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A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue
Ist Teil von
Methodology and computing in applied probability, 2016-03, Vol.18 (1), p.153-168
Ort / Verlag
New York: Springer US
Erscheinungsjahr
2016
Link zum Volltext
Quelle
Business Source Ultimate
Beschreibungen/Notizen
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
λ
i
when an external Markov process
J
(⋅) is in state
i
. It is assumed that molecules decay after an exponential time with mean
μ
−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
α
, for some
α
>0, whereas the arrival rates become
N
λ
i
, for
N
large. The main result of this paper is a functional central limit theorem (
F-CLT
) for the number of molecules, in that, after centering and scaling, it converges to an Ornstein-Uhlenbeck process. An interesting dichotomy is observed: (i) if
α
> 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
F-CLT
is the usual
N
, whereas (ii) for
α
≤1 the background process is relatively slow, and the scaling in the
F-CLT
is
N
1−
α
/2
. In the latter regime, the parameters of the limiting Ornstein-Uhlenbeck process contain the deviation matrix associated with the background process
J
(⋅).