  The Skorokhod map on the half-line has proved to be a useful tool for
studying processes with non-negativity constraints. In this work we introduce a
measure-valued analog of this map that transforms each element $\zeta$ of a
certain class of c\`{a}dl\`{a}g paths that take values in the space of signed
measures on the half-line to a c\`{a}dl\`{a}g path that takes values in the
space of non-negative measures on $[0,\infty)$ in such a way that for each $x >
0$, the path $t \mapsto \zeta_t[0,x]$ is transformed via a Skorokhod map on the
half-line, and the regulating functions for different $x > 0$ are coupled. We
establish regularity properties of this map and show that the map provides a
convenient tool for studying queueing systems in which tasks are prioritized
according to a continuous parameter. Three such well known models are the
earliest-deadline-first, the shortest-job-first and the
shortest-remaining-processing-time scheduling policies. For these applications,
we show how the map provides a unified framework within which to form fluid
model equations, prove uniqueness of solutions to these equations and establish
convergence of scaled state processes to the fluid model. In particular, for
these models, we obtain new convergence results in time-inhomogeneous settings,
which appear to fall outside the purview of existing approaches.
