In search of better statistics for traffic characterization

Abstract

We model and simulate stochastic traffic based on two established statistics: marginal distribution and autocorrelation function. Although apparently equivalent to each other in these aspects, the three different synthetic models produce notably different behavior in a G/D/1/<FONT FACE="Symbol">¥</FONT> queue. As an alternative modeling strategy, we measure the log-moment generating function of the synthetic models. Our results indicate that recent works related to Effective Bandwidth functions may provide the discriminatory statistic required to explain the noted inconsistency.

Effective Bandwidth; Traffic Modeling

In search of better statistics for traffic characterization

Tarkan Taralp, Michael Devetsikiotis and Ioannis Lambadaris

Department of Systems and Computer Engineering

Carleton University

Ottawa, Ontario K1S 5B6 Canada

{taralp,mike,ioannis}@sce.carleton.ca

Abstract

We model and simulate stochastic traffic based on two established statistics: marginal distribution and autocorrelation function. Although apparently equivalent to each other in these aspects, the three different synthetic models produce notably different behavior in a G/D/1/¥ queue. As an alternative modeling strategy, we measure the log-moment generating function of the synthetic models. Our results indicate that recent works related to Effective Bandwidth functions may provide the discriminatory statistic required to explain the noted inconsistency.

Keywords: Effective Bandwidth, Traffic Modeling

1 Introduction

Commercially available network modeling tools have become increasingly sophisticated and powerful, yet the results they provide are ultimately limited by the fidelity of the synthetic traffic which drive them. With the failure of Poisson modeling [16] the question arises as to which models should be adopted and more importantly against which criteria they are to be evaluated.

We model stochastic traffic by its arrival/flow rate with respect to time. The marginal distribution of the flow rate and its autocorrelation co-efficient function statistics are taken together to provide a distinguishing signature of that traffic for the purpose of characterization and synthesis. A third statistic based on "log-moment generating functions" is employed next and compared in effectiveness to the marginal-autocorrelation statistics in light of queue simulation experiments.

The kinds of stochastic traffic targeted in our approach are those which may be interpreted in terms of a flow rate, such as compressed video and aggregated data traffic. It is assumed that the discrete packet unit carrying the traffic is of a scale considerably smaller than the flow rate, for example the quasi-fluid like cell flow in ATM links. Methods used to discretize (ie. packetize) data is taken to be a subsequent and more technology specific modeling issue, outside the scope of this paper.

2 Traffic Characterization Criteria

We let the sequence X_k (k = 0, 1, 2, ...) denote a discrete-time traffic arrival process, with physical interpretation of bits per second or similar measure. The process X_k is taken to be time-stationary such that all its joint distributions of the form in (1) are invariant with respect to any time shift t :

The marginal distribution function Fx(x) (equivalently density function fx(x)) will act to capture the first-order characteristics of the traffic process. Referred to as the steady-state or marginal statistic, it provides an estimate of the probability (2). Statistical interdependence of X_k at different points in time k may be referred to as its "time structure" or "time dependence". Characterizing this aspect is the autocorrelation function rx(t) as defined in (3). Autocorrelation is considered to be a second-order statistic in the sense that it estimates the statistical relationship between two random variates X_k and X_k+tseparated by t units in time.

Empirical traffic streams and their synthetic model counterparts are usually evaluated with regard to these two functions, and the aim of most traffic models is to reproduce a given marginal distribution and autocorrelation structure in the traffic. One can informally refer to the development of synthetic models based on matching of marginal and autocorrelation characteristics as a "marginal-autocorrelation" modeling methodology. And while the use of such a methodology is arguably justifiable when considering Gaussian stochastic processes [12] it is probably not safe to rely on it in general.

