Real{Time Generation of Fractal ATM. Trac: Model, Algorithm, and. Implementation. Department of Electrical Engineering and

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1 Real{Time Generation of Fractal ATM Trac: Model, Algorithm, and Implementation Bong K. Ryu Mahesan Nandikesan Department of Electrical Engineering and Center for Telecommunications Research Columbia University New York, NY 127 Tel: (212) , Fax: (212) Abstract We design and implement a fractal trac generator module as a part of the realtime trac generation and monitoring system built at Columbia University. Fast generation of diverse types of synthetic fractal trac has been considered dicult to achieve with traditional fractal models. This fractal module employs a fractal trac model based on the superposition of i.i.d. fractal renewal point processes (Sup-FRP), a generation algorithm, and several approximation techniques used to meet the realtime constraint. This module can represent a trac source, real-time or not, exhibiting fractal behavior (such as long-range dependence) over a wide range of time scales, with tunable statistics such as mean, variance, and the Hurst parameter. The capability of the fractal module is fully analyzed by simulations and experiments over a broad range of model parameters. Numerical results show that (i) this generator produces fractal ATM cell streams at very high bit-rates (1-25 Mbps) in real time; and (ii) for a wide range of model parameters, the generator produces cell streams with desirable fractal properties and other statistics. Technical Subject Category: Modelling and Simulation Techniques 1

2 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 2 1 Introduction Unlike data networks, future multimedia networks running real-time video and voice applications would require quality of service (QOS) guarantees. Virtually all research in networking is now driven by this requirement. However, apart from theoretical and simulation results, little understanding has been gained on the impact of concurrent network load on QOS of individual connections in a real ATM network. We nd that real-time measurements of QOS taken from operational networks are fundamental for understanding and building robust networks with ecient QOS guarantees. In order to collect meaningful data, rst we need to be able to have some control over the statistical behavior of input trac and network load. Trac from actual sources (such as a camera or microphone) is neither exible enough to gain sucient data nor manipulative in terms of trac statistics. Thus, we need to supplement real trac with emulated trac. Such trac is generated using stochastic models such as the one we describe in this paper. Using these models, we can study the network response to changes in trac patterns. Therefore, one of the major challenges is to develop an advanced trac generation capability that can be used to study the QOS variations of real-time trac carried over integrated broadband networks. In the Center for Telecommunications Research at Columbia University, we have built a real-time trac generation and monitoring system on a Hewlett-Packard (HP) Broadband Series Test System (BSTS). This system permits real-time generation of ATM cell streams according to the statistics given to the comprising generator modules. Fractal behavior, better known through the self-similar nature of Ethernet trac [6] and long-range dependence of VBR video trac [1, 4], is a salient characteristic of real-world trac. As a result, incorporating a fast, accurate, and exible fractal trac generator into the BSTS has become urgent. To date, generating such type of trac has been limited, however, since (i) how individual arrivals give rise to fractal trac has not been fully understood until recently [14]; and (ii) currently used fractal models often require excessive amount of computational resources [4, 16]. Three requirements have been considered in selecting a fractal trac model for the BSTS. The rst, and perhaps most fundamental requirement is the generation speed; the fractal model must be suciently simple so that it is able to generate arrivals at the rates suitable for broadband application trac streams exhibiting fractal properties (usually several Mbps on the average). The second requirement which must be simultaneously considered is the accuracy; statistical properties of output trac stream must closely follow desired fractal

