Ilkka Norros

A storage model with self-similar input

A storage model with self-similar input process is studied. A relation coupling together the storage requirement, the achievable utilization and the output rate is derived. A lower bound for the complementary distribution function of the storage level is given.

On the use of fractional Brownian motion in the theory of connectionless networks

An abstract model for aggregated connectionless traffic, based on the fractional Brownian motion, is presented. Insight into the parameters is obtained by relating the model to an equivalent burst model. Results on a corresponding storage process are presented. The buffer occupancy distribution is approximated by a Weibull distribution. The model is compared with publicly available samples of real Ethernet traffic. The degree of the short-term predictability of the traffic model is studied through an exact formula for the conditional variance of a future value given the past. The applicability and interpretation of the self-similar model are discussed extensively, and the notion of ideal free traffic is introduced. >

An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions

The Radon-Nikodym derivative between a centred fractional Brownian motion Z and the same process with constant drift is derived by finding an integral transformation which changes Z to a process with independent increments. A representation of Z through a standard Brownian motion on a finite interval is given. The maximum-likelihood estimator of the drift and some other applications are presented.

The superposition of variable bit rate sources in an ATM multiplexer

When variable-bit-rate sources are multiplexed in an asynchronous transfer mode (ATM) network, there arise queues with a particular form of correlated arrival process. Such queues are analyzed by exploiting a result expressing the distribution of work in system of the G/G/1 queue originally derived by V.E. Benes (1963). A simple alternative demonstration of this result is analyzed and extended to the case of fluid input systems. The result is applied first to a queue where the arrival process is a superposition of periodic sources (the Sigma D/sub i//D/1 queue), and then to a variable-input-rate constant-output-rate fluid system. The latter is shown to model the so-called burst component of the considered superposition queuing process. The difference between this and the real queue, the cell component, can be evaluated by means of the results obtained for the Sigma D/sub i//D/1 queue. The relative importance of these two components is explored with reference to the particular case of a superposition of on/off sources. >

On a conditionally Poissonian graph process

Random (pseudo)graphs G N with the following structure are studied: first, independent and identically distributed capacities Λ i are drawn for vertices i = 1, …, N; then, each pair of vertices (i, j) is connected, independently of the other pairs, with E(i, j) edges, where E(i, j) has distribution Poisson(Λ i Λ j / ∑ k=1 N Λ k ). The main result of the paper is that when P(Λ1 > x) ≥ x −τ+1, where τ ∈ (2, 3), then, asymptotically almost surely, G N has a giant component, and the distance between two randomly selected vertices of the giant component is less than (2 + o(N))(log log N)/(-log (τ − 2)). It is also shown that the cases τ > 3, τ ∈ (2, 3), and τ ∈ (1, 2) present three qualitatively different connectivity architectures.

A most probable path approach to queueing systems with general Gaussian input

This paper develops a rather general and easy-to-use methodology how to obtain good approximations for queue length distributions in priority and generalized processor sharing systems where the cumulative input traffic streams are modeled as general Gaussian processes with stationary increments. The idea is to identify the most probable path in the threshold-exceeding event, or a heuristic approximation of it, and then to use probability estimates based on this path. The method is particularly useful for long-range dependent traffic and complicated traffic mixes, which are difficult to handle with traditional queueing theory.

Most probable paths and performance formulae for buffers with gaussian input traffic

[Abstract]: In this paper, performance formulae for a queue serving Gaussian traffic are presented. The main technique employed is motivated by a general form of Schilder’s theorem, the large deviation result for Gaussian processes. Most probable paths leading to a given buffer occupancy are identified. Special attention is given to the case where the sample paths of the Gaussian process are smooth. The performance approximations are compared with known analytical results or by means of simulation. The approximations appear to be surprisingly accurate.

Fluid queue driven by anM/M/1 queue

A fluid queue receiving its input from the output of a precedingM/M/1 queue is considered. The input can be characterized as a Markov modulated rate process and the well known spectral decomposition technique can be applied. The novel features in this system relate to the nature of the spectrum, which is shown to be composed of a continuous part and one or two discrete points depending on whether the load of the fluid queue is less or greater than the output to input rate ratio. Explicit expressions of the generalized eigenvectors are given in terms of Chebyshev polynomials of the second kind, and the resolution of unity is determined. The solution for the buffer content distribution is obtained as a simple integral expression. Numerical examples are given.

GPS schedulers and Gaussian traffic

This article considers Gaussian flows which are fed into a GPS (Generalized Processor Sharing) scheduler. The system is analyzed using a most probable path approach. This method gives quite good approximations for performance measures, like queue length distributions in the full range of queue levels. The approximations are based on the distinction whether it is more probable that an aggregated queue consists of traffic from one class only or whether it is a combination of several classes. The approximate queue length distribution for a specific flow is then calculated either using the Empty Buffer Approximation or the authors’ Rough Full Link Approximation, respectively.

Multifractal products of stochastic processes: construction and some basic properties

In various fields, such as teletraffic and economics, measured time series have been reported to adhere to multifractal scaling. Classical cascading measures possess multifractal scaling, but their increments form a nonstationary process. To overcome this problem, we introduce a construction of random multifractal measures based on iterative multiplication of stationary stochastic processes, a special form of T-martingales. We study the ℒ2-convergence, nondegeneracy, and continuity of the limit process. Establishing a power law for its moments, we obtain a formula for the multifractal spectrum and hint at how to prove the full formalism.