Internet-Type Queues with Power-Tailed Interarrival Times and Computational Methods for Their Analysis

Abstract

Internet traffic flows have often been characterized as having power-tailed (long-tailed, fattailed, heavy-tailed) packet interarrival times or service requirements. In this work, we focus on the development of complete and computationally efficient steady-state solutions of queues with power-tailed interarrival times when the service times are assumed exponential. The classical method for obtaining the steady-state probabilities and delay-time distributions for the G/M/1 (G/M/c) queue requires solving a rootfinding problem involving the Laplace-Stieltjes transform of the interarrival-time distribution function. Then the exponential service assumption is combined with the derived geometric arrival-point probabilities to get both the limiting general-time state and delay distributions. However, in situations where there is a power tail, the interarrival transform is typically quite complicated and never analytically tractable. In addition, it is possible that there is only a degenerate steady-state system-size probability distribution. Thus, an alternative approach to obtaining a steady-state solution is typically needed when power-tailed interarrivals are present. We exploit the exponentiality of the steady-state delay distributions for the G/M/1 and G/M/c queues, using level crossings and a transform-approximation method, to develop an alternative rootfinding problem when there are power-tailed interarrival times. Extensive computational results are given.

The origin of bursts and heavy tails in human dynamics

“…In some environments this second mechanism, whose origin is addressed in this paper, can be just as important as the much investigated first one. Given the differences in routing performance under Poisson and Pareto arrival time distributions [20,21,22], a queuing based model of human-driven arrival times could contribute to a better understanding of Internet traffic as well.Uncovering the mechanisms governing the timing of various human activities has significant scientific and commercial potential. First, models of human behavior are indispensable for large-scale models of social organization, ranging from detailed urban models [23,24], to modeling the spread of epidemics and viruses, the development of panic [25] or capturing financial market behavior [26].…”

Analysis of GI X /M(n)//N systems with stochastic customer acceptance policy

“…Furthermore, the matrix O = (o ij ) i,j ∈S where o ij = 1 {j =0} , with 1 {D} denoting the indicator function of statement D, is the smallest stochastic matrix in the Kalmykov ordering sense. These facts are on the basis to bound the matrix through (12) for some N ∈ N + , where the bounding matrices (N ) and¯ (N ) are stochastic matrices constructed using the expression (8) for as explained in the following definition [see ( [4], Theorem 5.8) for more details].…”

“…We have included the Pareto distribution in the list since its importance in queueing has increased recently with the finding that heavy tailed distributions are adequate to model Internet packet interarrival times [6,8,12,17]. The batch interarrival time distributions presented in this section have the following parameterizations with common positive mean λ −1 : deterministic with value λ −1 , exponential with rate λ, Erlang with four phases, i.e., E 4 ≡ E 4 (4λ); and the two parameter Pareto P(β, κ = (β − 1)/(λβ)), β > 1, with probability density function βκ β (κ + x) −(β+1) , for x > 0.…”

Bayesian Analysis of a Queueing System with a Long-Tailed Arrival Process

“…However, due to the features of internet data described above, classical queueing models cannot be applied in this context and new, more complex queueing models are demanded, see e.g. Fischer and Harris (1999), Greiner et al (1999), Harris et al (2000) and Fisher et al (2001). In particular, we note that if the interarrival distribution is heavy tailed, then it will often not possess a LT in closed form and therefore, the usual queueing theory techniques to calculate equilibrium distributions etc.…”

“…Fisher and Harris (1999) developed the Transform Approximation method or TAM, where the integral in the LT is substituted by a finite sum. Harris et al (2000) introduced an alternative way for directly finding the predictive equilibrium distributions, the Level Crossing method. Here we shall apply these two last methods and provide a comparison between them.…”