On Scaling Limits of Arrival Processes with Long-Range Dependence

Various classes of arrival processes in telecommunication traffic modeling based on heavy-tailed interarrival time distributions exhibit long-range dependence. This includes arrival rate processes of Anick-MitraSondhi (AMS) type where the rate process is an on/off-process with heavy-tailed on-period distribution and/or off-period distribution, as well as generalized Kosten type models (infinite source Poisson) with rate process given by the M/G/∞ queueing model with heavy-tailed service time distribution. The nature of such arrival processes can be studied by investigating rescaling limit processes that arise from using appropriate space-time scaling schemes. The typical behavior is the following dichotomy, which is described below: If the connection rate per time unit is fast then the generic limit process is fractional Brownian motion, whereas if the number of connections increases slowly relative to time then a stable Levy process appears as the approximating limit process. Such results have been obtained by Taqqu et al. (1997) and Levy and Taqqu (2000), for cases where the rescaling of time and space variables are performed in two separate steps. Similar results are obtained for joint scaling in time and space in Mikosh et al. (2002) for on-off processes and infinite source Poisson models and in Pipiras et al. (2002) for renewal rate processes. Gaigalas and Kaj (2002) consider an arrival process built from stationary renewal processes with heavy tailed interrenewal times. In addition to fast and slow connection rates an intermediate critical scaling regime is studied and a new limit process is established. This limiting process is neither Gaussian nor stable. It has continuous paths and stationary increments but is not self-similar. In this note we extend the results of Gaigalas and Kaj (2002) to a class of renewal reward processes and discuss the interpretation of the scaling limit process. Consider a sequence of independent, non-negative random variables U1, U2, . . . with distribution functions F1(t) = P (U1 ≤ t) and F (t) = P (Uk ≤ t), k ≥ 2. It is assumed that the distribution given by F (t) has finite expected value μ = E(U2) and that T1 is equipped with the equilibrium distribution function F1(t) = Feq(t) = 1 μ ∫ t 0 (1−F (s)) ds. LetNt, t ≥ 0, denote the stationary renewal counting process associated with the sequence of interrenewal times (Uk)k≥1 and, in addition, let (Xk)k≥1 be an i.i.d. sequence of random variables, independent of Nt, signifying rewards associated with each renewal event: