S. Q. Li and C. L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....output process, actually the autocorrelation function and the power density spectrum. In the last few years there have been some papers published which show that the autocorrelation function and the power density spectrum provide us with useful informations (see e.g. 2] 6] 8] 9] 11] [13], 16] 21] Fendick et al. 2] have shown that the assumption of uncorrelated processes can lead to inaccurately estimation of loss and delay of packets. Gihr and Tran Gia [3] have stated that the autocorrelation function is able to visualize process dependencies better than the Index of ....

....inaccurately estimation of loss and delay of packets. Gihr and Tran Gia [3] have stated that the autocorrelation function is able to visualize process dependencies better than the Index of Dispersion of Counts (IDC) Grunenfelder et al. 4] 5] 6] Helvik et al. 8] 9] Li et al. 11] 12] [13], and Ramamurthy and Sengupta [18] 19] have used the autocorrelation function for the description of more complex and realistic input traffi scenarios. In [5] the autocorrelation function of a video codec source was fitted into an Autoregressive Moving Average Model (ARMA) using the power ....

S. Li, C. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis", submitted for publication in IEEE/ACM Transaction on Networking to appear April 1992.

....service times [11] The fundamental reason for these suppositions is to be able to get relatively simple models from the analytical point of view. However, multimedia traffic, present in the highspeed networks, is characterized by high variability (burstiness) and strong positive correlation [10, 22], much more important than in the voice traffic. For this reason, the adequacy of the traditional models, and concretely of the independence hypothesis, is very criticized. Since Kleinrock s work by mid 60s [18] maybe the first one that investigated this issue seriously, the existence of time ....

S. Q. Li and C. L. Hwang. Queue response to input correlation functions: Continuous spectral analysis. IEEE/ACM Transactions on Networking, 1(6):678-- 692, December 1993.

.... employ an N state time continuous Markov chain, for which the generator matrix Q is of circulant type (i.e. each row is the previous row, shifted by one element) and within each state i the channel has a packet service rate of # i (channels currently subject to errors have a lower service rate) [11]. This matrix is computed from the power spectral density of a measured service rate process. Although this approach can match first and second order statistics of a measured trace, it is not easily adaptable to our methodology and notions. Another class of models are the Hidden Markov Models ....

San qi Li and Chia-Lin Hwang. Queue response to input correlation functions: Continuous spectral analysis. IEEE/ACM Transactions on Networking, 1:678-- 692, December 1993.

.... employ an N state time continuous Markov chain, for which the generator matrix Q is of circulant type (i.e. each row is the previous row, shifted by one element) and within each state i the channel has a packet service rate of i (channels currently subject to errors have a lower service rate) [11]. This matrix is computed from the power spectral density of a measured service rate process. Although this approach can match first and second order statistics of a measured trace, it is not easily adaptable to our methodology and notions. Another class of models are the Hidden Markov Models ....

San qi Li and Chia-Lin Hwang. Queue response to input correlation functions: Continuous spectral analysis. IEEE/ACM Transactions on Networking, 1:678- 692, December 1993.

.... employ an N state time continuous Markov chain, for which the generator matrix Q is of circulant type (i.e. each row is the previous row, shifted by one element) and within each state i the channel has a packet service rate of fi (channels currently subject to errors have a lower service rate) [24]. As input data they use the service rate process Rc(t) tc, obtained from measurements [25] They use the fact that the power spectral density function R(t) of the MMP generated by Q and (fro, fN ) can be explictly represented by the eigenvalues of Q. These values are chosen such that R(t) ....

San qi Li and Chia-Lin Hwang. Queue response to input correlation functions: Contin- uous spectral analysis. IEEE/ACM Transactions on Networking, 1:678-692, December 1993.

....in the experimental data. However, these models are analytically difficult to handle. Furthermore, they do not provide much insight into why they are meaningful on physical grounds. This explains in part why much modeling work still relies on more traditional multi state Markovian models (e.g. [24], 2] However, recent work has shown that the superposition of many on off sources with heavy tailed on and off periods results in aggregate traffic with LRD [36] 7] Furthermore, there is widespread evidence that human as well as computer sources of traffic do tend to behave as heavy tailed ....

....Indeed, several studies have used an approach where Markov models approximate traffic sources with long range dependence. This approach can be used to obtain accurate performance predictions since a power law decay can be approximated arbitrarily closely by enough exponential decay functions [24]. However, the resulting Markov models typically are complex multi state models that do not follow the principle of parsimonious modeling because every state added to such a model also adds several free parameters. This presents two problems, namely that of identifying the parameters (states and ....

S. Q. Li and C. L. Hwang. Queue Response to Input Correlation Function: Continuous Spectral Analysis. IEEE/ACM Trans. Networking, 1(3):678--692, 1993.

