Journal of Statistical Planning and Inference 137 (2007) 3939 – 3953 www.elsevier.com/locate/jspi
A comparison between several adjustment models to simulated teletraffic data F. Mallora,∗ , P. Mateob , J.A. Molera a Department of Statistics and Operative Research, Public University of Navarre, Campus Arrosadía, 31006 Pamplona, Spain b Department of Statistical Methods, University of Zaragoza, Pedro Cerbuna, 50009 Zaragoza, Spain
Received 1 October 2006; received in revised form 29 October 2006; accepted 6 April 2007 Available online 4 May 2007
Abstract In this paper we assess the suitability of the fractional Brownian motion and the Markovian modulated Poisson process to represent modern teletraffic data when we use them in dimensioning and quality of service studies. The procedure consists in adjusting a model of each family to a stream of data simulated with a physical model that emulates the real flow of data. Then, both models are profusely studied in order to establish which properties of the original data are captured. © 2007 Elsevier B.V. All rights reserved. MSC: 65C99; 60J65 Keywords: Simulation; Fractional Brownian motion; Markov modulated Poisson process; Telecommunications
1. Introduction The modern traffic in telecommunication networks (teletraffic in the sequel) has drastically changed during the past 20 years mainly because the transference of data, video, voice using the same support is allowed. The classical queue models which quite accurately reflected the behavior of telephony systems have become insufficient. So that, new mathematical models are required to forecast the capacity of the antennas (the servers of the system) and, then, the quality of service (QoS) that appears as a crucial measure for the success of a telecommunication company in the market. Unfortunately, there is not a consensus in the scientific community about the best family of models to fit the modern teletraffic and several families of models have been proposed in the specialized literature (see, for instance, Chapter 1 in Park and Willinger, 2000 and the references therein). In this paper, two of these models are selected in order to analyze their suitability to reflect modern teletraffic. These models are Markov modulated Poisson process, briefly MPPP (see, for instance. Heffes and Lucantoni, 1986) and fractional Brownian motion, briefly, FBM (see, for instance, Norros, 1994). They are well-known representatives of short range and long range dependent models, respectively. The study is carried out as follows. First, we simulate different sequences of teletraffic data by using a physical model based on on/off sources. Second, the teletraffic data are used to estimate the parameters of the MMPP and FBM models. Third, we generate data streams with the adjusted ∗ Corresponding author. Tel.: +34 948 16 92 15; fax: +34 948 16 92 04.
E-mail addresses:
[email protected] (F. Mallor),
[email protected] (P. Mateo),
[email protected] (J.A. Moler). 0378-3758/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2007.04.012
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models. Finally, these new streams are compared with the one generated by the physical model and the ability of each theoretical model to capture the properties of the original data is studied. This paper is organized as follows. In Section 2, we introduce some mathematical properties that hold in the modern teletraffic data and we justify the choice of the FBM and MMPP as representative models to adjust it. Moreover, we present some tools to adjust both theoretical models to teletraffic data. In Section 3, the procedures to generate the three kinds of streams of data are explained. Section 4 is devoted to the design of simulations and we present the conclusions obtained from the comparative study about the performance of the adjusted models. We conclude with some final remarks and future research in Section 5. 2. Teletraffic data: characteristics and models 2.1. Long range dependence We consider the description of the traffic that transits through a given link in a telecommunication network. For such purpose we define the stochastic process {St ; t ∈ R} which represents, for each t, the number of packets (or bytes) transferred up to time t. For each t, let Xt = St − St−1 , so that Xt is an incremental measure and indicates, for each t, the number of arriving data packets between the epochs t − 1 and t. The unit time is some fixed unity as seconds, milliseconds, etc. The notion of self-similarity is the characteristic that makes different the modern teletraffic, see, for instance, Leland et al. (1994). Intuitively, it means that in different unit times the process {Xt ; t ∈ R} presents the same behavior. Let {X(m) (i); i = 1, 2, . . .} be the aggregated process of X at level m, where X (m) (i) =
1 (X(i−1)m+1 + · · · + Xim ) m
then, when {Xt : t ∈ R} is self-similar, the time series plot of {X(m) (i); i 1} is similar to the one of {X (1) (i); i 1} for a wide range of different aggregation levels m. This characteristic does not hold for Poisson models. (m) (m) Let H be a real value in the interval (0, 1) and for each m, H and i 1, let XH (i) := X(m) (i)/mH −1 and XH := (m) {XH (i); i 1}. Following Cox (1984), we say that the process {X (1) (i); i 1} is an asymptotically second-order self-similar, with self-similarity parameter H , if (m)
lim Var(XH (i)) = 2 ,
m→∞
0<<∞
and (m) lim r (k) = 21 ((k m→∞ H
+ 1)2H − 2k 2H + (k − 1)2H ),
(m)
(m)
where rH (k) is the autocorrelation of lag k of the process XH , that is, (m)
rH (k) :=
(m)
(m)
Cov(XH (i), XH (i + k)) (m)
.
