  Ultra-reliable low-latency communication (URLLC) constitutes a key service
class of the fifth generation and beyond cellular networks. Notably, designing
and supporting URLLC poses a herculean task due to the fundamental need to
identify and accurately characterize the underlying statistical models in which
the system operates, e.g., interference statistics, channel conditions, and the
behavior of protocols. In general, multi-layer end-to-end approaches
considering all the potential delay and error sources and proper statistical
tools and methodologies are inevitably required for providing strong
reliability and latency guarantees. This paper contributes to the body of
knowledge in the latter aspect by providing a tutorial on several statistical
tools and methodologies that are useful for designing and analyzing URLLC
systems. Specifically, we overview the frameworks related to i) reliability
theory, ii) short packet communications, iii) inequalities, distribution
bounds, and tail approximations, iv) rare events simulation, vi) queuing theory
and information freshness, and v) large-scale tools such as stochastic
geometry, clustering, compressed sensing, and mean-field games. Moreover, we
often refer to prominent data-driven algorithms within the scope of the
discussed tools/methodologies. Throughout the paper, we briefly review the
state-of-the-art works using the addressed tools and methodologies, and their
link to URLLC systems. Moreover, we discuss novel application examples focused
on physical and medium access control layers. Finally, key research challenges
and directions are highlighted to elucidate how URLLC analysis/design research
may evolve in the coming years.