....function f(k) T k where T k is the interarrival time as de ned in Section 2.2. In Figure 5, we show the power spectrum for clipping levels 2, 6 and 11; we see that low frequency energy, i.e. around = 0, increases with the clipping level. This is an important fact, since San qi Li has shown in [9] that input power in the low frequency band has a dominent impact on queueing performance. 0.165 0.17 0.175 0.18 0.185 0.19 0.195 0.2 0.205 1 0.5 0 0.5 1 P(w) w 2 6 11 Figure 5: The power spectrum of MAPs for three di erent clipping levels E[N ] RE Simulation 27.7 2.3 ....

San-qi Li. Queue response to input correlation functions: Continuous spectral analysis. IEEE/ACM Transactions on Networking, 1(6):678-692, December 1993.

.... T i I r i r TS i V V TS , 1 0 if otherwise (20b) when A T i V ( is an integer, and where = TS r V r T r T # (21) The ES source can be now modeled as in [22] by an SBBP process matching the first and secondorder statistics [23] of an aggregate of N 1 M TS ( processes; let Q ES ( and B ES ( be the transition probability and the emission probability matrices of the ES model. So, the notation used in the following is: G TS ( and G ES ( the number of states of the underlying Markov chains of TS ....

S. Q. Li and C. L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis", IEEE/ACM Trans. Networking, Vol. 1, No. 6, pp. 678-692, Dec. 1993.

....models. This is equivalent to approximating a correlation function decaying as a power law by a sum of exponentials; although always possible, the number of parameters required in this approach will tend to infinity as the sample size increases. Such approaches are pursued, for example in [22] and [27, 28], and can be used successfully for solving certain queueing performance problems numerically. However, in this paper we argue strongly in favor of modeling LRD based on the principle of parsimony, also known as Occam s Razor (see for exam1 ple [21] The paper s second major contribution ....

.... original trace (A) fully shuffled trace (C) externally shuffled trace with block size m = 10 (E) and internally shuffled trace with block size m = 10 (F) Our experimental results are also qualitatively consistent with the results obtained from the frequency domain based approach considered in [27, 28] where it is noted that low frequencies in the power spectra dominate queueing behavior; recall that the LRD manifests itself as a sharp divergence in the low frequency region of the power spectrum. 4 Parsimonious Traffic Modeling One can conclude from the previous section that conventional ....

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S.Q. Li and C.L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis ", IEEE/ACM Trans. on Networking 1, pp. 678692, 1993.

....is to be evaluated, the Tagged Source, modeling the aggregate of the other sources loading the multiplexer by means of a single arrival process. In order to minimize the effects of the approximation, we model the aggregate traffic with an SBBP process matching its pdf and autocorrelation function [13]. The rest of the paper is organized as follows: in Section 2 a formal definition of skew is provided; in Sections 3 and 4 the monomedia and multimedia source models are respectively introduced and their statistics are derived. In order to make the skew performance analysis proposed here ....

....as A. In order to avoid the system state explosion occurring when A TS is modeled as the Markovian model superposition of a high number of multimedia sources, in this paper we model the A TS emission process with an SBBP process, n Z , matching the pdf and the autocovariance function of A TS [13]. Let us note that, in the case of multimedia sources with the same transition matrix ) W Q , the process modeling A TS can be achieved by applying the algorithm outlined in Fig. 3 as the superposition of 3 two state SBBP processes, and therefore ( n Z is an 8 state process, however many ....

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S. -q. Li, C. L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE Transactions on Networking, Vol. 1, No. 6, December 1993.

....is possible to approximate a heavy tailed distribution function by a superposition of exponential functions. One might then think of a LRD source as a simple generalization of the Markovian source. This approach, which amounts to mimicking LRD with Markovian models, has been taken for example in [13]. It can be used to obtain accurate approximate performance results since a power law decay can be approximated arbitrarily closely by enough exponential decay functions. However, the resulting Markovian models are complex multi state models. This presents two problems, namely that of identifying ....

S. Q. Li, C. L. Hwang, "Queue response to input correlation functions: continuous spectral analysis", IEEE/ACM Trans. Networking, vol. 1, no. 3, pp. 678-692, 1993.

....and r = 0 9 . see Fig. 8) Let us note that, as far as the aggregate of the other 10 multimedia sources, the External Sources, is concerned we used a model with 8 states, matching the autocorrelation function and the arrival probability density function of an aggregate of 10 multimedia sources [29]. 6. Conclusions In this paper we propose a model taking into account the intermedia time relationships existing in a multimedia traffic stream. The model is structured in two layers: an OBJECT layer, in which a multimedia source is seen as the superposition of mutually correlated monomedia ....

S. Q. Li and C. L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis",IEEE/ACM Trans. Networking, vol. 1, NO. 6, December 1993.

....monomedia traffic stream belonging to one particular multimedia source, here referred to as Tagged Source (TS) it is more efficient to model the aggregate of the remaining N 1 multimedia sources as a whole, that is, as a unique source, here referred to as A TS source. For this purpose, as in [42 43] it has been demonstrated 12 that the performance of a multiplexer is affected by only the first and second order statistics of the arrival process, we model the above A TS source by means of an SBBP process, n , matching the emission pdf and the autocorrelation function of the N 1 ....

S. Q. Li and C. L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis", IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993.