(m)
Var(XH (i))Var(XH (i + k)) (m)
(1)
The process {X (1) (i); i 1} is exactly second-order self-similar if for all m 1 Var(XH (i)) = Var(XH (i)) = 2 and (m)
rH (k) = 21 ((k + 1)2H − 2k 2H + (k − 1)2H ).
(1)
We will consider that the process {X (1) (i); i 1} exhibits long range dependence (LRD) if there exists a real value ∈ (0, 1) such that r(k) ∼ ck − as k → ∞ where c is a positive constant and r(k) is the autocorrelation coefficient for the kth lag. So that, LRD implies that ∞ k=1 r(k) = ∞, when this series converges, the process exhibits short range dependence (SRD). From (1), we have that r(k) ∼ ck −(2−2H ) , so that, when H lies in (0.5, 1) exactly second-order self-similarity implies LRD. In this case, the parameter H is known as the Hurst parameter and plays an important role characterizing the behavior of the data series. In fact, H determines the slow decay (hyperbolic) of the autocorrelation function (ACF).
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Fig. 1. Autocorrelation function of an LRD process (left) and an SRD process (right).
The difference between the behavior of an LRD process and an SRD process can be appreciated in Fig. 1 where the ACF is drawn for two data streams obtained with the physical model introduced in Section 3.1. The LRD data come from the aggregation of 100 on/off sources with Pareto distributed time and, then, the Hurst parameter is 0.9. The SRD data are the aggregation of 100 on/off sources with exponentially distributed time and, then, H = 0.5. 2.2. Stochastic models: FBM vs. MMPP As stated in the introduction of this work, we are going to study the suitability of two theoretical models, say the FBM and the MMPP, to capture the behavior of a simulated teletraffic data stream. In this section, we introduce formally both theoretical models and justify the choice of them as referential models. A standard FBM is a stochastic process {BH (t) : t ∈ R} such that the increments are stationary and normally distributed with mean 0 (BH (0) = 0) and variance t 2H , where H ∈ (0, 1). For being an exactly second-order selfsimilar process, when H > 0.5, it exhibits LRD. Observe that in practice we will fit non standard FBM, {B(t) : t ∈ R}, to a stream of data, and then we will have to estimate not only H , but a mean value, B0 = m, and a constant 2 such that, for each t > 0, Var[B(t)] = 2 t 2H . In Willinger et al. (1997), this family of models was proven useful to deal with teletraffic because the superposition of a large number of independent on/off sources with heavy tailed on and/or off periods leads to an aggregated process that behaves asymptotically like an FBM. Therefore, as the teletraffic data stream is generated by a physical model that aggregates several on/off sources of traffic, FBM processes appear as appropriate theoretical models. However, a queue system with an FBM as arrival process is difficult to handle and only asymptotic results for some performance measures of the system are available, see, for instance Norros (1994) and Chapter 8 in Park and Willinger (2000). An MMPP is a doubly stochastic Poisson process. Data follow an MMPP when the packet arrivals follow an Poisson distribution with a parameter that changes over time. This change is ruled according to a Markov process with L states that are the possible rates of the Poisson process. This family of processes appears as an extension of the classical models used in telephony systems, so they can take advantage of the well-studied results about the performance in classical systems by means of an appropriate extension, see, for instance, Lucantoni (1993). On the other hand, they are SRD processes. Consequently, the higher the number of states in the Markovian process, the softer their ACF will decrease and the more similar to the ACF of an LRD process will look. This entails that these models are highly parameterized to fit properly the teletraffic data. A pioneering and illustrative example is given in Heffes and Lucantoni (1986) where an MMPP/G/1/FIFO system is used to model packetized voice and traffic data with any number of sources. In Heyman and Lucantoni (2003), some references are given to show that MMPP have performed well in applications and are able to capture the self-similar behavior of the modern teletraffic. 3. Simulation: the model and the simulator In this section we deal with all the simulation and estimation aspects associated with this work. It is divided into three subsections. In the first one, we introduce the physical model considered to generate the teletraffic data stream.
