Fractional Brownian Input Research Articles

Statistical characteristics of queue with fractional Brownian motion input

The factional Brownian motion has attracted significant attention because it accurately represents Internet traffic characteristics and is amenable to analysis. A link between the probability density function of the steady-state queue size of a queue with a fractional Brownian input and the generalised Gamma distribution is discovered, and the mean, variance, third central moment and skewness are provided. New simulation results that validate the accuracy of these queueing statistics are also provided.

Performance evaluation and service rate provisioning for a queue with fractional Brownian input

The Fractional Brownian motion (fBm) traffic model is important because it captures the self-similar characteristics of Internet traffic, accurately represents traffic generated as an aggregate of many sources, which is a prevalent characteristic of many Internet traffic streams, and, as we show in this paper, it is amenable to analysis. This paper introduces a new, simple, closed-form approximation for the stationary workload distribution (virtual waiting time) of a single server queue fed by an fBm input. Next, an efficient approach for producing a sequence of simulations with finer and finer detail of the fBm process is introduced and applied to demonstrate good agreement between the new formula and the simulation results. This method is necessary in order to ensure that the discrete-time simulation accurately models the continuous-time fBm queueing process. Then we study the limitations of the fBm process as a traffic model using two benchmark models — the Poisson Pareto Burst Process model and a truncated version of the fBm. We determine by numerical experiments the region where the fBm can serve as an accurate traffic model. These experiments show that when the level of multiplexing is sufficient, fBm is an accurate model for the traffic on links in the core of an internet. Using our result for the workload distribution, we derive a closed-form expression for service rate provisioning when the desired blocking probability as a measure of quality of service is given, and apply this result to a range of examples. Finally, we validate our fBm-based overflow probability and link dimensioning formulae using results based on a queue fed by a real traffic trace as a benchmark and demonstrate an advantage for the range of overflow probability below 1% over traffic modelling based on the Markov modulated Poisson process.

Effective bandwidth for a single server queueing system with fractional Brownian input

The traffic patterns of today’s IP networks exhibit two important properties: self-similarity and long-range dependence. The fractional Brownian motion is widely used for representing the traffic model with the properties. We consider a single server fluid queueing system with input process of a fractional Brownian motion type. Packet-loss probability and mean delay are considered as QoS. We show that there is a scaling property among the stationary queue-length distributions of different input parameters and service rates. We also evaluate the scaling factor. From the scaling property, we drive formulas for the effective bandwidth to guarantee the QoS in a single source and multiple sources cases. The formulas indicate that it is essential to evaluate the distribution functions of a type of random variables. For multiple sources the shape of an admissible region and multiplexing gain are analyzed. Finally, numerical examples are shown to validate the proposed scheme.

Exact asymptotics for a queue with fractional Brownian input and applications in ATM networks

In this paper, a single channel FIFO fluid queue with an infinite buffer space and a long-range dependent input is studied. The input traffic is modeled by an average input rate plus a standard fractional Brownian motion as the fluctuation. Lower and upper bounds are derived for the tail distribution of the transient queue length at time T, which result in a logarithmic characterization of the asymptotic behavior of the tail distribution. Furthermore, the exact asymptotic is also obtained. It is observed that the transient queue length under fractional Brownian input does not suffer from the heavy-tail property as does the steady-state queue length. The results are used to compute the equivalent bandwidth requirement for ATM broadband connections with fractional Brownian traffic feed and finite connection holding time.

Exact asymptotics for a queue with fractional Brownian input and applications in ATM networks

In this paper, a single channel FIFO fluid queue with an infinite buffer space and a long-range dependent input is studied. The input traffic is modeled by an average input rate plus a standard fractional Brownian motion as the fluctuation. Lower and upper bounds are derived for the tail distribution of the transient queue length at time T, which result in a logarithmic characterization of the asymptotic behavior of the tail distribution. Furthermore, the exact asymptotic is also obtained. It is observed that the transient queue length under fractional Brownian input does not suffer from the heavy-tail property as does the steady-state queue length. The results are used to compute the equivalent bandwidth requirement for ATM broadband connections with fractional Brownian traffic feed and finite connection holding time.

Large buffer asymptotics for the queue with fractional Brownian input

In this paper, a strong asymptotic estimate for the queue content distribution of a fluid queue fed by a fractional Brownian input with Hurst parameter H ∊ [1/2,1[is studied. By applying general results on suprema of centred Gaussian processes, in particular, we show that for large x. Explicit formulae for constants κ, γ and L are given in terms of H and system parameters.

Large buffer asymptotics for the queue with fractional Brownian input

In this paper, a strong asymptotic estimate for the queue content distribution of a fluid queue fed by a fractional Brownian input with Hurst parameter H ∊ [1/2,1[is studied. By applying general results on suprema of centred Gaussian processes, in particular, we show that for large x. Explicit formulae for constants κ, γ and L are given in terms of H and system parameters.

Large buffer asymptotics for the queue with fractional Brownian input

In this paper, a strong asymptotic estimate for the queue content distribution of a fluid queue fed by a fractional Brownian input with Hurst parameter H ∊ [1/2,1[is studied. By applying general results on suprema of centred Gaussian processes, in particular, we show that for large x. Explicit formulae for constants κ, γ and L are given in terms of H and system parameters.