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

We show how a shot-noise fluid queue can be considered as the limiting case of a sequence of infinite-server queues with batch arrivals. The shot-noise queue we consider receives fluid amounts at the arrival times of a (time-inhomogeneous) Poisson process, the sizes of which are governed by some probability distribution that may also depend on time. The continuous rate at which fluid leaves the queue is proportional to the current content of the queue. Thus, intuitively, one can think of drops of fluid arriving in batches, which are taken into service immediately upon arrival, at an exponential service rate. 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.