M Meziane

Olivier Eichwald
email: eichwald@laplace.univ-tlse.fr

J.-P P Sarrette

Olivier Ducasse

Mohammed Yousfi

2D SIMULATION OF ACTIVE SPECIES AND OZONE PRODUCTION IN A MULTI-TIP DC AIR CORONA DISCHARGE

published or not. The documents may come

2D simulation of active species and ozone production in a multi-tip DC air corona discharge

Introduction

Non-thermal plasmas created by atmospheric pressure corona discharges are very efficient sources of active species like radicals or excited species. They are therefore used in many applications where chemical activation is needed such as flue gas pollution control [1][2], plasma ignition [3], bio-decontamination [4], medical treatment [5] or ozone production [6]. In this paper, we focus our study on ozone production by multipoints corona discharge. In such device, ozone is produced through the radical O that is created in time laps of about hundred of nanoseconds inside filamentary discharges. These radicals are then transported in the air flow and diffuse in the reactor volume where they interact more particularly with the background O 2 molecules to form O 3 . In the present paper we have developed a 2D simulation of these processes by coupling ten discharge / postdischarge phases until 1ms in a reactor having up to four anodic tips. Even if some previous works were already devoted to this kind of simulation [7][8], a complete 2D simulation simultaneously coupling all the energetic, chemical and hydrodynamics phenomena that take place inside a multi-tip corona reactor was never done. This perspective is yet possible because recent works have shown the ability of the commercial FLUENT software to follow the discharge and post-discharge phases in a mono-tip to plane configuration [9][10]. Section 2 details the used model and the main assumptions. Section 3 describes the simulation conditions and is followed by the description and the analysis of the obtained results.

Hydrodynamics model of the discharge and post-discharge phases

The hydrodynamics model is based on the system of non stationary Navier-Stokes equations for reactive gas mixture. It takes into account the conservation equations of mass, momentum, energy and density of each chemical species i, coupled with the conservation equation of the vibrational relaxation energy:

0 . = ∇ + ∂ ∂ v t r r ρ ρ (1) ⇒ ∇ - ∇ - = ∇ + ∂ ∂ τ ρ ρ . . r r r r r r P v v t v (2) i S J v m t m i i i i ∀ = ∇ + ∇ + ∂ ∂ r r r r .
. ρ ρ

(3)

t P T k v h t h ∂ ∂ + ∇ ∇ = ∇ + ∂ ∂ ) ( . . r r r r ρ ρ v v t i i i E j f h J v P v τ ε τ + + ∇ - ∇ + ∇ + ∑ ⇒ r r r r r r r r . . : . (4) v v v v v E j f v t τ ε ε ε - = ∇ + ∂ ∂ r r r r
. . (5) ρ is the mass density of the gas, the τ [11]. During each discharge phase, we inject inside the discharge volume, the radical and metastable source terms as well as the energetic source terms

E j f t r r
.

and

E j f v r r
. . The magnitude and the profile of the corresponding source terms are taken from a complete corona discharge simulation involving the primary and the secondary streamer propagation [12]. The magnitude and profile of the energetic source terms are taken after the pressure wave initiated by the first thermal shock [13].

Simulation conditions

The previous hydrodynamic model is solved using the commercial FLUENT Code. The 2D Cartesian computational domain is displayed in figure 1. It represents a (2cm x 1cm) tip-to-plane corona discharge reactor crossed by a lateral atmospheric air flow of 5ms -1 at 300K. A maximum of four tips can be used, distant from each other by 5mm. At each tip, a positive DC high voltage of magnitude equal to 7.2kV is applied. Under these conditions, the discharge phase duration is of around 150ns and the natural repetition rate of the discharge is of about 10kHz [12]. The density and energetic source terms profile and magnitude are calculated from a complete simulation of the discharge phase using an optimized 2D streamer corona discharge model [12]. The simulation involves 10 neutral chemical species

(N, O, O 3 , NO 2 , NO, O 2 , N 2 , N 2 (A 3 ∑ u + ), N 2 (a '1 ∑ u -
) and O 2 (a 1 ∆g)) reacting following 24 selected chemical reactions. Other species (eg, N 2 O, NO 3 , etc.) have not been considered in this simulation because of their low concentration in the ionized channel [8]. During each discharge phase, metastable excited species (N 2 (A 3 ∑ u + ), N 2 (a '1 ∑ u -) and O 2 (a 1 ∆g)), radicals (N and O) and electronic energy are injected for each tip inside a monofilamentary channel of 50µm of diameter and located in the gap space between the powered tip and the grounded plan.

Results and discussion

Figure 2 shows the cartography of the temperature at 150ns just after the first discharge phase when the four points are stressed by the DC high voltage. As in previous works [9], the temperature increase is confined inside the mono-filamentary discharge zones. The direct relaxation of the translational, rotational and excited energies (modelized by

E j f t r r
. term in equation ( 4)) induces a maximum temperature increase of 1200K just around the anodic tips. Figures 3 and4 show the cartography of the temperature and of the ozone density after 1ms (i.e after 10 discharge and post-discharge phases) when one, two, three or four tips are stressed by the DC high voltage. Figures 3a and4a show that for the mono tip case, the lateral air flow and the memory effect of the previous ten discharges leads to a wreath shape of the space distribution of the temperature and the ozone density. The temperature and the ozone maps are very similar. Indeed, both radical and energy source terms are higher near the tip (i.e. inside the secondary streamer area expansion [12]) and near the plane. Furthermore, the production of ozone is obviously sensitive to the gas temperature diminution since it is mainly created by the three body reaction (having a reaction rate inversely proportional to gas temperature). For more than one tip, the temperature and ozone wreaths interact each other and their superposition induces locally a rise of both the gas temperature and ozone density (see figures 3 and4). The local maximum of temperature is of around 325K for one tip case and increases up to 350K for four anodic tips. The average temperature in the whole computational domain remains quasi constant and the small variations show a linear behavior with the number of anodic tips (see figure 5). A linear increase with proportionality between the ozone density and the number of tips is also observed in figure 6. After 1ms, and for the four tips case, the mean ozone density inside the computational domain reaches 4x10 14 cm -3 . At the end of each discharge phase, the maximum of O radical production is located around the tip and reach about 10 17 cm -3 .We note that the atomic oxygen density decreases with the gas temperature to form ozone, and its population at 1ms is concentrated in the surrounding of each tip, with significant quantity of a 10 15 cm -3 in the reactor (see figure 7). Finally, in our simulation conditions, the energy injected per tip after 1ms is of around 0.1mJ. The mean quantity of O 3 created after 1ms and for 1 tip is of around 1.610 -9 g. Thus the production yield per tip can be estimated of around 16µgJ

M O M O O + → + + 3 2

Conclusion

The present work shows that the FLUENT software can be used to simulate in details the spatiotemporal evolution of an air flow stressed by repetitive corona discharges under mono and multitip anodic electrode configuration. In our simulation conditions, the 2D results show the strong correlation between the temperature decrease and the ozone production. As the O radical density and electronic energy are mainly deposed around the tips during the discharge phases, and because of the lateral air wind, both temperature and ozone concentration increase in a lateral wreath located near the tips and extending along the whole computational domain. Until 1ms and a maximum of four tips, the mean ozone production is proportional to the number of tips. Finally, one can notice that the complete simulation until 1ms and with four tips needs about 10 days of CPU time on the Hyperion machine [14].

  the static pressure and the stress tensor. For each chemical species i, n ⇒ τ i and m i are the density and the mass fraction, i J r the diffusive flux due to concentration and thermal gradients, S i the net rate of production per unit volume (due to chemical reactions). h is the static enthalpy, T the temperature, k the thermal conductivity and v thermal and vibrational energy. In the present study we assumed that the translational, rotational and electronic excitation energies relax immediately and that the vibrational energy stored during the discharge phases

Figure 1 :

 1 Figure 1: 2D Cartesian simulation domain of the multi-tip to plane corona discharge reactor.

Figure 2 :

 2 Figure 2: Temperature profile after the first discharge phase at t = 150ns.

Figure 3 :

 3 Figure 3: Temperature contours at 1ms i.e. after ten discharge and post-discharge phases. The number of high voltage tip is respectively (a) one, (b) two, (c) three and (d) four.

Figure 4 :

 4 Figure 4: Ozone density contours (cm -3 ) at 1ms i.e. after ten discharge and post-discharge phases. The number of high voltage tip is respectively (a) one, (b) two, (c) three and (d) four.

Figure 5 :

 5 Figure 5: Mean temperature increase inside the computational domain of figure 1 as a function of the number of tips

Figure 6 :

 6 Figure 6: Mean ozone density increase inside the computational domain of figure 1 as a function of the number of tips.

Figure 7 :

 7 Figure 7: Atomic oxygen density profile (cm -3 ) at 1ms, i.e. after ten discharge and post-discharge phases, in the case of four anodic tips.

Acknowledments:

TT his work was performed using HPC resources from CALMIP (Grant 2011-[P1053]) [14].


Vicky Fasen
email: fasen@ma.tum.de.

Modeling Network Traffic by a Cluster Poisson Input Process with Heavy and Light-Tailed File Sizes

Keywords: AMS 2000 Subject Classifications: primary: 90B22 secondary: 60F05, 60F10, 60F17, 60G52 cluster Poisson model, covariance function, cumulative input process, fluid queue, fractional Brownian motion, heavy tails, input model, long range dependence, self-similarity, stable Lévy motion, regular variation, scaling, teletraffic

We consider a cluster Poisson model with heavy-tailed interarrival times and cluster sizes as a generalization of an infinite source Poisson model where the file sizes have a regularly varying tail distribution function or a finite second moment. One result is that this model reflects long range dependence of teletraffic data. We show that depending on the heaviness of the file sizes, the interarrival times and the cluster sizes we have to distinguish different growths rates for the time scale of the cumulative traffic. The mean corrected cumulative input process converges to a fractional Brownian motion in the fast growth case. However, in the intermediate and the slow growth case we can have convergence to a stable Lévy motion or a fractional Brownian motion as well depending on the heaviness of the underlying distributions. These results are contrary to the idea that cumulative broadband network traffic converges in the slow growth case to a stable process. Furthermore, we derive the asymptotic behavior of the cluster Poisson point process which models the arrival times of data packets and the individual input process itself.

Introduction

The infinite source Poisson model is a fluid queue approximation of network data traffic that assumes that the transmission of data files to the system happens at the jump times of a Poisson process. The file sizes are assumed to form an iid (independently and identically distributed) sequence independent of the underlying Poisson process. This model is also called M/G/∞ model in the queueing literature. There is empirical evidence that the file sizes are heavy-tailed with an index of regular variation in the interval (1, 2) (cf. [START_REF] Crovella | Explaining world wide web traffic selfsimilarity[END_REF][START_REF] Crovella | Self-similarity in world wide web traffic: evidence and possible causes[END_REF][START_REF] Crovella | Heavy-tailed probability distributions in the world wide web[END_REF][START_REF] Hernandes-Campos | Variable heavy tailed durations in Internet traffic[END_REF][START_REF] Willinger | Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level[END_REF]). The infinite variance of the file size in the infinite source Poisson model causes long range dependence and self-similarity in the input process which are stylized facts of teletraffic data (cf. the data analysis in [START_REF] Crovella | Explaining world wide web traffic selfsimilarity[END_REF][START_REF] Crovella | Self-similarity in world wide web traffic: evidence and possible causes[END_REF][START_REF] Crovella | Heavy-tailed probability distributions in the world wide web[END_REF][START_REF] Leland | On the self-similar nature of Ethernet traffic (extended version)[END_REF]). However, this explains the long-range dependence of one individual source, but not that of the overall teletraffic. The teletraffic of the network was studied in the seminal work of Mikosch et al. [START_REF] Mikosch | Is network traffic approximated by stable Lévy motion or fractional Brownian motion?[END_REF]; see also the monographs of Resnick [START_REF] Resnick | Heavy-Tail Phenomena: Probabilistic and Statistical Modeling[END_REF] and Whitt [START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Applications to Queues[END_REF]. Guerin et al. [START_REF] Guerin | Empirical testing of the infinite source Poisson data traffic model[END_REF] applied the infinite source Poisson model to teletraffic data and realized some shortcomings in that model. They suggested as an alternative a cluster Poisson model to be more realistic.