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Fig. 2. Structural model for the traffic data.
In the second one, we indicate how to manage these data to estimate the parameters of the theoretical models, that is, we explain how to adjust the referential models to the teletraffic data stream. Finally, in the last one, we present the simulator that has been created in order to implement the procedures and techniques introduced in the previous subsections. Besides this, the simulation procedures that have been used to generate data streams from the adjusted models are explained. 3.1. A physical model The arriving traffic to a network link (as an antenna in mobile telephony) results from the aggregation of the data packets sent by many individual sources, for instance, each individual source might be a user of a cell phone. Then, each user is considered as a data source which is modelled according to an on/off alternating renewal process. The on periods correspond to periods of activity. We assume, for the sake of simplicity, that the data packets are sent at a constant rate. During the off periods, or inactivity periods, no packets are transmitted. In our simulation study we suppose that each user can be classified in k different types according to their bandwidth requirements. In this way, we can distinguish among SMS, voice, videoconference, photography, etc. Fig. 2 illustrates the model when three types of sources are considered. For each source i, i = 1.3, Ni represents the number of individual sources, Rate(i) is the transmission rate and Fi (x) and Gi (x) are the on and off period distribution functions, respectively. The resulting aggregated traffic is shown at the bottom of the figure. A crucial characteristic of this physical model is that the probability distributions involved are heavy tailed. A random variable X follows a heavy tailed distribution if P [X > x] ∼ cl(x)x − , as x → ∞, where 0 < , c is a constant and l(x) is a slowly varying function. Observe that when < 2, Var(X) = ∞. The simplest, and one of the most used heavy tailed distribution, is the Pareto distribution which has a density function f (x) = k x −−1 , for x k, where and k are real positive parameters. For more properties and applications of heavy tailed distributions, see Adler et al. (1998). The credibility of the physical model presented in this subsection relies on the existence of a link between the observed statistical properties of the aggregated traffic (LRD, mainly) and the statistical behavior of each individual connection (heavy tailed distributions for file sizes and connection times, mainly). This link is well established in the specialized literature. Evidences for a heavy tailed distribution of the data file sizes are presented in Arlitt and Williamson (1996) and Crovella and Bestavros (1996). In Duffy et al. (1994) distributions with infinite variance are considered the most appropriate to model the durations of phone calls. In Willinger et al. (1997), the data of Leland et al. (1994) are analyzed and it is concluded that the on/off periods have a heavy tailed distribution with a finite mean
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and an infinite variance. In Staehle et al. (2000), a review of models for internet, e-mail and ftp is presented, most of these models are based on sources of on/off type with heavy tailed distributions. But, Theorem 1 in Taqqu et al. (1997) is a fundamental result that establishes the relation between the aggregation of on/off sources and an LRD process. In the following lines we focus on this crucial result. Theorem 1 in Taqqu et al. (1997) shows that the aggregated total load, properly normalized, of M