  In a distributed algorithm, multiple processes, or agents, work toward a
common goal. More often than not, the actions of some agents are dependent on
the previous execution (if not also on the outcome) of the actions of other
agents. The resulting interdependencies between the timings of the actions of
the various agents give rise to the study of methods for timely coordination of
these actions.
  In this work, we formulate and mathematically analyze "Timely-Coordinated
Response" - a novel multi-agent coordination problem in which the time
difference between each pair of actions may be constrained by upper and/or
lower bounds. This problem generalizes coordination problems previously studied
by Halpern and Moses and by Ben-Zvi and Moses.
  We optimally solve timely-coordinated response in two ways: using a
generalization of the fixed-point approach of Halpern and Moses, and using a
generalization of the "syncausality" approach of Ben-Zvi and Moses. We
constructively show the equivalence of the solutions yielded by both
approaches, and by combining them, derive strengthened versions of known
results for some previously-defined special cases of this problem.
  Our analysis is conducted under minimal assumptions: we work in a
continuous-time model with possibly infinitely many agents. The general results
we obtain for this model reduce to stronger ones for discrete-time models with
only finitely many agents. In order to distill the properties of such models
that are significant to this reduction, we define several classes of
naturally-occurring models, which in a sense separate the different results. We
present both a more practical optimal solution, as well as a surprisingly
simple condition for solvability, for timely coordinated response under these
models.
  Finally, we show how our results generalize the results known for
previously-studied special cases of this problem.