....on a sound understanding of input traffic, modeling of VBR video traffic has received intense interest. Of input traffic statistics, the histogram (rate distribution) and the autocorrelation function (ACF) are considered of first importance in estimating network performances [13] 14] [20], 25] Recently, a number of empirical studies have demonstrated the existence of long range dependence (LRD) or self similarity in VBR video traffic [5] 11] 12] Various This work was done when H. Ahn, B. Kim, and B. D. Choi were with KAIST (Korea Advanced Institute of Science and ....

S.Q.Li and C. L. Hwang, "Queue response to input correlation functions: Continuous spectral analysis," IEEE/ACM Trans. Networking, vol. 1, no. 6, pp. 678-692, Dec. 1993.

....in the experimental data. However, these models are analytically difficult to handle. Furthermore, they do not provide much insight into why they are meaningful on physical grounds. This explains in part why much modeling work still relies on more traditional multi state Markovian models (e.g. [25, 2]) However, recent work has shown that the superposition of many on off sources with heavy tailed on and off periods results in aggregate traffic with LRD [37, 7] Furthermore, there is widespread evidence that human as well as computer sources of traffic do tend to behave as heavy tailed on off ....

....Indeed, several studies have used an approach where Markov models approximate traffic sources with long range dependence. This approach can be used to obtain accurate performance predictions since a power law decay can be approximated arbitrarily closely by enough exponential decay functions [25]. However, the resulting Markov models typically are complex multi state models that do not follow the principle of parsimonious modeling because every state added to such a model also adds several free parameters. This presents two problems, namely that of identifying the parameters (states and ....

S. Q. Li and C. L. Hwang. Queue Response to Input Correlation Function: Continuous Spectral Analysis. IEEE/ACM Trans. Networking, 1(3):678--692, 1993.

....the lack of such periodic components. A peak at frequency 0 (DC component) implies an asymmetric constant average level of the signal. The relevance of the frequency domain approach to random traffic offered to queueing systems has been recently highlighted by the work of S.Q. Li and coworkers [13, 14, 20, 21]. This work has demonstrated the importance of secondorder characterizations of input traffic on queueing, loss and output traffic statistics; more specifically, it suggests that the low frequencies of the spectrum dominate these performance measures. An application of the frequency domain ....

Li, S.Q. and Hwang, C.L., "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. on Networking, Vol. 1, No. 6, 1993.

....delays in a finite server group of exponential servers, based only on the traffic parameters ( A ; z A;exp ) 3. 8 FREQUENCY DOMAIN APPROACH TO TRAFFIC The Frequency Domain Approach (FDA) focuses on second order statistics of offered traffic and their effect on queue response to that traffic [61, 62, 86]; it has been motivated by the need to characterize multimedia traffic in high speed networks. FDA is distinguished by the fact that it directly utilizes the frequency domain (the traffic spectral functions) and advocates their use as a unified traffic measurement for analyzing and controlling ....

....to characterize binary sources (on off traffic) and to study the effect of their second order statistical properties on queue length and loss rate statistics. A modeling technique for constructing Markovian traffic processes, which match a prescribed power spectrum, is introduced in Li and Hwang [62]. Let Q be the infinitesimal generator (transition rate matrix) of an N state Markov chain, which modulates the Poisson rate of the traffic process with rate vector fl. It is shown that the eigenstructure of Q characterizes the effect of such traffic on queueing performance. Assume that Q is ....

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Li, S.Q. and Hwang, C.L., "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. on Networking 1:6 (1993), 678--692.

....statistics of the data. 1 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 traffic. 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 [11] This paper is concerned with the ....

....Assistant with the F.W.O. Fund for Scientific Research Flanders) x Senior Research Associate with the F.W.O. Fund for Scientific Research Flanders) cells that arrive per time unit. This sequence is measured and forms the input of our identification algorithm. The research of Li and Hwang [7] has demonstrated that only the first order and second order statistics have a significant impact on queueing performance. Therefore we will identify a model that has first and second order statistics that approximate those of the data as well as possible. The model should also fit into queueing ....

S.Q. Li and C.L. Hwang. Queue response to input correlation functions: continuous spectral analysis. IEEE/ACM Transactions on Networking, 1(6):678-- 692, 1993.

....a cumulative distribution function identical to that of the MMPP. Upper bound on the CMPP model order: Suppose that each number is represented in fixed point format with a digits after the point. It is then possible to write each ( M ) i as a rational number with denominator 10 a which implies ffi = 10 a . In the worst case, the greatest common divisor of all numerators is equal to 1. For this case, the order of the CMPP must be chosen to be equal to 10 a to model the same distribution function as the MMPP we started from. The risk to end up with a very high model order is avoided in the ....

Li S.Q. and Hwang C.L. (1993b) Queue response to input correlation functions: continuous spectral analysis. IEEE/ACM Transactions on Networking 1(6), 678-692.

