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1 Departement Elektrotechniek ESAT-SISTA/TR 98-3 Identication of the circulant modulated Poisson process: a time domain approach Katrien De Cock, Tony Van Gestel and Bart De Moor 2 April 998 Submitted for publication in Proceedings of MTNS'98. This report is available by anonymous ftp from ftp.esat.kuleuven.ac.be in the directory pub/sista/decock/reports/98-3.ps.gz 2 K.U.Leuven, Dept. of Electrical Engineering (ESAT), Research group SISTA, Kardinaal Mercierlaan 94, 300 Leuven, Belgium, Tel. 32/6/ , Fax 32/6/ , WWW: E- mail: katrien.decock@esat.kuleuven.ac.be, tony.vangestel@esat.kuleuven.ac.be, bart.demoor@esat.kuleuven.ac.be. Bart De Moor is a senior Research Associate with the F.W.O. (Fund for Scientic Research-Flanders). Tony Van Gestel is a Research Assistant with the F.W.O. (Fund for Scientic Research-Flanders). Katrien De Cock is a Research Assistant with the I.W.T. (Flemish Institute for Scientic and Technological Research in Industry). Work supported by the Flemish Government ( Administration of Science and Innovation (Concerted Research Action MIPS: Model-based Information Processing Systems, Bilateral International Collaboration: Modelling and Identication of nonlinear systems, IWT-Eureka SINOPSYS: Model-based structural monitoring using in-operation system identication), FWO-Vlaanderen: Analysis and design of matrix algorithms for adaptive signal processing, system identication and control, based on concepts from continuous time system theory and dierential geometry, Numerical algorithms for subspace system identication: Extension towards specic applications, FWO-Onderzoeksgemeenschappen: Identication and Control of Complex Systems, Advanced Numerical Methods for Mathematical Modelling); Belgian Federal Government( Interuniversity Attraction Pole IUAP IV/02: Modelling, Identication, Simulation and Control of Complex Systems, Interuniversity Attraction Pole IUAP IV/24: IMechS: Intelligent Mechatronic Systems); European Commission: (Human Capital and Mobility: SCIENCE-ERNSI: European Research Network for System Identication.)

2 Identication of the circulant modulated Poisson process: a time domain approach Katrien De Cock y, Tony Van Gestel z and Bart De Moor x K.U.Leuven, ESAT-SISTA, Kardinaal Mercierlaan 94, 300 Leuven, Belgium, Tel. 32/6/ , Fax 32/6/ , katrien.decock@esat.kuleuven.ac.be, tony.vangestel@esat.kuleuven.ac.be, bart.demoor@esat.kuleuven.ac.be. Keywords: trac modelling, circulant modulated Poisson process, ATM Abstract In this paper we discuss a new time domain method for the trac identication problem in ATM networks. Our identication approach for the circulant modulated Poisson process (CMPP) consists of two steps: the identication of the rst order parameters and the determination of the circulant stochastic matrix which matches the second order statistics of the data. Introduction Li et al. [6] have indicated that mathematical models can be used to perform several tasks in control mechanisms of ATM networks. The models they propose are measurement based and include the time correlation of trac. Whereas the approach of Li et al. [6, 7, 8] is mainly based on the frequency domain, this paper is concerned with measurement based parameter estimation in the time domain: the models match the cumulative distribution function and the autocorrelation function of the measured data as described in []. This paper is concerned with the modelling of the arrival process of cells in one node of the network. The arrival process a k (k = 0; ; 2; : : :) is the aggregated number of Work supported by the Flemish Government (BOF (GOA- MIPS), AWI (Bil. Int. Coll.), FWO (projects, grants, res. comm. (ICCoS)), IWT (IWT-VCST (CVT), ITA (ISIS), EU- REKA (Sinopsys))), the Belgian Federal Government (IUAP IV- 02, IUAP IMechS), the European Commission (TMR (Alapedes), ACTS (Aspect), SCIENCE (ERNSI)). y Research Assistant with the I.W.T. (Flemish Institute for Scientic and Technological Research in Industry). z Research Assistant with the F.W.O. (Fund for Scientic Research-Flanders). x Senior Research Associate with the F.W.O. (Fund for Scientic Research-Flanders). cells that arrive per time unit. This sequence is measured and forms the input of our identication algorithm. The research of Li and Hwang [7] has demonstrated that only the rst order and second order statistics have a signicant impact on queueing performance. Therefore we will identify a model that has rst and second order statistics that approximate those