John M. Noble

Optimal choice of checkpointing interval for high availability

Supporting high availability by checkpointing and switching to a backup upon failure of a primary has a cost. Trade-off studies help system architects to decide whether higher availability at the cost of higher response time is to strive for. The decision leads to configuring a fault-tolerant server for best performance. This paper provides a mathematical model employing queuing theory that helps to compute the optimal checkpointing interval for a primary-backup replicated server. The optimization criterion is system availability. The model guides towards the checkpointing interval that is short enough to give low failover time, but long enough to utilize most of the system resources for servicing client requests. The novelty of the work is the detailed modelling of service times, wait times for earlier calls in the queue, and priority of checkpointing calls over client calls within the queues. Studies on the model in Mathematica and validation of a modelling assumption through simulations are included.

The directed polymer in a random environment

A continuous space/time approximation of the well known ‘directed polymer’ problem is considered. Connection between the ‘Helmholtz Free Energy’ and the ‘Two Walker problem’ is shown. Rigorous proof of the superdiffusive mean squared displacement exponent of 4/3 is given when there is one space dimension and one time dimension. Asymptotically diffusive behaviour of c(k)tis shown when there are one ‘time’ and two ‘space’ dimensions. For higher dimensions, the behaviour is diffusive and the mean squared displacement is asymptotically t d. These results hold for all temperature, because the phase transition in the discrete model is no longer present in the continuous model; the renormalization procedure has set the transition temperature to k crit =0The joint distribution is also shown to be asymptotically sub-Gaussian for all dimensions and all temperatures (in the sense that the p thmoments as a function of pincrease more slowly than the moments of a Gaussian distribution). The ‘Helmholtz Free Energy’ is a...

Competition systems in a random environment: A convergence result

In this article, a competition system in a random environment is considered. There are two species of particles and each will propagate as follows. An individual particle will move according to a Poisson jump process on the lattice Z d , split into two at a rate which is random, depending on the environment and die off at a rate which is random, depending on the environment. The main result is that, under the mass/speed rescaling (the particles are of mass {\rm ε } while the reproduction and death rate are rescaled accordingly), as the mass of an individual particle tends to zero, the densities of the species are given precisely by the pair of coupled stochastic partial differential equations where Δ is the ‘Lattice Laplacian’. Here the {\rm κ } i are the diffusion rates of each species (which are assumed to be constant) and the {\rm γ } i are the parameters measuring the competitive effects of one species on the other. The quantities u and v denote the densities of the first and second species respectively. {\rm ξ } (1) and {\rm ξ } (2) denote ‘noise’ terms and are the rescaled differences between the natural birth rates and death rates respectively (i.e. the differences between the birth rates and the death rates in the absence of any other species). In the mass/speed rescaling, the variance of the densities of each species has vanished, so that these equations give the precise evolution of the zero-mass limit. *The work is based substantially upon the dissertation for the Master of Science degree written by Elke Thönnes under the supervision of John Noble while both authors were at the Department of Statistics, University College Cork, Ireland.

Magnetic field in a turbulent conducting fluid

The problem of magnetic field in conducting turbulent, incompressible fluid is considered. The velocity of the fluid is taken to be independent of the magnetic field and is described by a Gaussian field, ‘white noise’ in time with smooth space correlation. The main result is that no fast dynamo (by which is meant almost sure exponential growth of magnetic field) can exist for an incompressible fluid when the magnetic viscosity is positive. For d = 2, sharper results are obtained; the magnetic field dies out when the magnetic viscosity is strictly positive. Furthermore, when d = 2, existence and characterization of invariant measure are given for d = 2 when the magnetic viscosity is zero. The results are compared to those discussed by Baxendale and Rosovskii in [2]

The directed polymer in a random environment (Addendum)

This short note improves the result in the ‘Directed Polymer’ article by clarifying the calculations in the proof of the theorem following the sub‐Gaussianity result