3 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 3 behavior. The nal requirement is exibility; the model must be able to represent diverse fractal behavior which can be readily controlled by tunable parameters. As mentioned earlier, currently known self-similar models such as F-ARIMA (fractional autoregressive integrated moving average) processes [6] and Fractional Gaussian Noise processes [4] are generally computationally expensive, leading to a large amount of time for generation. New algorithms which permit fast generation of self-similar trac streams such as Random Midpoint Displacement algorithm [5] and Fast Fourier Transform method [1] have been excluded from consideration since those methods are limited to generating only a single type of fractal trac: Fractional Gaussian Noise. In [14], Ryu and Lowen have proposed fractal point processes as novel tools for modeling and analyzing various types of self-similar network trac. It has been found that the proposed models permit tunable statistics such as the Hurst parameter (degree of self-similarity), mean and variance of the arrival rate. Moreover, they permit fast generation of fractal time series on a modern workstation while exhibiting desired fractal behavior. Therefore, all of them are suitable for the BSTS. After considering the tradeo between speed and exibility, the Superposition of Fractal Renewal Processes (Sup-FRP) model has been selected as our fractal trac generator module for implementation on the BSTS. In this paper, we present an algorithm for generating cell arrival times of the Sup-FRP model. Then, we discuss several approximation techniques taken for expediting its generation speed on the BSTS. Numerical results show that the degradation of accuracy due to the approximations is negligible over a wide range of model parameters. This paper is organized as follows. Section 2 provides a brief description of the real-time trac generation and QOS monitoring system based on the BSTS. Section 3 provides the denition and statistical properties of the Sup-FRP model. Section 4 presents a generation algorithm based on the Sup-FRP model and approximation techniques. Section 5 presents numerical results. Finally, conclusions are drawn in the Section 6. 2 The Trac Generation and QOS Monitoring System At Columbia University, we have built a real{time trac generation and monitoring system on the BSTS. This system is capable of generating cell{level trac continuously in real{ time according to the statistics of any given trac model. Currently implemented trac generator modules in the system are Poisson process, two-state Markov modulated Poisson process, Multi-TES process [12], and this Sup-FRP model. The trac generator requests the

4 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 4 individual module for interarrival times and then schedules out cells on an ATM line at the appropriate epochs (Fig. 1). The maximum bit-rate attainable depends on the complexity Module 1 Module 2 MUX Transmitter Module N Figure 1: The block diagram of the real-time generator on the BSTS. of the individual module, which in turn depends on the underlying trac model. In every generated cell, a timestamp and sequence number are inserted before departure. These cells are then sent into an ATM network and looped back to the BSTS. Fig. 2 depicts this in a simplied diagram. On arrival at the BSTS, the cells are timestamped again. Each received cell thus has two timestamps and a sequence number. From this, QOS parameters such as delay, jitter and loss are computed in real{time. control & visualization traffic generation & monitoring loopback workstation CPP switch network (NYNET) Figure 2: Overview of the trac generation and QOS monitoring system implemented in an experimental ATM network (NYNET). The CPP (cell protocol processor) resides inside the BSTS.

5 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 5 The advantage of the BSTS over a regular workstation is that it can guarantee real{time delivery of cells. Moreover, its timestamp resolution of 1ns far exceeds that achievable on a workstation. The usefulness of this system rests in its ability to provide real{time visualization of the eect of changing various trac parameters. Thus, by turning a set of knobs, a user can play with the trac statistics and study the eect on the QOS provided to various streams. Experiments along these lines are being conducted at Columbia University. The results of these experiments, performed on NYNET 1, will be published in another paper. 3 The Sup-FRP Model 3.1 Statistical properties of fractal trac Recent measurement studies reveal that Ethernet and WAN trac exhibits sharply dierent statistical behavior from that of traditional trac models such as Poisson model [6, 11]. In those studies, it is argued that these types of trac are surprisingly well explained by the concept of self-similarity (fractal behavior), a term rst coined by Mandelbrot [9]. Formally, a wide sense stationary (WSS) process X = fx n ; n 2 Z + g is called asymptotically (secondorder) self-similar if X possesses a non-degenerate correlation structure, i.e., aggregating the original process X over larger and larger intervals does not change its correlation structure [2, 6]. R t Denote by dn(t) a point process and by N(t) = dn(s) the corresponding counting process, i.e., the number of arrivals up to time t. We then dene time series X = fx n ; n 2 Z + g as X n N[nT s ]? N[(n? 1)T s ]; (1) the number of arrivals during the n-th interval of duration T s. Thus the point process dn(t) serves as an underlying process for constructing X. We assume that the process X is widesense stationary (WSS) with mean E[X n ], variance 2 Var[X n ], and (normalized) autocorrelation function (ACF) r(k; T s ) Cov(X n ; X n+k )= 2. Denote by F (T ) the index of dispersion for counts (IDC), dened as F (T ) Var[N(T )]=E[N(T )], and dene S(f) as the power spectral density (PSD) of the point process dn(t). 2 1 An experimental wide-area network run by NYNEX. 2 We dene S(f) as the Fourier transform of the Coincidence Rate of a point process dn(t), not the Fourier transform of the ACF r(k; T s ) to capture the spectral behavior of dn(t) over all time scales and not only those longer than T s, and to be consistent with the analysis given in [14]. Both forms of PSDs are closely related by ltering and sampling operations and are virtually indistinguishable for time scales much larger than T s.