A cluster Poisson model is an obvious extension of the infinite source Poisson model. In the present paper we consider a cluster Poisson point process N modeling the arrival of data files sent to a server, i.e. we have a Poisson point process where at any jump time a cluster of data transmissions is initiated. The cluster sizes will have a finite first moment but an infinite second moment, and the interarrival times have an infinite second moment as well. This causes N to be long range dependent and makes N sufficiently irregular for our results (see Section 3).

On the other hand, we consider the amount of data which is transmitted to the server, the so called input process (A(t)) t≥0 . The file sizes are assumed to be regularly varying of index γ ∈ [START_REF] Anderson | A strong renewal theorem for generalized renewal functions in the infinite mean case[END_REF][START_REF] Baccelli | Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences[END_REF] or have a finite second moment. Hence, the present paper contains also the model of Fasen and Samorodnitsky [START_REF] Fasen | A fluid cluster Poisson input process can look like a fractional Brownian motion even in the slow growth aggregation regime[END_REF] and extends their results. In particular, we are interested in the characterization of the traffic generated by one source, but on the other hand we are also concerned with the overall traffic of the network and on which way the properties of an individual source are transferred to the traffic of the system. The teletraffic of the network is given by the cumulative input of all sources. Thus, let (A i ) i∈N be iid input processes; e.g. independent users in a computer pool generate independent traffic and A i models the traffic of user i. With n input processes at a time scale M , the mean corrected cumulative input is the stochastic process

S n,M (t) = n i=1 (A i (Mt) -E(A i (Mt))) for t ≥ 0.
(1.1)

The case where the time scale M = M n depends on the number of users n (bird's eye view) is the situation where the term fast growth and slow growth case comes in. Let (M n ) be a sequence of positive constants such that M n ↑ ∞ as n ↑ ∞. Then the fast growth case means n M n while the slow growth case means n M n . In this paper we will further use an intermediate growth case; see Section 5.2. Then (M n the fast growth case. In Gaigalas and Kaj [START_REF] Gaigalas | Convergence of scaled renewal processes and a packet arrival model[END_REF], who studied the renewal counting process of Pipiras and Taqqu [START_REF] Pipiras | Slow, fast and arbitrary growth conditions for renewalreward processes when both the renewals and the rewards are heavy-tailed[END_REF] with heavy-tailed inter-renewal times as a packet arrival model, an intermediate growth case also exist. However, their definition of intermediate growth case is different to ours.

For a cluster Poisson model we can interpret the different growth cases also as a change in the connection rate of one source, since the input of one cluster Poisson model with intensity λ(n) = nλ 0 has the same distribution as the cumulative input of n iid cluster Poisson sources with intensity λ 0 . Thus, Mikosch et al. [START_REF] Mikosch | Is network traffic approximated by stable Lévy motion or fractional Brownian motion?[END_REF] described their growth rates of (M n ) by a change in the connection rate rather than in the change of the number of users.

There are substantial reasons to separate the growth cases. If the number of sources n is very large, then the process in (1.1) is the sum of a very large number of iid terms that change relatively slowly. Hence, under general assumptions we can apply a central limit theorem and obtain convergence to a Gaussian process; see the interesting paper of Mikosch and Samorodnitsky [START_REF] Mikosch | Scaling limits for cumulative input processes[END_REF]. On the other hand, if the time scale M is very large, then the main phenomenon in (1.1) is, actually, in the deviations of the individual input processes from their means, A(Mt) -E(A(Mt)), for large M . Then it is not obvious which limit we will obtain. In Levy and Taqqu [START_REF] Taqqu | Using renewal processes to generate long-range dependence and high variability[END_REF][START_REF] Levy | On renewal processes having stable inter-renewal intervals and stable rewards[END_REF][START_REF] Levy | Renewal reward processes with heavy-tailed interrenewal intervals and stable rewards[END_REF] and Pipiras and Taqqu [START_REF] Pipiras | Slow, fast and arbitrary growth conditions for renewalreward processes when both the renewals and the rewards are heavy-tailed[END_REF] we find these different cases of first letting n and then M tend to infinity and opposite for a renewal-reward model. Their limit process not only depends on the order of the limits but also on whether the tails of the rewards are lighter or heavier than those of the renewals; fractional Brownian motion, stable Lévy motion as well as stable processes with dependent increments are possible limits (cf. Kaj and Taqqu [START_REF] Kaj | Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach[END_REF] for an extension of that idea). We will see a similar phenomena in our model.

We show that the asymptotic behavior of the variance of the input process and the asymptotic behavior of the mean corrected individual input process

A n (t) = A (0, M n t] -E(A (0, M n t]) a n , t ≥ 0, (1.2)
respectively, where (a n ) is a sequence of positive constants tending to ∞, is either determined by the cluster Poisson point process N , if γ > 3 -2H (γ > H -1 , respectively) and if the file sizes have a finite second moment, or by the file sizes if γ < 3 -2H (γ < H -1 , respectively), where H is a constant defined by the cluster Poisson model. We use these results on the traffic of one source to mathematically explain the behavior of the teletraffic in the system. Thus, we illustrate that the different regions for γ in the asymptotic behavior of the variance and the input process have also to be distinguish in the asymptotic behavior of the mean corrected cumulative input of the network. If γ > 3 -2H or if the file sizes have finite second moment, the process (S n,Mn (t)) t≥0 converges for any (sufficiently regular) sequence M n ↑ ∞ as n → ∞ to a fractional Brownian motion. This result extends the results of Fasen and Samorodnitsky [START_REF] Fasen | A fluid cluster Poisson input process can look like a fractional Brownian motion even in the slow growth aggregation regime[END_REF] who assumed finite second moments of the file sizes. On the one hand, we also allow heavy-tailed file sizes and on the other hand, our results include the fast growth case. However, the result is surprising. In the slow growth case of an ON/OFF and an infinite source Poisson model with heavy-tailed file sizes the limit is a stable Lévy motion (cf. Mikosch et al. [START_REF] Mikosch | Is network traffic approximated by stable Lévy motion or fractional Brownian motion?[END_REF]). Mikosch and Samorodnitsky [START_REF] Mikosch | Scaling limits for cumulative input processes[END_REF] discovered that the occurrence of a stable limit, in the slow growth case of a general telecommunication model, is not robust. Our result confirms that idea.

In contrast, for H -1 < γ < 3 -2H we have to distinguish three different growth rates of (M n ) where any growth case has a different scaling and limit distribution; a fractional Brownian motion with index (3-γ)/2 or H, or a γ-stable Lévy motion. Again we obtain in the slow growth case the convergence to a fractional Brownian motion and not to a stable process. Finally, the case γ < H -1 recovers the classical result of the ON/OFF and the infinite source Poisson model; in the fast growth case the convergence to a fractional Brownian motion and in the slow growth case to a γ-stable Lévy motion.

The paper is organized as follows. We start with some preliminaries in Section 2. In Section 2.1 we introduce a general cluster Poisson model and in Section 2.2 we present the model assumptions. As mentioned, a typical stylized fact of teletraffic data is long range dependence. Hence, in Section 3 we compute the asymptotic behavior of the variance and the covariance of the input process A and the point process N . On the one hand, we obtain that the variance behaves asymptotically like a power function of some index in (0, 1), and hence N and A exhibit long range dependence. On the other hand, we also see the influence of the underlying point process N and the file sizes Z on the dependence structure of A. Next, we study the asymptotic properties of

N (0, M n t] -E(N (0, M n t]) Var(N (0, M n ]) for t ≥ 0, (1.3)
in Section 4. The results are different to the conclusions for the cluster Poisson point process in Faÿ et al. [START_REF] Faÿ | Modeling teletraffic arrivals by a Poisson cluster process[END_REF] depending on their different model assumptions. Independent of the growth rate of (M n ) we obtain the convergence of (1.3) to a fractional Brownian motion. Finally, the asymptotic behavior of the input process (A n (t)) t≥0 and the properly normalized cumulative input (S n,Mn (t)) t≥0 are derived in Section 5. To increase the readability of the paper the proofs are postponed to Section 6. We will use the notation =⇒ for weak convergence, P -→ for convergence in probability, ν =⇒ for vague convergence, and FDD =⇒ for weak convergence of the finite dimensional distributions. For x ∈ R we write x + = max(0, x) and x = inf{k ∈ N : k ≥ x}. For two random variables X, Y the symbol X d = Y means that X has the same distribution as Y . Let (x n ) n∈N , (y n ) n∈N be sequences of constants then x n ∼ y n as n → ∞ iff lim n→∞ x n /y n = 1. For a distribution function F we denote by F = 1 -F the tail of F and finally, Leb denotes the Lebesgue measure.

Preliminaries

The cluster Poisson model

A cluster Poisson point process, which we use to model the arrival times of data files sent to the server, is defined as follows:

(i) The initial cluster points, denoted by . . . < Γ -1 < Γ 0 < 0 < Γ 1 < Γ 2 < . . . form a homogeneous Poisson process with rate λ 0 , the connection rate. (ii) Each cluster initiated at Γ m is of random size (K m + 1) with arrival points

Γ m,k = Γ m + T m,k for k = 0, . . . , K m ,
where

T m,k = k j=1
X m,j for k ∈ N and T m,0 = 0, and the within-cluster arrival times (X m,j ) j,m∈N are a sequence of iid positive random variables with common distribution F X independent of the sequence of iid positive cluster sizes (K m ) m∈Z with distribution F K .

Note that

K = K m = 0
λ = λ 0 (1 + E(K)). (2.2)
Mostly we will neglect the index m and write shortly

T k = k j=1 X j , k ∈ N, (2.3)
for an iid sequence (X j ) with distribution F X . Further, we assume:

(iii) the file sizes (Z m,k ) m,k∈N 0 are again an iid sequence of positive random variables with distribution function F Z and E|Z| < ∞;

(iv) (Γ m,k ), (K m ), (Z m,k ) and (X j,m ) are independent sequences.

Here,

N (M ) = m∈Z Km k=0 ε (Γ m,k ,Z m,k ) (2.4)
is a stationary marked point process. For details on stationary marked point processes we refer to Karr [START_REF] Karr | Point processes and their statistical inference[END_REF], Daley and Vere-Jones [START_REF] Daley | An Introduction to the Theory of Point Processes[END_REF][START_REF] Daley | An Introduction to the Theory of Point Processes[END_REF] and Faÿ et al. [START_REF] Faÿ | Modeling teletraffic arrivals by a Poisson cluster process[END_REF]. If (T l ) l∈Z denotes the ordered arrival times of (Γ m,k ) m∈Z,k=0,...,Km with corresponding file sizes (Z l ) l∈Z where (Z l ) l∈Z is again an iid sequence with distribution F Z , then the point processes N and N (M ) have the representation

N = l∈Z ε T l and N (M ) = l∈Z ε (T l ,Z l )
where . . . < T -1 < T 0 < 0 < T 1 < . . . as well. Now the amount of data brought to the server is modeled by the input process

A(t) = m∈Z Km k=0 [Z m,k ∧ (t -Γ m,k ) + -Z m,k ∧ (-Γ m,k ) + ] = R×R + f dN (M ) , ( 2.5)
where

f (s, z) = t 0 1 {s<x<s+z} dx for (s, z) ∈ R × R + .
With that representation we are able to compute the first and second moments of A(t) using the Poisson structure of the underlying point process N (M ) and hence, of A. This is presented in the next lemma whose proof is postponed to Section 6.

