Diffusion limits for networks of Markov-modulated infinite-server queues

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 Markov chain, called the background process. In previous papers on single-node queues with Markov modulation, two variants were distinguished. In the first variant the arrival rate and the server speed are modulated, whereas in the second variant the arrival rate and the service requirement are modulated. The setup of the present paper, however, is more general: we not only extend single-node systems to a network setting, but also 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. In particular, 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 multidimensional Ornstein–Uhlenbeck process. The proof of this result relies on weak convergence of stochastic integrals as well as continuous-mapping arguments.