In [14] the use of the marginal and autocorrelation pair is qualified by stating the need to consider "sample path appearance" in the exercise of model fitting. Other works consider the power spectral density function Sx(f) of the traffic process X_k particularly in the low frequency region, to be a more relevant descriptor of time dependence in the traffic [17]. A novel and relatively new approach has been put forth based on a treatment of traffic workload and the queue service rate generally referred to as the effective bandwidth statistic. Yet there does not appear to be enough independent study of this kind to come to consensus on the question: is marginal and autocorrelation enough to guarantee realistic models, or under what conditions is marginal-autocorrelation valid?

Our experimental work suggests that traffic description based on moment generating / effective bandwidth function methods have advantages over marginal-autocorrelation methods. The working hypothesis is that synthetic traffic models showing similar moment generating function signatures should produce similar performance in a queueing system, while those that rely on the marginal and autocorrelation can yield unpredictable results. As multimedia traffic in B-ISDN / IP networks becomes more structurally complex, the development of more sophisticated methodologies for traffic modeling has received renewed interest.

3 The Traffic Models

A set of traffic models, dissimilar to each other in mathematical construction, was chosen based on their ability to accurately capture and reproduce a given marginal-autocorrelation signature. The first model is in the ARM class, transforming a modulo-1 background state into its output process. The second model is based on independent non-exponential renewal periods. The third model distorts a background Gaussian process into a desired output process.

3.1 Extended-TES

The Extended-TES model is an Autoregressive Modular (ARM) Process, a family which also includes the TES and D-TES processes [13, 15]. An ARM process begins with an identically distributed input sequence {V₁, V₂, V₃,...} termed the innovation sequence, which is cumulatively summed in modulo-1 fashion to produce an ARM background sequence U_n The background sequence is then transformed by a distortion function D to yield the ARM foreground (output) sequence X_n.

The ARM⁺ category of models calculates the background sequence according to equation (4). The starting state U⁺₀ is chosen on [0,1) and the <x> indicates the operation x-[x].

A distortion function D can be composed of a stitching function Se in series with a cumulative distribution function H as shown in equation (5). It can be shown that the steady-state marginal distribution (7) of the output X_n is given wholly by H.

The TES model is a specialization of ARM such that its innovation sequence is defined to be IID. The Extended-TES (E-TES) model generalizes the function of TES by imparting statistical correlation in time upon the sequence using a finite impulse response filter of the form in equation (8). Thus Extended-TES is a model of class ARM (M A(q),D). The pre-innovation IID sequence Z_n feeds the FIR filter to produce the innovations which are then utilized in the same manner as in the TES model. The autocorrelation of a general ARM output sequence is given by equation (9). In [18] a numerical algorithm was developed to calculate the autocorrelation function given the distribution of Z the filter parameters {ao, a1,..., aq) and distortion function, for the E-TES model.

3.2 Spatial Renewal Process

The Spatial Renewal Process, introduced by Jelenkovic, Lazar and Semret in [8] efficiently models processes exhibiting arbitrary marginal distribution and aperiodically decaying autocorrelation. We consider a point process with IID inter-renewal time sequence {T_n} described by the CDF function F_t(t). This process plays the role of background process in the SRP model and is responsible for the time dependence structure. A second background process, independent of the first, is a sequence of IID values X_n which are distributed according to the desired steady-state marginal distribution of the traffic. The SRP process Y_t is composed of a chain of renewal periods where the nth period is T_n in length and the sample path during this period takes on the value of X_n .

The choice of a marginal distribution and of an autocorrelation are fully decoupled from each other in the SRP model. It follows from the independence of the X_n and T_n sequences that the steady-state marginal distribution of the output Y_t follows the distribution of X_n, which is completely arbitrary. The autocorrelation co-efficient function r(t) of Y_t is related to the SRP renewal distribution function F_T(t) using equation (10) or equivalently equation (11).

The autocorrelation function r(t) must be a decreasing and concave-up function (r´ (t) < 0, r´´(t) ³ 0, " t ³ 0) in order to ensure that the corresponding renewal distribution F_T(t) is indeed a valid CDF function. An autocorrelation function may be constructed of Pareto, exponential and piecewise linear segments, provided that the above conditions are met.