Lemma 2.1 (a) Let f : R × R + → R be measurable. Then

E R×R + f dN (M ) = λ R×R + f (s, z) F Z (dz) ds. (b) Let f, g : R → R + be measurable. Then Cov R f dN, R g dN = λ 0 ∞ k=-∞ c(|k|) R E(f (s)g(s + T k )) ds where c(k) = E(K -k + 1) + . (c) Let f, g : R × R + → R + be measurable and E|Z| < ∞. Then Cov R×R + f dN (M ) , R×R + g dN (M ) = λ 0 ∞ k=-∞ k =0 c(|k|) R×R 2 + E(f (s, z 1 )g(s + T k , z 2 )) F Z (dz 1 )F Z (dz 2 ) ds +λ R×R + f (s, z)g(s, z) F Z (dz) ds. (d) Let E|Z| < ∞ and h 1 , h 2 , h 3 ≥ 0, h 1 ≤ h 2 < h 3 . Then Cov(A(h 1 ), A(h 3 ) -A(h 2 )) = h 3 h 2 -h 1 (min(h 1 , h 3 -x) -(h 2 -x) + ) g(x) dx
where

g(t) = λ [E(Z 1 -t) + -E(Z 1 ∧ (Z 2 -t) + )] + R 2 1 {s 1 ≤0,s 2 ≤t} F Z (-s 1 )F Z (t -s 2 ) γ (N ) 2 (ds 1 , ds 2 ) (2.6) and γ N 2 (A × B) = λ 0 ∞ k=-∞ c(|k|) A P(s + T k ∈ B) Leb(ds) for A, B ∈ B(R + )
is the covariance measure of N .

A conclusion of Lemma 2.1 (a) and (2.5) is that A(t) has a finite mean under the assumption that Z has a finite mean and

E(A(t)) = λtE(Z) (2.7)
(cf. Mikosch and Samorodnitsky [START_REF] Mikosch | Scaling limits for cumulative input processes[END_REF], Lemma 2.2).

Model assumptions

In this paper the within-cluster interarrival times and the cluster sizes are assumed to be heavytailed which is modeled by regularly varying tail distribution functions. For a positive measurable function f we write

f ∈ R κ for κ ∈ R (f is regularly varying with index κ) if lim x→∞ f (tx) f (x) = t κ ∀ t > 0 .
More details on regular variation can be found, e.g., in Bingham et al. [START_REF] Bingham | Regular Variation[END_REF] and Resnick [START_REF] Resnick | Heavy-Tail Phenomena: Probabilistic and Statistical Modeling[END_REF].

Assumption A (a) The interarrival distribution function F X satisfies

F X ∈ R -1/β with β > 1.
(b) The cluster size distribution function F K satisfies

F K ∈ R -α with 1 < α < min(2, β).
Notice that Assumption A (a) assures that the within-cluster interarrival times have infinite mean; it also makes the arrival process sufficiently irregular for our results. Assumption A (b) makes sure that the file sizes transmitted within each cluster have infinite variance.

Assumption B

Assume that either of the following conditions is satisfied:

(a) β < 2 and lim sup x→∞ x F X (x) -F X (x + 1) F X (x) < ∞. (2.8) (b) F is arithmetic, with step size Δ > 0, and lim sup n≥0 n F X {nΔ} F X (nΔ) < ∞ . ( 2

Assumption C

The marks (Z l ) satisfy one of the following conditions:

(a) E|Z| 2 < ∞. Then we define γ := 2.

(b) F Z ∈ R -γ for some 1 < γ < 2.
Two conclusions from this assumption are important for our study. First, we have by Markov's inequality (in case (a)) and Karamata's Theorem (in case (b)) (cf. Resnick [START_REF] Resnick | Heavy-Tail Phenomena: Probabilistic and Statistical Modeling[END_REF], Theorem 2.1), respectively, the following upper bound of the tail of F Z :

P(Z > x) ≤ Cx -γ 1 for x ≥ 1, (2.10)
where γ 1 = 2 if γ = 2, and γ 1 < γ if 1 < γ < 2, respectively. In this paper γ 1 will be chosen sufficiently close to γ. Second, if we define

a(n) := n 1/2 if γ = 2 , and a(n) := F ← Z (n -1 ) if 1 < γ < 2,
then by Resnick [START_REF] Resnick | Heavy-Tail Phenomena: Probabilistic and Statistical Modeling[END_REF], Theorem 3.3 and Corollary 7.1 we have the central limit theorem

⎛ ⎝ 1 a(n) nt l=1 [Z l -E(Z)] ⎞ ⎠ t≥0 =⇒ (S γ (t)) t≥0 as n → ∞ in (D [0, ∞) , J 1 ), (2.11)
where (S γ (t)) t≥0 is a γ-stable Lévy process and for γ = 2 the limit process is a Brownian motion. The space D[0, 1] (D [0, ∞), respectively) denotes the space of right continuous functions on [0, 1] ([0, ∞), respectively) with left limits (shortly: càdlàg = continue à droite, limitée à gauche). The symbol J 1 reflects that D[0, 1] is provided with the Skorokhod J 1 topology. For completeness, C[0, 1] denotes the space of continuous functions on [0, 1] which we provide with the uniform metric; see Billingsley [START_REF] Billingsley | Convergence of Probability and Measures[END_REF].

Throughout the paper we use the following notation:

H = 2 + β -α 2β , ( 2.12
)

c(k) = E(K -k + 1) + , k ∈ N 0 , ( 2.13
)

G(t) = ∞ k=0 c(k)P(T k ≤ t), t ≥ 0, (2.14)
V (t) = D(α, β)t -1 (α -1) -1 F X (t) -2 P(K > F X (t) -1 ), t ≥ 0, (2.15)
where

D(α, β) = Γ(1 -β -1 ) 2-α Γ(3 -α) βΓ(2H) . (2.16)
Here, Γ denotes the Γ-function. We further use that under Assumption A the inequalities

1 2 < H < 1 and 1 < 2 -H < H -1 < 3 -2H < 2 (2.17)
hold. The next Lemma is crucial for the proof of our results.

Lemma 2.3

Let Assumption A hold. Then we have:

(a) c(k) ∼ (α -1) -1 kF K (k) ∈ R 1-α as k → ∞. (b) V (t) ∈ R 2H-2 as t → ∞, where 2H -2 ∈ (-1, 0). (c) G(t) ∼ (2H -1) -1 tV (t) ∈ R 2H-1 as t → ∞, where 2H -1 ∈ (0, 1). (d) Let additionally Assumption B hold. Then G(t) -G(t -1) ∼ V (t) ∈ R 2H-2 as t → ∞, where 2H -2 ∈ (-1, 0).

Asymptotic behavior of the variance

We start with the investigation of the asymptotic behavior of the variance of N and A. These results are essential to the understanding of the asymptotic behavior of these processes in Section 4 and 5. Similarly we derive now the asymptotic behavior of the covariance functions which show the long range dependence of N and A.

Proposition 3.1 Let Assumption A hold. Then Var(N (0, •]) ∈ R 2H and Var(N (0, t]) ∼ 1 H(2H -1) λ 0 t 2 V (t) as t → ∞. Proposition 3.2 Let Assumption A, B and C hold. (a) If γ > 3 -2H then Var(A(•)) ∈ R 2H , where 2H ∈ (1, 2), and
Var(A(t)) ∼ (E(Z)) 2 H(2H -1) λ 0 t 2 V (t) ∼ (E(Z)) 2 Var(N (0, t]) as t → ∞. (b) If γ < 3 -2H then Var(A(•)) ∈ R 3-γ , where 3 -γ ∈ (1, 2), and
Var(A(t)) ∼ 2λ (2 -γ)(3 -γ)(γ -1) t 3 F Z (t) as t → ∞.

Theorem 3.4 Let Assumption A, B and C hold. Define for

h ≥ 1, γ N (h) := Cov(N (0, 1] , N (h, h + 1]), γ A (h) := Cov(A(1), A(h + 1) -A(h)). Then γ N ∈ R 2H-2
, where 2H -2 ∈ (-1, 0), and

γ N (h) ∼ λ 0 V (h) as h → ∞. (3.1) (a) If γ > 3 -2H then γ A ∈ R 2H-2
, where 2H -2 ∈ (-1, 0), and

γ A (h) ∼ (E(Z)) 2 γ N (h) ∼ (E(Z)) 2 λ 0 V (h) as h → ∞. (3.2) (b) If γ < 3 -2H then γ A ∈ R 1-γ , where 1 -γ ∈ (-1, 0), and
γ A (h) ∼ λ γ -1 hF Z (h) as h → ∞. (3.3)

Asymptotic behavior of the number of packets

An interesting question itself is the behavior of the arrival process of data files modeled by the point process N . On the other hand, we require that result also for the derivation of the asymptotic behavior of the cumulative input to the server. (a) Then

N (0, M n t] -λM n t Var(N (0, M n ]) t≥0 =⇒ (B H (t)) t≥0 as n → ∞ in C [0, 1] ,
where (B H (t)) t≥0 is a standard fractional Brownian motion with Hurst index H given in (2.12).

(b) Let (N i ) be iid copies of N . Then

1 n Var(N (0, M n ]) n i=1 [N i (0, M n t] -λM n t] =⇒ (B H (t)) t≥0 as n → ∞ in C [0, 1] .
Astonishingly in contrast to the results in Faÿ et al. [START_REF] Faÿ | Modeling teletraffic arrivals by a Poisson cluster process[END_REF] the limit results in (a) and (b) do not depend on the growth rate of (M n ).

Asymptotic behavior of the input process

Let us start now with the asymptotic behavior of a mean corrected individual input process A. Afterwards we will investigate the cumulative input to the server.

Theorem 5.1 Let Assumption A, B and C hold and let

(M n ) be a sequence of positive constants such that M n ↑ ∞ as n → ∞. (a) Let γ > H -1 . Then A(M n t) -λM n tE(Z) Var(N (0, M n ]) t≥0 FDD =⇒ (E(Z)B H (t)) t≥0 as n → ∞,
where (B H (t)) t≥0 is a standard fractional Brownian motion with Hurst index H given in (2.12). In particular, if γ > 3 -2H then

A(M n t) -λM n tE(Z) Var(A (M n )) t≥0 =⇒ (B H (t)) t≥0 as n → ∞ in C [0, 1] . (b) Let γ < H -1 . Then A(M n t) -λM n tE(Z) a(M n ) t≥0 FDD =⇒ (λ 1 γ S γ (t)) t≥0 as n → ∞,
where (S γ (t)) t≥0 is the γ-stable Lévy motion given in (2.11).

Remark 5.2 (i)

In the case H -1 < γ < 3-2H we have only stated the convergence of the finite dimensional distributions. To show the tightness in C[0, 1] we can not use the classical arguments of Billingsley [START_REF] Billingsley | Convergence of Probability and Measures[END_REF] since the process does not converge in L 2 . However, we conjecture that the convergence holds in C[0, 1] by proving the L δ convergence for some δ ∈ (H -1 , γ). See also Remark 6.5 in Section 6.4.

