Point Process Approaches to the Modeling. and Analysis of Self-Similar Trac { Center for Telecommunications Research

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1 Proc. IEEE INFOCOM '96, San Francisco, CA, March 996. Point Process Approaches to the Modeling and Analysis of Self-Similar Trac { Part I: Model Construction Bong K. Ryu Steven B. Lowen Department of Electrical Engineering and Center for Telecommunications Research Columbia University, New York, NY 27 fryu,steve6g@ctr.columbia.edu Abstract We propose four fractal point processes (FPPs) as novel approaches to modeling and analyzing various types of self-similar trac: the fractal renewal process (FRP), the superposition of several fractal renewal processes (Sup-FRP), the fractal-shot-noise-driven Poisson process (FSNDP), and the fractal-binomialnoise-driven Poisson process (FBNDP). These models fall into two classes depending on their construction. Study of these models provides a thorough understanding of how self-similarity arises in computer network trac. We nd that (i) all these models are (second-order) self-similar in nature; (ii) the Hurst parameter alone does not fully capture the burstiness of a typical self-similar process; (iii) the heavy-tailed property is not a necessary condition to yield selfsimilarity; and (iv) these models permit parsimonious modeling (using only 2{5 parameters) and fast simulation. Simulation veries that these models exhibit fractal behavior over a wide range of time scales. Introduction It is expected that future high-speed networks will support a wide variety of services that exhibit extremely diverse trac characteristics. Characterizing the statistical behavior of a trac source is crucial to the proper design of high-speed networks, ensuring that they provide the pre-negotiated quality of service (QOS) to users while achieving high network utilization. Traditional trac models, most of which assume Markov characteristics, have been used extensively; in many cases they prove adequate for evaluating network performance (see, for example, [3, 5] and references therein). However, traditional Markov-based models often prove inadequate to the task of eciently characterizing the diverse trac now encountered over current teletrac networks [, 7, 8, 2, 22]. Recent analyses of high-quality trac measurements have revealed the prevalence of self-similarity (or long-range dependence) in data [7] and compressed video []. Despite the signicant attention self-similarity has received recently, a clear picture of its nature and importance has not yet emerged [3, 4, 6, 7, 7, 9]. Different manifestations and instances of self-similarity yield dierent results. This appears to stem from the nature of self-similar processes themselves. This study focuses on explaining the dynamics of individual arrivals giving rise to (second-order) selfsimilarity, using a point processes formulation. Thus, the fractal point processes (FPPs) introduced in this paper reveals how self-similarity arises from individual packet arrivals in the packet-switching networks. They also enjoy the properties of parsimonious modeling and fast simulation, and have direct applications in modeling Ethernet trac using TCP/IP (data [8] and video [2]) as well as long-range dependent VBR video trac over ATM [9]. We begin by introducing the family of fractal point processes (FPPs), whose increments processes are (second-order) self-similar. These models naturally fall into two classes, one based on renewal processes, and the other on doubly stochastic Poisson processes (DSPPs). Next, we construct and analyze four FPP models: the fractal renewal process (FRP), the superposition of several fractal renewal processes (Sup- FRP), the fractal-shot-noise-driven Poisson process (FSNDP), and the fractal-binomial-noise-driven Poisson process (FBNDP). Finally, we present simulation results for all FPPs, including comparison with theory. 2 Fractal Point Processes (FPPs) 2. Denition We call a point process fractal when a number of the relevant statistics exhibit scaling with related scaling exponents, indicating that the represented phenomenon contains clusters of points over all (or a relatively large set of) time or length scales [4, 5]. Such scaling leads mathematically to power-law dependencies in the scaled quantities [, 4]. Thus fractals and power-law forms of their statistics are closely related. Each statistic which scales will therefore provide an exponent; for a (mono-) fractal process all are simply related, yielding a single fractal exponent for the process. For a general point process, fractal scaling in one statistic does not necessarily imply fractal scaling in other statistics; if scaling exists in only one statistic, then we do not call this process fractal [, 4].