3.3 Distorted Gaussian

The Distorted Gaussian (DGauss) model begins with a Gaussian process with a given autocorrelation structure and maps it into an appropriate marginal distribution. Examples of this popular traffic generation technique include the Autoregressive-To-Anything Process [1] and the self-similar traffic model in [6]

Many techniques exist to generate Gaussian time series (Gaussian in the marginal distribution) with a wide range of autocorrelation decay characteristics. For the purposes of experimentation an appropriately modified version of the SRP-FGN model [19] was chosen due to its ease of fitting and generation speed. In this model, a background Gaussian process Z_k is imparted with an autocorrelation structure r´(t) and is run through a fitting function X_k= F_X^-1 (F_N(Z_k)) to map it values into an appropriate distribution. Due to the background-foreground transformations, pre-compensation is applied to the background autocorrelation r’ such that the resulting output autocorrelation r matches the desired specification [6].

4 Experiment Design

4.1 Traffic Signature

In keeping with the marginal-autocorrelation methodology, the three models were fit to a given set of histogram and autocorrelation statistics, illustrated in Figure 1. The histogram, from which the marginal was derived, was intentionally chosen to be highly non-Gaussian in shape, with tail extending much beyond the mean. The autocorrelation function was constructed of two segments, the first segment decaying according to c₁e^-0.0415k up to lag k=30, and the tail segment decaying at one-tenth the rate c₂e^-0.00415k.

Figure 1:
Marginal and Autocorrelation Traffic Specifications

4.2 Computer Simulation of G/D/1 Queue

Independent traces of synthetic traffic X_k were generated to a fixed length of 2 x 10⁵ samples. As pictured in Figure 2 each of the traffic models in turn was fed into a G/D/1/¥ Lindley queue system implemented by computer simulation. The queue discipline followed that of a discrete-time (slotted-time) Lindley queue [12] operating in "cut-through" mode according to:

where Q_k is the queue occupancy during time slot k, c is the service rate of the queue, and the <.>⁺operator indicates clipping of values less than zero up to zero.

Figure 2:
Experiment Block Diagram

4.3 Experimental Parameters

The input variables under study were the model type, the marginal distribution and the autocorrelation of the traffic. As far as practicable the latter two parameters were experimentally fixed, while the model type was varied among the different sources. The queue's service rate c was held at a constant value such that the server utilization ratio was 33%. The key output result was the buffer occupancy probability Pr[Q>y] of the queue, which measured the unconditional time-average probability of the queue size Q. The traffic stream was also characterized using the log-moment generating function to be introduced in a later section.

The experiment was designed to test the effectiveness of traffic modeling on the basis of the marginal-autocorrelation technique. In order to avoid the mitigating effects of statistical aggregation, multiplexing of sources was not considered and only individual streams of generated traffic were fed into the queue. Thirty independent runs of the simulation were performed for each traffic model. The independent replications approach for estimating mean statistics was used based on a Student's t-distribution with N=30 replications. Ninety-five percent confidence intervals on the measured traffic statistics and buffer occupancy are reported.

5 Experiment Results

5.1 Experiment Results - Input traffic

Marginal distribution (ie. histogram) and autocorrelation function statistics were measured for the synthetic traffic before entering the queueing system. Figure 3 compares the empirical marginal statistics to the specification, showing that the match by all three models is very good. The match on autocorrelation is also very tight. Figure 4 plots the upper and lower 95% confidence intervals of the empirically estimated curves with dash lines, and the specification curve in solid for comparison.

Figure 3:
Observed Input Traffic Marginal

Figure 4:
Observed Input traffic Autocorrelation

5.2 Experiment Results - Queue Behavior

Figure 5 plots the observed buffer occupancy probability Pr[Q>y] with respect to yThe dash lines on either side of the curves mark out the 95% confidence intervals in the observations. It is notable how greatly the queueing results vary, from the E-TES model with rapidly decaying occupancy, to the Dgauss model and the SRP model at the top end of the range. These results, despite a directed effort to produce three traffic models with the same marginal-autocorrelation signature, are somewhat unexpected. The observations as a whole suggest that two traffic processes possessing identical first- and second-order characteristics might in practice produce highly different queueing behavior and performance, and that modeling based on a marginal-autocorrelation methodology may not be reliable.