independent and identically distributed on/off sources with heavy tailed distributed length of on and/or off periods converges, when the number of sources goes to infinity (first) and then the time scale factor, T , tends to infinity, to an FBM. Although the theorem can be applied for other probability distributions, for the sake of brevity, we assume Pareto distributions for the length of on/off periods. We denote the parameters of the distributions for the length of on/off periods, respectively, as on , kon , off , koff . Assume that the rate transmission during the on periods is 1. Let {W (m) (t) : t 0} be the renewal process associated to the mth data source where, for each t 0, 1 if source m is in an on interval in time t, W (m) (t) = 0 if source m is in an off interval in time t. Then, we have 1 1 lim lim √ T →∞ M→∞ T H M where
=
0
Tt
M
(W (m) (s) − EW (m) (s)) ds = BH (t),
m=1
⎧ ⎪ ⎪ ⎪ ⎨
(on + off )3 (4 − off )
⎪ ⎪ ⎪ ⎩
22off aon
22on aoff
if off on , (2)
otherwise, (on + off )3 (4 − on ) ⎧ 3 − off ⎪ if off on , ⎨ 2 (3) H= ⎪ ⎩ 3 − on otherwise, 2 and, being () the gamma function, ⎧ 2on kon ⎪ ⎪ ⎨ (on − 1)2 (on − 2) if on 2, aon = on ⎪ ⎪ ⎩ Kon (2 − on ) otherwise, on − 1 ⎧ 2off koff ⎪ ⎪ ⎨ (off − 1)2 (off − 2) if on 2, aoff = off ⎪ ⎪ ⎩ Koff (2 − off ) otherwise. off − 1 If we extend the previous framework to consider the case when sources of R different types share the same transmission channel, we are in the conditions of Theorem 2 in Taqqu et al. (1997) and, then, we have that the aggregated cumulative traffic behaves statistically as a superposition of R FBMs with Hurst parameters H (r) , r = 1, . . . , R. Moreover, the term with the largest H (r) dominates the fluctuations of the traffic as T → ∞. 3.2. Estimation of the parameters of the adjusted models In order to evaluate the performance of the FBM and MMPP models to capture the behavior of a series of teletraffic data, we need to estimate previously the parameters of the models. The estimation of the parameters of the MMPP model is done following the lambda algorithm proposed in Heyman and Lucantoni (2003). The lambda algorithm divides the range of variation of data in several intervals, assigning each
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Fig. 3. Lambda algorithm for the MMPP estimation.
Fig. 4. The LRD as function of parameter a in the lambda algorithm.
one of them to the parameter (the intensity) of a Poisson process. First the peak rate ∗ of the data is determined and, for a value√a fixed by the designer, the largest parameter of the Poisson processes, 1 , is obtained by solving the equation √ 1 + a 1 = ∗ . All data in the interval 1 ± a 1 are assumed to be generated from a Poisson √ process with√intensity 1 . Now, we calculate the second largest parameter 2 as the solution of the equation 1 − a 1 = 2 + a 2 . This procedure is continued until the minimum rate ∗ , or the zero value, appears in one interval. Assume that N different rates of transference 1 , 2 , . . . , N have been estimated after the application of the algorithm. Then, Fig. 3 shows how the interval where the teletraffic data spread is divided after the application of the algorithm in N subintervals. Then, the lambda algorithm assumes that the data in each subinterval have been generated by a P P (i ), i = 1, . . . , N. For t = 1, . . . , T , let t = j , j = 1, . . . , N, when the traffic at time t is in the interval of values generated by a P P (j ). Then, the transition matrix, P = (pij ), of the embedded Markov chain of the Markov process that modulates the rate change is obtained as follows: pij =
number of transitions of{t } from i to j , number of transitions of{t } out of i
i, j = 1, . . . , N.
