4 research outputs found
Conservation Properties of the Hamiltonian Particle-Mesh method for the Quasi-Geostrophic Equations on a sphere
The Hamiltonian particle-mesh (HPM) method is used to
solve the Quasi-Geostrophic model generalized to a sphere, using
the Spherepack modeling package to solve the Helmholtz
equation on a colatitude-longitude grid with spherical harmonics.
The predicted energy conservation of a Poisson system is
shown to be approximately retained and statistical mean-eld
theory is veried
Time-scaling limits for Markov-modulated infinite-server queues
In this paper we study semi-Markov modulated M/M/ queues, which are to be understood as infinite-server systems in which the Poisson input rate is modulated by a Markovian background process (where the times spent in each of its states are assumed deterministic), and the service times are exponential. Two specific scalings are considered, both in terms of transient and steady-state behavior. In the former the transition times of the background process are divided by , and then is sent to ; a Poisson limit is obtained. In the latter both the transition times and the Poissonian input rates are scaled, but the background process is sped up more than the arrival process; here a central-limit type regime applies. The accuracy and convergence rate of the limiting results are demonstrated with numerical experiments
A functional central limit theorem for a Markov-modulated infinite-server queue
We consider a model in which the production of new molecules in a chemical reaction network occurs in a seemingly stochastic fashion, and can be modeled as a Poisson process with a varying arrival rate: the rate is when an external Markov process is in state . It is assumed that molecules decay after an exponential time with mean .
The goal of this work is to analyze the distributional properties of the number of molecules in the system, under a specific time-scaling. In this scaling, the background process is sped
up by a factor , for some , whereas the arrival rates become , for large.
The main result of this paper is a functional central limit theorem ({\sc f-clt}) for the number of molecules, in that the number of molecules, after centering and scaling, converges to an Ornstein-Uhlenbeck process. An interesting dichotomy is observed: (i)~if the background process jumps faster than the arrival process, and consequently the arrival process behaves essentially as a (homogeneous) Poisson process, so that the scaling in the {\sc f-clt} is the usual , whereas (ii)~for the background process is relatively slow, and the scaling in the {\sc f-clt} is In the latter regime, the parameters of the limiting Ornstein-Uhlenbeck process contain the deviation matrix associated with the background process
Time-scaling limits for Markov-modulated infinite-server queues
In this paper we study semi-Markov modulated M/M/ queues, which are to be understood as infinite-server systems in which the Poisson input rate is modulated by a Markovian background process (where the times spent in each of its states are assumed deterministic), and the service times are exponential. Two specific scalings are considered, both in terms of transient and steady-state behavior. In the former the transition times of the background process are divided by , and then is sent to ; a Poisson limit is obtained. In the latter both the transition times and the Poissonian input rates are scaled, but the background process is sped up more than the arrival process; here a central-limit type regime applies. The accuracy and convergence rate of the limiting results are demonstrated with numerical experiments