  In complex systems, external parameters often determine the phase in which
the system operates, i.e., its macroscopic behavior. For nearly a century,
statistical physics has extensively studied systems' transitions across phases,
(universal) critical exponents, and related dynamical properties. Here we
consider the functionality of systems, notably operations in socio-technical
ones, production in economic ones and, more generally, any schedule-based
system, where timing is of crucial importance. We introduce a stylized model of
delay propagation on temporal networks, where the magnitude of delay-mitigating
buffer acts as a control parameter. The model exhibits {\it timeliness
criticality}, a novel form of critical behavior. We characterize fluctuations
near criticality, commonly referred to as ``avalanches'', and identify the
corresponding critical exponents. The model exhibits timeliness criticality
also when run on real-world temporal systems such as production networks.
Additionally, we explore potential connections with the Mode-Coupling Theory of
glasses, the depinning transition and the directed polymer problem.