(ii) Since the left hand side in (b) has continuous sample paths but the right hand side has jumps, the convergence can not hold in (D[0, 1], J 1 ); cf. Whitt [START_REF] Whitt | Limits for cumulative input processes to queues[END_REF][START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Applications to Queues[END_REF]. However, we conjecture that with some technical effort we can extend the result to convergence in (D[0, 1], M 1 ) as in Resnick and Van den Berg [START_REF] Resnick | Convergence of high-speed network traffic models[END_REF]. Analog statements hold for further results presented in this paper; cf. Theorem 5.5 and Theorem 5.6.

For the asymptotic behavior of the mean corrected cumulative input process we have to distinguish again different cases depending on the heaviness of the file sizes. The reason for these cases is that we have on the one hand, the two different cases for the asymptotic behavior of the variance (γ > 3 -2H, γ < 3 -2H) and on the other hand, the two different cases for the asymptotic behavior of a mean corrected input process (γ > H -1 , γ < H -1 ). These cases result also in different cases for the asymptotic behavior of the mean corrected cumulative input process;

3 -2H < γ ≤ 2, H -1 < γ < 3 -2H and 1 < γ < H -1 . Recall that 1 < H -1 < 3 -2H < 2.

3 -2H < γ ≤ 2

Theorem 5.3 Let Assumption A, B and C hold. Further, let (M n ) be a sequence of positive constants such that M n ↑ ∞ as n → ∞, and either

(a) lim inf n→∞ n -1 γ-1 +η M n > 0 for every η > 0, or (b) lim n→∞ n -1 γ-1 +η M n = 0 for some η > 0.
Further, let (S n,Mn (t)) t≥0 be given as in (1.1) and suppose γ > 3 -2H. Then

[n Var(A(M n ))] -1/2 S n,Mn (t) t≥0 =⇒ (B H (t)) t≥0 as n → ∞ in C [0, 1] .
In particular, if

lim n→∞ n -1 γ-1 M n = C ∈ (0, ∞]
holds, then (a) is valid. The conditions (a) and (b) imply a kind of continuity condition on (M n ) which is natural. However, if E(Z 2 ) < ∞ we can neglect the continuity condition; see Theorem 5.4 below.

In the classical results as the ON/OFF and the infinite source Poisson model with heavytailed file sizes we have contrary to Theorem 5.3 to distinguish between fast and slow growth rates of (M n ). In the slow growth case we obtain the convergence to a γ-stable Lévy motion. Hence, our general Poisson cluster model shows that this is not necessarily the case if we have heavy-tailed file sizes.

Theorem 5.4 Let Assumption A, B and C hold. Further, let (M n ) be a sequence of positive constants such that M n ↑ ∞ as n → ∞, and let (S n,Mn (t)) t≥0 be given as in (1.1). Suppose that

E(Z 2 ) < ∞. Then [n Var(A(M n ))] -1/2 S n,Mn (t) t≥0 =⇒ (B H (t)) t≥0 as n → ∞ in C [0, 1] .

H -1 < γ < 3 -2H

Theorem 5.5 Let Assumption A, B and C hold and let H -1 < γ < 3 -2H. Further, let (M n ) be a sequence of positive constants such that M n ↑ ∞ as n → ∞, and (S n,Mn (t)) t≥0 be given as in (1.1).

(a) Let (M n ) satisfies the fast growth condition and

lim n→∞ n -1 γ-1 +η M n = 0 for some η > 0. Then [n Var(A(M n ))] -1/2 S n,Mn (t) t≥0 =⇒ (B 3-γ 2 (t)) t≥0 as n → ∞ in C [0, 1] .
lim n→∞ n 2-γ 2Hγ-2 -η M -1 n = ∞ (5.2)
for some η > 0. Then

a(nM n ) -1 S n,Mn (t) t≥0 FDD =⇒ (λ 1 γ S γ (t)) t≥0 as n → ∞. (c) Let (M n ) satisfies the slow growth condition lim n→∞ n 2-γ 2Hγ-2 +η M -1 n = 0
for some η > 0. Then

[n Var(N (0, M n ])] -1/2 S n,Mn (t) t≥0 FDD =⇒ (E(Z)B H (t)) t≥0 as n → ∞.

1 < γ < H -1

Theorem 5.6 Let Assumption A, B and C hold and let 1 < γ < H -1 . Further, let (M n ) be a sequence of positive constants such that M n ↑ ∞ as n → ∞, and (S n,Mn (t)) t≥0 be given as in (1.1).

(a) Let (M n ) satisfies the fast growth condition

lim n→∞ n -1 γ-1 +η M n = 0
for some η > 0. Then

[n Var(A(M n ))] -1/2 S n,Mn (t) t≥0 =⇒ (B 3-γ 2 (t)) t≥0 as n → ∞ in C [0, 1] . (b) Let (M n ) satisfies the slow growth condition lim n→∞ n 1 γ-1 +η M -1 n = 0 for some η > 0. Then a(nM n ) -1 S n,Mn (t) t≥0 FDD =⇒ (λ 1 γ S γ (t)) t≥0 as n → ∞.
Remark 5.7 For the ON/OFF and the infinite source Poisson model with heavy-tailed file sizes the growth rates of (M n ) can be equivalently described by the asymptotics of the covariance function γ A . Here, this is only possible for 1 < γ < H -1 .

QUES9196_source.tex; 24/09/2010; 15:40 p. 14 6 Proofs

Proofs of Section 2

Proof of Lemma 2.1 (a) Consider the marked point process

N * = ∞ k=-∞ ε (Γ k ,K k ,(X j,k ) j∈N ,(Z j,k ) j∈N 0 ) in R × N 0 × R N + × R N 0 + ,
which is a Poisson random measure with intensity measure

m * = (λ 0 × Leb) × F K × ∞ j=1 F X ∞ l=0 F Z . Let f * : R × N 0 × R N + × R N 0 + → R be defined as f * (s, k, (x j ), (z j )) = k l=0 f ⎛ ⎝ s + l j=1 x j , z l ⎞ ⎠ .
Then

R×R + f dN (M ) = R×N 0 ×R N + ×R N 0 +
f * dN * . Thus, by Campbell's formula (cf. Baccelli and Brémaud [START_REF] Baccelli | Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences[END_REF], p. 18)

E R×R + f dN (M ) = E R×N 0 ×R N + ×R N 0 + f * dN * = R×N 0 ×R N + ×R N 0 + f * dm * = λ 0 R×N 0 ×R N + ×R N 0 + k l=0 f ⎛ ⎝ s + l j=1 x j , z l ⎞ ⎠ ∞ l=0 F Z (dz l ) ∞ j=1 F X (dx j ) F K (dk) ds = λ 0 (E(K) + 1) R×R + f (s, z) F Z (dz) ds. (b) The covariance measure γ N 2 of N is given by γ (N ) 2 (A × B) = Cov(N (A), N(B)) = λ 0 R γ * ((s + A) × (s + B)) ds (6.1)
where

γ * (A × B) = E K k=0 K m=0 1 {T k ∈A,Tm∈B} for A, B ∈ B(R); see Faÿ et al. [14], Proposition 2.3. Let γ * (A) := E K k=0 K m=0 1 {T k -Tm∈A} = E K k=0 (K -k + 1) 1 {T k ∈A} + E K k=1 (K -k + 1) 1 {-T k ∈A} = ∞ k=1 c(k) [P(T k ∈ A) + P(-T k ∈ A)] + c(0) 1 {0∈A} .
Then by Campbell's formula

Cov R f dN, R g dN = R 2 f (s 1 )g(s 2 ) γ (N ) 2 (ds 1 , ds 2 ) = λ 0 R 2 f (s)g(s + v) ds γ * (dv)
for measurable functions f, g : R → R. In the case f, g positive this can be rewritten as

Cov R f dN, R g dN = λ 0 ∞ k=-∞ c(|k|) R E(f (s)g(s + T k )) ds. (6.2) (c) Finally, the covariance measure γ (M ) 2
of the marked point process N (M ) is given by

γ (M ) 2 (ds 1 , ds 2 , dz 1 , dz 2 ) = γ (N ) 2 (ds 1 , ds 2 )F Z (dz 1 )F Z (dz 2 ) + γ (d) (ds 1 , ds 2 , dz 1 , dz 2 ) (6.3)
where γ (d) is the signed diagonal measure

γ (d) (A 1 × A 2 × B 1 × B 2 ) = λ Leb(A 1 ∩ A 2 ) [F Z (B 1 ∩ B 2 ) -F Z (B 1 )F Z (B 2 )] (6.4) for A 1 , A 2 ∈ B(R), B 1 , B 2 ∈ B(R + ). This means Cov(N (M ) (A 1 × B 1 ), N (M ) (A 2 × B 2 )) = γ (M ) 2 (A 1 × A 2 × B 1 × B 2 ).
For more details we refer to Mikosch and Samorodnitsky [START_REF] Mikosch | Scaling limits for cumulative input processes[END_REF], p. 895. Again Campbell's formula gives

Cov R×R + f dN (M ) , R×R + g dN (M ) = R 2 ×R 2 + f (s 1 , z 1 )g(s 2 , z 2 ) γ (M ) 2
(ds 1 , ds 2 , dz 1 , dz 2 ) (6.5)

for measurable functions f, g : R × R + → R. If f, g are positive the last equation reduces by (6.3) and (6.4) to

Cov R×R + f dN (M ) , R×R + g dN (M ) = λ 0 ∞ k=-∞ k =0 c(|k|) R×R 2 + E(f (s, z 1 )g(s + T k , z 2 )) F Z (dz 1 ) F Z (dz 2 ) ds +λ R×R + f (s, z)g(s, z) F Z (dz) ds =: V 1 (t) + V 2 (t). (6.6) (d) Let us define f (s, z) := [0,h 1 ] 1 {s<x<s+z} dx and f (s, z) := [h 2 ,h 3 ] 1 {s<x<s+z} dx for s ∈ R, z ≥ 0. Then A(h 1 ) = R×R + f dN (M ) and A(h 3 ) -A(h 2 ) = R×R + f dN (M ) .
Thus, a conclusion of (6.5) is

Cov(A(h 1 ), A(h 3 ) -A(h 2 )) = R 2 ×R 2 + f (s 1 , z 1 ) f (s 2 , z 2 ) γ (M ) 2 (ds 1 , ds 2 , dz 1 , dz 2 ) = R 2 ×R 2 + [0,h 1 ] 1 {s 1 <x<s 1 +z 1 } dx [h 2 ,h 3 ] 1 {s 2 <y<s 2 +z 2 } dy γ (M ) 2 (ds 1 , ds 2 , dz 1 , dz 2 ) = [0,h 1 ]×[h 2 ,h 3 ] R 2 ×R 2 + 1 {s 1 <x<s 1 +z 1 ,s 2 <y<s 2 +z 2 } γ (M ) 2
(ds 1 , ds 2 , dz 1 , dz 2 ) dx dy.

If we define

g(x, y) = R 2 ×R 2 + 1 {s 1 <x<s 1 +z 1 ,s 2 <y<s 2 +z 2 } γ (M ) 2 (ds 1 , ds 2 , dz 1 , dz 2 ), then
Cov(A(h 1 ), A(h 3 ) -A(h 2 )) = [0,h 1 ]×[h 2 ,h 3 ]
g(x, y) dx dy.

