  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.
