James B. Martin

Stationary distributions of multi-type totally asymmetric exclusion processes

We consider totally asymmetric simple exclusion processes with n types of particle and holes (n-TASEPs) on Z and on the cycle Z N . Angel recently gave an elegant construction of the stationary measures for the 2-TASEP, based on a pair of independent product measures. We show that Angels construction can be interpreted in terms of the operation of a discrete-time M/M/1 queueing server; the two product measures correspond to the arrival and service processes of the queue. We extend this construction to represent the stationary measures of an n-TASEP in terms of a system of queues in tandem. The proof of stationarity involves a system of n 1-TASEPs, whose evolutions are coupled but whose distributions at any fixed time are independent. Using the queueing representation, we give quantitative results for stationary probabilities of states of the n-TASEP on Z N , and simple proofs of various independence and regeneration properties for systems on Z.

Limiting shape for directed percolation models

We consider directed first-passage and last-passage percolation on the nonnegative lattice Z d + , d ≥ 2, with i.i.d. weights at the vertices. Under certain moment conditions on the common distribution of the weights, the limits g(x) = lim n→ ∞ n- 1 T(|nx|) exist and are constant a.s. for x ∈ R d + , where T(z) is the passage time from the origin to the vertex z ∈ Z d + . We show that this shape function g is continuous on R d + , in particular at the boundaries. In two dimensions, we give more precise asymptotics for the behavior of g near the boundaries; these asymptotics depend on the common weight distribution only through its mean and variance. In addition we discuss growth models which are naturally associated to the percolation processes, giving a shape theorem and illustrating various possible types of behavior with output from simulations.

Gravitational instability of miscible fluids in a Hele-Shaw cell

We revisit the Rayleigh–Taylor instability when the two fluids are miscible and in the geometry of a Hele-Shaw cell. We provide analytical dispersion relations for the particular cases of either a sharp front between the two fluids or of a uniform density gradient stratification and for various fluid flow models, including an unbounded geometry, a two-dimensional gap-averaged Navier–Stokes–Darcy equation, and an effective porous medium. The results are compared to three-dimensional lattice BGK simulations, based on which the relevance of the various models in different wavelength regimes is discussed.

Large Tandem Queueing Networks with Blocking

Systems consisting of many queues in series have been considered by Glynn and Whitt (1991) and Baccelli, Borovkov and Mairesse (2000). We extend their results to apply to situations where the queues have finite capacity and so various types of “blocking” can occur. The models correspond to max-plus type recursions, of simple form but in infinitely many dimensions; they are related to “percolation” problems of finding paths of maximum weight through a 2-dimensional lattice with random weights at the vertices. Topics treated include: laws of large numbers for the speed of customers progressing through the system; stationary behaviour for systems with external arrival processes; a functional central limit theorem describing the behaviour of the “front of the wave” progressing through a system which starts empty; stochastic orderings for waiting times of customers at successive queues. Several open problems are noted.

MARC: A Many-Core Approach to Reconfigurable Computing

We present a Many-core Approach to Reconfigurable Computing (MARC), enabling efficient high-performance computing for applications expressed using parallel programming models such as OpenCL. The MARC system exploits abundant special FPGA resources such as distributed block memories and DSP blocks to implement complete single-chip high efficiency many-core micro architectures. The key benefits of MARC are that it (i) allows programmers to easily express parallelism through the API defined in a high-level programming language, (ii) supports coarse-grain multithreading and dataflow-style fine-grain threading while permitting bit-level resource control, and (iii) greatly reduces the effort required to re-purpose the hardware system for different algorithms or different applications. A MARC prototype machine with 48 processing nodes was implemented using a Virtex-5 (XCV5LX155T-2) FPGA for a well known Bayesian network inference problem. We compare the runtime of the MARC machine against a manually optimized implementation. With fully synthesized application-specific processing cores, our MARC machine comes within a factor of 3 of the performance of a fully optimized FPGA solution but with a considerable reduction in development effort and a significant increase in retarget ability.

