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1 Departement Elektrotechniek ESAT-SISTA/TR 98- Stochastic System Identication for ATM Network Trac Models: a Time Domain Approach Katrien De Cock and Bart De Moor April 998 Accepted for publication in roceedings of EUSICO { 98 This report is available by anonymous ftp from ftpesatkuleuvenacbe in the directory pub/sista/decock/reports/98-psgz KULeuven, Dept of Electrical Engineering (ESAT), Research group SISTA, Kardinaal Mercierlaan 94, 00 Leuven, Belgium, Tel /6/ 7 09, Fa /6/ 9 70, WWW: katriendecock@esatkuleuvenacbe, bartdemoor@esatkuleuvenacbe Bart De Moor is a senior Research Associate with the FWO (Fund for Scientic Research-Flanders) Katrien De Cock is a Research Assistant with the IWT (Flemish Institute for Scientic and Technological Research in Industry) Work supported by the Flemish Government ( Adistration of Science and Innovation (Concerted Research Action MIS: Model-based Information rocessing Systems, Bilateral International Collaboration: Modelling and Identication of nonlinear systems, IWT-Eureka SINOSYS: Modelbased structural monitoring using in-operation system identication), FWO- Vlaanderen: Analysis and design of matri 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: Etension towards specic applications, FWO- Onderzoeksgemeenschappen: Identication and Control of Comple Systems, Advanced Numerical Methods for Mathematical Modelling); Belgian Federal Government( Interuniversity Attraction ole IUA IV/0: Modelling, Identication, Simulation and Control of Comple Systems, Interuniversity Attraction ole IUA IV/4: IMechS: Intelligent Mechatronic Systems); European Commission: (Human Capital and Mobility: SIMONET: System Identication and Modelling Network, SCIENCE-ERNSI: European Research Network for System Identication)

2 STOCHASTIC SYSTEM IDENTIFICATION FOR ATM NETWORK TRAFFIC MODELS: A TIME DOMAIN AROACH Katrien De Cock and Bart De Moor KULeuven, ESAT-SISTA, Kardinaal Mercierlaan 94, 00 Leuven, Belgium, Tel /6/ 9 5, Fa /6/ 9 70, WWW: katriendecock@esatkuleuvenacbe, bartdemoor@esatkuleuvenacbe ABSTRACT In our paper we discuss a new time domain approach to the trac identication problem for ATM networks The Markov modulated oisson process is identied in two steps By applying a nonnegative least squares algorithm we obtain in a very fast way a description of the rst order statistics of the data This rst order characterisation includes also an estimate of the model order Consequently, we are able to identify a Markov modulated oisson process without a priori knowledge of the model order The identication of the second order statistics is based on unconstrained optimisation algorithms Keywords: trac identication, Markov modulated oisson process, ATM INTRODUCTION Our 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; ; ; : : : ) is the aggregated number of cells that arrive per time unit This sequence is measured and forms the input of our identication algorithm An important innovation described in this paper is the fast characterisation of the rst order statistics with automatic deteration of the model order In Section we discuss the identication problem, our identication method is eplained in Section and we end with some conclusions in Section 4 K De Cock is Research Assistant with the IWT (Flemish Institute for Scientic and Technological Research in Industry), B De Moor is Senior Research Associate with the FWO (Fund for Scientic Research-Flanders) This work was supported by the Flemish Government (BOF (GOA-MIS), AWI (Bil Int Coll), FWO (projects, grants, res comm (ICCoS)), IWT (IWT-VCST (CVT), ITA (ISIS), EUREKA (Sinopsys))), the Belgian Federal Government (IUA IV-0, IUA IMechS), the European Commission (HCM (Simonet), TMR (Alapedes), ACTS (Aspect), SCI- ENCE (ERNSI)), NATO (CRG) and industry (Electrabel) THE IDENTIFICATION ROBLEM In this section we state the identication problem First, we show why the Markov modulated oisson process is a good model for the arrival process of cells in an ATM network (Section ) Then, in Section, we eplain the structure of the Markov modulated oisson process and we give the epressions for its rst and second order statistics Model Choice The task of trac identication is to construct stochastic models, the statistics of which match those of the data In addition, the identied model should t into queueing analysis which is used to evaluate the network performance The research of Li and Hwang [6] has demonstrated that only the rst order and second order statistics have a signicant impact on queueing performance Classic queueing theories have generally ignored the second order statistics entirely By using a Markov modulated oisson process with rst order and second order statistics matching those of the measured trac, the correlation in multimedia trac can be captured and eploited The Markov Modulated oisson rocess Model Structure A Markov modulated oisson process (MM) of order N consists of an N-state Markov chain in which each state i (i ; ; : : : ; N) represents a oisson process with rate i In other words, a Markov modulated oisson process is a oisson process the rate of which is changed (modulated) according to a Markov chain In this paper, we only consider discrete time Markov chains Let S k (k 0; ; ; : : : ) be the stochastic process that denotes the state moving at discrete time instants between a nite number of states,, : : :, N The transition probabilities depend only on the previous state, so that S k (k 0; ; ; : : : ) is a Markov process Let the matri [p ij ] be the transition matri: p ij rfs k+ j j S k ig :