2 All the FPP models we describe in this paper are stationary, and we therefore consider only stationary point processes in the following analysis. 2.2 Statistical measures Important second-order statistics for an FPP are the power spectral density (PSD), coincidence rate (CR), index of dispersion for counts () [also called the Fano factor], and count-based covariance function (COV). We employ the PSD, S(!), derived directly from the point process itself, and not from the sequence of arrivals recorded in adjacent counting windows of equal duration. The CR measures the correlation between pairs of arrivals (events) with a specied time delay between them, regardless of intervening arrivals, and is related to the autocorrelation function used with continuous processes [2]. It is dened as PrfE(; ) and E(; + )g G() lim ; ()! 2 where E(s; t) denotes the occurrence of at least one event of the point process in the half-open interval [s; t). A particularly useful statistic is the F (T ), which is dened as the variance of the number of arrivals in a specied time window of width T divided by the mean number of arrivals. Finally, the COV C(k; T ) is dened as the covariance between the number of arrivals in two counting windows of counting time T and separation kt. If we dene X k N(kT )? N[(k? )T ], where N(t) represents the number of arrivals up to time t, then C(k; T ) Cov(X n ; X n+k ): (2) These four second-order statistics (, PSD, COV, CR) may be obtained from each other by means of the following relations [4, 2] Z T F (T ) = (T )? (T? jj) G()? 2 d?t Z S(!) =? G()e?j! d (3) C(k; T ) = Z T?T (T? jj) G(kT + )? 2 d; valid for any regular point process, where is the expected rate. 2.3 Fractal nature of FPPs For fractal point processes with fractal exponent < <, the second-order statistics will have special forms, and in the case of a purely fractal process all of the following relationships hold [4, 2]: G()= 2 = + (jj= )? + ()= S(!)= = + (j!j=! )? + (!=2) F (T ) = + (T=T ) (4) + (T=T ) C(k; T ) = T k = ; (T=T ) r 2 (k + )=2 k > ; with! T = cos(=2)?( + 2)? T = ( + )=2; (5) where R (x) is the Dirac delta function,?(x) e?t t x? dt is the gamma function, and r 2 (f(k)) f(k + )? 2f(k) + f(k? ) is the second central difference operator. The second line of (4) implies =f noise, and the last line of (4) also implies that the autocorrelation function () r(k; T ) is given by r(k; T ) C(k; T )=C(; T ) T = T + T 2 r2 (k + ) (k > ); (6) making the process X = fx n g long-range dependent with g(t ) T =(T + T ) and H = ( + )=2 [2]. For very small T, i.e. T T, the prefactor g(t ) approaches zero, so that the LRD property becomes negligible at this time scale. For large values of T, on the other hand, g(t ) approaches unity, increasing the degree of the long-range dependence. Thus (6) shows how the time scale aects the LRD property of a process X; the Hurst parameter H alone is insucient for characterizing the long-range dependence of an FPP. Broadly speaking, the time T marks the lower limit for signicant scaling behavior in the and. For this reason, we call this parameter the fractal onset time. We note that a purely fractal process, one which exhibits scaling over all time and frequency ranges as in (4), presents mathematical diculties; these processes have innite power, for example. This diculty is obviated in practice by restricting the validity of (4) to a nite range of times and frequencies, which must be the case for all experimental data. In this case, if any one of the relationships in (4) holds over the relevant time and frequency ranges then the other three must also, with the parameters again given by (5). Thus for FPPs all the second-order statistics exhibit power-law scaling with closely related exponents. This means that to second order, for < <, and over the specied ranges, three parameters suce to specify an FPP: the mean rate, the fractal exponent, and the fractal onset time T. [Equivalently, either w and can be specied instead of T, since any one species the other two via (5).] 2.4 Self-similar nature of FPPs As a further illustration of the fractal qualities of FPPs, we show that they are indeed (second-order) self similar for fractal exponents in the range < <. Let X n (m) m P? km n=(k?)m+ X n (concatenating adjacent counting intervals). Then we have C (m) (k; T ) = m?2 C(k; mt ), yielding r (m) (k; T ) = r(k; mt ): (7) Combining the results from (6) and (7) results in r (m) (k; T ) = (mt ) T + (mt ) 2 r2 (jkj + ): (8)