Figure 5:
Observed Buffer Occupancy Probability

The description of autocorrelation as a second-order statistic is apt in that it characterizes the probabilistic relationship between pairs (X_k, X_k+t) of variates. But this characterization is incomplete in that only the linear correlation between the two variates is estimated. Furthermore, joint probabilities with higher dimensionality of the form in (1) hint at the use of higher order measures such as "tri-spectrum" [7]. Autocorrelation as a measurement of the input traffic emphasizes the power spectral nature of a traffic flow, lending itself more to signal theory rather than queueing analysis. There is no simple conceptual connection between the spectral density of a traffic flow and workload input into the queueing system except for a few rules of thumb. For example, it is commonly accepted that the lower frequency components in the autocorrelation function are primarily responsible for buffer occupancy in a queue system [17]. Or, that a long and heavy positive tail on the autocorrelation implies greater burstinessand buffer occupancy as compared to traffic with rapidly decaying autocorrelation, all other things being equal.

Original work carried out by others [17, 15] proposes the use of the power spectrum function S_X(f) in place of autocorrelation. Furthermore it characterizes the mean service time and finite buffer size of the queue in an analysis of traffic and queue as one system, as opposed to characterizing the input traffic in isolation.

6 Moment Generating Functions

In a further departure away from the traditional characterization of stochastic traffic, an approach closer to queueing principles has attracted interest in recent years. The notion of an "effective bandwidth function" in the characterization and application of traffic to queueing systems was put forth by papers from Kelly and Gibbens [5, 9, 10] and others. Asymptotic time scale versions of effective bandwidth are also investigated in [2, 3, 4, 11] . Traffic is conceptualized as a fluid flow source feeding into an infinite queue with constant rate server as depicted in Figure 6. The probability of a traffic burst at constant rate r (r > c) over a duration of time T relates to the probability of reaching some level b in the queue.

Figure 6
: Model or Fluid Flow Queue

In the asymptotic version, the effective bandwidth function of the traffic is related to the queue by setting a*(q) = c and solving for the q parameter. The buffer occupancy probability is then given by Pr[Q > b] » e^{-q b} [4, 2, 3 11]. An advantageous property of effective bandwidth is that it lends itself to circuit-switching like techniques for performing network analysis [11] However the technique is analytical in nature and difficult to apply empirically [4]. It provides asymptotic results in the sense that the time scales under consideration approach infinity (T ® ¥ ). The practical implication is that the buffer occupancy relationship Pr[Q > b] » e^{-q b} is accurate only for b chosen sufficiently large and for Pr[Q > b] very small.

Our approach, being empirical in nature, assumes the two parameter (q,T) form of effective bandwidth with finite time scale parameter T. In (13) partial-sum sequences A_k^T of the empirical traffic sample X_k are computed for a set of time scales T. In (14) the characterization function denoted B is defined by making use of the aggregated sequences from (13). The expected value expression is estimated numerically from the family of A_k^T

The two-dimensional surface described by B(q,T) is evaluated over a range of values for q Î {...10^-1, 10⁰, 10¹,...} and for T Î {1, 2, 4, 8,...}. The T parameter acts to define the time scale over which the traffic X_k and its partial sums A_k^T are aggregated and averaged. The q parameter acts to modify the weighting of measurement between that of mean value and maximum value detection due to the exponential behavior of the moment generating function. As q ® 0 the expression (14) converges to the mean P {A^T}/T, and as q ® ¥ it converges to the peak value P {A^T}/T. The value produced by B{q ,T} may be viewed as the traffic or workload flow rate. Mean estimates of the function value, measured from 30 runs of each synthetic model, produced the curved surfaces shown in Figures 7 a-c.

