Queueing networks are stochastic systems formed by interconnected resources
routing and serving jobs. They induce jump processes with distinctive
properties, and find widespread use in inferential tasks. Here, service rates
for jobs and potential bottlenecks in the routing mechanism must be estimated
from a reduced set of observations. However, this calls for the derivation of
complex conditional density representations, both over the stochastic network
trajectories and rates, and it is considered an intractable problem. Numerical
simulation procedures designed for this purpose do not scale due to high computational costs; furthermore, variational approaches relying on approximating
measures and full independence assumptions are unsuitable. In this paper, we
offer a probabilistic interpretation of variational methods applied to inference
tasks with queueing networks, and show that approximating measure choices
routinely used with jump processes yield ill-defined optimization problems. Yet,
we demonstrate that it is still possible to enable a variational inferential task,
by considering a novel space expansion treatment over an analogue counting
process for job transitions. Overall, we present and compare exemplar use
cases with practical queueing networks, and show that our framework offers
an efficient and improved alternative where existing variational or numerically
intensive solutions fail.