In Fig. 4, we represent for different values of a, the ACF of the data series generated by the adjusted MMPP to an LRD data series. Observe that the smaller the value of a, the better the LRD property of the original series is captured. To adjust an FBM to a series of traffic data we have to estimate three parameters: the mean, the standard deviation and the Hurst parameter. The accumulated traffic until time t is expressed as √ B(t) = mt + aMBH (t), √ where m is the data mean, aM is the standard deviation of data and BH (t) is the standard FBM which is completely determined by the value of H . There exist several statistical methods to estimate the Hurst parameter. Some of them
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are based on the adjustment of straight lines, as the aggregated variance method, the R/S methods or the periodogram method. Or they can be based on some well-known estimation methods, as the Whittle estimator, see a deeper discussion about the statistical procedures to estimate the Hurst parameter in Leland et al. (1994). In this work, the lcost method, based on the level 0 crossings of increments of a fractional Brownian motion, has been used (see Coeurjolly, 2000). In our study we also consider the FBM obtained by applying the limit result of Taqqu et al. (1997) introduced in Section 3.1. The process X(t) describing the arrival of packet data at time t behaves as √ on X(T t) ∼ M + T H MBH (t), on + off where on and off are the expected values of the time distribution in the on/off periods, respectively, is a constant and BH (t) represents the increments of a standard FBM. Following Taqqu et al. (1997), when the on/off periods are Pareto distributed the values of H and are obtained from (3) and (2), respectively. 3.3. The simulator This subsection is devoted to explain the three distinguishable modules that compound the simulator designed for this work. In the first module, a data stream from the physical model introduced in Section 3.1 is simulated. First, the procedure establishes the sequence of times for the on/off periods (by using, for example, the inversion method in the Pareto case) for each one of the considered sources. Each type of source is characterized by its name, the number of individual sources of this type, the distribution family for the on/off periods, the values of the parameters that define both distributions and the rate of transmission in the on periods. The distribution for the on/off periods allowed are Pareto, truncated Pareto, Exponential, Inverse Gamma, Inverse Weibull and Log-Normal. The simulation starts assigning to each individual source the on state with probability p = on /(on + off ) and the off state with probability 1 − p. The parameters used to specify the simulation model as well as the full sequence of the simulated data can be saved. In the second module, an MMPP is adjusted to a stream of data with the lambda algorithm; the estimated parameters can be saved in a file. These estimated parameters, or others fixed by the user, can be applied to simulate a data stream generated by the associated MMPP. The corresponding code is designed with standard simulation procedures. In the third module, two FBMs can be adjusted to a stream of data in two different ways. The difference between the adjusted FBMs relies on the procedure followed to estimate the parameters. As stated in Section 3.2, in one case the mean and variance of the FBM are obtained from the stream of data and the Hurst parameter from the lcost method. In the other case, the limit results stated in Taqqu et al. (1997) are considered. This module also simulates a stream of data for each FBM adjusted. The literature offers several exact methods to simulate an FBM with a trade off between the length of the trajectories and the required computational complexity. We refer to the work of Bardet et al. (2003) for a survey of such methods. In this work, the method of Levinson has been used (see, for instance, Brockwell and Davis, 1991) and the code was obtained from Coeurjolly (2001). Other capabilities of the simulator are the calculus and graphical representation of the ACF of the simulated data, the calculus and graphical representation of the mean value and, also, the standard deviation and the Hurst parameter of consecutive batches of simulated data, which are useful to study the evolution as the simulation time increases. A Java code of this simulator can be obtained from the authors upon request. 4. Conclusions from simulation experiments We carried out a detailed simulation study where a high number of simulation runs were performed. For the sake of brevity, we report some outstanding conclusions and illustrate them with tables and graphics obtained from the simulation. (1) The theoretical mean traffic underestimates or overestimates the observed average traffic We generate the traffic from the aggregation of M on/off sources, with a rate of transmission r(i), i = 1, . . . , M, in on periods and expected traffic: M i=1
on (i) × r(i). on (i) + off (i)
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Fig. 5. The observed mean underestimates (left picture) or overestimates (right picture) the expected traffic.
Fig. 6. Simulated traffic for a single source.
But, from the simulation study we have that when the tail of the on period is heavier than the tail of the off period, then the observed mean is less than the expected traffic. Alternatively, when the tail of the off is heavier than the tail of the on period, the observed mean is greater than the expected traffic.
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Fig. 7. Slow approach of the data mean to expected mean when difference between parameters is big.
We illustrate this fact in Fig. 5. On the left side, we represent the teletraffic generated with the first module of the simulator when 100 Pareto distributed on/off sources are aggregated, with a transmission rate of 1 and a heavier tail for the on period distribution (on = 1.2, off = 1.5). As on = off = 6, the expected traffic is 50. However, the smoothed mean, calculated as the average of the observed data in batches of 2 × 105 time units, is always under the expected one (represented in the picture by the thick horizontal line). The opposite phenomenon is observed in the right picture, where 150 Pareto distributed sources with a heavier tail for the off period distribution have been simulated. We observe that the stream data, and its batched means, are less than the expected traffic that equals 270. The explanation for this bias comes from the underestimation of the expected traffic for the sum of observations with a heavy tailed distribution. It is known that when a random variable, say X, follows a heavy tailed distribution with index then the following central limit theorem holds: ¯ n1−1/ (X(n) − E(X)) → S , ¯ where X(n) represents the sample mean obtained from n data and S is an stable distribution, which is not symmetric and gives more probability to values lower than the expected one. It is well known that the heavier the tail of a distribution, the more skewed the limit distribution. Then, the average of observed on/off periods have high probability of being less than the expected ones and this fact is more extreme when the tail becomes heavier. This behavior can be observed in Fig. 6 where a trajectory of a single on/off source is drawn and, provided that the simulated time is long
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Fig. 8. Fast approach of the data mean to expected mean when difference between parameters is small.