Figure 7:
Observed Input Traffic B Function

Additionally, a horizontal plane representing the constant service rate c of the queue is superimposed on the curves. Lindley's queue equation in (12) implies that bursts of traffic with mean rates r £ c are fully absorbed by the service capacity c and do not impact on the occupancy state Q whereas bursts r > c cause a net buildup of workload. The superposition of the traffic flow and queue service c-plane taken together with their physical interpretations led to a reasonable hypothesis: that traffic showing a similar B (q,T) signature in a "critical region" above the c-plane will produce similar effects in a queueing system, or at least that the critical region will better predict such queue effects.

6.1 Experiment Results

As was seen, the SRP and DGauss models produced similarly high buffer occupancy in the queue, which was reflected in the similar shapes of the B(q ,T) traffic measurements. The E-TES traffic measurement is visually very different than the two others and has appeared to reflect the difference in queueing results. The greater amount of volume shown in the SRP case as compared to the DGauss case also corresponds to the level of buffer occupancy experienced in the simulation.

It was desired to make some numerical comparisons of the surfaces with particular attention given to the volume above the c-plane. A function B^* was defined as shown in (15) for this purpose.

The following two tables present a rudimentary numerical comparison of the measured B* surfaces of the traffic. Table 2 calculates the average difference between the surfaces B*_x-B*_y in a pair-wise comparison of the various traffic. The order and magnitude of the numerical values seem to imply that SRP traffic produce the highest buffer occupancy followed closely by DGauss traffic, and that much lower buffer occupancy be expected for E-TES. The root-mean-square (RMS) differences between the B* surfaces are also reported in Table 3. The closer values between SRP and DGauss are consistent with the observed buffer results in that they imply a similarity between the two traffics, and provide a contrast to the E-TES traffic, also consistent with the observed buffer results. There appears to be a strong correlation or relationship between the traffic characterization surface B (q,T) on the input side and the buffer occupancy probability on the output, regardless of stochastic model type. The autocorrelation function and its relationship to the buffer occupancy was seen to be highly sensitive to the type of synthetic model used, and was clearly not robust in the face of dissimilar traffic models.

7 Conclusion

Empirical measurements of B (q , T) appear to characterize traffic for queueing analysis more robustly than the marginal-autocorrelation pair. The choice of dissimilar traffic models and the definition of a traffic with highly non-Gaussian distribution were intentionally employed to test the limitations of marginal-autocorrelation modeling. The intent of our work was not to abandon the use of marginal-autocorrelation, but to show how issues in this area still need to be addressed.

The traffic under study in this paper was characterized by na asymmetrical heavy tailed marginal distribution function. It exhibited a rapid decay in the early autocorrelation lags followed by a slower decay in the tail. Compressed video and LAN/WAN traffic exhibit similar features. But the general validity of the B (q , t) approach can only be determined by further experimentation with a wider range of traffic. For example the characterization did not describe traffic with periodic time structure (see [9]) such as MPEG video. Also, the packetization of data was not considered to be the modeling issue, and while reasonable at the ATM scale of cell traffic, modeling large-packet based traffic of the IP scale needs further consideration.

With regard to numerical efficiency, estimates of the B (q, T) surface are computationally more demanding than the measurement of marginal and autocorrelation for a given empirical traffic trace. Nevertheless, the approach is promising in that it can visualize discrepancies between stochastic traffic that are not detected by other (eg. marginal-autocorrelation) means. The experiments also suggest the existence of a third statistic, which together with marginal and autocorrelation may provide the necessary characterization for good traffic models.

8 Acknowledgements

This research was supported by grants from the Telecommunications Research Institute of Ontario and by the Natural Sciences and Engineering Research Council of Canada.

References

[1] C.Cario and B.Nelson. Numerical Methods for Fitting and Simulation Autoregressive-To-Anything Processes. Technical report, Delphi Packard Electric Systems, Ohio, USA, August 1996. Preprint.