....statistics. 2.1 Model Choice The task of traffic identification is to construct stochastic models, the statistics of which match those of the data. In addition, the identified model should fit into queueing analysis which is used to evaluate the network performance. The research of Li and Hwang [6] has demonstrated that only the first order and second order statistics have a significant impact on queueing performance. Classic queueing theories have generally ignored the second order statistics entirely. By using a Markov modulated Poisson process with first order and second order statistics ....

S.Q. Li and C.L. Hwang. "Queue Response to Input Correlation Functions: Continuous Spectral Analysis", IEEE/ACM Transactions on Networking, vol. 1, no. 6, December 1993, pp. 678-692.

....its time complexity and accuracy with the method of Li and Hwang [8] I. INTRODUCTION The circulant modulated Poisson process (CMPP) is a restricted version of the Markov modulated Poisson process (MMPP) which is known to be a good model for the arrival processes in telecommunication networks [6, 7, 11]. Applications of these models for ATM networks are found in [6, 4, 10] A real traffic stream a(t) arriving at a queueing system is generally described by a train of impulses corresponding to message arrivals. In this paper, we consider the stochastic process a k (k = 1; 2; where a k is ....

....at a queueing system is generally described by a train of impulses corresponding to message arrivals. In this paper, we consider the stochastic process a k (k = 1; 2; where a k is the number of cells that arrive during the time interval (k Gamma 1; k ] The research of Li and Hwang [7] has demonstrated that only the first order and second order statistics of the arrival process have a significant impact on queueing performance. The task of traffic identification is to construct stochastic models, the first and second order statistics of which match those of the data. In this ....

S.Q. Li and C.L. Hwang. "Queue Response to Input Correlation Functions: Continuous Spectral Analysis", IEEE/ACM Transactions on Networking, vol. 1, 1993, pp. 678-692.

....modulated Poisson process, ATM Summary Li et al. 1] 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 traffic. Whereas the approach of Li et al. [1, 2, 3] is mainly 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) EUREKA (Sinopsys) the Belgian Federal Government (IUAP IV 02, IUAP IMechS) the European Commission (HCM (Simonet) TMR ....

....should fit into queueing analysis which is used to evaluate the network performance. The circulant modulated Poisson process (CMPP) is a restricted version of the Markov modulated Poisson process (MMPP) which is known to be a good model for the arrival processes in telecommunication networks [1, 2, 6]. The MMPP and the CMPP are model classes which can be incorporated in queueing analysis. The CMPP has several computational advantages compared with the MMPP (see e.g. 5] The circulant modulated Poisson process is a Poisson process the rate of which is changed (modulated) according to a ....

S.Q. Li and C.L. Hwang. Queue Response to Input Correlation Functions: Continuous Spectral Analysis. IEEE/ACM Transactions on Networking, vol. 1, no. 6, Dec. 1993, pp. 678-692.

....that is to be presented at the IEEE Infocom 96 Conference in San Franscisco in March 1996. QoS) guarantees for VBR video is non trivial in packet switched networks. For instance, correlated traffic with heavy tail distribution dramatically increases the queue length statistics at a multiplexer [1, 12, 13, 14]. Supporting VBR video traffic at a deterministic fixed service, not close to the peak, usually results in large buffers, large delay, and large delay jitter. Although the larger the correlation, the larger the queue length statistics at a multiplexer. The opposite is true for the errors of ....

S. Q. Li and C. L. Hwang., " Queue Response to Input Correlation Functions: Continuous Spectral Analysis, ` ` IEEE/ACM Trans. on Networking, Vol. 2, No. 6, Dec. 1994, pp. 678-692.

....affected by other marginal distributions, though future work might be needed to support this. 6. 2 Multiple time scale based traffic analysis There is a significant amount of interest in capturing the time scale at which key statistics of traffic to network performance are to be evaluated [11, 12, 13, 16]. Our conclusions clearly indicate that traffic behavior after certain time scale (i.e. CTS) is not relevant to network performance such as CLR. As discussed in [16] the CTS is closely related with the cutoff frequency c introduced in [11, 12, 13] Note that a practical buffer size is about one ....

....to network performance are to be evaluated [11, 12, 13, 16] Our conclusions clearly indicate that traffic behavior after certain time scale (i.e. CTS) is not relevant to network performance such as CLR. As discussed in [16] the CTS is closely related with the cutoff frequency c introduced in [11, 12, 13]. Note that a practical buffer size is about one frame duration (30 msec or so) whereas the time scale at which the LRD property of VBR video traffic begins to appear is an order of tens, hundreds, or even thousands frames. Further work is currently under way on finding CTS of various types of ....

S. Q. Li and C. L. Hwang. Queue response to input correlation functions: Continuous spectral analysis. IEEE/ACM Trans. Net, 1:678--692, 1993.

....of a slowly decaying auto correlation structure [4, 5, 6, 16] Providing efficient transport and Quality of Service (QoS) guarantees for VBR video is nontrivial in packet switched networks. For instance, correlated traffic dramatically increases the queue length statistics at a multiplexor [1, 11, 12, 13]. Supporting VBR video traffic at a deterministic fixed service, not close to the peak, usually results in large buffers, large delay, and large delay jitter. Because bandwidth in ATM networks can be allocated on demand, dynamic bandwidth allocation and re negotiation during the connection ....

