Poisson Shot Noise Traffic Model and Approximation of Significant Functionals

and Applied Analysis 3 we propose the followingmodel describing the CIP over [0, t] by a “moving average type”:


Introduction
The aim of this paper is to present a method for approximating the traffic that accumulates in an Internet server.We suppose that a unique server deals with an infinite sized source that sends data over independent transmissions to the server according to a stationary Poisson process.We are interested in the cumulative input of work to the server (also called total accumulated work).This stochastic process generated by the traffic of the transmission over an interval of time [0, ] is denoted by   and called cumulative input process (briefly CIP).The terminology work could be understood as a portion of bandwidth that has to be allocated in order to evacuate the traffic.We also used on purpose the terminology transmission because we will see that the approximation we propose could be applied for the traffic of packets upstream and/or packets downstream, or even to more global objects related to the traffic such as downloads from the Internet.The approximation we propose works for fixed and single type of transmissions.Different types of transmissions simply induce the superposition of their corresponding CIP.
In the specialized literature, it seems that the rules are to assume a mechanism of evolution in time of this CIP [1][2][3][4][5][6].The inconvenience of these assumptions on the evolution in time is that they are quite impossible to check.
In [7], we presented a model based on a general Poisson shot noise representation without imposing any such a mechanism.With minimal technical assumptions we showed that the CIP can be approximated in a certain sense by a nicely tractable stochastic process, namely, a stable process with only positive jumps.The sense is the weak convergence in law of a modified version of the CIP to the stable process.This sense is extremely useful because it allows some functionals of the CIP to converge to appropriate functionals of the stable process.This will be the central key for forecasting the bandwidth allocation via stable processes.A widespread literature is devoted to stable processes and powerful statistical methods are available; see [8].The procedure of approximation for the CIP is resumed as follows: within an interval of time [0, ] large enough, we observe a certain number of transmissions ordered according to the first arrived.In Internet traffic each transmission can be traced by its related packets and several data can be extracted for each transmission.We showed in [7] with Theorem 1, stated below, that the nice approximation of the CIP holds weakly in law with limit , a strict stable process totally skewed to the right, for some correcting terms   ,   .The latter holds under these three easily checkable properties: (1) the arrivals of the transmissions form a Poisson process (eventually with intensity increasing at large time scales); (2) the size of each transmission is heavy tailed with infinite variance.In probabilistic terminology, the right tail of their distribution is regularly varying with index − for some  ∈ (0, 2).This parameter of heaviness gives the index of self-similarity of the stable process; (3) the length of each transmission has finite expectation (other alternative weaker assumptions are feasible).
A comparative study with several of the existing models is also provided in our paper [7] and illustrates why our assumptions are weaker and more tractable.
In the present paper we start in Section 2 by presenting the model for the CIP.In Section 3 we give its corresponding limit theorem.In Section 4 an approximation procedure for functionals of the CIP, namely, (1) the supremum process corresponding to the maximum input of work; (2) the right inverse, corresponding to the first passage time process over critical barriers; (3) the storage mapping, corresponding to the process solution of a storage equation.
We will see that we obtain limit theorems similar to those obtained in [7] and no additional assumption is needed for achieving this goal.The following result is stated in Theorem 3 under the same formalism than Theorem 1, take  any of the mappings behind the previous functionals.Then, there exist correcting terms  ⋆  ,  ⋆  and a companion mapping Section 5 is devoted to the proofs.The tools we used there are based on the monograph of Whitt [9].