enough, large periods with extremely high values for the variable appear. These periods are known as burstiness periods and its existence explains the high instability of the data stream average. The means of the on/off periods considered in Fig. 6 are 11 and 3.6, respectively. Due to the presence of burstiness periods, observed in this case after 3 million time units, the smoothed mean, calculated as the average of the observed data in batches of 2 × 105 time units, evolves as follows. Before the burstiness period does not reach the expected traffic, say 0.753, in the burstiness period increases up to a value greater than 0.9 and, after the burstiness period decreases again and takes value below the expected traffic. (2) Convergence speed toward the expected teletraffic can be very slow This statement is also a consequence of the analysis done in conclusion (1) about the central limit theorem for heavy tailed distributions. What we can state after the simulation study is that the speed of convergence of the estimated mean to the expected one depends on the difference in the tail heaviness of the on and off period distributions. In fact, if the difference between the heaviness of the tails is large and the Hurst parameter takes a high value, then the initial difference between the observed mean and the expected traffic is large too and, then, the approach happens slowly. In other words, the transitory periods can be quite large. The previous discussion is illustrated in Figs. 7–9. In all of them we draw the simulated aggregated traffic that results from 600 Pareto distributed on/off periods but with different parameter values. In the former, a big difference between the on and the off parameters is considered, the parameters chosen are on = 1.1, kon = 2, off = 1.96, koff = 30, giving an expected traffic of 158.6. Observe that the aggregated traffic takes values under the expected mean and even after 6 × 106 time units the observed mean remains under the value 140. In the second, a smaller difference between
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Fig. 9. No tendency in the data mean in the case of equal parameters.
the on and the off parameters is considered, concretely we take on = 1.25, kon = 2, off = 1.75, koff = 6, giving an expected traffic of 250. In this case we have simulated the stream up to 105 time units and the process starts closer to the expected mean and approaches it quicker. In the latter, we consider equal values for on and off . In this case, data fluctuate around the expected traffic and there is not any tendency in the smoothed mean. (3) When there exist transitory periods both families of models show a bad performance When there exists a very long transitory period, as one could expect, the adjusted FBM and MMPP models do not fit well the data because the transitory period is not captured by these models. Fig. 10 illustrates this situation. We have simulated during 1 million time units the aggregation of 5000 Pareto distributed on/off sources. The aggregated traffic is represented at the top of the figure, and, below at left the two graphics are data streams generated with the adjusted MMPP (above) and FBM (below) models to the aggregated traffic. Pictures on the right show Q–Q plots: first column taking all data and second column after deleting the half initial data. When we consider all simulated data both models provide a bad fit but when we remove the data in the transitory period (those in which we observe how the aggregated traffic is still growing) and consider the data in the stable situation the goodness of fit improves and furthermore, we observe that the adjusted FBM performs better. (4) Theoretical models may not be suitable for assessing the QoS From this simulation study we have observed some flimsy results provided by the adjusted models when compared with that obtained directly with the teletraffic data stream. The following example is quite illustrative. We consider an antenna with capacity of 60 kb/s and a buffer of 30 kb. The traffic comes from the aggregation of 100 on/off sources
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Fig. 10. Goodness of fit of FBM and MMPP with and without transitory data.
