Modeling teletraffic arrivals by a Poisson cluster process

Abstract

Abstract. In this paper we consider a Poisson cluster process N as a generating process for the arrivals of packets to a server. This process generalizes in a more realistic way the infinite source Poisson model which has been used for modeling teletraffic for a long time. At each Poisson point Γj, a flow of packets is initiated which is modeled as a partial iid sum process Γj + P k i=1 Xji, k ≤ Kj , with a random limit Kj which is independent of (Xji) and the underlying Poisson points (Γj). We study the covariance structure of the increment process of N . In particular, the covariance function of the increment process is not summable if the right tail P (Kj > x) is regularly varying with index α ∈ (1, 2), the distribution of the Xji's being irrelevant. This means that the increment process exhibits long-range dependence. If var(Kj) < ∞ long-range dependence is excluded. We study the asymptotic behavior of the process (N (t)) t≥0 and give conditions on the distribution of Kj and Xji under which the random sumsXji have a regularly varying tail. Using the form of the distribution of the interarrival times of the process N under the Palm distribution, we also conduct an exploratory statistical analysis of simulated data and of Internet packet arrivals to a server. We illustrate how the theoretical results can be used to detect distributional characteristics of Kj , Xji, and of the Poisson process.

Scaling Limits for Cumulative Input Processes

“…If the process N c is a randomly stopped renewal process, we obtain a model that we will here call a renewal Poisson cluster process. This model was studied in Faÿ et al [11], see also Hohn and Veitch [18] and Hohn et al [19] for some empirical studies. It can be explicitly constructed as follows.…”

“…We refer the reader to Faÿ et al [11] for the details on the above statements and on more information on the behavior of the ground process and its possible limits.…”

“…Bounds on the tail of S L are readily available in many standard cases; see, for example, Embrechts et al [10,Theorem A3.20] or Faÿ et al [11].…”

“…We give a limit theorem in one such situation, when the tails of K dominate those of the marks. Such a model was studied in Faÿ et al [11] and applied to real-life and simulated data. In this case the scaling limit is determined by the tail of K, as the following result shows.…”

Prediction in a Poisson cluster model

“…More recently, shot noise processes have been used for modeling large computer networks such as the Internet; see for example Konstantopoulos and Lin [12], Kurtz [13] for some early work. In the context of the workload of large computer networks, shot noise processes arise as aggregated versions of the ON/OFF or infinite source Poisson models, also known as M/G/∞ model; see for example Levy and Taqqu [16], Pipiras and Taqqu [24], Mikosch et al [22] and for further extensions Faÿ et al [2], Mikosch and Samorodnitsky [23]. Other applications include finance (Samorodnitsky [25], Klüppelberg and Kühn [8]) and physics (Giraitis et al [3]).…”