  We define the positive diameter of a finite group $G$ with respect to a
generating set $A\subset G$ to be the smallest non-negative integer $n$ such
that every element of $G$ can be written as a product of at most $n$ elements
of $A$. This invariant, which we denote by $\diam_A^+(G)$, can be interpreted
as the diameter of the Cayley digraph induced by $A$ on $G$.
  In this paper we study the positive diameters of a finite abelian group $G$
with respect to its various generating sets $A$. More specifically, we
determine the maximum possible value of $\diam_A^+(G)$ and classify all
generating sets for which this maximum value is attained. Also, we determine
the maximum possible cardinality of $A$ subject to the condition that
$\diam_A^+(G)$ is "not too small".
  Conceptually, the problems studied are closely related to our earlier work
and the results obtained shed a new light on the subject. Our original
motivation came from connections with caps, sum-free sets, and quasi-perfect
codes.