A Three State Hard-Core Model on a Cayley Tree

Abstract We consider a nearest-neighbor hard-core model, with three states , on a homogeneous Cayley tree of order k (with k + 1 neighbors). This model arises as a simple example of a loss network with nearest-neighbor exclusion. The state σ(x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson flow of calls of rate λ at each site x, each call has an exponential duration of mean 1. If a call finds the node in state 1 or 2 it is lost. If it finds the node in state 0 then things depend on the state of the neighboring sites. If all neighbors are in state 0, the call is accepted and the state of the node becomes 1 or 2 with equal probability 1/2. If at least one neighbor is in state 1, and there is no neighbor in state 2 then the state of the node becomes 1. If at least one neighbor is in state 2 the call is lost. We focus on ‘splitting’ Gibbs measures for this model, which are reversible equilibrium distributions for the above process. We prove that in this model, ∀ λ > 0 and k ≥ 1, there exists a unique translationinvariant splitting Gibbs measure *. We also study periodic splitting Gibbs measures and show that the above model admits only translation - invariant and periodic with period two (chess-board) Gibbs measures. We discuss some open problems and state several related conjectures.

Pattern of reaction diffusion fronts in laminar flows.

Autocatalytic reaction between reacted and unreacted species may propagate as solitary waves, namely, at a constant front velocity and with a stationary concentration profile, resulting from a balance between molecular diffusion and chemical reaction. The effect of advective flow on the autocatalytic reaction between iodate and arsenous acid in cylindrical tubes and Hele-Shaw cells is analyzed experimentally and numerically using lattice Bhatnagar-Gross-Krook simulations. We do observe the existence of solitary waves with concentration profiles exhibiting a cusp and we delineate the eikonal and mixing regimes recently predicted.

A PHASE TRANSITION FOR COMPETITION INTERFACES

We study the competition interface between two growing clusters in a growth model associated to last-passage percolation. When the initial unoccupied set is approximately a cone, we show that this interface has an asymptotic direction with probability 1. The behavior of this direction depends on the angle ? of the cone: for ??180°, the direction is deterministic, while for ?<180°, it is random, and its distribution can be given explicitly in certain cases. We also obtain partial results on the fluctuations of the interface around its asymptotic direction. The evolution of the competition interface in the growth model can be mapped onto the path of a second-class particle in the totally asymmetric simple exclusion process; from the existence of the limiting direction for the interface, we obtain a new and rather natural proof of the strong law of large numbers (with perhaps a random limit) for the position of the second-class particle at large times.

Normal stress measurements in sheared non-Brownian suspensions

Measurements in a cylindrical Taylor–Couette device of the shear-induced radial normal stress in a suspension of neutrally buoyant non-Brownian (noncolloidal) spheres immersed in a Newtonian viscous liquid are reported. The radial normal stress of the fluid phase was obtained by measurement of the grid pressure Pg, i.e., the liquid pressure measured behind a grid which restrained the particles from crossing. The radial component of the total stress of the suspension was obtained by measurement of the pressure, Pm, behind a membrane exposed to both phases. Pressure measurements, varying linearly with the shear rate, were obtained for shear rates low enough to insure a grid pressure below a particle size dependent capillary stress. Under these experimental conditions, the membrane pressure is shown to equal the second normal stress difference, N2, of the suspension stress whereas the difference between the grid pressure and the total pressure, Pg−Pm, equals the radial normal stress of the particle phase, Σr...

The threshold of the instability in miscible displacements in a Hele–Shaw cell at high rates

For sufficiently large viscosity ratios and injection rates, miscible displacements in a vertical Hele–Shaw cell at high rates become unstable, leading to three-dimensional (3D) fingering patterns. Below the instability threshold, the base state is 2D in the form of a “tongue” of constant thickness. We apply the long wave Saffman–Taylor stability analysis to find an expression for the threshold of instability as a function of the viscosity ratio and the injection rate. The results are in agreement with the experimental data.