A Fluid-Diffusion-Hybrid Limiting Approximation for Priority Systems with Fast and Slow Customers

The paper “Fluid-Diffusion-Hybrid (FDH) Approximation” proposes a new heavy-traffic asymptotic regime for a two-class priority system in which the high-priority customers require substantially larger service times than the low-priority customers. In the FDH limit, the high-priority queue is a diffusion, whereas the low-priority queue operates as a (random) fluid limit, whose dynamics are driven by the former diffusion. A characterizing property of our limit process is that, unlike other asymptotic regimes, a non-negligible proportion of the customers from both classes must wait for service. This property allows us to study the costs and benefits of de-pooling, and prove that a two-pool system is often the asymptotically optimal design of the system.

On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients

UDC 519.21 In this paper we solve a selection problem for multidimensional SDE where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane It is assumed that the drift has a Hoelder asymptotics as approaches and the limit ODE does not have a unique solution.We show that if the drift pushes the solution away from then the limit process with certain probabilities selects some extremal solutions to the limit ODE. If the drift attracts the solution to then the limit process satisfies an ODE with some averaged coefficients. To prove the last result we formulate an averaging principle, which is quite general and new.

The Limit of Stationary Distributions of Many-Server Queues in the Halfin–Whitt Regime

We consider the so-called GI/GI/N queue, in which a stream of jobs with independent and identically distributed service times arrive as a renewal process to a common queue that is served by [Formula: see text] identical parallel servers in a first-come, first-served manner. We introduce a new representation for the state of the system and, under suitable conditions on the service and interarrival distributions, establish convergence of the corresponding sequence of centered and scaled stationary distributions in the so-called Halfin–Whitt asymptotic regime. In particular, this resolves an open question posed by Halfin and Whitt in 1981. We also characterize the limit as the stationary distribution of an infinite-dimensional, two-component Markov process that is the unique solution to a certain stochastic partial differential equation. Previous results were essentially restricted to exponential service distributions or service distributions with finite support, for which the corresponding limit process admits a reduced finite-dimensional Markovian representation. We develop a different approach to deal with the general case when the Markovian representation of the limit is truly infinite dimensional. This approach is more broadly applicable to a larger class of networks.

Functional Limit Theorems for Shot Noise Processes with Weakly Dependent Noises

We study shot noise processes when the shot noises are weakly dependent, satisfying the ρ-mixing condition. We prove a functional weak law of large numbers and a functional central limit theorem for this shot noise process in an asymptotic regime with a high intensity of shots. The deterministic fluid limit is unaffected by the presence of weak dependence. The limit in the diffusion scale is a continuous Gaussian process whose covariance function explicitly captures the dependence among the noises. The model and results can be applied in financial and insurance risks with dependent claims as well as queueing systems with dependent service times. To prove the existence of the limit process, we employ the existence criterion that uses a maximal inequality requiring a set function with a superadditivity property. We identify such a set function for the limit process by exploiting the ρ-mixing condition. To prove the weak convergence, we establish the tightness property and the convergence of finite dimensional distributions. To prove tightness, we construct two auxiliary processes and apply an Ottaviani-type inequality for weakly dependent sequences.

A functional limit theorem for general shot noise processes

By a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally Hölder continuous Gaussian limit process, and that the response function is regularly varying at infinity, we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process of shots is a Brownian motion, then the limit process for the general shot noise process is a Riemann–Liouville process. We specialize our result for five particular counting processes. Also, we investigate Hölder continuity of the limit processes for general shot noise processes.

Berry–Esseen Type Estimate and Return Sequence for Parabolic Iteration in the Upper Half-Plane

We prove a Berry–Esseen type theorem for the convergence rate in the monotone central limit theorem. When the underlying measure is singular relative to Lebesgue measure, this central limit process is viewed as an infinite measure-preserving dynamical system and we prove that it has a regularly varying return sequence of index $1/2$.

On a selection problem for small noise perturbation in the multidimensional case

The problem on identification of a limit of an ordinary differential equation with discontinuous drift that perturbed by a zero-noise is considered in multidimensional case. This problem is a classical subject of stochastic analysis, see, for example, [6, 29, 11, 20]. However the multidimensional case was poorly investigated. We assume that the drift coefficient has a jump discontinuity along a hyperplane and is Lipschitz continuous in the upper and lower half-spaces. It appears that the behavior of the limit process depends on signs of the normal component of the drift at the upper and lower half-spaces in a neighborhood of the hyperplane, all cases are considered.

Averaging principles for SPDEs driven by fractional Brownian motions with random delays modulated by two-time-scale Markov switching processes

This work considers stochastic partial differential equations (SPDEs) driven by fractional Brownian motions (fBm) with random delays modulated by two-time scale Markov switching processes leading to a two-time scale formulation. The two-time scale Markov chains have a fast-varying component and a slowly evolving component. Our aim is to obtain an averaging principle for such systems. Under suitable conditions, it is proved that there is a limit process in which the fast changing “noise” is averaged out. The slow component has a limit that is an average with respect to the stationary distribution of the fast component. The limit process is substantially simpler than the original system leading to reduction of the computational complexity.

Functional laws for trimmed Lévy processes

Two different ways of trimming the sample path of a stochastic process in 𝔻[0, 1]: global ('trim as you go') trimming and record time ('lookback') trimming are analysed to find conditions for the corresponding operators to be continuous with respect to the (strong)J1-topology. A key condition is that there should be no ties among the largest ordered jumps of the limit process. As an application of the theory, via the continuous mapping theorem, we prove limit theorems for trimmed Lévy processes, using the functional convergence of the underlying process to a stable process. The results are applied to a reinsurance ruin time problem.