S. Q. Li and C. L. Hwang., " Queue Response to Input Correlation Functions: Continuous Spectral Analysis,` ` IEEE/ACM Trans. on Networking, Vol. 2, No. 6, Dec. 1994, pp. 678-692.

....are given in [27] Our results indicate that rapid high frequency variations in the connection arrival process can largely be ignored and that it is the low frequency variations that have the greatest impact on the performance of a loss system. This is consistent with the results presented in [20, 21], which show that input power in the low frequency band has the dominant impact on performance in a system with queueing, whereas high frequency power can largely be neglected. We note that the cutoff frequency above which the high frequencies can be ignored was found to decrease with increases in ....

S. Li and C. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Transactions on Networking, Vol. 1, No. 6, pp. 678-692, December, 1993.

....is more interesting for bandwidth efficiency. This has not been studied before. We use a 5 state Markov Chain to represent the fluctuations in the available capacity. Its transition rate matrix has a single complex eigenvalue to characterize the time varying scales of the service process ([23]) 2 . Our goal is to study the interplay between the time scales of the service process and those of the feedback process. As in the previous cases, we have two sources with different round trip delays. The roundtrip delays are in the ratio 1:4. The sources have the same behaviour as in the ....

S. Q. Li and C. L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

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S. Q. Li and C. L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

....be x with real b and real A; f(x) is limited to a convolved binomial function [4] They are certainly insufficient to capture the diversed traffic correlation and burstiness behavior, especially the strong pseudoperiodic nature as found in MPEG video traces and feedback controlled ABR traffic. In [5, 6], Li and Hwang proposed a structure of circulant modulated Poisson process (CMPP) whose R(r) has the form be k with real b but complex A and whose cumulative function F(x) f f(u)du is expressed as a piece wise step function. It is obvious that the sum of complex exponentials represent a much ....

....frequency Im A) and half power bandwidth 2Re At) with average power bl. Every pair of complex conjugate eigenvalues then contributes two bells which are symmetric at the central frequency Since lower frequency power has much more impact on the queueing performance than higher frequency power [5], it is more convenient to use the power spectrum tha the correlation function for the second order input statistical measurement. This is true especially since we are interested in the power spectral matching in the low frequency region. For simplicity, in the following two examples we assume ....

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S. Q. Li and C. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

....modeling and single queue analysis. Based on statistical measurement and classification, each multimedia traffic can be represented by two statistical functions: rate histogram and power spectrum We say these two statistical functions to be important statistics for queueing analysis since, in [18], Li showed that the queueing performance such as mean queue, queue variation, and cell loss is essentially determined by and of the input traffic, whereas the influence of higher order input statistics such as bispectrum and trispectrum is negligible. In a recent work [19] Hajek also studied the ....

....in Figs. 5(b) and 6(b) It is clear that the MPEG video has much less power in the low frequency band than the JPEG video. The less the low frequency power, the better the queueing performance will be since the low frequency power is equivalent to long term correlations in the time domain [18]. CHONG AND LI: PBC BASED CONNECTION CONTROL 1079 (a) b) c) d) Fig. 6. JPEG Star Wars modeling by CMPP and queueing performance. a) Rate cumulative distribution. b) Power spectrum. c) Mean and standard deviation of queue. d) Loss rate. Next, we study the effect of traffic smoothing on ....

S. Q. Li and C. L. Hwang, "Queue response to input correlation functions: Continuous spectral analysis," IEEE/ACM Trans. Networking, vol. 1, pp. 678--692, Dec. 1993.

....mean rate. Therefore it improves traffic predictability. The conclusions in this section will be verified later by quantitative analysis of real traffic traces. C. Roles of Traffic Properties in Traffic Predictability What traffic properties characterize the inherent traffic predictability By ([10], 11] traffic statistics of orders higher than two has little effect on queueing performance. Therefore, we will use the Gaussian traffic process of a bell shaped power spectrum for analysis in that it provides a clear picture about traffic statistical structure. With this picture we show that ....

....error. Among 1st order traffic statistics, the CDF tail behavior accounts most for traffic predictability, whereas the variance coefficient reflects predictability in a straightforward way. Next is the analysis. Assume the power spectrum and auto correlation function of fY (t)g are respectively [10]: P ( 2 2 ffi( N Gamma1 X l=1 l ( Gamma2 l ) 2 l 2 ; R(t) 2 N Gamma2 X l=1 l e l jtj ; where N is the eigenvalue number; 2 is the DC term of a zero eigenvalue; l is the corresponding power vector of eigenvalue l . Each pair of ( l ; l ) forms a ....

