  This paper develops tools to obtain robust probabilistic estimates for
queueing models at the large deviations (LD) scale. These tools are based on
the recently introduced robust R\'enyi bounds, which provide LD estimates (and
more generally risk-sensitive (RS) cost estimates) that hold uniformly over an
uncertainty class of models, provided that the class is defined in terms of
R\'enyi divergence with respect to a reference model and that estimates are
available for the reference model. One very attractive quality of the approach
is that the class to which the estimates apply may consist of hard models, such
as highly non-Markovian models and ones for which the LD principle is not
available. Our treatment provides exact expressions as well as bounds on the
R\'enyi divergence rate on families of marked point processes, including as a
special case renewal processes. Another contribution is a general result that
translates robust RS control problems, where robustness is formulated via
R\'enyi divergence, to finite dimensional convex optimization problems, when
the control set is a finite dimensional convex set. The implications to
queueing are vast, as they apply in great generality. This is demonstrated on
two non-Markovian queueing models. One is the multiclass single-server queue
considered as a RS control problem, with scheduling as the control process and
exponential weighted queue length as cost. The second is the many-server queue
with reneging, with the probability of atypically large reneging count as
performance criterion. As far as LD analysis is concerned, no robust estimates
or non-Markovian treatment were previously available for either of these
models.