6 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 6 If a WSS process X is asymptotically second-order self-similar, then X exhibits at least the following three characteristics over a wider range of time or frequency scales: 3 F (T ) a 1 T (slowly decaying variance), r(k; T s ) a 2 k?(1?) (long-range dependence), (2) S(f) a 3 f? (1=f noise), for < < 1 4 and some positive constants a i ; i = 1; 2; 3 [3, 8, 14, 15]. The constant is called the fractal exponent and related to the Hurst parameter H by = 2H? 1. The relations in (2) are all of power-law form, since scaling and fractals are closely related [14, references therein]. If the process X is constructed from an FPP model, then X is asymptotically second-order self-similar, since F (T ) = 1 + (T=T ) (T ), T r(k; T s h +1i s ) = 1 (k + 1) +1? 2k +1 + (k? 1) (k = 1; 2; : : :), (3) T s + T 2 S(f) = [1 + (f=f )? ] (f > ), where T and f are positive constants, and E[N(T )]=T is the average rate [14]. In the next section, we describe one such fractal point process, the Sup-FRP model, in more detail. 3.2 Denition of the Sup-FRP model The Sup-FRP model is dened as the superposition of M independent and probabilistically identical fractal renewal processes (FRPs) [7]. Therefore, constructing the Sup-FRP model is quite simple as schematized in Fig. 3. Since each FRP stream is a renewal point process, the Sup-FRP model is completely characterized by M and the common interarrival probability density function (pdf) p(). We use the following pdf [14] p() =( A?1 e? =A for A, e? A?(+1) for > A, (4) where = 2?. The parameter A serves as a threshold between exponential behavior and power-law behavior of interarrival times. Indeed, (4) indicates that the source of fractal behavior of the Sup-FRP model is the very large interarrival times spanning several orders of magnitude with signicant probability due to the \heavy tail" of p(). As a result, the Sup-FRP model exhibits fractal behavior over the range A T [14]. Thus this model can 3 It is easy to show that the variance-time plot [6] and the IDC plot are identical. 4 Values of in excess of unity are possible, although of these statistics only S(f) scales for > 1.