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S. Q. Li and C. Hwang, Queue Response to Input Correlation Functions: Continuous Spectral Analysis, IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

....; B 1 = 2(ff fi) oe 2 fl = M ffl(1 Gamma ffl)fl 2 on : 13) The impulse term in (12) 2fl 2 ffi( represents the DC which is contributed by the non zero average arrival rate fl. The constant term, fl, corresponds to the white noise effect of Poisson local dynamics in packet generation [12]. The last term in (12) which is our major interest, has the bell shape centered at zero frequency with its half power bandwidth given by B 1 and the average power equal to the arrival rate variance oe 2 fl . Three conditions are required for the two state Markov chain design and so to fix the ....

S.Q. Li and C.L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Transactions on Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

....; B 1 = 2(ff fi) oe 2 fl = M ffl(1 Gamma ffl)fl 2 on : 13) The impulse term in (12) 2fl 2 ffi( represents the DC which is contributed by the non zero average arrival rate fl. The constant term, fl, corresponds to the white noise effect of Poisson local dynamics in packet generation [21]. The last term in (12) which is our major interest, has the bell shape centered at zero frequency with its half power bandwidth given by B 1 and the average power equal to the arrival variance oe 2 fl . Three conditions are required for the two state Markov chain design and so to fix the ....

S.Q. Li and C.L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Transactions on Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

....for the general construction of a(t) and s(t) In consequence, two state Markov chains are frequently used to construct different processes with limited statistic properties. Our work in the past several years focused on the development of fast algorithms for both modeling and queueing analysis [1, 2, 3, 4, 5, 6, 7, 8]. SMAQ tool (Statistical Match And Queueing tool) naturally grew out of this development for the integration of traffic service modeling and queueing analysis. A fundamental distinction of our work from others is that SMAQ tool is measurement based. In our view, both a(t) and s(t) can be composed ....

....linear system theory and digital signal processing, except that in our case both a(t) and s(t) must be nonnegative and without the limit of normal distribution to their first order statistics. In our framework, this is achieved through the introduction of circulant modulated Poisson process (CMPP) [2], whose inherent properties provide us a great freedom to match CDF and PSD in complex form and lead to successful development of fast algorithms for matched model construction. The third component provides various numerical solutions about a finite buffer system involved with a(t) and 2 s(t) ....

[Article contains additional citation context not shown here]

S. Q. Li and C. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

....power spectral function P ( and its steady state statistics by input rate distribution function f(x) For the following two reasons we consider only funtions P ( and f(x) of the input process. First, the queueing performance is found to be much less dependent on higher order input statistics [1, 2]. Second, in signal processing area, only P ( and f(x) are likely to be measured in practice [4] In other words, instead of considering the queue response to the original random input process, here we measure only the queue response to second order and steady state input statistics. In queueing ....

....and steady state input statistics. In queueing analysis one may construct a Markov modulated Poisson process (MMPP) to match with P ( and f(x) of the original input. The power spectrum of MMPP can be expressed in a rational function form and its distribution is a discrete probability function [2, 3]. The complexity of MMPP construction, however, lies in solving a so called inverse eigenvalue problem, which is rather difficult, if possible, in general [2] In this paper, we use the superposition of heterogeneous 2 state Markov chains (MC) for the construction of P ( and f(x) The ....

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S. Q. Li and C. L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

....modeling and VBR traffic characterization. The linear dynamic model is developed with the help of recent research results on frequency domain queueing analysis of multimedia traffic. It is shown that congestion occurs whenever the low frequency input traffic rate exceeds the link capacity ([4]) In other words, we only need to adapt the ABR traffic to the slow varying VBR traffic. The state variable of the dynamic model can then be defined as the low frequency unused link capacity; the input variables are the rate of each ABR connection. We end up with a multiple input single output ....

....(or short term moving average operation) There are two fundmental reasons behind the proposed slow time adaptation. One reason is for congestion control effectiveness. Recent multimedia traffic queueing analyses indicate that queue congestion is mainly captured by low frequency traffic behavior [4, 5]. In other words, network congestion occurs whenever the filtered input rate of aggregate traffic exceeds the nodal link capacity. Hence, the slow adaptation of ABR traffic to compensate the low frequency variation of VBR traffic is most effective to prevent network congestion. Refer to [4, 5, 10] ....

[Article contains additional citation context not shown here]

S. Q. Li and C. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

....4 Input Process Construction The N state MMPP input process is constructed such that Q is diagonalizable with distinct eigenvalues [0; 1 ; N Gamma1 ] Denote the right column and the left row eigenvector of the l th eigenvalue by g l and h l , respectively. Then as demonstrated in [11, 12], each eigenvalue contributes a complex exponential term to the input rate autocorrelation R i ( fl(t)fl(t ) fl 2 N Gamma1 X l=1 l e l j j (19) with l = P k P j k fl k fl j g lk h lj and mean input rate fl. Correspondingly, each eigenvalue contributes a bell shaped ....

....R i ( is real and P i ( is symmetric. In the numerical study, we construct an MMPP input process in which the modulating chain is of circulant form. Each row of Q = circ( a) circ(a0 ; a1 ; aN Gamma1 ) is a forward shift permutation of the previous row and thus must be real [12]. To easily isolate individual parameter effects on the jitter correlation, we consider an input process containing a single real eigenvalue or a single pair of complex eigenvalues ( single eigenvalue , or single bell power spectrum) Processes with a single real eigenvalue include the ....