of the data as well as possible. The model should also t into queueing analysis which is used to evaluate the network performance. A Markov modulated Poisson process (MMPP) satises both conditions, but a more tractable model structure is the circulant modulated Poisson process (CMPP) which is a restricted version of the MMPP. 2 The circulant modulated Poisson process In this section we rst explain the structure of a circulant modulated Poisson process (CMPP). Then we discuss some important properties of the CMPP with respect to our identication method. Because we want to approximate the rst and second order statistics of the data with those of a CMPP, we give the expressions for the rst and second order statistics of the model. 2. The model structure A CMPP of order N C consists of an N C -state circulant Markov chain in which each state i (i = ; 2; : : : ; N C ) represents a Poisson process with rate i. In this paper we only consider discrete time Markov chains. The circulant Markov chain is characterised by a circulant stochastic transition matrix Q: Q = q q 2 : : : q NC q NC q : : : q NC? ; q 2 q 3 : : : q

3 where XN C i= q i 0; for i = ; 2; : : : ; N C () q i = : (2) Throughout this paper, we will deal exclusively with the steady state case, where the state distribution is time independent: T = T Q : So is the left eigenvector of the circulant stochastic matrix Q corresponding to eigenvalue. The Poisson parameters of the CMPP are arranged in a column vector 2 R NC, where each element i (i = ; 2; : : : ; N C ) is a positive real number, representing the arrival rate of the Poisson process associated with state i. At time step k, the CMPP (Q; ) will emit a k arrivals generated by the Poisson process with the arrival rate i if the state of the Markov chain is i. 2.2 Properties of a CMPP Circulant matrices have some important properties because of their special structure (see e.g. []). A circulant stochastic matrix Q is always diagonalisable. Its eigenvalue decomposition can be computed very eciently. If Q = V V? is the eigenvalue decomposition of the circulant stochastic matrix Q, then we have: V is a Fourier matrix: p?2? V (i; j) = p exp (i? )(j? ) NC N C V? = V H ; where V H denotes the complex conjugate transpose of V. The eigenvectors of a circulant matrix are independent of the entries of that matrix, they only depend on the dimension of the matrix. The product of a Fourier matrix V and a vector x is equal to the discrete Fourier transform of that vector x. This discrete Fourier transform can be computed with a fast Fourier transform (fft). The left eigenvector of a circulant stochastic matrix Q of dimension N C corresponding to the eigenvalue is a uniform vector T C = : : : (3) N C N C N C because circulant stochastic matrices are double stochastic. As a consequence, the state distribution of a CMPP is independent of the entries of the transition matrix. The eigenvalues i of a circulant matrix Q can be computed as follows: p T T : : : NC = N C V q : : : q NC T = fft q : : : q NC : (4) The vector of eigenvalues of a circulant matrix Q is equal to the fft of a row of Q. 2.3 The rst and second order statistics of a CMPP As a description for the rst order statistics of the cell arrival process a k, we use the cumulative distribution function F d, dened as F d (l) = P rfa k lg; l 2 N: The cumulative distribution function of a CMPP is given by F C (l) = N C XN C j= e?(c)j lx i=0 i! ( C) i j : (5) It does not depend on the transition matrix Q in contrast with the cumulative distribution function of a MMPP. Therefore we can uncouple the identication of the rst order and of the second order parameters. The second order statistics of the data a k are described by the autocorrelation function R(n) = Efa k a k+n g: The autocorrelation function of a CMPP can be computed as follows: R C (n) = T?Q n?e = C T N V n V H C = N 2 C fft( T C ) diag ([ n i ]) fft( C ) ; (6) where is obtained as in (4) and fft( C ) is the complex conjugate of fft( C ). Only two fft's are needed to compute the autocorrelation function of a CMPP. 3 The identication of a CMPP In this section our new identication method for the CMPP is discussed. Because of the special structure of the CMPP, the identication of the rst and second order parameters can be uncoupled. In Figure an overview is given of the complete identication method. First, the cumulative distribution function of the data is used to nd the rst order parameters of an MMPP. Those rst order MMPP parameters are then translated to CMPP parameters taking into account the restrictions of the circulant structure. This is the rst step of our identication method and it is explained in more detail in Section 3.. The second step consists of the identication of the circulant stochastic matrix by unconstrained optimisation which is discussed in Section 3.2. In Section 3.3 an example of our identication procedure is given. 3. The identication of the rst order parameters The cumulative distribution function of the data must be approximated by the cumulative distribution function of

