ABSTRACT. 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 α , 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 α is observed. For α ≤ 1 the parameters of the limiting process contain the deviation matrix associated with the background process.

In this paper we consider an Ornstein-Uhlenbeck (OU) process (M(t)) t≥0 whose parameters are determined by an external Markov process (X(t)) t≥0 on a finite state space {1, . . . , d}; this process is usually referred to as Markov-modulated OrnsteinUhlenbeck. We use stochastic integration theory to determine explicit expressions for the mean and variance of M(t). Then we establish a system of partial differential equations (PDEs) for the Laplace transform of M(t) and the state X(t) of the background process, jointly for time epochs t = t 1 , . . . , t K . Then we use this PDE to set up a recursion that yields all moments of M(t) and its stationary counterpart; we also find an expression for the covariance between M(t) and M(t +u). We then establish a functional central limit theorem for M(t) for the situation that certain parameters of the underlying OU processes are scaled, in combination with the modulating Markov process being accelerated; interestingly, specific scalings lead to drastically different limiting processes. We conclude the paper by considering the situation of a single Markov process modulating multiple OU processes.

This paper studies the diffusion limit for a network of infinite-server queues operating under Markov modulation (meaning that the system's parameters depend on an autonomously evolving background process). In previous papers on (primarily single-node) queues with Markov modulation, two variants were distinguished: one in which the server speed is modulated, and one in which the service requirement is modulated (i.e., depends on the state of the background process upon arrival). The setup of the present paper, however, is more general, as we allow both the server speed and the service requirement to depend on the background process. For this model we derive a Functional Central Limit Theorem: we show that, after accelerating the arrival processes and the background process, a centered and normalized version of the network population vector converges to a multivariate Ornstein-Uhlenbeck process. The proof of this result relies on expressing the queueing process in terms of Poisson processes with a random time change, an application of the Martingale Central Limit Theorem, and continuous-mapping arguments.

This paper focuses on an infinite-server queue modulated by an independently evolving finite-state Markovian background process, with transition rate matrix Q ≡ (qij) d i,j=1 . Both arrival rates and service rates are depending on the state of the background process. The main contribution concerns the derivation of central limit theorems for the number of customers in the system at time t ≥ 0, in the asymptotic regime in which the arrival rates λi are scaled by a factor N , and the transition rates qij by a factor N α , with α ∈ R + . The specific value of α has a crucial impact on the result: (i) for α > 1 the system essentially behaves as an M/M/∞ queue, and in the central limit theorem the centered process has to be normalized by √ N ; (ii) for α < 1, the centered process has to be normalized by N 1−α/2 , with the deviation matrix appearing in the expression for the variance.

A. This paper studies a system with multiple in nite-server queues which are modulated by a common background process. If this background process, being modeled as a nite-state continuous-time Markov chain, is in state j, then the arrival rate into the i-th queue is λi,j, whereas the service times of customers present in this queue are exponentially distributed with mean µ −1 i,j ; at each of the individual queues all customers present are served in parallel (thus re ecting their in nite-server nature). Three types of results are presented: in the rst place (i) we derive di erential equations for the probability generating functions corresponding to the distributions of the transient and stationary numbers of customers (jointly in all queues), then (ii) we set up recursions for the (joint) moments, and nally (iii) we establish a central limit theorem in the asymptotic regime in which the arrival rates as well as the transition rates of the background process are simultaneously growing large. K. Markov-modulation in nite-server queues central limit theorems