Consider a GI/GI/1 queue operating under shortest remaining processing time with preemption. To describe the evolution of this system, we use a measure valued process that keeps track of the residual service times of all jobs in the system at any given time. Of particular interest is the waiting time for large jobs, which can be tracked using the frontier process, the largest service time of any job that has ever been in service. We propose a fluid model and present a functional limit theorem justifying it as an approximation of this system. The fluid model state descriptor is a measure valued function for which the left edge of the support is the fluid analog for the frontier process.
Under mild assumptions, we prove existence and uniqueness of fluid model solutions.
Furthermore, we are able to characterize the left edge of fluid model
solutions as the right continuous inverse of a simple functional of the initial condition,
arrival rate, and service time distribution. When applied to various examples, this
characterization reveals the dependence on service time distribution of the rate at which the
left edge of the fluid model increases.

The totally asymmetric simple exclusion process (TASEP) is one of the
simplest models of interacting particle systems on the one-dimensional
lattice. It is equivalent to a random growth model from the
Kardar-Parisi-Zhang universality class. We focus on fluctuations of the
particle positions for a nonequilibrium TASEP that starts from certain
deterministic initial conditions. We (rigorously) derive the scaling
exponents 1/3 and 2/3, and identify the limit laws as those of Gaussian
Orthogonal and Unitary ensembles of the random matrix theory.

The ``random sampling'' in the title has nothing whatsoever to
do with statistics. Instead, it refers to the perfect reconstruction of
a band-limited function from samples, a classical problem of
Fourier analysis and signal processing. With deterministic sampling,
almost everything is known in one dimension and almost nothing
is known in higher dimensions. It turns out that one loses very little in
efficiency by using random sampling, and in return one can use
probabilistic techniques to get some interesting theoretical results.
This is joint work with Karlheinz Gr\"ochenig.

Imaging of obscured targets in random media is a difficult and important problem. One of the central questions is that of stability which is particularly relevant to imaging in stochastic media. The main goal of the research is to develop a general criterion for multiple-frequency array imaging of multiple targets in stochastic media. An important feature of the cluttered media we considered here is that the coherent or mean signals do not vanish. It is called Rician fading medium in communication. Foldy-Lax formulas are used to simulate the exact wave propagations in the random media. In the talk, I will propose two models: passive array model and active array model. Then the stabilities of the imaging function with multiple point targets are given in both cases, followed by the numerical simulations to show the consistency with the analysis. If time permits, I will give some preliminary results about the stability conditions of the multiple extended targets, as well as the simulations.