3 As m!, the second term of the denominator of (8) will dominate, resulting in lim m! r(m) (k; T ) = 2 r2 (jkj + ): Comparing this with (6), it is easy to see that X has a non-degenerate correlation structure and therefore is asymptotically (second-order) self-similar. In addition, since C (m) (; T ) = Var(X (m) ), we have i Var(X (m) ) = T hm? + (T=T ) m?(?) ; so that the variance of X (m) varies as m?(?) for large m. Thus the process X has the slowly-decaying variance property, another mathematically equivalent manifestation of self-similarity [7]. Therefore, the increments process X constructed from all FPPs with < < are asymptotically second-order self-similar with Hurst parameter H = ( + )=2. 3 Two FPP Construction Methods FPPs are characterized by power-law behavior in their second-order statistics, as shown in (4). Analytical relations exist to generate all these statistics from each other, as given by (3); thus mathematically all are equivalent. We employ the coincidence rate in the following, since this proves easier analytically; we reiterate the form for a FPP given in the rst line of (4): G() = 2 [ + (jj= )? ] + (): (9) 3. Renewal point process method A renewal point process by denition has i.i.d. interarrival times; thus the interarrival time pdf completely species the process. If this distribution is heavy tailed, then the coincidence rate G() will also decay in a power law form, as given by (9). This yields the fractal renewal point process (see Sec. 4); the superposition of a number of independent and identical realizations of this process also has a coincidence rate of the same form, and therefore also belongs to the FPP family of processes (see Sec. 5). Since the behavior of the tail of the interarrival time pdf determines the power-law form of the CR and hence the fractal nature of the FPP, the behavior of the interarrival time pdf at short times is arbitrary. Thus a wide variety of renewal-based FPPs exist for a given average rate and power-law exponent. 3.2 Doubly stochastic Poisson process (DSPP) method The doubly stochastic Poisson point process (DSPP) method derives from the similarity between the CR of a DSPP and the autocorrelation function of its rate. To show this, let I(t) denote the stationary stochastic rate of a DSPP, and R I () denote the autocorrelation function of this rate. Then, for 6= we have PrfE(; ) and E(; + )g G() lim ;! 2 E [PrfE(; ) and E(; + )j Ig] = lim! 2 I(t)I(t + ) = lim E! = E[I(t)I(t + )] R I (): 2 Thus if a stationary continuous-time stochastic process I(t) with an autocorrelation function R I () having the form of (9) serves as the rate for a DSPP, the result will be a FPP. In the following we consider two such continuous-time stochastic processes: Fractal Binomial Noise (FBN) and Fractal Shot Noise (FSN). FBN consists of the superposition of several independent and identical fractal ON/OFF processes whose sojourn times are heavy-tailed (see Sec. 6). FSN is a type of shot noise [6] where the linear lter assumes a power-law decaying form (see Sec. 7). In the following we present several FPP models constructed by the above approaches which may be employed to generate data mimicking self-similar trac. For each model we rst identify the relevant parameters, then analyze how these parameters determine the three fundamental quantities:, H, and T. In particular, we determine whether all three of these quantities can be independently specied. Finally, we consider the remaining parameters (those in excess of the three needed to specify, H, and T ), if any, and their eects on the character of the FPP. Such eects involve statistics over short time scales and clustering property over all (or relatively large range of) time scales, a salient feature of FPPs 4 Fractal Renewal Process (FRP) Perhaps the simplest FPP is the standard fractal renewal process [3]. For the standard FRP, the times between adjacent arrivals are independent random variables drawn from the same pdf. Fig. -(a) provides a schematic representation of this point process. In particular, the pdf decays as a power law p 2 (t) = kt?(+) for A < t < B, () otherwise, where A and B are cuto parameters, is a fractal exponent ( < < 2), and k is a normalizing constant determined by the requirement R p 2(t) dt =. For < < this process is fully fractal: the power spectral density, coincidence rate, index of dispersion for counts, and even the interarrival time pdf all exhibit power-law scaling as in (4) over time scales lying between A and B, and with related power-law exponents completely determined by =. For < < 2 the PSD,, CR, and (but not the interarrival time pdf) still exhibit power-law scaling of the form of (4), but with an associated exponent given by = 2?. For gamma 2 the process no longer has fractal second-order statistics. Thus is limited to a range between zero and unity, and for each value of there corresponds two values of. In practice, however, the range < < 2 proves