7 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 7 represent a trac source, real-time or not, exhibiting fractal behavior (such as long-range dependence) over the range A T with tunable statistics such as the mean, variance, and Hurst parameter. Note that the renewal property is lost as a result of superposition. FRP (1) τ i (1) FRP (2) τ i (2) FRP (3) τ i (3) Sup-FRP τ i Figure 3: A realization of the Sup-FRP point process dn(t) with M = 3. For each j = 1; 2; : : : ; M, (j) i represents the i-th interarrival time of j-th FRP stream drawn from the common pdf p(). 3.3 Statistics of the Sup-FRP model The Sup-FRP model has three parameters, ( < < 1), A, and M. If the WSS process X dened in (1) is constructed from the Sup-FRP model, then its Hurst parameter H, mean, and variance 2 are given by H = ( + 1)=2; = E[X n ] = E[N(T s )] = T s ; 2 = Var[X n ] = F (T s ) = (1 + (T s =T )) T s ; (5) where [7] = 2? ; = M [1 + (? 1)?1 e? ]?1 A?1 ; T = 2?1?2 (? 1)?1 (2? )(3? )e? [1 + (? 1)e ] 2 A : The Hurst parameter H species the degree of long-range dependence (1=2 < H < 1), the average arrival rate of the (stationary) Sup-FRP process, and T the fractal onset time [14]. Since M can take only positive integer values, arbitrary values of H,, and T (or equivalently, H,, and 2 ) cannot be achieved exactly. In practice, we specify and H, and adjust M to approximate the desired T (or 2 ) from (6). The integer M is of the order T ; for most trac data this quantity greatly exceeds unity, so that the nite resolution (6)

8 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 8 of M does not pose a signicant problem. Once T determined by (3). and are given, the ACF r(k; T s ) is The parameter M controls the burstiness of the Sup-FRP model; for xed and H, trac generated with smaller M exhibits burstier behavior, and therefore larger variance; see Fig Fractal Trac Generator Module In this section, we provide an algorithm to build a fractal trac generator module based on the stationary Sup-FRP model described earlier. Then, the module is designed by applying several approximations to the algorithm: integer, linear interpolation, and table lookup. These approximations are discussed below in more detail. 4.1 The Algorithm Denote by S (j) the elapsed time of the j-th FRP stream, i.e., S (j) = (j) + (j) (j) k for some k and j = 1; 2; : : : ; M. To generate the stationary Sup-FRP, each (j) is generated from the equilibrium distribution (or forward recurrence time distribution) G(t) given by Z 1 G(t) Z t = p()ddu u ( [1 + (? 1)?1 e? ]?1 (1? e?t=a ) for t A, 1? [1 + (? 1)e ]?1 (t=a) 1? for t > A. (7) where p() and are from (4) and (3). Let U be an i.i.d. random variable uniformly distributed over the unit interval [,1). For each j, set G( (j) ) = 1? U and then dene V 1 + (? 1)e U: (8) This yields (j) =(??1 A ln [U(V? 1)(V? U)?1 ] for V 1, AV 1=(1?) for V < 1. (9) Similarly for the interarrival times (j) i, set P ( (j) i ) Z (j) i p(u)du = 1? U; which yields (j) i =(??1 A ln[u] for U e?, e?1 AU?1= for U < e?. (1)

9 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 9 Let S represent a simulation clock and S (j) represent the elapsed time of the FRP stream j. Then, the interarrival times j are generated by the following algorithm: 1. Choose the values of H,, and M and determine the model parameters. 2. For each j = 1; 2; : : : ; M, generate (j) from (8) and (9) and set S (j) = (j). 3. Find j such that j = argmin j fs (j) g. 4. Output = S (j ). 5. Advance the simulation clock: S S (j ). 6. Generate a new interarrival time from (1) and S (j ) S (j ) Find a new j such that j = argmin j fs (j) g. 8. Output i = S (j )? S. 9. Advance the simulation clock: S S (j ). 1. Repeat the steps 6 { 9 until the specied number of arrivals are generated. 4.2 Approximations The algorithm given above indicates that much computing resource is devoted for evaluating (8) { (1), which involve computationally expensive functions such as logarithmic, exponential, and power functions. We have sought algorithms to eliminate such operations while maintaining the accuracy of the output. Having carefully examined the above algorithm, we devised approximations which enable the BSTS to maximize its capacity for real-time delivery of scheduled cells. This section explains how the fractal trac module benets from these approximations Integer approximation The most obvious way to enhance the generation speed is to change oating-point based operations into integer based ones since integer-based operations are faster. In our fractal trac generator module, all the dimensionless oating-point variables and constants (e.g., H) are scaled to integers, whenever necessary, by multiplying an appropriate constant C and taking only the integer part. The scaling factor C is of the form 2 m, m 2 Z +. Clearly, the larger the scaling factor, the smaller the truncation error. What we must ensure is that the