with Pareto distributed periods. The parameters chosen provide an expected traffic of 64.7 kb/s and Hurst parameter H = 0.95. In Fig. 11, the probability of buffer overflow over time in a log–log scale is represented. Surprisingly, there exists a very low probability of overflow in the short term, however, the expected input traffic is greater than the service capacity. The probability of overflow grows to 1 as the time increases but the time required is quite long. In fact, if we consider the time unit in milliseconds, after more than 2 h (8 millions of milliseconds) the probability of overflow is 0.03. We can explain this behavior from the conclusions (1) and (2) in this section. So that our simulation study gives a negative answer to the comment in Section 1.2.3 in Park and Willinger (2000), “It is unclear whether the asymptotic formulas—beyond their qualitative relevance—are also practically useful as resource provisioning and traffic engineering tools”. (5) The Hurst parameter H of the aggregated process is affected by all type of sources Theorem 2 in Willinger et al. (1997) states that when we aggregate several types of on/off sources with different Hurst parameters, the corresponding aggregated traffic behaves as the stream with highest Hurst parameter. However, we observe in this simulation study that it depends on the mix of sources and their rates. Fig. 12 shows simulation results for different portfolios of three types of on/off sources with 500 individual on/off sources each one. Each row of the table corresponds to a simulation experiment in which columns S1–S3 provide the rate of packet transmission for the types of sources 1–3, respectively. Each source has Pareto distributed on/off periods, with the parameter values indicated at the bottom of the figure. Observe that the Hurst parameters for the sources are 0.95, 0.75 and 0.65 for types 1, 2 and 3, respectively, but the estimated Hurst parameter for the aggregated data in some cases is far from the maximum one, 0.95.
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antenna buffer overflow probability for 100 aggregated ON/OFF sources Antenna capacity 60 Kb/s Buffer capacity 30 Kb ON Pareto(1.1, 1) OFF Pareto(1.5, 2). Expected traffic 64.7 Kb/s
Log-Probability
0
Time Prob. 10 4
0
5 10 4 10 10 5 10 10 6 10 2 10 6 5 10 4 10 6 0.02 8 10 6 0.03
4.0
4.5
5.0
5.5 Log-Time
6.0
6.5
7.0
Probabilities and time are in log. 10 scale
Fig. 11. Example of QoS. The expected input to the antenna is greater than the capacity of service.
S1 1 1 15 1 1 1 3
Traffic Rate S2 S3 1 1 5 15 5 1 5 20 2 2
100 1 3 1
a=1.5 3 7 8 20 11 4 4
MMPP a=1.0 4 11 12 30 16 5 6
a=0.5 7 21 23
Mean 535 6000 5937.2
St.Dev. 15.6 173.6 172.4
H 0.869 0.726 0.877
60 32 10 11
31542 5380.6 1673.8 1666.2
1095 223.5 41.3 40.8
0.667 0.818 0.835 0.843
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Fig. 12. Table showing the Hurst parameter of the traffic process when several types of sources are aggregated.
We seize this figure to observe that the number of parameters of the adjusted MMPP model (number of states in the Markov process) highly depends on the variability of the traffic rates among types of sources. The number of estimated parameters are presented for different values of the parameter a in the columns a = 1.5, and 0.5, respectively. (6) FBM reproduces the “shape” and the self-similarity of the teletraffic data better than the MMPP In Fig. 13 we picture three data streams. In the first row, the simulated teletraffic data obtained from the aggregation of on/off sources is drawn. We adjust the theoretical models with the second and third modules of the simulator, and we represent in the second row the data stream generated by the adjusted FBM and in the third row, a data stream simulated from the adjusted MMPP. If we observe the shape of the three graphics, we realize that the adjusted FBM
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performs better the aggregated teletraffic than the adjusted MMPP. Similarly, in Fig. 14 we represent in the first column the on/off aggregated traffic, in the second, the adjusted FBM and in the third column the adjusted MMPP. On the other hand, each row represents the data at different time units, in the first row we consider the data in the initial unit time, in the second data are grouped in batches of size 5 and in the third row data are grouped in batches of size 25. Once again, data of the adjusted FBM reflect self-similarity but not that obtained from the MMPP. 5. Concluding remarks In our work we have developed an useful simulation software that allows us to generate teletraffic data by means of a physical model that aggregates individual on/off sources. These data have been used to test the validity of some theoretical results as well as the ability of the FBM and MMPP models to fit the data. We have seen how the data behavior strongly depends on the tail heaviness of the on/off period distributions. The more shocking situations appear when the heaviness of the tails is higher. As we have seen, this situation can lead to obtain wrong results with the adjusted models, like those showed in conclusion (4).
