  Quantum walks constitute an important tool for designing quantum algorithms
and information processing tasks. In a lackadaisical walk, in addition to the
possibility of moving out of a node, the walker can remain on the same node
with some probability. This is achieved by introducing self-loops,
parameterized by self-loop strength $l$, attached to the nodes such that large
$l$ implies a higher likelihood for the walker to be trapped at the node. In
this work, {\it directed}, lackadaisical quantum walks is studied. Depending on
$l$, two regimes are shown to exist -- one in which classical walker dominates
and the other dominated by the quantum walker. In the latter case, we also
demonstrate the existence of two distinct scaling regimes with $l$ for quantum
walker on a line and on a binary tree. Surprisingly, a significant
quantum-induced speedup is realized for large $l$. By tuning the initial state,
the extent of this speedup can be manipulated.