3 If (k) is the column vector with i-th entry rfs k ig, then (k + ) T (k) T : Throughout this paper, we will deal eclusively with the steady state case, where the state distribution is time independent: T T : Matri is a N N stochastic matri, that is R NN N ; p ij 0 and p ij The nonnegative vector is the left eigenvector of the stochastic matri corresponding to eigenvalue for which holds N i i : The oisson parameters are arranged in a column vector The ith element of, i 0 is the arrival rate of the oisson process associated with state i At time step k, the MM (; ) will emit a k arrivals generated by the oisson process with the arrival rate i if the state of the Markov chain is i The aim of trac identication is to nd and from arrivals a k, so that they form an MM with rst order and second order statistics that match those of the measured trac a k Afterwards, and can be used in queueing analysis dierence is that we have uncoupled the identication of the rst order and the second order statistics The rst order parameters, and, and the model order N are detered by solving a nonnegative least squares problem [4] (see Section ) The transition matri is obtained by unconstrained optimisation This is briey eplained in Section Identication of the First Order Statistics The identication of the rst order parameters of an MM boils down to the deteration of N, and so that F M approimates the cumulative distribution function of the data, F d, as well as possible The cumulative distribution function of the data sequence is a staircase function It is computed as follows: F d (l) m l m j0 k (a k ; j) ; where (i; j) is the Kronecker delta and m is the number of data The rst and second order statistics of arrivals As already mentioned, only the rst and second order statistics of arrivals a k are considered The rst order statistics are described by the cumulative distribution function F, dened as F (l) rfa k lg : For an MM, we have F M (l) N i N i i e?i l j0 j Sfrag replacements i j! i F i (l) () where F i (l) is the cumulative distribution function of the oisson process with parameter i The second order statistics of a k are characterised by the autocorrelation function R, dened as R(n) Efa k a k+n g : The autocorrelation of an MM is given by: RM (0) T (? +? )e R M (n) T? n ()?e; for all n N0 ; K 0 N g 0 Deteration of the components of a miture distribution The cumulative distribution function of the data, F d, must be approimated by the cumulative distribution F d F d 6 4W qr d (0) DD W Rd () corr st i 0 N; ; reduce a k compute distribution and autocorrelation R d W v; W R where? diag() and e [ : : : ] T (see [7] for proof) N red ; red ; red K Wv v? red + W R Rd? R M THE IDENTIFICATION METHOD Our identication approach is summarised in Figure It is mainly due to Yi and De Moor [8] The main 0 0 diag() N Figure : Identication of the MM where the cumulative distribution function is matched by solving the nonnegative least squares problem (5) and the autocorrelation is matched by unconstrained optimisation The MM is found without a priori knowledge of the model order N The contribution of the user is indicated by the oval

4 function of an MM This implies that F d must be approimated by a nonnegative linear combination of cumulative distributions of oisson processes (see ()): F d ; () where (F d ) l F d (l? ) and F d R L+ (L is the maimum number of arrivals of the given set of data) The columns of the matri D R (L+)N correspond to the cumulative oisson distribution functions F i (i ; ; : : : ; N) If the states (ie, the i 's) that form part of the MM which models a given data set, are given, then the matri D is ed and can be detered by solving the system of equations () Without knowledge of the states or the number of states, the same approach yields N, and if in () D is replaced by an enlarged version D This matri D has much more columns than D and the same number of rows Each column D j of D represents a possible state or possible oisson cumulative distribution In fact, the domain of possible i 's, eg, [0:00; L], is discretized with a step h in order to get a broad choice of states: j (j? )h The rst oisson parameter,, has to be chosen small enough in order to incorporate possible zero-valued a k 's The computed cumulative distribution function F d must be reconstructed as a nonnegative linear combination of the columns of D: kf d? Dk ; subject to i 0 : (4) Let be the optimal value of The number of nonzero components of gives the MM model order N The indices of the non-zero components of give the i 's The i 's are given by the values of the non-zero components of Fitting the autocorrelation function at n 0 and n Because the autocorrelation function R M (n) of an MM at n 0 and n only depends on the rst order parameters and : 8 >< >: R M (0) R M () N j ( j + j ) N j j A it is very important that these vectors are chosen in such a way that the autocorrelation function can be matched Therefore, we add the following two rows to the matri D W D corr ( ) W ( + ) : : : W ( L+ L+) W W : : : W L+ andp the vector F d is etended with W R d (0) and W Rd (), where W and W are weighting factors Solving the nonnegative least squares problem 4 F d W Rp d (0) W Rd () 5? D D corr ; subject to i 0 (5) will then give the model order N, the oisson parameters and the state distribution vector that approimate the cumulative distribution function of the data and the autocorrelation function at n 0 and n Reduction of the Model Order The computational compleity of the identication of the second order statistics (see Section ) and the compleity of the queueing analysis highly depend on the model order For the identication of the autocorrelation O(N ) operations are needed per function evaluation in the optimisation algorithm and the computational compleity of the eisting queueing analysis techniques is at least proportional to N [, ] Therefore it is advantageous to reduce the model order In our approach it is possible to reduce the model order after the identication of the rst order parameters, ie before the stochastic matri is detered The number of MM states is reduced by replacing similar states (with oisson parameters i close to each other) by one state This new state is characterised by ( red ) i n i j j and its contribution to the MM is given by ( red ) i n i j ; where n i is the number of states ( j ; j ) that are grouped The value of n i is detered by the number of states that are present (the states for which i 6 0) in a certain -range This range depends on the value of the j 's The smaller a certain j, the smaller the region in which we look for states similar to the state characterised by that j, because the variance of a oisson distribution function with parameter j is equal to j In Figure we show an eample of this reduction approach The circles are the results of (5) They represent the values of as a function of the proposed 's The model order is equal to 6, the number of non-zero components of After reduction, we have 8 states, represented by the dash-dotted lines and the stars The cumulative distribution functions of both models are shown in the second picture of Figure The full line is the cumulative distribution function of the 6th order model, the dash-dotted line represents the cumulative distribution function of the reduced model