4 R (a) Fractal Renwal Process (b) Fractal ON/OFF Process Figure : The standard FRP and alternating FRP (fractal ON/OFF process) Models. superior. For < < the resulting simulations are extremely bursty and do not mimic practical traf- c data, and sample statistics do not reliably follow the analytic forms of (4) except for prohibitively large simulation lengths. Another advantage of employing > is that it renders the outer cuto B unnecessary; setting B! leads to a positive mean rate, in contrast to the < case. Eliminating the outer cuto also yields better power-law behavior in the PSD and, and simplies simulation [3]. The probability density function then assumes the simpler form for t A, p 2 (t) = A t?(+) () for t > A. Choosing in the range < < 2 proves far superior to < < for the same desired value of, but the form of the interarrival pdf in () can be further improved. In particular, the resulting F (T ) has a dip near T = T, caused by the abrupt cuto in the interarrival time pdf that still remains; furthermore, the power spectral density exhibits excessive oscillations for the same reason. Improvement results from employing a smoother pdf [4] A p 2 (t) =? e?t=a for t A, e? A t?(+) (2) for t > A, which is continuous for all t. The practical FRP model as presented in (2) has only two parameters: and A; thus this model cannot fully specify the set of fundamental quantities, H, and T. Thus the FRP model only proves useful for simulating data with T of the order of unity. Using the relation = 2? we obtain H = ( + )=2; = + (? )? e?? A? ; T = 2??2 (? )? (2? )(3? )e? [ + (? )e ] 2 A : (3) 5 Superposition of Fractal Renewal Processes (Sup-FRP) A dierent fractal point process results from the superposition of a number of independent and identical FRPs. Although the resulting FPP (Sup-FRP) is no longer renewal, the marginal distribution of the interarrival times is still heavy-tailed [2]. Furthermore, the corresponding PSD, CR,, and retain their scaling behavior, albeit over a somewhat reduced range of times and frequencies. With each component FRP described by the interarrival time pdf given in (2), three parameters characterize the Sup-FRP: and A from the individual FRPs, and M, the number of FRPs superposed. In this case, the three fundamental quantities become [3] H = ( + )=2 = M + (? )? e?? A? ; T = 2??2 (? )? (2? )(3? )e? [ + (? )e ] 2 A : (4) with = 2?. The only dierence between (4) and (3) is a factor of M in the expression for ; the quantities H and T remain unchanged. The Sup- FRP model indeed has three parameters, although M can take only positive integer values. Thus arbitrary values of, H, and T cannot be achieved exactly. In practice, we specify and H, and adjust M to approximate T as closely as possible. The integer M is of the order T ; for most trac data this quantity greatly exceeds unity, so that the nite resolution of M does not pose a signicant problem. 6 Fractal-Binomial-Noise-Driven Poisson Process (FBNDP) The fractal renewal process (FRP) described previously is recast as a real-valued process which alternates between two values, such as zero and R (> ). This alternating fractal renewal process (AFRP) would then start at a value of zero (\OFF"), and then switch to a value of R (\ON") at a time corresponding to the rst event in the FRP. At the second such event, the AFRP would switch back to zero, and would proceed to switch back and forth at every successive event of the FRP. Thus for the AFRP, all ON/OFF periods are i.i.d. with the same heavy-tailed distribution as in the FRP. Figure -(b) illustrates such a process. As with the Sup-FRP, M independent and identical fractal ON/OFF processes may be added together, yielding fractal binomial noise (FBN) with the same fractal exponent as the single fractal ON/OFF process [3]. Let I(t) denote the resulting FBN process. The autocorrelation function of I(t) is merely a scaled version of the of the individual AFRP processes, which in turn follow the decaying power-law form of (9) [3]. Thus I(t) can serve as a stationary stochastic rate function for a Poisson process, resulting in the fractal-binomial-noise-driven Poisson point process (FBNDP). The construction of the FBNDP is schematized in Fig. 2. Since I(t) is a binomial process, and the ON and OFF periods have identical mean values, we have PrfI(t) = nrg = 2?M M! n!(m? n)! (5) for n = ; ; 2; : : :M. The FBNDP has four free parameters (A,, R, and M) which determine, H and T as follows, where we

