Shot-noise fluid queues and infinite-server systems with batch arrivals

We show how to obtain the partial differential equation for (the Laplace-Stieltjes transform of) the queue content at time t, as well as its solution, from the corresponding infinite-server systems by taking appropriate limits. Also, for the special case of a time-homogeneous arrival process, we show that the scaled number of occupied servers in the infinite-server system converges as a process to the shot-noise queue content, implying that finite-dimensional distributions also converge.