  We consider the leader election problem in population protocol models. In
pragmatic settings of population protocols, self-stabilization is a highly
desired feature owing to its fault resilience and the benefit of initialization
freedom. However, the design of self-stabilizing leader election is possible
only under a strong assumption (i.e. the knowledge of the \emph{exact} size of
a network) and rich computational resources (i.e. the number of states).
Loose-stabilization, introduced by Sudo et al [Theoretical Computer Science,
2012], is a promising relaxed concept of self-stabilization to address the
aforementioned issue. Loose-stabilization guarantees that starting from any
configuration, the network will reach a safe configuration where a single
leader exists within a short time, and thereafter it will maintain the single
leader for a long time, but not forever. The main contribution of the paper is
a time-optimal loosely-stabilizing leader election protocol. While the shortest
convergence time achieved so far in loosely-stabilizing leader election is
$O(\log^3 n)$ parallel time, the proposed protocol with design parameter $\tau
\ge 1$ attains $O(\tau \log n)$ parallel convergence time and
$\Omega(n^{\tau})$ parallel holding time (i.e. the length of the period keeping
the unique leader), both in expectation. This protocol is time-optimal in the
sense of both the convergence and holding times in expectation because any
loosely-stabilizing leader election protocol with the same length of the
holding time is known to require $\Omega(\tau \log n)$ parallel time.