Since N (M ) is stationary, g depends only on the difference |y -x|. Thus, let g(v) := g(0, v). Then

Cov(A(h 1 ), A(h 3 ) -A(h 2 )) = h 1 0 h 3 -x h 2 -x g(x, x + v) dv dx = h 1 0 h 3 -x h 2 -x g(v) dv dx = h 3 h 2 -h 1 [min(h 1 , h 3 -v) -(h 2 -v) + ] g(v) dv.
Furthermore, (6.3) and (6.4) give

g(v) = R 2 ×R 2 + 1 {s 1 <0<s 1 +z 1 ,s 2 <v<s 2 +z 2 } γ (M ) 2 (ds 1 , ds 2 , dz 1 , dz 2 ) = λ [E(Z 1 -v) + -E(Z 1 ∧ (Z 2 -v) + )] + R 2 1 {s 1 ≤0,s 2 ≤v} F Z (-s 1 )F Z (v -s 2 ) γ (N ) 2 (ds 1 , ds 2 ).
Proof of Lemma 2.3. (a) follows from Karamata's theorem and (b) by the definition of V . (c) is a conclusion from Omey [START_REF] Omey | Multivariate Regular Variation and its Applications in Probability Theory[END_REF] and (d) from Anderson and Athreya [START_REF] Anderson | A strong renewal theorem for generalized renewal functions in the infinite mean case[END_REF], Theorem 2, if Assumption B (a) holds and from Doney [START_REF] Doney | One-sided local large deviation and renewal theorems in the case of infinite mean[END_REF], Theorem 3, if Assumption B (b) holds. By Zolotarev [START_REF] Zolotarev | Mellin-Stieltjes transforms in probability theory[END_REF], Theorem 3, the constants in [START_REF] Anderson | A strong renewal theorem for generalized renewal functions in the infinite mean case[END_REF] and [START_REF] Doney | One-sided local large deviation and renewal theorems in the case of infinite mean[END_REF] are equal.

Proofs of Section 3

Proof of Proposition 3.1.

If we insert f = g = 1 (0,t] in Lemma 2.1 (b) we obtain Var(N (0, t]) = λ 0 ∞ k=-∞ c(|k|) t 0 P(0 < s + T k ≤ t) ds = λt + 2λ 0 ∞ k=1 c(k) t 0 P(T k ≤ v) dv = λt + 2λ 0 t 0 G(v) dv. Since G ∈ R 2H-1 , 2H -1 > 0,
h(t) ≤ Ct 1-γ 1 , t ≥ 1.

(c) Let Assumption C hold, and let

h(t) = ∞ 0 F Z (x)F Z (x + t) dx. Then there exists a C > 0 such that h(t) ≤ Ct -γ 1 , t ≥ 1.

(d) Let Assumption A and C hold, and let

h(t) = ∞ k=1 c(k) ∞ 0 F Z (x)P(Z > x + t + T k ) dx. Then there exists a C > 0 such that h(t) ≤ Ct 2H-γ 1 -1 , t ≥ 1.

(e) Let Assumption A, B and C hold, and let

h(t) = ∞ 0 F Z (x) ∞ k=1 c(k)P(T k < x + t ≤ T k + Z) dx. Then h(t) ∼ (E(Z)) 2 V (t) as t → ∞. Proof. (a) By Lemma 2.3 we know that G(t) -G(t -1) ∼ V (t) as t → ∞. Hence, there exist C 1 , t 0 > 0 such that G(x + t) -G(x + t -1) ≤ C 1 V (x + t) for x ≥ 0, t ≥ t 0 . Let z ≤ t/2 and t ≥ 2t 0 . Then G(x + t) -G(x + t -z) ≤ G(x + t) -G(x + t -z ) = z j=1 [G(x + t -j + 1) -G(x + t -j)] ≤ C 1 z j=1 V (x + t -j + 1) ≤ C 1 z sup 1≤j≤t/2 V (x + t -j + 1) ≤ C 2 (z + 1)V (t),
where we used V (•) ∈ R 2H-2 and Theorem 1.5.3 of Bingham et al. [START_REF] Bingham | Regular Variation[END_REF]. (b) Inequality (2.10) gives

E(Z -t) + = ∞ t P(Z > x) dx ≤ C 3 ∞ t x -γ 1 dx = C 3 γ 1 -1 t 1-γ 1 .
(c) We have again by (2.10)

h(t) ≤ C 4 ∞ 0 F Z (x)(x + t) -γ 1 dx ≤ C 4 t -γ 1 ∞ 0 F Z (x) dx = C 4 E(Z)t -γ 1 . (6.7) (d) Note that h(t) = ∞ 0 F Z (x) ∞ k=1 c(k)P(Z > x + t + T k ) dx = ∞ 0 F Z (x) ∞ x+t ∞ k=1 c(k)P(T k < z -x -t) F Z (dz) dx ≤ ∞ 0 F Z (x) ∞ x+t G(z -x -t) F Z (dz) dx.
Since G is non-decreasing we have

h(t) ≤ ∞ 0 F Z (x) ∞ t G(z) F Z (dz) dx.
Keep in mind that G ∈ R 2H-1 , F Z ∈ R -γ , then Potter's Theorem and Karamata's Theorem give

h(t) ≤ C 5 E(Z)t 2H-γ 1 -1 .
(e) We can write

h(t) = ∞ 0 F Z (x) ∞ 0 ∞ k=1 c(k)P(x + t -z ≤ T k < x + t) F Z (dz) dx = ∞ 0 F Z (x) ∞ 0 [G(x + t) -G(x + t -z)] F Z (dz) dx = ∞ 0 F Z (x) t/2 0 + ∞ t/2 [G(x + t) -G(x + t -z)] F Z (dz) dx =: h 1 (t) + h 2 (t). (6.8)
Thus, we obtain by dominated convergence, (a) and Lemma 2.3 (d)

lim t→∞ h 1 (t) V (t) = ∞ 0 F Z (x) ∞ 0 zF Z (dz) dx = (E(Z)) 2 . (6.9)
Further,

h 2 (t) = ∞ 0 F Z (x) ∞ t/2 [G(x + t) -G(x + t -z)] F Z (dz) dx ≤ ∞ 0 F Z (x)G(x + t)F Z (t/2) dx = F Z (t/2) t 0 + ∞ t F Z (x)G(x + t) dx =: h 21 (t) + h 22 (t). (6.10)
Then (2.10), Potter's Theorem and Lemma 2.3 (c) give for some 0 < < 1

h 21 (t) ≤ C 6 t -γ 1 t 0 F Z (x)(x + t) 2-α β + dx = C 6 t 2-α β + -γ 1 t 0 F Z (x) x t + 1 2-α β + dx ≤ C 7 E(Z)t 2-α β + -γ 1 .
Next,

h 22 (t) ≤ C 8 t -γ 1 ∞ t x -γ 1 (x + t) 2-α β + dx = C 8 t -γ 1 ∞ t x 2-α β + -γ 1 1 + t x 2-α β + dx ≤ C 9 t -γ 1 t 2-α β + -γ 1 +1 . Finally, since 2-α β + -γ 1 = 2H -1 + -γ 1 ≤ 2H -2 for small we have h 2 (t) ≤ C 10 t 2-α β + -γ 1 = o(h 1 (t)) as t → ∞. (6.11)
Thus, (6.8)-(6.11) result in the proof of (d).

Lemma 6.2 Let Assumption A, B and C hold, and g be defined as in (2.6).

(a) If γ > 3 -2H then g(t) ∼ λ 0 (E(Z)) 2 V (t) ∈ R 2H-2 as t → ∞. (b) If γ < 3 -2H then g(t) ∼ λ γ -1 tF Z (t) ∈ R 1-γ as t → ∞.
Proof. We use the decomposition

g(t) = λ [E(Z 1 -t) + -E(Z 1 ∧ (Z 2 -t) + )] + R 2 1 {s 1 ≤0,s 2 ≤t} F Z (-s 1 )F Z (t -s 2 ) γ (N )
2 (ds 1 , ds 2 ) =: g 1 (t) + g 2 (t).

(6.12)

From Lemma 6.1 (b) we conclude that

g 1 (t) ≤ 2λE(Z -t) + ≤ Ct 1-γ 1 . (6.13)
Further, we have (cf. [START_REF] Willinger | Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level[END_REF] of Mikosch and Samorodnitsky [START_REF] Mikosch | Scaling limits for cumulative input processes[END_REF])

g 2 (t) = λh 1 (t) + λ 0 h 2 (t) + λ 0 h 3 (t) (6.14)
where

h 1 (t) = ∞ 0 F Z (x)F Z (x + t) dx, h 2 (t) = ∞ k=1 c(k) ∞ 0 F Z (x)P(Z > x + t + T k ) dx, h 3 (t) = ∞ k=1 c(k) ∞ 0 F Z (x)P(T k < x + t ≤ T k + Z) dx. Note that V (•) ∈ R 2H-2 by Lemma 2.3 and 2H -2 ∈ (-1, 0). Then Lemma 6.1 (c)-(e) give g 2 (t) ∼ λ 0 h 3 (t) ∼ λ 0 (E(Z)) 2 V (t) as t → ∞. (6.15)
Let γ > 3 -2H. Then (6.12), (6.13) and (6.15) result in (a).

If γ < 3 -2H then F Z ∈ R -γ and g 1 (t) ∼ λ γ -1 tF Z (t) as t → ∞. (6.16)
The reason is that

E(Z 1 ∧ (Z 2 -t) + ) ≤ E(Z 1 )P(Z 2 > t) = o(tF Z (t)) as t → ∞,
and by Karamata's Theorem

E(Z -t) + ∼ 1 γ -1 tF Z (t) as t → ∞.
Thus, (b) follows from (6.12), (6.15) and (6.16).

Proof of Proposition 3.2.

We have by Lemma 2.1 (d) that

Var(A(t)) = t -t [min(t, t -x) -(-x) + ]g(x) dx = 2 t 0 (t -x)g(x) dx. (6.17)
Let γ > 3-2H. Then from Lemma 6.2 we know that g ∈ R 2H-2 . Hence, by Karamata's Theorem

Var(A(t)) ∼ 2 2H -1 t 2 g(t) - 1 H t 2 g(t) ∼ (E(Z)) 2 (2H -1)H λ 0 t 2 V (t) as t → ∞
follows. In the case γ < 3 -2H the same arguments lead to the result.

Proof of Theorem 3.4. First, we show (3.1). By Lemma 2.1 (b) we have with f = 1 (0,1] , g = 1 (h,h+1] (cf. Faÿ et al. [START_REF] Faÿ | Modeling teletraffic arrivals by a Poisson cluster process[END_REF], equation (2.9)) that

γ N (h) = λ 0 1 0 ∞ k=1 c(k) [P(T k ≤ v + h) -P(T k ≤ v + h -1)] dv = λ 0 1 0 [G(v + h) -G(v + h -1)] dv. Since G(u) -G(u -1) ∼ V (u) ∈ R 2H-2 as u → ∞
γ N (h) ∼ λ 0 V (h) as h → ∞.
Next, we investigate the asymptotic behavior of γ A . Here, Lemma 2.1 (d) gives

γ A (h) = h+1 h-1 [min(1, h + 1 -x) -(h -x) + ]g(x) dx = 1 -1 g(z + h)(1 -|z|) dz. Further, g ∈ R 2H-2 if γ > 3 -2H and g ∈ R 1-γ if γ < 3 -2H
, respectively (by Lemma 6.2) such that by Embrechts et al. [START_REF] Embrechts | Modelling Extremal Events for Insurance and Finance[END_REF], Lemma 1.3.5 (a), we obtain

γ A (h) ∼ g(h) 1 -1 (1 -|z|) dz = g(h) as h → ∞,
which proves the Theorem.

