction error. Furthermore, let us make the reasonable assumption that the prediction error ðnÞ has a symmetric zero-mean distribution. Such an assumption is quite logical and is verified in experiments (see Fig. 6). In this hypothesis, half of the time the worst case delay would be larger than the target value d if the service rate is computed as in (17). To sort out this problem, let us correct (17) by adding at the numerator a positive constant n:
cðn þ 1Þ ¼
^xðn þ 1Þ þ n d
ð21Þ
This allows to account for positive value of the prediction error ðnÞ (negative values of ðnÞ would give D < d even in the ideal case (17)). The new worst case delay experienced by packets becomes
D¼
^xðn þ 1Þ þ ðnÞ d ^xðn þ 1Þ þ n
ð22Þ
for which, now, D 6 d if and only if ðnÞ 6 n. In other words, the original problem involving packet delay has been translated by contraction into the following problem involving the prediction error ðnÞ:
PrfD P dtg ¼ PrfðnÞ P ng
ð23Þ
To estimate (23), we use the well known Chebyshev inequality which assures that any random variable X with mean value lX and finite variance r2X satisfies
PrfjX lX j P krX g 6
1 2
k
ð24Þ
By (24) and because of the assumed symmetry of the probability density function of ðnÞ, we have then
PrfðnÞ P kr g 6
1 2
2k
ð25Þ
and, by selecting n ¼ kr
PrfD P dg 6
1 2k
2
¼p
AT p ¼ cðn þ 1Þ T p
AT p ¼ cðn þ 1Þ T p ¼ xðn þ 1Þ
(1) receives as inputs parameter the target delay d and the violation probability p as well asqthe xðn þ 1Þ; ffiffiffiffi predicted value ^ 1 ; (2) computes the parameter k ¼ 2p ^ ðnÞ; (3) estimates r ^ ; (4) computes the parameter n ¼ kr ^ ; (5) calculates the service rate cðn þ 1Þ ¼ xðnþ1Þþn d (6) selects the queueing service rate accordingly. Let us give a brief discussion on the value of the sampling time T p to be used. One more time, let us suppose the prediction algorithm to be exact, that is ^ xðn þ 1Þ ¼ xðn þ 1Þ with service rate
Tp P xðn þ 1Þ d
ð29Þ
and, thus, the constraint
Tp P d
ð30Þ
must be satisfied. 5.2. Constant server utilization (CSU) algorithm The performance of the SDB algorithm will be compared to that proposed in [14], where resources are still allocated on the basis of traffic prediction although considering the server utilization as the project parameter. The allocation algorithm is very simple and the service rate is calculated by keeping constant the server utilization q, defined as the ratio between the average incoming traffic rate and the service rate. In other words, by defining the estimated arrival rate as
^rðn þ 1Þ ¼
^xðn þ 1Þ Tp
ð31Þ
the service rate is computed in order to maintain the ratio
^rðn þ 1Þ ¼q cðn þ 1Þ
ð32Þ
constant. 6. Performance evaluation The performance of the above proposed allocation strategies has been assessed by discrete event simulations driven by the Table 4 Aggregation time T p ¼ 100 ms d (s)
p
EðDÞ (s)
EðqÞ
PrfD P dg
0.025
0.1 0.05 0.01
0.0022 0.0013 0.0004
0.178 0.152 0.095
0.015 0.006 0.00012
0.05
0.1 0.05 0.01
0.0342 0.0165 0.0024
0.355 0.304 0.191
0.12 0.06 0.007
0.075
0.1 0.05 0.01
0.193 0.098 0.012
0.522 0.452 0.287
0.265 0.0175 0.033
ð27Þ
In summary, at any time interval n T p , the algorithm acts as follows:
ð28Þ
which should be greater than xðn þ 1Þ to leave the queue empty at the beginning of the next time slot. This condition gives
ð26Þ
The last equation involves the evaluation of r , for which we use the classical estimator of the sample variance taken over samples of length N, as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u 1 X r^ ðnÞ ¼ t 2 ðn iÞ N 1 i¼1
cðn þ 1Þ ¼ xðn þ 1Þ=d. The total amount of traffic that can be processed by the queue in the time interval T p is then
Table 5 Aggregation time T p ¼ 500 ms d (s)
p
EðDÞ (s)
EðqÞ
PrfD P dg
0.125
0.1 0.05 0.01
0.0023 0.0014 0.0006
0.176 0.155 0.103
0.0004 0.0004 0.0003
0.25
0.1 0.05 0.01
0.033 0.018 0.0028
0.351 0.309 0.206
0.035 0.017 0.0005
0.375
0.1 0.05 0.01
0.228 0.11 0.016
0.522 0.461 0.308
0.132 0.075 0.008