5 Fractal-Binomial-Noise-Driven Poisson Process (FBNDP) R R Poisson Generator N(t) I(t) Figure 2: The FBNDP Model. FBN homogeneous, but rather reects the variations of the fractal-shot-noise driving process. Thus the two forms of randomness inherent in the DSPP are, in the particular case of the FSNDP, two separate Poisson processes, linked by a power-law-decaying linear lter. As a result of this lter, the FSNDP exhibits fractal behavior of the form of (4) for time scales in the range A T B. The FBNDP and the FSNDP thus both belong to the fractal DSPP family, comprising a realvalued fractal rate function driving a Poisson point process; they dier only in how the fractal rate functions are constructed. The FSNDP model does not involve the heavy-tailed property in the same manner that the FRP, Sup-FRP, and FBNDP models do, since the interarrival time statistics of both Poisson processes in the FSNDP model have exponential tails. use the same density as in (2) for the ON and OFF periods of the individual AFRP processes [2]: H = ( + )=2; = RM=2; T = ( + )(2? )? R? A? [(? )e 2? + ]: (6) Thus, H, and T can all be specied, and in fact an extra parameter exists, implying that dierent FB- NDPs can be constructed with the same, H, and T. For example, decreasing M (while increasing R to keep the overall rate constant) will increase the probability that the rate becomes zero [see (5)], during which no arrivals can occur. Since the OFF period is also heavy-tailed, the resulting FBNDP exhibits a high degree of clustering, especially for the limiting case M =. For this value of M, heavy-tailed periods of arrivals will alternate with heavy-tailed inter-burst quiescent periods. Similarly, increasing the value of M reduces the degree of clustering. 7 Fractal-Shot-Noise-Driven Poisson Process (FSNDP) The fractal-shot-noise-driven Poisson point process (FSNDP) [2] is a special case of the doubly stochastic Poisson process (DSPP). For the FSNDP, the rate of the inhomogeneous Poisson process is fractal shot noise [], which is itself a ltered version of a different, homogeneous Poisson point process. Figure 3 schematically illustrates the FSNDP as a two-stage stochastic process. The rst stage is a homogeneous Poisson process (HPP) with constant rate. Its output M(t) becomes the input to a linear lter with a power-law decaying impulse response function h(t) = c=t?=2 for A < t < B, otherwise, (7) with, A, and B dened as following (), and c a positive amplitude constant. This lter produces fractal shot noise I(t) at its output, which then becomes the stochastic rate for the last stage, a second Poisson point process. The resulting process N(t) is not Rate (µ) Poisson Generator Linear Filter h(t) Fractal-Shot-Noise-Driven Poisson Process (FSNDP) N(t) M(t) I(t) Poisson Generator t Homogeneous Poisson Process Fractal Shot Noise FSNDP Events Figure 3: The FSNDP Model. The FSNDP model has ve parameters: A, B,, k, and, which determine H,, and T as follows [2]: H = ( + )=2 = 2? cb =2 T ( + )?(? =2) = 2?( + =2)?(? ) c? B =2 : t t t (8) Since A does not appear in (8), in many applications it can be set to zero, which results in a rate function I(t) with innite variance. This proves not be problematic, however, since an integrated version of the shot noise (which necessarily has a nite variance), is more directly involved in FSNDP statistics. Four parameters remain, so that as with the FBNDP a free parameter exists which determines the clustering characteristics of the process; we focus on the product B. For small values of this quantity, successive impulse response functions rarely overlap, leading to large gaps between impulse response functions where FSNDP arrivals cannot occur. Thus the process appears more clustered. 8 Simulation of FPPs We performed simulations on the FRP, Sup-FRP, FSNDP, and FBNDP models; for the FBNDP we employed two sets of parameters. For each FPP model, we set = arrivals/sec and H = :9; we also set

6 ( x ) 2 4 ( x ) 2 4 ( x ) 2 4 ( x ) 2 4 ( x ) 2 4 FRP Sup-FRP with M = FSNDP with B = ^3 sec FSNDP with B = ^5 sec FBNDP with M = FBNDP with M = ( x ) 2 ( x ) 2 ( x ) 2 ( x ) 2 ( x ) 2 FRP Sup-FRP with M = FSNDP with B = ^3 sec FSNDP with B = ^5 sec FBNDP with M = FBNDP with M = ( x ) ( x ) Figure 4: Sample path comparison of FPPs. Parameters as in Table except for the values of B in FSNDP simulations. Left: counting time T = : sec, Right: T = sec. Model A (msec) B (sec) M R FRP.48.8 Sup-FRP FSNDP?4 7.8 FBNDP # FBNDP # Table : Parameter values of the resulting FPP model for each of the ve FPPs simulated from the specication = (points/sec), H = :9, and T = msec. For all simulations, the simulation length L is 5 3 sec, yielding an expected number of arrivals of 5 6. T = msec, or as close as possible to that value. For the FRP, setting and H as above results in T = :2 msec. Since the parameter M in the Sup-FRP can take only integer values, the closest value in this case is T = 9:6 msec, achieved with M = 8. For the FSNDP, we set A =?7 sec and B = 7 sec to minimize the eects of fractal cutos on the estimated fractal exponent [2], resulting = 9:6?5 and c = 6:6 3 from (8). The value of B in particular must exceed the simulation length; this value provides a comfortable margin of 2, leading to only small deviations from ideal fractal behavior [2]. Finally, for the FBNDP we simulated two cases: M = 2 (FBNDP #) and M = (FBNDP #2). Table presents the parameter values derived from the simulation