Queueing models for cable access networks

abstract in thesi

Triangular M/G/1-type and tree-like QBD Markov chains

In applying matrix-analytic methods to M/G/1-type and tree-like QBD Markov chains, it is crucial to determine the solution to a (set of) nonlinear matrix equation(s). This is usually done via iterative methods. We consider the highly structured subclass of triangular M/G/1-type and tree-like QBD Markov chains that allows for an efficient direct solution of the matrix equation

Spectral gap of the Erlang A model in the Halfin-Whitt regime

We consider a hybrid diffusion process that is a combination of two Ornstein-Uhlenbeck processes with different restraining forces. This process serves as the heavy-traffic approximation to the Markovian many-server queue with abandonments in the critical Halfin-Whitt regime. We obtain an expression for the Laplace transform of the time-dependent probability distribution, from which the spectral gap is explicitly characterized. The spectral gap gives the exponential rate of convergence to equilibrium. We further give various asymptotic results for the spectral gap, in the limits of small and large abandonment effects. It turns out that convergence to equilibrium becomes extremely slow for overloaded systems with small abandonment effects

Asymptotic inversion of the Erlang B formula

The Erlang B formula represents the steady-state blocking probability in the Erlang loss model or M=M=s=s queue. We derive asymptotic expansions for the offered load that matches, for a given number of servers, a certain blocking probability. In addressing this inversion problem we make use of various asymptotic expansions for the incomplete gamma function. A similar inversion problem is investigated for the Erlang C formula

Equidistant sampling for the maximum of a Brownian motion with drift on a finite horizon

A Brownian motion observed at equidistant sampling points renders a random walk with normally distributed increments. For the difference between the expected maximum of the Brownian motion and its sampled version, an expansion is derived with coefficients in terms of the drift, the Riemann zeta function and the normal distribution function

Equidistant sampling for the maximum of a Brownian motion with drift on a finite horizon

A Brownian motion observed at equidistant sampling points renders a random walk with normally distributed increments. For the difference between the expected maximum of the Brownian motion and its sampled version, an expansion is derived with coefficients in terms of the drift, the Riemann zeta function and the normal distribution function

Grootschalige interactie met wiskunde

In ons dagelijks leven hebben we te maken met allerlei netwerken. Je kunt daarbij denken aan netwerken voor nutsvoorzieningen als gas, water en elektriciteit, maar bijvoorbeeld ook aan netwerken van mensen. Sinds de komst van internet spelen datanetwerken een belangrijke rol in ons leven en maken we steeds meer gebruik van sociale netwerken zoals Twitter en Facebook. Hoe kun je in deze hedendaagse netwerken de capaciteit zo goed mogelijk afstemmen op de vraag van de gebruikers? Johan van Leeuwaarden, hoogleraar stochastische netwerken, zet in zijn oratie, uitgesproken op 20 september 2013 aan de Technische Universiteit Eindhoven, uiteen hoe hiervoor stochastische modellen gebruikt kunnen worden

Relaxation time for the discrete D/G/1 queue

When queueing models are used for performance analysis of some stochastic system, it is usually assumed that the system is in steady-state. Whether or not this is a realistic assumption depends on the speed at which the system tends to its steady-state. A characterization of this speed is known in the queueing literature as relaxation time. The discrete D/G/1 queue has a wide range of applications. We derive relaxation time asymptotics for the discrete D/G/1 queue in a purely analytical way, mostly relying on the saddle point method. We present a simple and useful approximate upper bound which is sharp in case the load on the system is not very high. A sharpening of this upper bound, which involves the complementary error function, is then developed and this covers both the cases of low and high loads. For the discrete D/G/1 queue, the stationary waiting time distribution can be expressed in terms of infinite series that follow from Spitzer’s identity. These series involve convolutions of the probability distribution of a discrete random variable, which makes them suitable for computation. For practical purposes, though, the infinite series should be truncated. The relaxation time asymptotics can be applied to determine an appropriate truncation level based on a sharp estimate of the error caused by truncating