4 minimise k 0 F d solve min kf x d? Dxk 2 s.t. x i 0 NM M M a k compute distribution and autocorrelation translate NC C MMPP into CMPP R d min k kr d? R Ck 2 2 step : step 2: identication identication of the of the Q rst order parameters circulant transition matrix Figure : Identication of the CMPP where the cumulative distribution function is matched by solving a nonnegative least squares problem followed by a translation of MMPP parameters to CMPP parameters. The autocorrelation function is matched by unconstrained optimisation. a CMPP. This implies that F d must be reconstructed as a linear combination of cumulative distributions of Poisson processes in which each coecient is equal to N C (see (5)). Our new method for the identication of the rst order parameters consists of two parts: ) Based on the cumulative distribution function of the data a k, we determine which Poisson processes are involved in the arrival process. The distribution of these N M N C Poisson processes can then be viewed as the state distribution vector M of an MMPP, which is not uniform. The Poisson parameters form the MMPP parameter M. In this rst part we approximate the cumulative distribution function of the data with a nonnegative linear combination of cumulative Poisson distributions (which in fact is the cumulative distribution of an MMPP): F d D ; where (F d ) l = F d (l? ) and F d 2 R L+ (L is the maximum number of arrivals of the given set of data). Each column D j of D represents a possible state or possible Poisson cumulative distribution. In fact, the domain of possible i 's is discretized in order to get a broad choice of candidate states. The discretisation step is not chosen to be constant but it increases linearly because the variance of a Poisson process is equal to its rate. In order to determine which of the proposed cumulative Poisson distributions take part in F d we solve min x kf d? Dxk 2 ; subject to x i 0 : We have used the algorithm of Lawson and Hanson [5] to solve this problem. Let x be the optimal value of x. The number of non-zero components of x gives the MMPP model order N M. The indices of the nonzero components of x give the ( M ) i 's. The ( M ) i 's are given by the values of the non-zero components of x. For more details we refer to [2, 4]. 2) In order to obtain a uniform distribution of the Poisson processes (see (3)) we take as many copies of each state as needed to approximate each component ( M ) i of the non-uniform distribution vector M with multiples of NC. The total number of states is then equal to N C. In Figure we refer to this second part as the translation of MMPP parameters into CMPP parameters. More details are found in [2, 3]. By imposing a certain model order N C we introduce a modelling error. However, we can determine a bound on this error as a function of N C, so that we can choose the minimal model order that still guarantees a given accuracy of the cumulative distribution function. 3.2 The identication of the circulant transition matrix For the identication of the circulant transition matrix Q we solve the following optimisation problem: min Q kr d? R C k 2 2 ; where R d is the autocorrelation function of the data and R C is the autocorrelation function of the CMPP. In order to obtain an unconstrained optimisation problem we use the parameterisation of Q, proposed by Yi and De Moor []: 8 >< >: q i = q NC = P ki 2 N + C? j= kj 2 ; for i = ; : : : ; N C? P N + C? j= kj 2 : This parameterisation takes the constraints on the circulant stochastic matrix into account (see () and (2)). The only variable in the optimisation problem is now k 2 R NC?. During the optimisation we keep the Poisson parameters C which are found in the rst step, constant. The circulant structure is exploited during the evaluation of the autocorrelation function (see (6)). Because of the ef- cient computation of the eigenvalue decomposition, the complexity of this optimisation step is O(N C log(n C )) per function evaluation.