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Concerning to the question of which family of methods is better, our answer should be that the FBM model is able to perform better the aggregated traffic, although not always provides a good fit, specially in those cases where exist a long transient period. But we have also showed how the FBM obtained from the theoretical limit result of Taqqu et al. (1997) may be very far from the data and then not useful for real practical studies. Similar shortage is found for the limit result about the Hurst parameter when aggregating different types of sources. About the MMPP model we have observed the same shortcoming with the transient periods. Fitting the shape of the teletraffic data by a theoretical model appears as a challenging and ambitious problem, as we have seen in this work. Future research can be focused on partial problems as comparing the main performance measures among several theoretical models and the teletraffic data stream, in the way presented in conclusion (5). In this new framework, parsimonious theoretical models allow the analytical calculation of the performance measures. However, in some cases (see conclusion (5) in Section 4), the lambda algorithm requires a high number of parameters to adjust the teletraffic data and then, the suitability of MMPP models can be unfairly reduced. So that, new strategies to adjust MMPP must be considered. Acknowledgments The authors acknowledge the suggestions made by two anonymous referees who helped to clarify the first version of this paper. The authors are also indebted to Dr. E. Morozov for interesting suggestions and discussions during his visits to Pamplona and Zaragoza (Spain). References Adler, R. J, Feldman, R.E., Taqqu, M.S. (Eds.) 1998. A Practical Guide to Heavy Tails. Statistical Techniques and Applications. Birkhäuser, Basel. Arlitt, M.F., Williamson, C.L., 1996. Web server workload characterization: the search for invariants, In: Proceedings of the ACM SIGMETRICS, vol. 96, pp. 126–137. Bardet, J.M., Lang, G., Oppenheim, G., Philippe, A., Taqqu, M.S., 2003. Generators of Long-Range Dependent Processes: A Survey. In: Theory and Applications of Long Range Dependence. Birkhäuser, Basel, pp. 579–623. Brockwell, P.J., Davis, R.A., 1991. Time Series: Theory and Methods. second ed.. Springer, New York. Coeurjolly, 2001. http://www.jstatsoft.org/v05/i07/ . Cox, D.R., 1984. Long-range dependence: a review. In: David, H.A., David, H.T. (Eds.), Statistics: An Appraisal. Iowa State University Press, Ames, IA, pp. 55–74. Crovella, M., Bestavros,A., 1996. Self-similarity in World Wide Web traffic: evidence and possible causes. In: Proceedings of theACM SIGMETRICS, vol. 96, pp. 151–160. Duffy, D.E., McIntosh, A.A., Rosenstein, M., Willinger, W., 1994. Statistical analysis of CCSN/SS7 traffic data from working CCS subnetworks. IEEE J. Select. Areas Comm. 12 (3), 544–551. Heffes, H., Lucantoni, D.M., 1986. A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance. IEEE J. Select. Areas Comm. SAC-4 (6), 856–868. Heyman, D.P., Lucantoni, D., 2003. Modelling multiple IP traffic streams with rate limits. IEEE/ACM transactions on networking. 11 (6), 948–958. Leland, W.E., Taqqu, M.S., Willinger, W., Wilson, D.V., 1994. On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Trans. Networking 2, 1–15. Lucantoni, D., 1993. The BMAP/G/1 queue: a tutorial In: Donatiello, L., Nelson, R., (Eds.), Performance Evaluation of Computer and Communication Systems. Springer, Berlin, pp. 330–358. Norros, I., 1994. A storage model with self-similar input. Queueing Systems 16, 387–396. Park, K., Willinger, W., 2000. Self-Similar Network Traffic and Performance Evaluation. Wiley, New York. Staehle, D., Leibnitz, K.Y., Tran-Gia, P., 2000. Source traffic modelling of wireless applications. Technical Report, Institute of Computer Science, University of Würzburg, Germany. Taqqu, M.S., Willinger, W., Sherman, R., 1997. Proof of a fundamental result in self-similar traffic modeling. Comput. Comm. Rev. 27 (2), 5–23. Willinger, W., Taqqu, M., Sherman, R., Wilson, D., 1997. Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level. IEEE/ACM Trans. Networking 5 (1), 71–86.