The Modified M/G/1 queue
The MX/M/1 queue with working breakdown
In this paper, we consider a batch arrival MX/M/1 queue model with working breakdown. The server may be subject to a service breakdown when it is busy, rather than completely stoping service, it will decrease its service rate. For this model, we analyze a two-dimensional Markov chain and give its quasi upper triangle transition probability matrix. Under the system stability condition, we derive the probability generating function (PGF) of the stationary queue length, and then obtain its stochastic...
The ODE method for some self-interacting diffusions on ℝd
The aim of this paper is to study the long-term behavior of a class of self-interacting diffusion processes on ℝd. These are solutions to SDEs with a drift term depending on the actual position of the process and its normalized occupation measure μt. These processes have so far been studied on compact spaces by Benaïm, Ledoux and Raimond, using stochastic approximation methods. We extend these methods to ℝd, assuming a confinement potential satisfying some conditions. These hypotheses on the confinement...
The one dimensional annealed δ-Lyapounov exponent
The Optimal Control in Batch Arrival Queue With Server Vacations, Startup and Breakdowns
The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent
In this paper we study the parabolic Anderson equation , , , where the -field and the -field are -valued, is the diffusion constant, and is the discrete Laplacian. The -field plays the role of adynamic random environmentthat drives the equation. The initial condition , , is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump...
The parabolic-parabolic Keller-Segel equation
I present in this note recent results on the uniqueness and stability for the parabolic-parabolic Keller-Segel equation on the plane, obtained in collaboration with S. Mischler in [11].
The PH/M/c queue with varying environment
The Pólya-Aeppli process and ruin problems.
The posterior metric and the goodness of gibbsianness for transforms of Gibbs measures.
The principle of variation for relativistic quantum particles
The proportional likelihood ratio order and applications.
In this paper, we introduce a new stochastic order between continuous non-negative random variables called the PLR (proportional likelihood ratio) order, which is closely related to the usual likelihood ratio order. The PLR order can be used to characterize random variables whose logarithms have log-concave (log-convex) densities. Many income random variables satisfy this property and they are said to have the IPLR (increasing proportional likelihood ratio) property (DPLR property). As an application,...
The quenched invariance principle for random walks in random environments admitting a bounded cycle representation
We derive a quenched invariance principle for random walks in random environments whose transition probabilities are defined in terms of weighted cycles of bounded length. To this end, we adapt the proof for random walks among random conductances by Sidoravicius and Sznitman (Probab. Theory Related Fields129 (2004) 219–244) to the non-reversible setting.
The queue-length in GI/G/ queues.
The recurrent event process of a success preceded by a failure and its application.
The regenerative process with rare event appears on the second phase.
The reliability of an element with alternating failure rate
The renewal theorem for random walk in two-dimensional time
The renormalization transformation for two-type branching models
This paper studies countable systems of linearly and hierarchically interacting diffusions taking values in the positive quadrant. These systems arise in population dynamics for two types of individuals migrating between and interacting within colonies. Their large-scale space–time behavior can be studied by means of a renormalization program. This program, which has been carried out successfully in a number of other cases (mostly one-dimensional), is based on the construction and the analysis of...