10 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 1 intermediate values during the operations of multiplication or division should not exceed the largest integer representable by the system. To illustrate this, let a; b 2 R + and assume we want to compute r = ab. In the module, what is computed is r C = [ac][bc]=c [(ab)c] where r C is a scaled value of r representing rc and [x] represents the nearest integer from x.. If we use C = 2 16 and ab > 1, then the intermediate number (ab)c 2 would exceed the maximum size, resulting in undesired and fatal truncation (from the left) since our system has a 32-bit word length. As a result, r C would not represent rc correctly. By keeping track of each operation and using a proper choice of C depending on the range of each variable, this problem is readily resolved. The output i will be appropriately scaled down to ensure the proper time unit Interpolation and Table approximations We approximate the logarithmic and exponential functions [ln() and exp()] used in (9) and (1) by a combination of interpolation and table lookup. 5 Let us rst determine the range of possible values that the argument of ln() takes on in our algorithm by closely examining (9) and (1). For V 1 and U > e?, it is easy to see that both arguments, U(V? 1)(V? U)?1 and U, appearing in (9) and (1), lie between zero and unity since 1 < < 2 and < U < 1. For V < 1 and U < e?, (9) and (1) reduce to (j) = AV 1=(1?) = Ae 1=(?1)(? ln V ) ; (11) (j) i = e?1 AU?1= = e?1 Ae 1=(? ln U ) : (12) Thus, in all cases we need to evaluate ln() only in the range (; 1). We approximate ln() as follows. First, the unit interval (; 1) is divided equally into 2 n subintervals where n is a positive integer. We call n the quantization parameter. As will be shown later, larger values of n result in greater accuracy at the expense of correspondingly larger table size (more memory) since this parameter determines the size of the table for approximating the exponential function. We consider n = 8; 12; 16 and study the tradeo between accuracy and table size. Let x j = j=2 n, j = 1; 2; 3; : : : ; 2 n. For any x 2 (; 1], there exists a corresponding j such that 2 n x j after the integer approximation discussed earlier. Let r 2 f1; 2; 3; : : : ; ng such that 2?r x j < 2?(r?1). Then, using a simple linear interpolation technique,? ln x is approximated as? ln x? ln x j r? (j 2?(n?r)? 1): (13) 5 Note that the power function can be expressed by a combination of the log and exp, i.e., a x = e x ln a.

11 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 11 Fig. 4 schematizes this approximation. Since lim x! +(? ln x) = 1, we choose the maximum value of ln(), L max as L max? ln(x 1 =2) = (n + 1) ln 2. r r - (j/ 2 (n-r) - 1) r log 2 x 2 -r j/2 n 2 -(r-1) 1 x Figure 4: Approximation of the log function based on linear interpolation. The gure was drawn using the log base 2 to show the approximation more clearly. Larger values of n provide greater accuracy. Unlike ln(), we use a pre-computed table for approximating exp() since the required table size turns out to be relatively small. But there exists a key challenge in this approximation; (11) implies that the argument of exp() is indeed [; 1) since 1 < < 2! This appears problematic since the range of exp() is also unbounded. However, based on the fact that only the rst arrival ( (j) ) of each FRP stream is generated from (9), we resolve this problem by using (12) to limit the exp table. Let min be the minimum value of that can be allowed in the generator. For example, min = 1:1 results in max = :99 (H max = :995). Denote by x max the maximum possible argument of exp() and let E max e xmax, the corresponding maximum value of the table for exp(). From (12), we have x max =?1 minl max =?1 min(n + 1) ln 2; (14) and therefore E max = e xmax = e?1 min (n+1) ln 2 : (15) Denote by max the maximum interarrival time of individual FRP streams that can be generated by (12). Substituting (15) into (12) yields max = e?1 AE max = Ae?1 min (n+1) ln 2?1 : (16) We use this value whenever (j) generated by (11) exceeds it.

