The production of molecules in a chemical reaction network is modelled as a
Poisson process with a Markov-modulated arrival rate and an exponential decay
rate. We analyze the distributional properties of $M$, the number of molecules,
under specific time-scaling; the background process is sped up by $N^{\alpha}$,
the arrival rates are scaled by $N$, for $N$ large. A functional central limit
theorem is derived for $M$, which after centering and scaling, converges to an
Ornstein-Uhlenbeck process. A dichotomy depending on $\alpha$ is observed. For
$\alpha\leq1$ the parameters of the limiting process contain the deviation
matrix associated with the background process.Comment: 4 figure

Anderson, D.

Blom, J.

de Turck, K.

Mandjes, M.

Thorsdottir, H.

English

arXiv

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 $\\lambda_i$ when an external Markov process $J(\\cdot)$ is in state $i$. It is assumed that molecules decay after an exponential time with mean $\\mu^{-1}$. 
\nThe 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 
\nup by a factor $N^{\\alpha}$, for some $\\alpha>0$, whereas the arrival rates become $N\\lambda_i$, for $N$ large.
\nThe 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 $\\alpha>1$ 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 $\\sqrt{N}$, whereas (ii)~for $\\alpha\\leq1$ the background process is relatively slow, and the scaling in the {\\sc f-clt} is $N^{1-\\alpha/2}.$ In the latter regime, the parameters of the limiting Ornstein-Uhlenbeck process contain the deviation matrix associated with the background process $J(\\cdot)$.
\n
\n
\

Anderson, D.F.

Blom, J.G. (Joke)

Mandjes, M.R.H. (Michel)

Thorsdottir, H. (Halldora)

deTurck, K.E.E.S

CWI's Institutional Repository

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 $\lambda_i$ when an external Markov process $J(\cdot)$ is in state $i$. It is assumed that molecules decay after an exponential time with mean $\mu^{-1}$. 

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 $N^{\alpha}$, for some $\alpha>0$, whereas the arrival rates become $N\lambda_i$, for $N$ 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 $\alpha>1$ 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 $\sqrt{N}$, whereas (ii)~for $\alpha\leq1$ the background process is relatively slow, and the scaling in the {\sc f-clt} is $N^{1-\alpha/2}.$ In the latter regime, the parameters of the limiting Ornstein-Uhlenbeck process contain the deviation matrix associated with the background process $J(\cdot)$

Blom, Joke

Mandjes, Michel

Thorsdottir, Halldora

NARCIS 

A functional central limit theorem for a Markov-modulated
  infinite-server queue

A functional central limit theorem for a Markov-modulated infinite-server queue

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