The Model
Through all the following, we assume that a unique server deals with an infinite sized source.Transmissions arrive to the server according to a homogeneous Poisson process on R labeled by its points (  ) ∈Z so that ⋅ ⋅ ⋅  −1 <  0 < 0 <  1 ⋅ ⋅ ⋅ and hence {− 0 ,  1 , ( +1 −   ,  ̸ = 0)} are i.i.d.exponentially distributed random variables with parameter    is considered as the time of initiation of the th transmission.
Let the counting measure (d) = ∑ ∈Z    (d) and define the Poisson process  by The quantity   represents the number of transmissions started between time  = 0 and time  = .We are interested in the cumulative input process (CIP) generated over an interval of time [0, ] and denoted by   .It corresponds to the size of the files transmitted by the source.There are many ways to model it (from the most trivial way to the most sophisticated).Specification of the source behavior could be taken into account adding more and more parameters.In order to avoid this intricacy, most of the authors [2][3][4][5][6] have a macroscopic approach strongly connected with times of initiation of the transmissions, their duration, and their rate.As we will see, this paper confirms the pertinence of this approach, and we show that it is sufficient for having the required control on the cumulative input process.
Our aim is to describe, in the more general setting, the CIP and to give an approximation of the law of functionals of this process.Notice that the CIP describes the work generated over the interval [0, ].Time  = 0 is when our "observation starts" and  =  is when our "observation finishes." Observe that times of initiation of transmissions are either positive or negative (before or after our observation starts).The th transmission starts at time   and continues over the interval of time [  , +∞).
Suppose we observed the transmissions since ever and until time  and we want to calculate the work generated by the th transmission.This work holds over the random interval (−∞, ] ∩ [  , +∞).The length of this interval is the r.v.
We deduce that the work generated by the th transmission is given by a quantity which depends on the length ( −   ) + .We will denote this work by where (  ()) ∈R is a stochastic process; the random variable   () is an increasing function of , vanishing if  ≤ 0 and describing the quantity of work that could be generated by the th transmission over an interval of time of length .If we had observed only over the interval (  , ] (instead of (−∞, ]), the work that could be generated by the th transmission should be written as the difference we propose the following model describing the CIP over [0, ] by a "moving average type": The processes (  ) ∈Z are assumed to be an i.i.d.sequence independent of the arrival process (  ) ∈Z .For each  ∈ Z, the process   has right continuous left limited sample paths, vanishing on the negative real axis and increasing to a finite r.v.  (∞).
We will see in the sequel that the asymptotic distributional behavior of the process (  ) ≥0 is the same as the one of its "finite memory" part which is a Poisson shot noise (see [10,11]): Observe that the process  has stationary increments, while  has not.It is then natural to introduce the process which characterizes the total work required by all the transmissions started within the interval [0, ].This process enjoys a very special property: it is a Lévy process; that is, it has independent and stationary increments, and is a compound Poisson process (see the appendix and [12] for more account on Lévy processes).This process turns out to be the principal component of the processes  and , a component which will give the right approximation by a stable process, as stated in Theorem 1 below.Another special process is defined by This process is special because it is stationary.Contrary to the others, it has problems of definitions.Namely, the contribution of the past (before time 0) could make the random variables   be infinite.The same problem can occur for the process , but it is actually finite under our assumptions.The processes  and  are very well defined because they are finite sums.This is the reason why we will only consider the processes , , and .These processes are increasing.Because these processes are closely connected to the load of the transmissions, we will call them load processes.We will see in next section that, when adequately modified, they satisfy a weak limit theorem and share a common limiting process which is a totally skewed to right stable process.

Forecasting the CIP
We present here the main result obtained in [7].This result may appear highly theoretical.The assumptions under which the result works are actually easy to check on real internet data.We state the result under its theoretical form and give right after the indications to its best exploitation.

Technical Assumptions for the Approximation of the CIP.
The next theorem shows that the load processes  = , , or , after being correctly drifted and normalized, are approximated in law by a strict stable process totally skewed to the right (see [7,12] for more account on stable processes).
The main keys for proving these results were (1) the infinite divisibility property of the load processes ; (2) the stationary increments property of the CIP  and the fact that it shares the same finite-dimensional expectations as the total work process  : (3) the stationary and independent increments property of the process .

The Assumptions.
Recall that the processes   are i.i.d.Let and the stopping time  defined by The r.v. ∞ and  are actually versions of the size and the length of any transmission.

Behavior of the Size of a
Transmission.The r.v. ∞ is finite and has a regularly varying tail of index −, with  ∈ (0, 2); that is, there exists a deterministic increasing function (), such that lim Condition ( 13) is equivalent to where  is a slowly varying function (i.e., for all  > 0, lim  → ∞ ()/() = 1).Actually, the function () is simply the quantile function () = inf{ : P( ∞ > ) −1 ≥ }, that is, the generalized inverse of the function   /().It is known that () =  1/ () for some slowly varying function  and we can always choose an increasing version of .For more account on regular variation theory, the reader is referred to [13].
3.1.3.Assumption on the Arrivals.Notice that the intensity parameter  of the Poisson arrival process  is not necessarily fixed.It may depend on a scale  and  =   may go to infinity with  ↑ ∞ as studied by Maulik and Resnick [14] and Kaj [1].Through all the following, we are in the situation where   =  0 is fixed or   is increasing to a value  0 ∈ (0, ∞] when  goes to infinity.

Technical Assumptions on the Length of the Transmissions: Connection with the Intensity and with the Size.
Actually, at large time scales, we do not really need to distinguish between the two cases of finite intensity (  =  0 or   ↑  0 ∈ (0, ∞)) and infinite intensity (  ↑ ∞), with the latter being obviously more technical.We will only need this assumption: In case of finite intensity at large scales,  =  0 or  =   ↑  0 ∈ (0, ∞), it is also obvious that condition ( 15) is equivalently expressed with   = 1.Moreover, in this case, it is simply implied by E[] < ∞.See [7] for more comments on assumption (15).Theorem 1 presented below is proved in [7].It generalizes many existing models (see [7] for a comparison) and treats the complete infinite variance case  ∈ (0, 2) but also provides more powerful approximations (the weak convergence in law) with less conditions than what was used to be assumed in the literature.It states that the processes , , and , when correctly drifted and normalized, are attracted in law by a common non-Brownian -stable process  (the Case  = 2 corresponding to the Standard Brownian motion will be explicitly excluded in the sequel).