5 lacements F Figure : Reduction of a 6th order model (full line and circles) to a model of 8th order (dash-dotted line and stars) In the second picture the cumulative distribution function of both models are shown The dierence between both distribution functions is given by the dotted line of order 8 The dierence between both distribution functions is given by the dotted line It is clear that the reduction step does not introduce large errors in the cumulative distribution function Identication of the Second Order Statistics For the identication of the second order statistics we use the following parameterisation from [8] : R N()! : K! (K) with (K) 64 k kn k j k Nj l : : : : : : k () k N() k j k Nj k j k Nj in order to obtain an unconstrained optimisation problem The matri K [k ij ] R N() is the variable in the optimisation algorithm We want the autocorrelation function of the model, which is dependent on, and (see equation ()), to be as close as possible to the autocorrelation function of the data Because the vectors and are already known from (5), only the stochastic matri has to be detered To ensure that the resulting matri has (or a vector close to ) as its left eigenvector corresponding to eigenvalue, the following unconstrained 75 optimisation problem is solved: KR N() W v k? vk + W R kr d? R M k ; where v is the left eigenvector of (K) corresponding to eigenvalue and W v and W R are weighting factors In this optimisation the state parameters remain constant The essence of this method is that we keep the rst order parameters from (5) and impose to be the left eigenvector of corresponding to eigenvalue The resulting method is summarised in Figure By eliating the optimisation of the rst order statistics we obtain an algorithm that is faster than that of Yi and De Moor [8] For more details on the computational compleity of our method in comparison with the method of Yi and De Moor, we refer to [] 4 CONCLUSIONS In this paper we have discussed a new method for the identication of a Markov modulated oisson process The rst order statistics are matched by solving a nonnegative least squares problem and the second order statistics are identied by unconstrained optimisation An important advantage of this method is the automatic deteration of the model order which can be reduced if necessary REFERENCES [] C Blondia and O Casals \Statistical Multipleing of VBR Sources: A Matri-Analytic Approach", erformance Evaluation, vol 6, 99, pp 5-0 [] 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/97-90, 997 [] SQ Li and CL Hwang \On the Convergence of Traf- c Measurement and Queueing Analysis: A Statistical- Matching and Queueing (SMAQ) Tool", IEEE/ACM Transactions on Networking, vol 5, 997, pp 95-0 [4] CL Lawson and RJ Hanson Solving Least Squares roblems, hiladelphia: SIAM, Classics in Applied Mathematics series, vol 5, 995 [5] SQ Li and CL Hwang \Queue Response to Input Correlation Functions: Discrete Spectral Analysis", IEEE/ACM Transactions on Networking, vol, 99, pp 5-5 [6] SQ Li and CL Hwang \Queue Response to Input Correlation Functions: Continuous Spectral Analysis", IEEE/ACM Transactions on Networking, vol, no 6, December 99, pp [7] Van Mieghem \The Mapping of Trac arameters into a MM(N)", Technical report, Alcatel Corporate Research Center, Antwerpen, Belgium, 995 [8] C Yi and B De Moor \Trac Identication of ATM Networks with Optimization Algorithms", roceedings of the 5th IEEE Conference on Decision and Control, Kobe, Japan, December 996, pp 77-8 This report is available by anonymous ftp from ftpesatkuleuvenacbe in the directory pub/sista/decock/reports/97-90psgz

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x 104 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