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aggregate LAN traffic measured at Bellcore in 1989 [24]. The prediction algorithm was an NLMS of order 20 (as in Section 4.3) such a choice proved to be a reasonable trade-off between computational complexity and prediction accuracy. Moreover, several values of the time quantum T p have been considered. In the experiments, the target delay bound d and the violation probability p are the input of the system. Incoming traffic is measured over time interval T p of 100 ms, 500 ms and 1 s, respectively. Tables 4–6 show the results obtained for the above listed values of T p by reporting the average measured delay, the mean server utilization and the measured violation probability against several values of target delay bound and violation probability. Table 4 shows that the project requirements are reasonably met for a target delay smaller than 12 T p . Smaller target delay results in average experienced delay and actual violation probability significantly lower than the project parameters. Moreover, the server utilization drops to very low levels. This is due to the very strict delay constraint which requires a large amount of resource to be allocated which, in turn, produces network under-utilization. The server utilization increases for target delay values larger than 12 T p ; unfortunately, in this case, project requirements are not met any longer. In this case, due to prediction error, the hypothesis that at the end of any time interval nT p the queue is empty is not always verified, and the buffer backlog builds-up determining higher packet propagation delays.
Tables 5 and 6 present similar results, while it can be noticed that project requirements are met even for target delays larger that 1 T . The higher delay constraint, indeed, makes possible queueing 2 p backlog phenomena less critical with nearly no impact on the overall performance. For the sake of completeness, Fig. 7 shows the delay complementary probability in the case of SDB running with T p ¼ 500 ms and project constraints d ¼ 250 ms and p ¼ 5%. Analogous experiments have been carried out for the CSU algorithm. As above mentioned, the scope of the CSU strategy is to take advantage of traffic prediction to achieve high network utilization. To assess the performance, the algorithm has been tested with value of T p ¼ 100 ms, T p ¼ 500 ms and T p ¼ 1 s and for different values of target utilization. Since results are similar among each other, for the sake of conciseness we report here the case of T p ¼ 1 s only. Table 7 shows the nominal and the actual server utilization, together with the average packet propagation delay and its 1% quantile. The results show a fair agreement between nominal and actual server utilization. Notice that high values of network load are achieved at the cost of extremely large values of the 1% quantile of the queueing delay. This confirms the rationale behind the algorithm proposed in this paper in that, even in CSU, reasonable values of queueing delay are obtained at (low) utilization values. 6.1. Final remarks
Table 6 Aggregation time T p ¼ 1 s d (s)
p
EðDÞ (s)
EðqÞ
PrfD P dg
0.25
0.1 0.05 0.01
0.003 0.0022 0.0012
0.179 0.159 0.109
0.00126 0.0012 0.0009
0.5
0.1 0.05 0.01
0.0353 0.0191 0.0043
0.356 0.316 0.217
0.017 0.0065 0.0013
0.75
0.1 0.05 0.01
0.257 0.13 0.02
0.530 0.473 0.324
0.088 0.048 0.006
The performance analysis of the proposed algorithm shows that when T p is in the range [0.1 s, 1 s], and the target maximum queueing delay is less than T p , the target performance is satisfied. As mentioned in Section 2, such a range is typical of application targeted for some cellular systems (where the prediction horizon is of the order of a few tens of ms) or dynamics bandwidth allocation in GEO satellite networks (where a prediction horizon of about 500 ms is a reasonable value). In wired networks, although in some cases the short time scale of predictions does not realistically enable end-to-end re-computation of resources (such as LSP remodulation in DiffServ over MPLS [17]), one may consider that on a 1 Gbps link, 100 ms are equivalent to about 8000–1500 bytes long packets. Thus, on a local basis, the prediction horizon looks
1 Delay = 250 ms, p=0.05
Complementary Probability
0.1
0.01
0.001
1e-04
1e-05
1e-06 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Packet Delay Fig. 7. Complementary probability distribution of the packet propagation delay with parameter T p ¼ 500 ms, d ¼ 250 ms, p ¼ 5%.