specication for each of the ve models simulated. Fig. 4 shows the sample path behavior of these processes at two dierent time scales. For the FSNDP we use B = 3 sec and B = 5 sec to make shot-noise uctuations more visible. Notice the sharp dierence in burst structure among models. This further illustrates that the Hurst parameter alone does not capture the burstiness of LRD trac. Model b (points/sec) b H CPU (min) FRP Sup-FRP FSNDP FBNDP # FBNDP # Table 2: Estimated parameters from the FPP simulations, and the averaged time required to perform each run. 95% condence intervals are included for the rate estimate. For each of the ve FPPs simulations listed in Table we performed independent runs. Fig. 5 presents the resulting estimated F(T b ) and br(k; T ) for each FPP model; means and 95% condence intervals are shown. We chose a counting time of T = msec (= T ) for the estimated br(k; T ), so that power-law behavior would be readily apparent at this time scale. The analytical values F (T ) and r(k; T ) are also included, obtained from (4) and (6). Table 2 lists the three estimated fundamental parameters corresponding to these ve FPP simulations. The

7 estimated fractal exponent b was obtained by a leastsquares t to a doubly logarithmic plot of the, log[ F(T b )? ] vs. log(t ), between counting times of T = : msec and T = sec, with two samples per decade. We obtained the estimated Hurst exponent by the simple relation H b = (b + )=2. The simulation results support the notion that the four models indeed generate fractal point processes with the desired rate of arrivals and the Hurst parameter. Table 2 shows that the estimated rate follows the desired value of = closely and with little variation for all ve simulated FPP models. The estimated Hurst parameter H b also closely follows the original value.9. However, Fig. 5 illustrates that some deviation between simulated and analytical results occurs in the and. The dierence in the is systematic with all simulated points falling below the analytical curve, and roughly paralleling it. The simulated FPP results from all ve plots also differ from the analytically predicted results, as shown in Table 2, although they all follow the form of (4). This bias appears to be intrinsic to the fractal nature of the processes [4], varying among the FPP models and even between the two versions of the FBNDP simulated. See [2] for detailed analysis of Fig. 5 and the bias. Despite the various biases and variances, all FPP models simulated indeed generate stochastic point processes which are fractal, and thus exhibit power-law behavior. Each model has its own distinctive characteristics, and may prove best in simulating a particular fractal trac source; all are useful. Finally, we consider the eciency of these ve FPP simulation methods. The last column of Table 2 shows the average CPU times for each FPP simulation of 5 million arrivals on an SGI Challenger XL running IRIX 5.3. Three of the simulations require about two minutes, or 24 sec per event. The FBNDP # requires the longest time to simulate, for an extremely large number of power-law computations are required due to the small value of A. Thus, FPP models can generate large sets of fractal data eciently on modern workstations. 9 Concluding Remarks We have introduced four fractal point processes (FPPs) which provide an improved understanding of self-similar trac observed in current computer networks: the Fractal Renewal Process (FRP), Superposition of i.i.d. FRPs (Sup-FRP), the Fractal- Shot-Noise-Driven Poisson Process (FSNDP), and the Fractal-Binomial-Noise-Driven Poisson point Process (FBNDP). Some of these models appear to exhibit greater variability than earlier models [8, 23, 24]. We have shown that these models fall into two classes, renewal-based and DSPP-based, depending on their construction, and how the heavy-tailed property need not relate to the fractal nature of these models. Perhaps most importantly, the Hurst parameter H alone does not suce in describing the burstiness and clustering in a typical self-similar process, despite assertions to the contrary [7]. The FPP models presented in this paper cover a Analysis Simulation (a) FRP Model Lag k (Time Unit T =. sec) (b) Sup-FRP Model Lag k (Time Unit T =. sec) (c) FSNDP Model.... (d) FBNDP Model # (M = 2).... (e) FBNDP Model #2 (M = ) Lag k (Time Unit T =. sec) Lag k (Time Unit T =. sec) Lag k (Time Unit T =. sec) Figure 5: Results of simulations of ve FPP models. Left: doubly logarithmic plot of the estimated bf(t ) vs. counting time T. Right: plot of the estimated br(k; T ) vs. lag k. The counting time T is : sec. Mean values obtained from the simulations are indicated by a dot, with 95% condence intervals delineated by error bars. Analytic values are included for comparison.

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