  Model checking for Halpern and Shoham's interval temporal logic HS has been
recently investigated in a systematic way, and it is known to be decidable
under three distinct semantics. Here, we focus on the trace-based semantics,
where the infinite execution paths (traces) of the given (finite) Kripke
structure are the main semantic entities. In this setting, each finite infix of
a trace is interpreted as an interval, and a proposition holds over an interval
if and only if it holds over each component state (homogeneity assumption). In
this paper, we introduce a quantitative extension of HS over traces, called
parametric HS (PHS). The novel logic allows to express parametric timing
constraints on the duration (length) of the intervals. We show that checking
the existence of a parameter valuation for which a Kripke structure satisfies a
PHS formula (model checking), or a PHS formula admits a trace as a model under
the homogeneity assumption (satisfiability) is decidable. Moreover, we identify
a fragment of PHS which subsumes parametric LTL and for which model checking
and satisfiability are shown to be EXPSPACE-complete.