12 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 12 Indeed, the value max upper-limits the fractal behavior of the fractal module since no interarrival times larger than this value can be produced. As a result, the fractal module exhibits fractal behavior over the range A T max. The normalized value log 1 [ max =A] = (?1 min (n + 1) ln 2? 1) log 1 e (17) provides information on the range of time scales over which one can observe the fractal behavior from the arrivals generated by the module. For this reason, we call (17) the Normalized Range of Time Scales (NRTS). In practice, it is common to see that real-world trac shows fractal behavior over several orders of time scale, usually NRTS 3 [6, 13]. Therefore, one must choose suciently large n to obtain large NRTS to generate realistic fractal trac. n L max x max E max NRTS Memory (Kbytes) , , Table 1: Required memory size for storing the table for exp() for dierent values of the quantization parameter n. The corresponding NRTS is included. min = 1:1 and the size of the each table element is 4 byte. We make the table for exp() as follows: First, we divide each unit subinterval in [; x max ] equally with the length of 2?8, requiring about x max Kbytes of the memory. Table 1 shows the maximum values (L max and E max ) of the two functions after approximations for dierent values of the quantization parameter n. It also includes the corresponding memory size required to store the table and the corresponding NRTS given by (17). It shows that the table size increases linearly with n, and approximately 12 Kbytes (n = 16) of memory suces to observe fractal behavior over about 3 to 4 orders of time scales. 5 Numerical Results Before implementing the fractal module on the BSTS, we examine its capability in terms of speed and accuracy. First, we study how the model parameters aect the generation speed. Second, we investigate the accuracy of the module in terms of rst-and second-order statistics. Finally, we implement the module into the BSTS and present results which show that cells generated from the fractal module on the BSTS exhibit the desired fractal behavior.

13 Real{Time Generation of Fractal ATM Trac: B. K. Ryu Generation speed In general, the generation speed of the fractal module is aected by M and. Recall that for each and every new interarrival time, the algorithm given in x4.1 requires nding the F RP (j ) stream such that j = argmin j fs (j) g which would produce the next output = S (j )? S. Since, on the average, it requires about M=2 comparisons to nd the minimizer j, it would take more time to generate the same number of arrivals with larger M. Recall also that generating i (j) > A requires more operations than generating i (j) A. For example, interarrival times produced by (12) [ i (j) > A] need to perform linear interpolation and to access the table whereas those produced by exponential distribution (the rst line of (9) and (1) each) need to perform the interpolation only. Therefore, it will take less time to generate the same number of arrivals with the higher average rate (with other parameters held xed) since the number of arrivals generated by (12) are likely to be smaller with the higher average rate. 6 Parameters Gen. Time (sec) Gen. Rate (Mbps) H M w/o appr. w/ appr. w/o appr. w/ appr , ( 4 Kbps) , ( 4 Kbps) , ( 4 Mbps) , ( 4 Mbps) Table 2: Eect of M and on the generation speed and the speed improvement of our fractal module due to the approximations. The mean CPU times are obtained by using the UNIX time command and averaging 1 iterations for each case of producing cells. The variation around the mean value was within 1% and the 53-byte cell size was assumed. Table 2 shows how M and aect the generation speed. The mean CPU times taken to produce cells are obtained by using the UNIX time command and averaging 1 iterations for each case. It also includes the speed improvement of our fractal module due to the approximations applied. We set H = :9 and vary M and such that the dierence in the generation speed for each case becomes noticeable. Three observations are made from this table: 1. The fractal trac generator based on the Sup-FRP model not only exceeds the target rate (specied by ) but also achieves very high generation speeds in real time, ranging 1-25 Mbps on the average. 6 When M and H are xed, the change of results in the change of A by (6). However, the eect of such change on the number of arrivals generated by the power-law function is negligible due to the (right) \heavy" tail of the pdf p() in (4).