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Table 7 Aggregation time T p ¼ 1 s
q
EðqÞ
EðDÞ (s)
1%-quantile (s)
0.95 0.8 0.5 0.3
0.903 0.789 0.517 0.316
11.46 3.387 0.26 0.028
43 21 3.5 0.6
sufficient to change scheduling configurations to either allow an extra budget of bandwidth to high priority traffic that is foreseen to increase its bitrate or to grant extra resources to low priority traffic (such as TCP) upon a short time decreasing activity of valuable traffic is predicted. Furthermore, in other contexts, even short prediction times may be of interest, in particular when dealing with multimedia traffic. This is typically the case of a sudden increase of the number of VoIP calls or audio/video streaming flows on a backbone link as a consequence of emergencies or occasional flash crouds events (such as free tickets to the first caller, sport events like penalty kicks on a soccer game, etc.). In all those cases, even a time scale of a second might be critical in that network nodes build-up a considerable backlog that has to be served in the following seconds. The availability of an effective prediction technique may allow to catch-up the growth of traffic volumes by adaptively remodulating resources; on the contrary, the amount of voice and video packets stuck in the buffers will suffer from intolerable delays (if not discarded at all), and this phenomenon may persist for a significant amount of time thereafter. 7. Conclusion The paper addresses the design of two modules of a resource allocation system based on traffic prediction. In particular, the paper compares different prediction algorithms to be used in the system, and proposes a resource allocation algorithm, named SDB, aimed at guaranteeing a probabilistic bound on the maximum packet queueing delay. Three main prediction techniques have been analyzed in the paper: the NLMS predictor, the first order predictor and the radial basis function predictor. Performance of the three algorithms has been compared in terms of their accuracy in capturing actual traffic behavior, their implementing complexity and their responsiveness. The results show the high complexity of the RBFP, while a very low computation time is required by the first order predictor. On the contrary, RBFP presents higher accuracy and responsiveness with respect to the first order predictor. Differently from these two prediction schemes, the NLMS permits to achieve an adequate trade-off among complexity and accuracy–responsiveness. Indeed, the analysis has shown that the NLMS has a medium complexity, but permits to achieve the same performance of the RBFP in terms of accuracy and responsiveness. Upon the above results, the paper presents the novel SDB resource allocation strategy based on linear traffic prediction (implemented via the NLMS algorithm) and on the estimation of queueing delay through the Chebishev bound. The scope of the algorithm is to determine the minimum service capacity to be allocated in order to guarantee a probabilistic bound on the maximum delay experienced by packets traversing the resource. The results show that the project constraints are met as long as the values of delay and aggregation time satisfy the underlying analytic hypotheses. Such a technique candidates for managing quantitative QoS requirements in that it guarantees limited packet delays with assigned violation probability.