Importance of the Mode of Convergence.
The convergence in Theorems 1 and 3 holds in the weak mode in the space D = D(R+, R) of Càdlàg functions endowed with the  1 -Skorohod topology.We will not get into details about Skorohod topologies; we just say that, according to [15, 3.20 page 350], a sequence (   ) ≥0 of stochastic processes is said to converge in law to a process (  ) ≥0 , and we denote if and only if the limit process  is well identified via finite-dimensional convergence; that is, for all  ≥ 1 and  1 ,  2 , . . .,   we have convergence in distribution of the R  random variable and the sequence of processes   must be tight, where tightness is a technical criterion (strongly related to the modulus of continuity of the topology) ensuring the existence of the limit.This paper mostly used the powerful tools on  1 topology presented in the book of Jacod and Shiryaev [15].This Topology is nicely tractable, since many important functionals are continuous and preserve convergence.This is the aim of the next section.We stress that all convergences in Theorem 3 also hold in the space (D,  1 ).

Approximation of the CIP by a Stable
Limit.Notice that stable processes and their functionals are very studied in the probability theory literature (see [8]).We recall that it is defined as a Lévy process (see [12] for more account), with no positive jumps and whose Laplace transforms are given for every ,  ≥ 0 by E[ −  ] =  Ψ() : where  ∈ (0, 2) is the index of stability and Theorem 1.The processes , , and  are attracted in law by a common stable process [7].Assume (13) and (15).Let  = , , or  and  be a strict -stable process totally skewed to the right.Define for each  ≥ 0 where the function  is given by (13), the function  is defined by and the function ℎ is given by (19).Letting  ↑ ∞, one gets the weak convergence in law

Forecasting the Bandwidth
The main idea of this work is to extend convergence ( 22) and give its equivalent on functionals of the process .
The approximated functionals will enable to forecast the badwidth allocation in order to avoid congestion.For this purpose, we present three natural and important functionals that are continuous on (D,  1 ) and also weak-convergencepreserving there.Theorem 3 provides limit theorems for functionals of the processes , , and .
4.1.Some Important Continuous Mappings on the Space (D, 1 ).Let D ↑ the subset of processes that are in D and are increasing, D  the subset of processes in D in that are unbounded above and null in 0 and D ↑  the subset of processes  in that are in D ↑ ∩ D  and satisfy  (−1) (0) = 0. Let a process  in D and consider for all  ≥ 0: (1) The right inverse or first passage time process  (−1) is defined by The random variable  (−1) () is simply the first time that the stochastic process  crosses some critical barrier  and then describes the congestion time.
(2) The supremum and the infimum processes  ↑ and  ↓ are defined by The r.v  ↑ () and  ↓ () are, respectively, the maximum and the minimum of the values of the process  over the interval of time [0, ] and then, they describe the extreme workloads.
More could be said on the reflection mapping.The pair of processes ( ∨ , ) is said to solve the Skorohod problem associated with the process  with  ∨ () ≥ 0, () is nondecreasing, (0) = 0, and the reflection condition ∫ ∞ 0  ∨ ()d() = 0 holds.See [16, page 375] for more details on Skorohod problem.For the sake of illustration of this problem, consider the problem of a server dealing with an input process  and serving at a constant rate  > 0.Then, the buffer content process denoted by  ,∨ has positive paths and solution of the following storage equation with service rate : Rewriting the storage equation d() = d(  − ) + d() with d() = 1 (()=0) d, it is easy to see that which corresponds to the reflection mapping of the drifted process (  − ) ≥0 .For this reason we call   →  ,∨ the storage mapping which coincides with   →  ∨ when  = 0.

The Result.
Having justified the importance of such functionals of the CIP, we will apply them on  = , , or .We will show that, correctly normalized as in ( 22), these functionals of  are approximated by a companion functional of the limiting -stable process .Of course the approximation depends on the index of stability .This fact is explained by Theorem 3.
Recall the functions , ℎ, and  given, respectively, by ( 13), (21), and (19).When 1 <  < 2, we obviously have lim For more readability, the proofs of the last lemma and the following theorem are postponed to Section 5. Now, recall the first passage time processes  (−1) is given by (23), the supremum  ↑ is given by (24), and the reflection mapping process  ∨ is given by (25).Theorem 3. Functionals of the CIP are attracted in law by functionals of the stable process.