Proofs of Section 4

For the proof of Theorem 4.1 we use the following notation. Let (6.18) where V and D(α, β) are given in (2.15) and (2.16), respectively, and

b n := nM n F (M n ) -2 P(K > F X (M n ) -1 ) = 1 D(α, β)(α -1) -1 nM 2 n V (M n ), n ∈ N,
σ 2 := 2λ 0 2 + β -α ∞ 0 y -(2+β-α)/β P(S 1/β (1) ≤ y) dy+ (6.19) +λ 0 ∞ 0 E 2 2 -α I(w + 1) 2-α + 2 α -1 I(w)I(w + 1) 1-α - 2 (2 -α)(α -1) I(w) 2-α dw,
where (S 1/β (t)) t≥0 is a 1/β-stable Lévy process which satisfies

nt k=1 X k F ← X (n -1 ) t≥0 =⇒ (S 1/β (t)) t≥0 as n → ∞ in (D [0, ∞) , J 1 ), and
I(w) = inf{t ≥ 0 : S 1/β (t) > w} for w > 0.
Furthermore, for the proof of Theorem 4.1 we require the next Lemma.

Lemma 6.3 Let the assumptions and the notation of Theorem 4.1 hold. Further, let (b n ) be given as in (6.18) and σ 2 be given as in (6.19), respectively. Then

n Var(N (0, M n ]) ∼ σ 2 b 2 n as n → ∞,
and in particular,

σ 2 = λ 0 (2H -1)H D(α, β)(α -1) -1 = Γ(1 -β -1 ) 2-α Γ(3 -α) β(α -1)(2H -1)HΓ(2H)
.

Proof. We decompose N (0, M n t] in the independent components

N (0, M n t] = N + (0, M n t] + N -(0, M n t]
where N + (0, M n t] is the number of arrivals of packets in (0, M n t] belonging to a cluster initiated in (0, M n t] and N -(0, M n t] is the number of arrivals of packets in (0, M n t] belonging to a cluster initiated in (-∞, 0], respectively, i.e.

N + (0, M n t] = #{(m, k) ∈ Z 2 : Γ m ∈ (0, M n t] , Γ m,k ∈ (0, M n t] , k ∈ {0, . . . , K m }}, N -(0, M n t] = #{(m, k) ∈ Z 2 : Γ m ∈ (-∞, 0] , Γ m,k ∈ (0, M n t] , k ∈ {0, . . . , K m }}.
Furthermore,

N ± (0, M n t] -E(N ± (0, M n t]) b n
are infinitely divisible with characteristic triplet (0, 0, ν ± n,t ) where for A ∈ B(R):

ν + n,t (A) = λ 0 Mnt 0 P b -1 n K k=0 ε T k [0, u] ∈ A du, ν - n,t (A) = λ 0 ∞ 0 P b -1 n K k=0 ε T k (u, u + M n t] ∈ A du;
for more details see Fasen and Samorodnitsky [START_REF] Fasen | A fluid cluster Poisson input process can look like a fractional Brownian motion even in the slow growth aggregation regime[END_REF], p. 405. Thus, Sato [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF], Example 25.12, and the independence of N + and N -result in

Var N (0, M n t] b n = Var N + (0, M n t] b n + Var N -(0, M n t] b n = ∞ 0 x 2 ν + n,t (dx) + ∞ 0 x 2 ν - n,t (dx).
Define ν ± n,t := n ν ± n,t . Now, Lemma 5.1 and Lemma 5.2 of Fasen and Samorodnitsky [START_REF] Fasen | A fluid cluster Poisson input process can look like a fractional Brownian motion even in the slow growth aggregation regime[END_REF] (which are also valid for = ∞) give

n b 2 n Var(N (0, M n t]) = ∞ 0 x 2 ν + n,t (dx) + ∞ 0 x 2 ν - n,t (dx) ∼ t 2H σ 2 + + t 2H σ 2 - as n → ∞, where σ 2 + + σ 2 -= σ 2 .
Proof of Theorem 4.1. We start with the proof of (b). (b) Fasen and Samorodnitsky [START_REF] Fasen | A fluid cluster Poisson input process can look like a fractional Brownian motion even in the slow growth aggregation regime[END_REF], Proposition 5.1, says that

1 b n n i=1 [N i (0, M n t] -λM n t] FDD =⇒ (σB H (t)) t≥0 as n → ∞, ( 6.20)
which gives together with Lemma 6.3 the convergence of the finite dimensional distributions in (b). For the proof of the tightness we use Billingsley [START_REF] Billingsley | Convergence of Probability and Measures. 1st edn[END_REF], Theorem 12.3. Therefore, it is sufficient to prove

E|N n (t)| 2 ≤ Ct 2H-δ for t ∈ [0, 1], (6.21)
some C > 0, 0 < δ < 2H -1, where

N n (t) := (n Var(N (0, M n ])) -1 2 n i=1 [N i (0, M n t] -λM n t] for t ∈ [0, 1],
is stationary. Note that

E|N n (t)| 2 = Var(N (0, M n t]) Var(N (0, M n ]) . Proposition 3.1 gives Var(N (0, •]) ∈ R 2H .
Thus, Potter's Theorem (Bingham et al. [START_REF] Bingham | Regular Variation[END_REF], Theorem 1.5.6) and t ∈ [0, 1] result in (6.21).

(a) Let ν n,t be the Lévy measure of N n (t), and ν n,t be the Lévy measure of N (0, M n t] -λM n t Var(N (0, M n ]) .

Then

ν n,t (•) = 1 n ν n,t 1 √ n • .
We have to prove

(i) ν n,t (•) = 1 n ν n,t ( 1 √ n •) ν =⇒ 0 on (0, ∞] as n → ∞, (ii) lim n→∞ {|x|≤1} x 2 ν n,t (dx) = lim n→∞ {|z|≤n -1/2 } z 2 ν n,t (dz) = t 2H , (iii) lim n→∞ {|x|>1} x ν n,t (dx) = lim n→∞ 1 √ n {|z|>n -1/2 } z ν n,t (dz) = 0,
which reflects the convergence of the characteristic triplet of the infinitely divisible random variable N n (t) to the characteristic triplet of a Gaussian random variable with mean 0 and variance t 2H . Then the claim follows by Kallenberg [START_REF] Kallenberg | Foundations of Modern Probability[END_REF], Theorem 15.14. The proof of (i)-(iii) can be done step by step as in Fasen and Samorodnitsky [START_REF] Fasen | A fluid cluster Poisson input process can look like a fractional Brownian motion even in the slow growth aggregation regime[END_REF], Lemma 5.1 and Lemma 5.2 using lim n→∞ n -1 2 F (M n )b n = ∞ and Lemma 6.3.

QUES9196_source.tex; 24/09/2010; 15:40 p. 24

Proofs of Section 5

For the proof of Theorem 5.1 we require the following Lemma.

Lemma 6.4 Let Assumption A, B and C hold, and define

A (1) (t) = l∈Z 1 {T l ≤0} ((T l + Z l ) + ∧ t), t ≥ 0, A (2) (t) = l∈Z 1 {0<T l ≤t} (T l + Z l -t) + , t ≥ 0. (a) Suppose E(Z 2 ) < ∞. Then 0 ≤ E(A (1) (t)) = E(A (2) (t)) ≤ λ 2 E(Z 2 ), t ≥ 0. (b) Suppose 1 < γ < 2. Then E(A (1) (t)) = E(A (2) (t)) ∼ λ(2 -γ) -1 (γ -1) -1 t 2 F Z (t) as t → ∞. Proof. Let f (s, z) = 1 {s≤0} ((s + z) + ∧ t).
Then we obtain by Lemma 2.1 (a)

E(A (1) (t)) = λ ∞ 0 0 -∞ ((s + z) + ∧ t) ds F Z (dz) = λ ∞ 0 0 -∞ t 0 1 {x<z+s} dx ds F Z (dz) = λ t 0 ∞ x F Z (s) ds dx. (a) If E(Z 2 ) < ∞ then E(A (1) (t)) = λ ∞ 0 F Z (s) s∧t 0 dx ds ≤ λ ∞ 0 F Z (s)s ds = λ 2 ∞ 0 F Z ( √ z) dz = λ 2 E(Z 2 ). (b) Define h(x) = ∞ x F Z (s) ds.
Then by Karamata's Theorem we know that h ∈ R 1-γ and h(x) ∼ (γ -1) -1 xF Z (x) as x → ∞. Thus, again with Karamata's Theorem we have

E(A (1) (t)) = λ t 0 h(x) dx ∼ λ(2 -γ) -1 th(t) ∼ λ(2 -γ) -1 (γ -1) -1 t 2 F Z (t) as t → ∞.
The equality E(A (1) (t)) = E(A (2) (t)) was proven in Mikosch and Samorodnitsky [START_REF] Mikosch | Scaling limits for cumulative input processes[END_REF], p. 905.

Proof of Theorem 5.1. We use the decomposition

A(M n t) = N (0,Mnt] l=1 Z l + l∈Z 1 {T l ≤0} ((T l + Z l ) + ∧ M n t) - l∈Z 1 {0<T l ≤Mnt} (T l + Z l -M n t) + =: A (0) (M n t) + A (1) (M n t) -A (2) (M n t),
where A (0) (M n t) represents the amount of data whose transmission is initiated in (0, M n t]. On the other hand, A (1) (M n t) reflects the data in (0, M n t] whose transmission was initiated before 0.

Finally, A (2) (M n t) is the data transfer going on after M n t but which was started in (0, M n t]. Then (1) (M n t) -A (2) (M n t)] =: S 1 (M n t) + S 2 (M n t). (6.22) On the one hand, applying Theorem 4.1 we know that

A(M n t) -λM n tE(Z) = ⎡ ⎣ N (0,Mnt] l=1 Z l -λM n tE(Z) ⎤ ⎦ + [A
N (0, M n t] -λM n t Var(N (0, M n ]) t≥0 =⇒ (B H (t)) t≥0 as n → ∞ in C [0, 1] . (6.23)
On the other hand, by (2.11)

Mnt l=1 [Z l -E(Z)] a(M n ) t≥0 =⇒ (S γ (t)) t≥0 as n → ∞ in (D[0, 1], J 1 ). (6.24)
(a) Furthermore, by Proposition 3.1 and Potter's Theorem there exists for some 0

< < H -γ -1 a C 1 > 0 such that for M n large a 2 (M n ) Var(N (0, M n ]) ≤ C 1 M 2 γ + n M 2H- n = C 1 M 2 γ -2H+2 n which tends to 0 as n → ∞. Hence, Mnt l=1 [Z l -E(Z)] Var(N (0, M n ]) t≥0 P -→ (0) t≥0 as n → ∞ in (D[0, 1], J 1 ). (6.25)
Then (6.23), (6.25) and Whitt [START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Applications to Queues[END_REF], Corollary 13.3.2 result in

S 1 (M n t) Var(N (0, M n ]) t≥0 =⇒ (E(Z)B H (t)) t≥0 as n → ∞ in (D [0, 1] , J 1 ). (6.26)
Note that 2γ -H < 0 since γ > H -1 > 1. Moreover, by Proposition 3.1, Lemma 6.4 and Potter's Theorem there exists for some 0 < < H+γ-2 2 a constant C 2 > 0 such that for t > 0 and M n large

E(A (j) (M n t)) Var(N (0, M n ]) ≤ C 2 (M n t) 2-γ+ M H- n ≤ C 2 M 2-γ-H+2 n , j = 1, 2,
which tends to 0 as n tends to ∞. Hence, Markov's inequality gives for any δ > 0,

lim n→∞ P A (j) (M n t) Var(N (0, M n ]) > δ = 0, which means S 2 (M n t) Var(N (0, M n ]) P -→ 0 as n → ∞.
(6.27)

The conclusion follows then from (6.22), (6.26) and (6.27).

