  For a commutative cancellative monoid $M$, we introduce the notion of the
length density of both a nonunit $x\in M$, denoted $\mathrm{LD}(x)$, and the
entire monoid $M$, denoted $\mathrm{LD}(M)$. This invariant is related to three
widely studied invariants in the theory of non-unit factorizations, $L(x)$,
$\ell(x)$, and $\rho(x)$. We consider some general properties of
$\mathrm{LD}(x)$ and $\mathrm{LD}(M)$ and give a wide variety of examples using
numerical semigroups, Puiseux monoids, and Krull monoids. While we give an
example of a monoid $M$ with irrational length density, we show that if $M$ is
finitely generated, then $\mathrm{LD}(M)$ is rational and there is a nonunit
element $x\in M$ with $\mathrm{LD}(M)=\mathrm{LD}(x)$ (such a monoid is said to
have accepted length density). While it is well-known that the much studied
asymptotic versions of $L(x)$, $\ell (x)$ and $\rho (x)$ (denoted
$\overline{L}(x)$, $\overline{\ell}(x)$, and $\overline{\rho} (x)$) always
exist, we show the somewhat surprising result that $\overline{\mathrm{LD}}(x) =
\lim_{n\rightarrow \infty} \mathrm{LD}(x^n)$ may not exist. We also give some
finiteness conditions on $M$ that force the existence of
$\overline{\mathrm{LD}}(x)$.