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S.Q. Li and C.L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678--692.

....scales. Of course, strong correlation over a larger time scale simply means more power in the low frequency band of the power spectrum. It is a well known fact that when the arrival process shows several different time scales of variation, the largest time scales dominate the queueing behaviour ([20]) 3 Transient Queue Distributions Consider an arrival process described by an MMPP with its transition rate matrix Q and the input rate vector fl. For simplicity in the queueing analysis, the service process is considered to be exponential with mean service rate s . Thus the queueing model ....

....0 ; a 1 ; aN Gamma1 ] be the first row of the transition rate matrix Q. For the circulant, each row in Q is shifted to the right to form the next row, denoted by Q = circ( a) The circulant has several advantages over other Markov Chains in the modeling of correlated arrival rate processes ([20]) Initially we chose a circulant whose rate matrix had a single eigenvalue so that we could exactly identify the effect of an isolated time scale range on the transient queue response. For the sake of completeness however, we have also included some examples of circulants with multiple time scale ....

[Article contains additional citation context not shown here]

S. Q. Li and C. L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

....Q to be decomposable into a set of independent MC elements, where the eigenvalues of each element must be given in explicit analytical form. Typically, the size of each MC element has to be no more than four states, which substantially limits the range of arrival MC s for source modeling [31] [35]. By contraries, the Folding algorithm only requires the QBD structure, and therefore can handle any type of Markovian source models. Note that we use the two state Markovian source models here simply because they are available. It is shown in [10] that the results based on both Queue Occupancy ....

S.Q. Li and C.L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," presented at the 7th IEEE Computer Communication Workshop in Hilton Head Island, SC, Oct. 1992 (also submitted to IEEE/ACM Trans. on Networking, July 1992).

....discrete state Markov chain (MC) and fl = fl 0 ; fl 1 ; fl M Gamma1 ] is an input rate vector characterizing the Poisson arrival rate in each state. The autocorrelation function R( of the corresponding bandwidth process is given by R( Effl(t)fl(t )g = fi fl)e Qj j fl T [12], where fi denotes the element by element product of two vectors and = 0 ; 1 ; is the stationary probability vector of Q, i.e. i = lim t 1 Pr[fl(t) fl i ] Assume Q to be diagonalizable and can therefore be expressed in the form of Q = P M Gamma1 l=0 l g l h l , where l ....

....bandwidth process of the ith input source by P i ( Its D.C. component is equal to 2 fl 2 i ffi( where fl i is the average input rate of the source. Except for the D.C. term, the power spectrum P a ( of the aggregated traffic is the direct summation of the individual component spectra [12], i.e. P a ( N X i=1 P i ( 8 6= 0 (10) and the D.C. term of P a ( is given by 2 fl 2 ffi( with fl = P i fl i . The p.d.f. f a (x) dFa (x) dx of the aggregated bandwidth process is the convolution of the p.d.f. s, denoted by f i (x) 1 i N , of the individual ....

[Article contains additional citation context not shown here]

S. Q. Li and C. L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

....functions in traffic measurement are rate distribution f(x) histogram) and power spectrum P ( equivalently, autocorrelation function) The former describes steady state statistics; the latter captures second order statistics. The queueing performance is largely dependent on f(x) and P ( [6, 7]. Recently, Hajek and He observed that the the queueing behavior cannot be predicted solely based on the mean and P ( of the arrival process [8] They also stressed the importance of f(x) in determining queueing behavior. In our measurement architecture, traffic is further decomposed into three ....

....ignore the traffic measurement and its associated modeling. Otherwise, the measurement of HF traffic flow would unnecessarily complicate network designs in terms of real time operation and hardware support. Similarly in theoretical analysis, the HF traffic modeling is much difficult to achieve [7]. In the LF region, one only needs to measure the peak rate steady state behavior of the traffic (i.e, the tail portion of its distribution) In both LF and HF regions, the second order statistics have no effect on the link bandwidth allocation. Equivalently in the time domain, the timescale of LF ....

[Article contains additional citation context not shown here]

S. Q. Li and C. L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

....the system phenomena observed in this paper, affected by the individual effect of binary source dynamics, a unique observation is obtained with the power spectrum in frequency domain. That is, the larger the input powers in low frequencies, the longer the queue and the higher the loss rate [20] [21]. Note that most correlated traffic queueing analyses so far have emphasized performance studies of a single separate link. There is a strong need to extend our effort to modeling of a network wide traffic integrations. The analytical bottleneck for such extension is how to characterize the output ....

....b l ( to P ( Graphically, each such component will represents a bell shape curve located at the central frequency Imf l g with its half power bandwidth equal to Gamma2Ref l g. Note that the input power spectrum is always additive for the superposition of independent sources. Refer to [20][21] for detail analysis of input power spectrum. The work in [20] indicates that the queueing performance is essentially dominated by the power spectrum in low frequency band. In principle, the lower the frequency, the more the inputs are autocorrelated, and so the longer the queue. Now it is of ....