14 Real{Time Generation of Fractal ATM Trac: B. K. Ryu The approximations applied for the fractal module result in signicant speed improvement (15{25%). 3. The generation speed is reduced by increasing M and decreasing. Note that for xed M and, the eect of H on the generation speed is negligible. 5.2 Accuracy As mentioned earlier, the accuracy of the fractal module is aected by the approximations (integer, interpolation, and table lookup). Therefore, the arrivals produced by the module are subject to the quantization error caused by the approximations. We characterize this error by comparing several statistics of the time series produced by the fractal module to those produced by another module which employs no approximations sample path by sample path. By sample-path-by-sample-path comparison we mean that each generation is performed using the same initial seed number so that both modules use the same set of uniformly distributed i.i.d. random numbers for each experiment. Table 3 shows the parameters used to conduct this sample-path-by-sample-path comparison. Note that max limits the maximum time scale at which fractal behavior disappears. Therefore, the IDC F (T ) will saturate (at) as T! max and the ACF r(k; T s ) will drop to zero as kt s! max. (cells/sec) H M (slots) A (slots) max (1 6 slots) 5, , , Table 3: Parameter specication. represents the analytical value of the mean inter-cell time in units of slots. max is computed using the quantization parameter n = 16. M = 1 M = 5 M = 1 original real-time original real-time original real-time Table 4: Comparison of the mean inter-cell time in unit of slots based on 2 replications. ( = 46:3) Fig. 5 shows the sample path behavior of the Sup-FRP model produced by both modules using the parameters given in Table 3. While the overall behavior matches closely, a closer look reveals that the detailed behavior diers fairly. This implies that the quantization error is small enough so that overall statistical behavior is not aected by the approximations. This

15 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 15 M = 1, without approximations M = 1, with approximations Count 5 Count Bin M = 5, without approximations Bin M = 5, with approximations Count 5 Count Bin M = 1, without approximations Bin M = 1, with approximations Count 5 Count Bin Bin Figure 5: Visual comparison of the sample paths produced by the original and real-time generators with = 5 cells/sec, H = :9, T s = 1 slots (bin size), and M = 1; 5; 1.

16 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 16 is reected by Fig. 6 where the IDC and ACF curves are compared sample-path-by-samplepath using 9 dierent seeds. For each replication (sample path), 1 6 arrivals are generated, the IDC F (T ) is estimated over three orders of time scale (T min = 1 2 T 1 5 = T max slots), and the ACF r(k; T s ) is estimated at the time scale (bin size) T s = 1 slots = 4.32 msec. Fig. 7 supports this where the IDC and ACF curves are plotted by averaging the results based on 2 replications with 2 dierent seeds. The case of M = 1 shows, however, that the IDC of the fractal module with approximations begins to fall down near T = T max. This is attributed to the fact that max is not suciently larger than T max ; from the Table 3, we have max < 1T max. As a result, the ACF curve of the fractal module with approximations is below that of the other. Therefore, in this case the discrepancy between the original and approximated modules is mainly caused by the limitation of the table size, rather than by the quantization error. On the other hand, both M = 5; 1 show that the error caused by the limitation of the table size disappears as we have suciently large max compared to T max ( max 5T max ); see Table 3. As a result, the IDC and ACF curves of both modules exhibit close behavior, implying that the quantization error is negligible. Therefore, we suggest that parameters of the Sup-FRP model be carefully chosen such that max T max. By doing so, the fractal module for the BSTS can produce arrivals with desirable fractal properties with negligible error. Finally, Table 4 shows that indeed the average inter-cell times of both modules closely follow the desirable value. All of these results convincingly show that the fractal module designed is able to accurately emulate fractal trac at the very high rates. -

17 Real{Time Generation of Fractal ATM Trac: B. K. Ryu (a) IDC (b) ACF Figure 6: Sample-path-by-sample-path comparison of the IDC and ACF for M = 5 using 9 dierent seeds.

