  The {\em height} of a trace is the height of the corresponding heap of pieces
in Viennot's representation, or equivalently the number of factors in its
Cartier-Foata decomposition. Let $h(t)$ and $|t|$ stand respectively for the
height and the length of a trace $t$. Roughly speaking, $|t|$ is the
`sequential' execution time and $h(t)$ is the `parallel' execution time. We
prove that the bivariate commutative series $\sum_t x^{h(t)}y^{|t|}$ is
rational, and we give a finite representation of it. We use the rationality to
obtain precise information on the asymptotics of the number of traces of a
given height or length. Then, we study the average height of a trace for
various probability distributions on traces. For the uniform probability
distribution on traces of the same length (resp. of the same height), the
asymptotic average height (resp. length) exists and is an algebraic number. To
illustrate our results and methods, we consider a couple of examples: the free
commutative monoid and the trace monoid whose independence graph is the ladder
graph.