[Article contains additional citation context not shown here]

S.Q. Li and C.L. Hwang, "Queue response to input correlation functions: continuous spectral analysis," presented at the 7th IEEE Computer Communication Workshop in Hilton Head Island, SC, Oct. 1992.

....blocking or excessive delay. This extreme case is difficult to predict due to its infrequent occurrence and its dependence on individual sources. In practice, the transmission bandwidth requirement needs to be assessed using on line traffic measurement, particularly in live services. The study in [3] indicates that one of the most important input statistics to measure for queueing analysis is power spectrum. Two basic concepts were discovered in [4] on examining input traffic in the frequency domain. First, the transmission bandwidth in a finite buffer system is essentially determined by the ....

S. Q. Li and C. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis, " IEEE/ACM Trans. on Networking, Vol. 2, No. 6, pp. 678-692, Dec. 1993.

....worst case input scenario in order to avoid buffer blocking. This extreme case is difficult to predict due to its infrequent occurrence and its dependence on individual sources. In practice, the transmission bandwidth requirement needs to be assessed using on line traffic measurement. The study in [3] indicates that the most important input statistics to measure for queueing analysis is the power spectrum. Two basic concepts were discovered in [4] by describing the input traffic in the frequency domain. First, the effective bandwidth in a finite buffer system is essentially determined by the ....

S.Q. Li and C. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," presented at 7th IEEE Computer Commun. Workshop, Oct. 1992 (also submitted to IEEE/ACM Trans. on Networking).

....such delayed detection followed by the unnecessary extension of congestion periods significantly increases the oscillation period as well as the oscillation magnitude of ABR traffic within the network, causing a large consumption of buffer resources. Recent multimedia traffic queueing analyses [5] indicate that the queue congestion is mainly captured by low frequency traffic behavior. Based on the same principle, the congestion can be early detected through the observation of filtered input rate. This is best described through a simulation example given in Fig. 1 where a sample path of ....

S. Q. Li and C. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

....controller in [5] is designed only with the requirement of closed loop stability, whereas no other performance criteria such as steady state error are included. In this paper, we propose a new explicit rate control scheme based on the results of frequency domain analysis of multimedia traffic [6, 7]. It was found that the link capacity required by input traffic at each node is essentially captured by its low frequency characteristics. In other words, no congestion occurs at the node if the control design guarantees that the aggregate CBR VBR ABR traffic rate, filtered in a low frequency ....

S. Q. Li and C. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

....is the average input rate; C 1 is the coefficient of variation of the input rate process; BW is the half power bandwidth of the single bell power spectrum of each individual 2 state MMPP. Their aggregation yields another MMPP which is used to model various input traffic environment as outlined in [21, 22]. The average input rate of each 2 state MMPP is fixed by the overall average input rate, i.e. fl 1 = fl 50 . We further choose the variation coefficient C 1 = 1:23. Hence, the input power spectrum is changed by the half power bandwidth BW without affecting the input steady state statistics. ....

S. Q. Li and C. L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, pp. 678-692.

....signal processing techniques are available to measure traffic statistics. In particular, second order statistics can be measured by many sophisticated software packages or in hardware chips. The concept of spectral representation of multimedia traffic in queueing analysis was first introduced in [5, 6, 7]. In [7] we used a special class of Markov chain, called a circulant, to construct input processes. One significant advantage of using circulant is to identify the impact of power spectrum, bispectrum, trispectrum and distribution of the input process on the characteristics of queue and loss rate. ....

....techniques are available to measure traffic statistics. In particular, second order statistics can be measured by many sophisticated software packages or in hardware chips. The concept of spectral representation of multimedia traffic in queueing analysis was first introduced in [5, 6, 7] In [7] we used a special class of Markov chain, called a circulant, to construct input processes. One significant advantage of using circulant is to identify the impact of power spectrum, bispectrum, trispectrum and distribution of the input process on the characteristics of queue and loss rate. The ....

[Article contains additional citation context not shown here]

S. Q. Li and C. L. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. Networking, Vol. 1 , No. 5, Oct. 1993, pp. 522-533.

....in order to avoid buffer blocking and excessive delay. This extreme case is difficult to predict due to its infrequent occurrence and its dependence on individual sources. In practice, the transmission bandwidth requirement needs to be assessed using on line traffic measurement. The study in [3] indicates that the most important input statistics to measure for queueing analysis is the power spectrum. Two basic concepts were discovered in [4] by describing the input traffic in the frequency domain. First, the effective bandwidth in a zero loss finite buffer system is essentially ....

S. Q. Li and C. Hwang, "Queue Response to Input Correlation Functions: Continuous Spectral Analysis," IEEE/ACM Trans. on Networking, Vol. 2, No. 6, pp. 678-692, Dec. 1993.

No context found.

S.-Q. Li and C.-L. Hwang. Queue response to input correlation functions: Continuous spectral analysis. IEEE/ACM Transactions on Networking, pages 678-- 692, Dec. 1993.

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