  We define and study a variant of the \emph{Stanley depth} which we call
\emph{total depth} for partially ordered sets (posets). This total depth is the
most natural variant of Stanley depth from $\llbracket S_k\rrbracket$ -- the
poset of nonempty subsets of $\{1,2,\dots,k\}$ ordered by inclusion -- to any
finite poset. In particular, the total depth can be defined for the poset of
nonempty submultisets of a multiset ordered by inclusion, which corresponds to
a product of chains with the bottom element deleted. We show that the total
depth agrees with Stanley depth for $\llbracket S_k\rrbracket$ but not for such
posets in general. We also prove that the total depth of the product of chains
$\bm{n}^k$ with the bottom element deleted is $(n-1)\lceil{k/2}\rceil$, which
generalizes a result of Bir{\'{o}}, Howard, Keller, Trotter, and Young (2010).
Further, we provide upper and lower bounds for a general multiset and find the
total depth for any multiset with at most five distinct elements. In addition,
we can determine the total depth for any multiset with $k$ distinct elements if
we know all the interval partitions of $\llbracket S_k\rrbracket$.