18 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 18 M = 1 w approximations M = 5 w approximations M = 1 w approximations M = 1 M = 5 M = 1 r(k) w approximations r(k) w approximations r(k) w approximations Figure 7: Comparison of the IDC and ACF curves averaged over 2 replications for M = 1; 5; 1 with = 5 cells/sec and H = :9. r(k) = r(k; T s ) with T s = 1 slots (bin size). 5.3 Experiments on the BSTS We nally implement and test the fractal module on the BSTS and present results. Fig. 8 compares the sample path behavior of fractal and Poisson trac generated on the BSTS with the same average rate. Clearly, the fractal trac exhibits much burstier behavior. The IDC and ACF curves of both types of trac are compared in Fig. 9. These statistics are based on about 13, cells (several seconds long) captured as described in Section 2. Even with this small number of data, the trac emulated by the fractal module exhibits the desired fractal behavior over a wide range of time scales; it is easily distinguished from Poisson behavior. During the entire generation period (about 1 hour), no cell was dropped due to the failure to transmit cells at the scheduled time, verifying that real-time generation of fractal trac at the very high rates on the BSTS is successful.

19 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 19 Fractal (Sup-FRP) Poisson Arrivals per time unit Arrivals per time unit time (sec) time (sec) Figure 8: Comparison of Fractal trac and Poisson trac generated on the the BSTS. The time unit (bin size) is 1ms. Both models have the same average rate = 2; (cells/sec) 8 Mbps. For fractal trac, H = :9 and M = 5. IDC ACF Fractal (Sup-FRP) Poisson r(k) Fractal (Sup-FRP) Poisson T (usec) 5 1 Figure 9: IDC and ACF curves of fractal and Poisson trac based on about 13, arrivals for each model generated on the BSTS. The ACF r(k) was obtained using the bin size 1 ms. Both generators have the same average rate.

20 Real{Time Generation of Fractal ATM Trac: B. K. Ryu 2 6 Conclusions This work presented a trac model (Sup-FRP), a generation algorithm, and several approximation techniques for building a real-time fractal trac generator module in an ATM environment. Such a generator is essential for assessing useful information on the QOS variations of real fractal trac under diverse scenarios of network load. The capability of this generator was fully analyzed by simulations and experiments on the BSTS over a broad range of model parameters. Results show that (i) this generator produces fractal cell arrivals with the rate ranging 1-25 Mbps in real time; (ii) the mean, variance, and Hurst parameter can be independently controlled; and (iii) for a wide range of model parameters, the generator produces cell arrivals with desirable fractal properties and the average rate. This work provides evidence that generating synthetic fractal trac can be done not only in a computationally ecient way but also in real time. As a result, it opens a way to conduct QOS-monitoring experiments for estimating QOS variations of a fractal trac source under various network load scenarios. Acknowledgement The work reported here was funded in part by Hewlett-Packard. We are grateful to Prof. Aurel Lazar and Dr. Giovanni Pacici for their comments and suggestions, and Brian Smith, David Smith and Dr. Nicholas Malcolm of Hewlett-Packard for helping us with the BSTS. References [1] J. Beran, R. Sherman, M. S. Taqqu, and W. Willinger. Long-range dependence in variable-bit-rate video trac. IEEE Trans. Comm., 43:1566{1579, [2] D. R. Cox. Long-range dependence: A review. In H. A. David and H. T. Davis, editors, Statistics: An Appraisal, pages 55{74. The Iowa State University Press, Ames, Iowa, [3] A. Erramilli and W. Willinger. Fractal properties in packet trac measurements. In Proc. St. Petersburg Regional ITC Seminar, [4] M. W. Garrett and W. Willinger. Analysis, modeling and generation of self-similar VBR video trac. In Proc. ACM SIGCOMM '94, London, 1994.

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