5 R F placements x Figure 2: The cumulative distribution function (top) and the autocorrelation function (bottom) of the ethernet data (full line) and of the identied CMPP (dash-dotted line). 3.3 Example We give an example of our identication procedure where a CMPP is identied starting from measured ethernet data. The data are measurements of one hour of internet trac between the Lawrence Berkeley laboratory and the rest of the world. The traces were made by Vern Paxson [0] and they are available from the Internet Traf- c Archive of the Lawrence Berkeley National Laboratory [9]. From the lbl-pkt database we have used the measurements lbl-pkt-4. The data give the number of bytes that was sent per microsecond. We have used intervals of 0.25 seconds in order to avoid too many zeros in the data sequence. All bytes that were sent in an interval [t i ; t i + 0:25s] are grouped. Our specic dataset is available from our database for identication DAISY. It consists of data points. The order of the identied CMPP was chosen to be equal to 28. In Figure 2 we compare the cumulative distribution function (top picture) and the autocorrelation function (bottom picture) of the data with those of the identied model. The full line represents the statistics of the data and the dashdotted line gives the statistics of the CMPP. 4 Conclusions In this paper we have presented a new time domain identication algorithm for ATM network trac. The algorithm identies the circulant modulated Poisson process (CMPP). Only the rst order and second order statistics are matched, respectively the cumulative distribution function and the autocorrelation function. We identify the CMPP in two steps. The rst order l n statistics are identied very eciently by solving a nonnegative least squares problem followed by a translation of the parameters to the restricted CMPP format. The circulant stochastic transition matrix is identied by unconstrained optimisation. Due to lack of space we could not include the complete analysis of the computational complexity of our CMPP identication method. The interested reader is referred to [2]. References [] A. Berman en R.J. Plemmons. Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics series, vol. 9. SIAM, Philadelphia, 994. [2] K. De Cock, T. Van Gestel and B. De Moor. Stochastic system identication for ATM network trac models: a time domain approach. Internal report ESAT-SISTA/ , K.U.Leuven, Belgium, 997. [3] K. De Cock and B. De Moor. Identication of the rst order parameters of a circulant modulated Poisson process. Internal report ESAT-SISTA/ , K.U.Leuven, Belgium, 997. Accepted for publication in Proceedings of International Conference on Telecommunications (ICT'98). [4] K. De Cock and B. De Moor. Stochastic system identication for ATM network trac models: a time domain approach. Internal report ESAT-SISTA/ , K.U.Leuven, Belgium, 998. Accepted for publication in Proceedings of EUSIPCO-98 [5] C.L. Lawson and R.J. Hanson. Solving Least Squares Problems. Classics in Applied Mathematics series, vol. 5. SIAM, Philadelphia, 995. [6] S.Q. Li and C.L. Hwang. Queue response to input correlation functions: discrete spectral analysis. IEEE/ACM Transactions on Networking, (5):522{533, 993. [7] S.Q. Li and C.L. Hwang. Queue response to input correlation functions: continuous spectral analysis.ieee/acm Transactions on Networking, (6):678{ 692, 993. [8] S.Q. Li and C.L. Hwang. On the convergence of traf- c measurement and queueing analysis: a statisticalmatching and queueing (SMAQ) Tool. IEEE/ACM Transactions on Networking, 5():95{0, 997. [9] The internet trac archive, [0] V. Paxson en S. Floyd. The failure of Poisson modeling. IEEE/ACM Transactions on Networking 3(3):226{244, 995. [] C. Yi and B. De Moor. Trac identication of ATM networks with optimization algorithms. Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan, pp. 277{282, This report is available by anonymous ftp from ftp.esat.kuleuven.ac.be in the directory pub/sista/decock/reports/

Optimal trac light control for a single intersection Bart De Schutter and Bart De Moor y ESAT-SISTA, KU Leuven, Kardinaal Mercierlaan 94, B-3 Leuven (

Katholieke Universiteit Leuven Departement Elektrotechniek ESAT-SISTA/TR 97- Optimal trac light control for a single intersection Bart De Schutter and Bart De Moor Proceedings of the 997 International