To properly assess the performance of the SDB algorithm, a second strategy (CSU) aiming at maintaining a given level of server utilization is considered. Such a requirement is well met in our experiments and, thanks to predictions, high values of server utilization are achieved even in the presence of VBR traffic. Unfortunately, this is obtained at the cost of very large queueing delays, which can be decreased by paying off in terms of a significant reduction of the network utilization. Indeed, at low traffic loads, results obtained under CSU and under the SDB technique are consistent, and prove the effectiveness of the latter in providing statistical guarantees on the packet delay introduced by network resources. Possible drawbacks of the SDB algorithm can be summarized into two points. The first one comes from the use of the Chebyshev bound that, in some cases, may be pretty loose and may results in bandwidth overprovisioning. The second drawback is the one already pointed out in Section 6, as we noticed that for target delays larger than half of the sampling time T p , the project requirement might not be met; indeed, in these cases, the hypothesis of empty queue at the end of any time interval may not be verified, and the buffer backlog builds-up determining higher packet propagation delays. However, this problem reduces as T p increases since, in this case, queueing backlog phenomena become less critical. Acknowledgment This work has been sponsored by the Italian MIUR research program MIMOSA. References [1] S. Blake, et al., An Architecture for Differentiated Services, IETF RFC, 2475, December 1998. [2] B.E. Carpenter, K. Nichols, Differentiated services in the Internet, Proceedings of the IEEE 90 (9) (2002). [3] MPLS Resource Center, http://www.mplsrc.com/. [4] D. Awduche, et al., Overview and Principles of Internet Traffic Engineering, IETF RFC 3272, 2002. [5] H.L. Lu, I. Faynberg, An architectural framework for support of quality of service in packet networks, IEEE Communications Magazine (2003). [6] K. Park, W. Willinger (Eds.), Self-Similar Network Traffic and Performance Evaluation, John Wiley and Sons, 2000. [7] F. Le Faucheur (Ed.), Protocol Extensions for Support of Diffserv-Aware MPLS Traffic Engineering, IETF RFC 4124, June 2005. [8] S.S. Oruganti, M. Devetsikiotis, Analyzing robust active queue management schemes: a comparative study of predictors and controllers, in: Proceedings of ICC 2003, Anchorage, Alaska, May 2003. [9] Y.H. Chan, T. Randhawa, S. Hardy, Traffic prediction based access control using different video traffic models in 3G CDMA high speed data networks, in: IWCMC ’06: Proceedings of the 2006 International Conference on Wireless Communications and Mobile Computing, Vancouver, British Columbia, Canada, 2006, pp. 227–232. [10] P. Koutsakis, On providing dynamic resource allocation based on multimedia traffic prediction in satellite systems, Computer Communications 30 (2) (2007) 404–415. [11] F. Chiti, R. Fantacci, F. Marangoni, Advanced dynamic resource allocation schemes for satellite systems, in: Proceedings of IEEE ICC 2005, vol. 3, Seoul, Korea, pp. 1469–1472. [12] R.G. Garroppo, S. Giordano, S. Lucetti, G. Procissi, Chaotic prediction and application to resource allocation strategies, in: Proceedings of the IEEE ICC 2004, 2004. [13] A.M. Adas, Using adaptive linear prediction to support real-time vbr video under rcbr network service model, in: Proceedings of the IEEE/ ACM Transaction on Networking, vol. 6, no. 5, October 1998, pp. 635– 644. [14] S. Chong, S.-Q. Li, J. Gosh, Predictive dynamic bandwidth allocation for efficient transport of real-time VBR video over ATM, IEEE Journal of Selected Area in Communications 13(1) (1995). [15] A. Sang, S. Li, A predictability analysis of network traffic, in: Proceedings of INFOCOM, 2000, pp. 342–351. [16] G.Carrozzo, N. Ciulli, S. Giordano, Multi access inter domain architecture for QoS provisioning in MPLS/DiffServ networks, in: TANGO Project Symposium, January 2004. [17] R.G. Garroppo, S. Giordano, F. Mustacchio, F. Oppedisano, G. Procissi, A flexible measurement system for supporting traffic engineering in DiffServ/MPLS networks, in: Workshop on Traffic Engineering, Protection and Restoration for NGI, Lund, Sweden, May 27–28, 2004.
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