  Motivated by applications that involve setting proper staffing levels for
multi-server queueing systems with batch arrivals, we present a thorough study
of the queue-length process $\{Q(t); t \geq 0\}$, departure process $\{D(t); t
\geq 0\}$, and the workload process $\{W(t); t \geq 0\}$ associated with the
M$_{t}^{B_{t}}$/G$_{t}$/$\infty$ queueing system. With two fundamental
assumptions of (non-stationary) Poisson arrivals and infinitely many servers,
we otherwise maintain a highly general model, in which the service duration and
batch size distributions may depend on time and, moreover, where the service
durations within a batch may be arbitrarily dependent. Nevertheless, we find
that the Poisson and infinite server assumptions are enough to show that for
each $t > 0$, the law of $Q(t)$ is that of a weighted sum of mutually
independent Poisson random variables. We further invoke this type of
decomposition to derive various joint Laplace-Stieltjes transforms associated
with the queue-length and departure processes. Next, we study the
time-dependent behavior of the workload process, and we conclude by
establishing almost sure convergence of the queue-length and workload processes
(when properly scaled) to two different shot-noise processes, elevating the
weak convergence results shown previously.
