P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods. New York: Springer-Verlag, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the traffic, the optimal linear predictor can be formulated as (7) where is a covariance stationary stochastic process with zero mean, variance , and autocovariance function , and is the vector of stored traffic measurements . is the memory length of the predictor. The solution is given by [13] (8) where is the covariance matrix, is values of the autocovariance function starting at lag . The variance of the prediction errors is given by (9) III. IMPLEMENTATION OF THE RATE CONTROL ALGORITHM FOR SELF SIMILAR TRAFFIC The rate based control algorithm is implemented as follows. At the ....

P. Brockwell and R. Davis, Time Series: Theory and Methods, 2nd ed. New York: Springer-Verlag, 1991.

....This result shows that is asymptotically zero mean Gaussian. As for , it can be expressed in the following form: Lemma 1 shows that converges almost surely to zero. Since is a component of the vector , it is asymptotically Gaussian. This implies that almost surely converges to zero [5]. According to (19) we deduce the following result. where Using Propositions 1 and 2 as well as (18) we finally obtain the main result of this section. Theorem 2: where . We have thus proved the asymptotic normality of , and that the convergence rate of is as in standard frequency ....

....denote the unconjugated conjugated asymptotic covariance matrices In order to express and in terms of and , we note that since it follows that Substituting from (20) we obtain that From (21) it follows that is a deterministic term converging to zero, as . From this, we deduce further that [5] (27) 28) Plugging (27) 28) 25) and (26) back into (24) yields (29) Finally, by combining all the previous results, we obtain the following compact and simple expression for the asymptotic variance: 30) where IV. CHOOSING THE WEIGHTING MATRIX AND PARAMETER We now exploit (30) to study ....

P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods. New York: Springer-Verlag, 1991.

....that converges in distribution to a zero mean Gaussian distribution. This result shows that is asymptotically zero mean Gaussian. Moreover, Lemma 1 shows that converges almost surely to zero. As is asymptotically Gaussian, it is bounded in probability. Therefore, m converges almost surely to zero [19]. Theorem 3 follows now immediately from (19) 4 The notations e[ and =m[ denote the real and imaginary parts of a complex number, respectively. 138 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 1, JANUARY 2002 APPENDIX D PROOF OF ASYMPTOTIC NORMALITY OF Let cum denote the th order ....

....follows that cum which implies that converges to a Gaussian distribution. APPENDIX E PROOF OF THEOREM 4 As converges almost surely toward and converges in distribution to a centered normal distribution of variance , converges to a normal distribution of zero mean and standard deviation (see [19]) In order to complete the proof of the theorem, we still need to establish the expression of . According to Theorem 3 and (20) it is easy to check that (24) In order to evaluate , we need to obtain a closed form expression for . For this, one can observe that the matrix can be expressed as ....

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P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods. New York: Springer-Verlag, 1991.

....and Watts (1968) 17 See Jenkins and Watts (1968, p. 437) 9 where # represents the coherence between series X and series Y at frequency #, and m denotes the window width; m = N B W #, with N equaling the number of observations, and B W is the bandwidth used, B W = 0.0171. According to Brockwell and Davis (1991), the coherence is distributed as the square of the multiple correlation coefficient, and thus Y # F(2, 4m ) under the hypothesis ## XY (#)# = 0. 16 [SEE TABLE 2] The resultant F values from (2) in Table 2 strongly reject Hypothesis 3, since the null hypothesis that the peak coherence equals ....

Brockwe ll, P.J., and Davis, R.A., Time Series: Theory and Methods, 2nd Ed., New York: Springer-Verlag, 1991.

....#46# describes a stationary, second order Gaussian process. Given parameter values # =## 1 ; # p # and # =## 1 ; # q #, and a seriesx 1 ; x t , it is straightforward to make predictions from #46# to times t 1;t 2; conditional on the #rst t data points. For example, following Brockwell and Davis #1991, pp.256#,x t 1 has a Gaussian distribution with mean x t 1 and variance # 2 r t which are calculable from the recursive formulae: # x t 1 = P t i=1 # it #x t 1,i , x t 1,i #; 1 # t#max#p; q# x t 1 = # 1 x t : # p x t 1,p P q i=1 # it #x t 1,i , x t 1,i #; t # ....

Brockwell, P. J. and Davis, R. A. #1991#.Time series: theory and methods. New York: Springer-Verlag.

.... M = 5 (Controller 3) Maximum network delay: d = 10 (Controller 3) Controller gain: 1= 1:2 M ( d 1) 0:0152 (Controller 3) For Controllers 1 and 2, the nominal service rate and the AR process parameters i are estimated online using the Yule Walker algorithm [26], assuming that the order of the AR process is 8: This is discussed in the next subsection. For Controller 3, we take to be equal to 4500 cells per time unit. The propagation delay from one node to the next is 1.6 time units. Note that the actual delay is variable since the cells go through node ....

....but is never actually used in the control algorithms, as it is not needed. Tuning the parameters of the AR process and nding the best possible value of p is a challenging task in general. Several methods exist to calculate the parameters, i , once p is xed. We use the Yule Walker algorithm [26] to determine p and i s from the data that is observed online. Let T be the time interval over which we attempt to t an AR model to the available capacity. Then one criterion to determine the best order for an AR process is the nal prediction error (FPE) de ned by FPE = 2 T p T p ....

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P.J. Brockwell and R.A. Davis, Times Series: Theory and Methods. New York: Springer Verlag, 1991.

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P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods. New York: Springer-Verlag, 1991.

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P.F. Brockwell and R.A. Davis. Time Series: Theory and Methods. New York: Springer-Verlag, New York, 1998.

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P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods. New York: Springer-Verlag, 1991.

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P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods. New York: Springer-Verlag, 1987.

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Brockwell, P.J. and R.A. Davis (1991): Time Series: Theory and Methods, 2nd ed., New 21 York: Springer-Verlag.

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Brockwell,P.J. and Davis,R.A.(1991a).Time Series: Theory and Methods.New York: Springer Verlag.

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P.J. Brockwell and R.A. Davis, Time Series: Theory and Methods, 2nd Edition. New York: Springer-Verlag, 1991.

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