In the case γ > 3 -2H we can prove the convergence in C[0, 1]. Thus, we show that (A n (t)) t∈ [0,[START_REF] Anderson | A strong renewal theorem for generalized renewal functions in the infinite mean case[END_REF] , where

A n (t) = A(M n t) -λM n tE(Z) Var(A(M n )) ,
is tight by Billingsley [START_REF] Billingsley | Convergence of Probability and Measures. 1st edn[END_REF], Theorem 12.3. As in Theorem 4.1 it is sufficient to prove that there exists a 0 < δ < 2H -1 such that

E|A n (t)| 2 ≤ C 3 t 2H-δ for t ∈ [0, 1] (6.28)
and some C 3 > 0. We have

E|A n (t)| 2 = Var(A(M n t)) Var(A(M n )) .
Since Proposition 3.2 says that Var(A(•)) ∈ R 2H , Potter's Theorem shows (6.28) as well.

(b) Here we have again by Potter's Theorem that for some constant C 4 > 0:

Var(N (0, M n ]) a 2 (M n ) ≤ C 4 M 2H+ n M 2 γ - n = C 4 M 2H-2 γ +2 n , which tends to 0 as n → ∞. Hence, N (0, M n t] -λM n t a(M n ) t≥0 =⇒ (0) t≥0 as n → ∞ in C [0, 1] . ( 6.29)
Again Whitt [START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Applications to Queues[END_REF], Corollary 13.3.2, (6.24) and (6.29) give

S 1 (M n t) a(M n ) t≥0 =⇒ (λ 1 γ S γ (t)) t≥0 as n → ∞ in (D [0, 1] , J 1 ). (6.30)
Finally, by Lemma 6.4 and Potter's Theorem there exists for some 0 < < (γ-1)

2 2γ a constant C 5 > 0 such that E(A (j) (M n t)) a(M n ) ≤ C 5 M 2-γ+ n M 1 γ - n = C 5 M - (γ-1) 2 γ +2 n -→ 0 as n → ∞, j = 1, 2.
Thus, for any t ∈ [0, 1],

S 2 (M n t) a(M n ) P -→ 0 as n → ∞, ( 6.31)
and we obtain the result by (6.22), (6.30) and (6.31).

Remark 6.5 In both (a) and (b) we saw that the properly normalized process (S 1 (M n t)) t≥0 converges in (D[0, 1], J 1 ). However, the proof of the tightness of the normalized process (S 2 (M n t)) t≥0 can be difficult independently of the topology in D[0, 1]. Although the tightness of the scaled process (A (1) (M n t)) is by A (1) (M n t) ≤ A (1) (M n ), 0 ≤ t ≤ 1, obvious, the tightness of (A (2) (M n t)) is involved.

Proof of Theorem 5.3.

(a) We start to prove the convergence of the finite dimensional distributions. Therefore, we use for i = 1, . . . , n the decomposition

A i (t) = N i (0,t] l=1 Z (i) l + l∈Z 1 {T (i) l ≤0} ((T (i) l + Z (i) l ) + ∧ t) - l∈Z 1 {0<T (i) l ≤t} (T (i) l + Z (i) l -t) + =: A (0) i (t) + A (1) i (t) -A (2) i (t). Then S n,Mn (t) = n i=1 [A i (M n t) -λM n tE(Z)] = n i=1 ⎡ ⎣ N i (0,Mnt] l=1 Z (i) l -λM n tE(Z) ⎤ ⎦ + n i=1 [A (1) i (M n t) -A (2) i (M n t)] =: S (1) n,Mn (t) + S (2)
n,Mn (t).

(6.32)

First, we investigate the asymptotic behavior of (S

n,Mn (t)) t≥0 . A conclusion of Theorem 4.1 and Proposition 3.

2 is that n i=1 [N i (0, M n t] -λM n t] n Var(A (M n )) t≥0 =⇒ (E(Z) -1 B H (t)) t≥0 as n → ∞ in C[0, 1]. (6.33)
Next, note that

⎛ ⎝ n i=1 ⎡ ⎣ N i (0,Mnt] l=1 Z (i) l -λM n tE(Z) ⎤ ⎦ ⎞ ⎠ t≥0 d = ⎛ ⎜ ⎝ n i=1 N i (0,Mnt] l=1 Z l -λnM n tE(Z) ⎞ ⎟ ⎠ t≥0 , ( 6.34)
and that by (2.11) nMnt l=1

[Z l -E(Z)] a(nM n ) t≥0 =⇒ (S γ (t)) t≥0 as n → ∞ in (D[0, 1], J 1 ). (6.35)
By assumption (a) we have that there exists a constant C 1 ∈ (0, ∞) such that

n 1 γ-1 -M -1 n ≤ C 1 for n ∈ N. (6.36)
Moreover, by Proposition 3.2 and Potter's Theorem we have that for any > 0 there exist C 2 > 0 and n 0 ∈ N such that for n ≥ n 0 :

a(nM n ) 2 n Var(A (M n )) ≤ C 2 (nM n ) 2 γ + nM 2H- n = C 2 (n 1 γ-1 -M -1 n ) 2Hγ-2 γ -2 n 3-2H-γ γ-1
+ , (6.37) where = 2 γ-1 + 2Hγ-2 γ -2 2 + . For small the right hand side of (6.37) tends to 0 by (6.36).

Hence, Corollary 13.3.2 of Whitt [START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Applications to Queues[END_REF], (6.33), (6.35) and (6.37) result in

⎛ ⎜ ⎜ ⎜ ⎜ ⎝ n i=1 N i (0,Mnt] l=1 Z l -λnM n tE(Z) n Var(A(M n )) ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ t≥0 =⇒ (B H (t)) t≥0 as n → ∞ in (D [0, 1] , J 1 ),
and finally by (6.34) in

⎛ ⎝ S (1) n,Mn (t) n Var(A(M n )) ⎞ ⎠ t≥0 =⇒ (B H (t)) t≥0 as n → ∞ in (D [0, 1] , J 1 ). (6.38) Since γ > 3 -2H is equivalent to - 1 2 (γ -1) > 2 -H -γ,
there exists an > 0 such that

- 1 2 (γ -1) 1 1 -(γ -1) -2 > 2 -H -γ.
Finally, by Potter's Theorem, Proposition 3.2, Lemma 6.4 and (6.36) we have for some C 3 , C 4 > 0 and n large

nE(A (j) (M n t)) n Var(A (M n )) ≤ C 3 n(M n t) 2-γ+ (nM 2H-2 n ) 1 2 ≤ C 3 (n 1 γ-1 -M -1 n ) γ-1 2(1-(γ-1)) M - n t 2-γ+ ≤ C 4 M - n , j = 1, 2, (6.39)
which tends to 0 as n → ∞. Then Markov's inequality and (6.39) result in

lim n→∞ P n i=1 A (j) i (M n t) n Var(A (M n ))
> δ = 0 for any δ > 0, and consequently, as n → ∞,

1 n Var(A (M n )) S (2) n,Mn (t) = 1 n Var(A (M n )) n i=1 A (1) i (M n t) -A (2)
i (M n t) P -→ 0. (6.40) Thus, (6.32), (6.38), (6.40) and Theorem 25.4 of Billingsley [START_REF] Billingsley | Probability and Measure[END_REF] give

1 n Var(A(M n )) n i=1 [A i (M n t) -λM n tE(Z)] t≥0 FDD =⇒ (B H (t)) t≥0 as n → ∞.
The tightness follows as in Theorem 5.1 by Proposition 3.2 and Billingsley [START_REF] Billingsley | Convergence of Probability and Measures. 1st edn[END_REF], Theorem 12.3.

(b) In the fast growth case the result follows from 3 -2H > γ, Lemma 6.2 (a), and Mikosch and Samorodnitsky [START_REF] Mikosch | Scaling limits for cumulative input processes[END_REF], Theorem 4.2, where we have E|M (0)| 2+ < ∞ for some > 0 by Lemma 4.1 of Mikosch and Samorodnitsky [START_REF] Mikosch | Scaling limits for cumulative input processes[END_REF] and Faÿ et al. [START_REF] Faÿ | Modeling teletraffic arrivals by a Poisson cluster process[END_REF], p. 124. Remark 6.6 If F Z ∈ R -γ with 1 < γ < 2 then similar computations as in Lemma 6.7 show that for some constant C > 0:

Var(A (1) (t)) ∼ Ct 3 F Z (t) as t → ∞.
However, the second moment of A (2) (t) does not exist (V 2 (t) = ∞ in (6.6)) such that with our approach we can not neglect the continuity condition in Theorem 5.3 in general.

Proof of Theorem 5.4. The proof follows as in Theorem 5.3 (a). The only difference is that we use instead of (6.37),

a(nM n ) 2 n Var(A (M n )) ≤ C nM n nM 2H- n = CM 1-2H+ n -→ 0 as n → ∞,
which does not depend on the growth rate, and instead of (6.39) that

nE(A (j) (M n t) -E(A (j) (M n t))) 2 n Var(A(M n )) = Var(A (j) (M n t)) Var(A(M n )) ≤ Ct M n M 2H- n n→∞ -→ 0
by an application of Lemma 6.7 below and E(A (1) (M n t)) = E(A (2) (M n t)).

Lemma 6.7 Let Assumption A, B and C hold and

E(Z 2 ) < ∞. (a) Let A (1) (t) = l∈Z 1 {T l ≤0} ((T l + Z l ) + ∧ t), t ≥ 0. Then there exist C, t 0 > 0 such that Var(A (1) (t)) ≤ Ct for t ≥ t 0 . (b) Let A (2) (t) = l∈Z 1 {0<T l ≤t} (T l + Z l -t) + , t ≥ 0. Then there exist C, t 0 > 0 such that Var(A (2) (t)) ≤ Ct for t ≥ t 0 .
Proof.

(a) We use the decomposition (6.6) of the variance in

Var(A (1) (t)) = V 1 (t) + V 2 (t),
where f (s, z) = g(s, z) = 1 {s≤0} ((s + z) + ∧ t). Without loss of generality we can assume λ 0 = 1.

