  A queue layout of a graph $G$ consists of a vertex ordering of $G$ and a
partition of the edges into so-called queues such that no two edges in the same
queue nest, i.e., have their endpoints ordered in an ABBA-pattern. Continuing
the research on local ordered covering numbers, we introduce the local queue
number of a graph $G$ as the minimum $\ell$ such that $G$ admits a queue layout
with each vertex having incident edges in no more than $\ell$ queues. Similarly
to the local page number [Merker, Ueckerdt, GD'19], the local queue number is
closely related to the graph's density and can be arbitrarily far from the
classical queue number.
  We present tools to bound the local queue number of graphs from above and
below, focusing on graphs of treewidth $k$. Using these, we show that every
graph of treewidth $k$ has local queue number at most $k+1$ and that this bound
is tight for $k=2$, while a general lower bound is $\lceil k/2\rceil+1$. Our
results imply, inter alia, that the maximum local queue number among planar
graphs is either 3 or 4.
