A functional central limit theorem for Markov additive arrival processes and its applications to queueing systems

We prove a functional central limit theorem for Markov additive arrival processes where the modulating Markov process has the transition rate matrix scaled up by \(n^{\alpha }\) (\(\alpha >0\)) and the mean and variance of the arrival process are scaled up by n. It is applied to an infinite-server queue and a fork–join network with a non-exchangeable synchronization constraint, where in both systems both the arrival and service processes are modulated by a Markov process. We prove functional central limit theorems for the queue length processes in these systems joint with the arrival and departure processes, and characterize the transient and stationary distributions of the limit processes. We also observe that the limit processes possess a stochastic decomposition property.