Then, with Fubini's Theorem

V 1 (t) = ∞ k=-∞ k =0 c(|k|) R×R 2 + E 1 {s≤0,s+T k ≤0} ((s + z 1 ) + ∧ t)((s + T k + z 2 ) + ∧ t) F Z (dz 1 )F Z (dz 2 ) ds = 2 -1 k=-∞ c(-k) R -×R 2 + [0,t]×[0,t] E 1 {x<s+z 1 } 1 {y<s+T k +z 2 } dx dy F Z (dz 1 )F Z (dz 2 ) ds = 2 ∞ k=1 c(k) R + [0,t]×[0,t] F Z (x + v)P(y < -v -T k + Z) dx dy dv.
By the monotonicity of F we have

V 1 (t) ≤ 2t t 0 ∞ 0 F Z (v) ∞ k=1 c(k)P(Z > y + v + T k ) dv dy. A conclusion of Lemma 6.1 (d) with γ 1 = 2 is V 1 (t) ≤ 2t t 0 0 + t t 0 ∞ 0 F Z (v) ∞ k=1 c(k)P(Z > y + v + T k ) dv dy ≤ 2tt 0 (E(Z)E(G(Z)) + C 1 t 2H-2+ ) ≤ C 2 t, t ≥ 1. p. 30
Note, min(x, y) 2 ≤ xy for x, y ≥ 0. Hence,

V 2 (t) = R + R 1 {s≤0} ((s + z) + ∧ t) 2 ds F Z (dz) ≤ R + 0 -∞ (s + z) + t ds F Z (dz) = t R + 0 -z (s + z) ds F Z (dz) = t R + z 2 2 F Z (dz) = t E(Z 2 ) 2 .
(b) With the notation of (6.6) and f (s, z) = g(s, z) = 1 {0<s≤t} (s + zt) + we have for v ≥ v 0 and some v 0 > 1. Since G ∈ R 2H-1 we obtain by Karamata's Theorem and (6.41), (6.42), (6.43), (6.47), We use now the fact that if γ > H -1 then γ -1 (2Hγ -2) > 0. Moreover, Var(N (0, •]) ∈ R 2H by Proposition 3.1. Thus, by Potter's Theorem for any > 0 there exist C 1 > 0 and n 0 ∈ N such that for n ≥ n 0 we have n Var(N (0,

Var(A (2) (t)) = V 1 (t) + V 2 (t). First, V 1 (t) = ∞ k=-∞ k =0 c(|k|) R×R 2 + E 1
F Z (x + v) dx ≤ E(Z 2 ) ∞ 0 1 (x + v) 2 dx ≤ C 1 v -1 . ( 6
V 1 (t) = 2 v 0 0 + t v 0 R + F Z (x + v) dx R + ∞ k=1 c(k)P(y + v -Z < T k ≤ v) dy dv ≤ C 8 + C 9 t v 0 G(v)v -1 dv ∼ C 10 G(t)
M n ]) a(nM n ) 2 ≤ C 1 nM 2H+ n (nM n ) 2 γ - = C 1 n -2-γ γ M 2Hγ-2 γ n n M 2 n = C 1 n 2-γ 2Hγ-2 M -1 n -2Hγ-2 γ -2 n +2
(nM n ) 1 γ - = C 2 n γ-1 γ M -1 γ (γ-1) 2 n n M 2 n t 2-γ+ = C 2 n + 2 γ-1 (n 1 γ-1 M -1 n ) (γ-1) 2 γ
-2 t 2-γ+ , (6.54) which tends to 0 as n → ∞ for small enough by (5.1). Thus, Markov's inequality gives

P 1 a(nM n ) n i=1 A (j) i (M n t) > δ ≤ nE(A (j) (M n t))
a(nM n ) -→ 0 as n → ∞ for any δ > 0, and finally for t > 0, S

n,Mn (t) a(nM n ) P -→ 0 as n → ∞. (6.55)

The statement in (b) is then a conclusion of (6.32), (6.53) and (6.55).

(c) The conclusion follows from (6.32), (6.35), (6.50), (6.54), Whitt [START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Applications to Queues[END_REF], Corollary 13. which tends to 0 as n → ∞ for small enough by the small growth condition.

Proof of Theorem 5.6.

(a) We use the arguments as in Theorem 5.6 (a). (b) On the one hand, γ < 2 implies 1 -2γ -1 < 0. On the other hand, since γ < H -1 we have 2 Hγ -1 < 0. Hence, again by Potter's Theorem for any > 0 there exist C 1 > 0 and n 0 ∈ N such that for n ≥ n 0 n Var(N (0,

M n ]) a(nM n ) 2 ≤ C 1 nM 2H+ n (nM n ) 2 γ - = C 1 n 1-2 γ + M 2(H-γ -1 )+2 n (6.56)
which tends to 0 as n → ∞ for small enough. Note, that here we not require the slow growth condition. Then the conclusion follows from (6.32), (6.35), (6.50), (6.54), Whitt [START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Applications to Queues[END_REF], Corollary 13.3.2 and (6.56).

n

  ) are both sequences of positive constants tending to ∞ as n → ∞ with M (1) n << M (2) n where n << M (1) n reflects the slow, M (1) n << n << M (2) n the intermediate and n >> M (2)

Remark 3. 3 (a) If γ > 3 -

 33 2H the asymptotic behavior of the variance is completely determined by the point process N . The file sizes Z have no influence on the rate of increase of the variance.(b) In contrast, if γ < 3-2H, the rate of increase of the variance comes only from the file sizes.If we have an infinite source Poisson model with F Z ∈ R -γ for some 1 < γ < 2 then we have the same asymptotic behavior of the variance (cf. Mikosch and Samorodnitsky[START_REF] Mikosch | Scaling limits for cumulative input processes[END_REF], p. 895).

Theorem 4 . 1

 41 Let Assumption A and B hold, and let (M n ) be a sequence of positive constants such that M n ↑ ∞ as n → ∞.

1 γ

 1 Let (M n ) satisfies the intermediate growth condition lim n→∞ n

2 [ 1 y+v/ 2 y 0 [J 1 1 1 y 2 dy + 1 1 01

 2120111 )P(y + v -Z < T k ≤ v) )P(y + vz < T k ≤ v) F Z (dz) + ∞ y+v ∞ k=1 c(k)P(T k ≤ v) F Z (dz) = G(v) -G(y + vz)] F Z (dz) + ∞ y+v G(v) F Z (dz) =: J 1 (y, v) + J 2 (y, v) + J 3 (y, v). (6.43) Since G ∈ R 2H-1we have for large v by Lemma 2.3 and Lemma 6.1 (a) thatJ 1 (y, v) ≤ C 2 G(v)v -(zy) F Z (dz) ≤ C 2 G(v)v -1 v 2 P(Z > y) ≤ C 3 G(v)y -2 1 {y≥1} +C 3 G(v) 1 {0≤y≤1} . (6.44)Note that by the monotonicity of G and Markov's inequality we obtainJ 2 (y, v) ≤ G(v)P(Z > y + v/2) ≤ C 4 G(v)(y + v/2) -2 . (6.45) Finally, J 3 (y, v) = G(v)P(Z > y + v) ≤ C 5 G(v)(y + v) -2 . (6.46) Thus, (6.44)-(6.46) result in ∞ (y, v) + J 2 (y, v) + J 3 (y, v)] dy ≤ C 6 G(v) ∞ dy ≤ C 7 G(v) (6.47)

  3.2, anda(nM n ) 2 n Var(N (0, M n ]) ≤ C 3 n

  gives us the infinite source Poisson model. The times (Γ m,k ) correspond in a telecommunication model with the arrival times of data files, denoted by (Z m,k ), at the server. The cluster Poisson point process describing the arrivals of data files is then

		Km
	N =	ε Γ m,k	(2.1)
	m∈Z	k=0
	where ε denotes the dirac measure. This means N (s, t] counts the number of arrivals of data
	packets in the time interval (s, t]. If E(K) < ∞ then by Faÿ et al. [14], Proposition 2.1, the
	intensity of N is given by

  by Lemma 2.3 (c), we conclude with Karamata's Theorem,

	Var(N (0, t]) ∼	1 H(2H -1)	λ 0 t 2 V (t) as t → ∞.
	The proof of Proposition 3.2 is based on some auxiliary Lemmatas.
	Lemma 6.1
	(a) Let Assumption A and B hold, and let h(z, t) = G(x + t) -G(x + t -z) for x ≥ 0. Then
	there exist t 0 , C > 0 such that
	h(z, t) ≤ C(z + 1)V (t)
	for x > 0, z < t/2, t ≥ t 0 .

(b) Let Assumption C hold, and let h(t) = E(Zt) + . Then there exists a C > 0 such that

  by Lemma 2.3 (d) we obtain by Embrechts et al. [13], Lemma 1.3.5 (a), that

  {0<s≤t,0<s+T k ≤t} (s + z 1t) + (s + T k + z 2t) + F Z (dz 1 )F Z (dz 2 ) ds {s+T k ≤t} 1 {x<s+z 1 -t} 1 {y<s+T k +z 2 -t} ) dx dy F Z (dz 1 )F Z (dz 2 ) ds

		∞
	= 2	k=1	c(k)	[0,t]×R 2 + R 2
		t
	= 2			F
		0	R +

+ E(1 Z (x + v) dx R + ∞ k=1 c(k)P(y + v -Z < T k ≤ v) dy dv. (

6

.41) Markov's inequality gives ∞ 0

  dsF Z (dz) = tλE(Z 2 ). (6.49) Hence, (6.48) and (6.49) complete the proof. Proof of Theorem 5.5. (a) follows again by Lemma 6.2 (b), and Mikosch and Samorodnitsky [26], Theorem 4.2. (b) We use again the decomposition given in (6.32). On the one hand, applying Theorem 4.1 we have that

			as t → ∞.	(6.48)
	Finally,
	V 2 (t) = λ			R +	0	t	z 2 (6.50)
	Furthermore, note that γ < 3 -2H is equivalent to
	2 -γ 2Hγ -2	>	1 γ -1	.

R + R 1 {0<s≤t} (s + zt) 2 + ds F Z (dz) ≤ λ n i=1 [N i (0, M n t] -λM n t] n Var(N (0, M n ]) t≥0 =⇒ (B H (t)) t≥0 as n → ∞ in C [0, 1] .

  → ∞ in (D [0, 1] , J 1 ).(6.53) Return to mind that E(A (j) (•)) ∈ R 2-γ by Lemma 6.4, j = 1, 2. Thus, by Potter's Theorem for any > 0 there exist C 2 > 0 and n 0 ∈ N such that for n ≥ n 0 we have

	QUES9196_source.tex; 24/09/2010; 15:40	p. 32
	A conclusion of (6.34) and (6.52) is that
		⎛ ⎝	S a(nM n ) (1) n,Mn (t)	⎞ ⎠	t≥0	=⇒ λ	1 γ S γ (t)
		nE(A (j) (M n t)) a(nM n )	≤ C 2	n(M n t) 2-γ+
							2-γ
							2Hγ-2	(6.51)
	which tends to 0 as n → ∞ for small enough by (5.2). Hence, Corollary 13.3.2 of Whitt [34],
	(6.50), (6.35) and (6.51) result in
	⎛ ⎜ ⎜ ⎜ ⎜ ⎝	n i=1 N i (0,Mnt] l=1 a(nM n ) Z l -λnM n tE(Z)	⎞ ⎟ ⎟ ⎟ ⎟ ⎠

t≥0

=⇒ λ 1 γ S γ (t) t≥0 as n → ∞ in (D [0, 1] , J 1 ). (6.52) t≥0 as n

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QUES9196_source.tex; 24/09/2010; 15:40 p.[START_REF] Fasen | A fluid cluster Poisson input process can look like a fractional Brownian motion even in the slow growth aggregation regime[END_REF]

QUES9196_source.tex; 24/09/2010; 15:40 p.[START_REF] Kaj | Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach[END_REF]

QUES9196_source.tex; 24/09/2010; 15:40 p.[START_REF] Kallenberg | Foundations of Modern Probability[END_REF]

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QUES9196_source.tex; 24/09/2010; 15:40 p.[START_REF] Levy | On renewal processes having stable inter-renewal intervals and stable rewards[END_REF]

QUES9196_source.tex; 24/09/2010; 15:40 p.[START_REF] Pipiras | Slow, fast and arbitrary growth conditions for renewalreward processes when both the renewals and the rewards are heavy-tailed[END_REF]

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