[2] C. S. Chang. Stability, Queue Length, and Delay of Deterministic and Stochastic Queueing Networks. IEEE Transactions on Automatic Control, 39(5):913-931, May 1994.

[3] C. S. Chang, P. Heidelberger, S. Juneja, and P. Shahabuddin. The Application of Effective Bandwidth to Fast Simulation of Communication Networks. Technical report, IBM Research Division, Yorktown Heights, NY, USA, May 1993. RC 18877.

[4] C. S. Chang and J. A. Thomas. Effective Bandwidth in High-Speed Digital Networks. IEEE Journal on Selected Areas in Communications, 13(6):1091-1099, August 1995.

[5] R. J. Gibbens. Traffic Characterisation and Effective Bandwidths for Broadband Network Traces. In Stochastic networks: Theory and Applications, Royal Statistical Society Lecture Note Series 4, pages 169 - 179. Oxford University Press, 1996.

[6] C. Huang, M. Devetsikiotis, I. Lambadaris, and A. R. Kaye. Modeling and Simulation of Self-Similar Variable Bit Rate Compressed Video: A Unified Approach. In Proceedings of ACM Sigcomm 95, pages 114-125, 1995.

[7] C. Hwang and S. Q. Li. On the Convergence of Traffic Measurement and Queueing Analysis: A Statistical Match Queueing SMAQ Tool. In Proc. IEEE Infocom, pages 602-612, 1995.

[8] P. R. Jelenkovic, A. A. Lazar, and N. Semret. Multiple Time Scales and Subexponentiality in MPEG Video Streams. In Proceedings of International IFIP-IEEE Conference on Broadband Communications, pages 64-75, 1996.

[9] F. P. Kelly. Modelling Communication Networks, Present and Future. In Proc. R. Soc. Lond. A (1995) 444, The Clifford Paterson Lecture. The Royal Society, 1995.

[10] F. P. Kelly. Notes on Effective Bandwidths. In Stochastic networks: Theory and Applications, Royal Statistical Society Lecture Note Series 4, pages 141-168. Oxford University Press, 1996.

[11] G. Kesidis, J. Walrand, and C. S. Chang. Effective Bandwidths for Multiclass Markov Fluids and Other ATM Sources. IEEE ACM Transactions on Networking, pages 424 - 428, August 1993.

[12] A. Leon-Garcia. Probability and Random Processes for Electrical Engineering, pages 336-337. Addison-Wesley Publishing Company, Don Mills, Ontario, 2nd edition, May 1994.

[13] B. Melamed. Autoregressive Modular (ARM) Processes. Technical report, Enterprise Technology Solutions, Inc., Warren, New Jersey, 1996. Draft Copy.

[14] B. Melamed and J. R. Hill. A Survey of TES Modeling Applications. Technical report, Rutgers Center for Operations Research, New Jersey, USA, 1995. Http://rutcor.rutgers.edu/ .

[15] B. Melamed, Q. Ren, and B. Sengupta. Modeling and Analysis of a Single Server Queue with Autocorrelated Traffic. In Proc. IEEE Infocom, pages 634-642, 1995.

[16] V. Paxson and S. Floyd. Wide Area Traffic: The Failure of Poisson Modeling . IEEE Transactions on Networking, 3(3):226-244, June 1995.

[17] H. D. Sheng and S. Q. Li. Second Order Effect of Binary Sources on Characteristics of Queue and Loss Rate. In Proc. IEEE Infocom, pages 18-27, 1993.

[18] T. Taralp. Towards a New Method of Traffic Characterization based on the Moment Generating Function with Application to Queueing . Master's thesis, Carleton University, Ottawa, Canada, October 1997.

[19] T. Taralp, M. Devetsikiotis, I. Lambadaris, and A. Bose. Efficient Fractional Gaussian Noise Generation Using the Spatial Renewal Process. In Proc. IEEE ICC}, June 1998.

Publication Dates

• Publication in this collection
  31 July 2000
• Date of issue
  Feb 1999