The two-parameter Poisson-Dirichlet distribution, denoted
$\mathsf{PD}(\alpha, \theta)$ is a probability distribution on the set of
decreasing positive sequences with sum 1. The usual Poisson-Dirichlet
distribution with a single parameter $\theta$, introduced by Kingman, is
$\mathsf{PD}(0, \theta)$. Known properties of $\mathsf{PD}(0, \theta)$,
including the Markov chain description due to Vershik, Shmidt and Ignatov, are
generalized to the two-parameter case. The size-biased random permutation of
$\mathsf{PD}(\alpha, \theta)$ is a simple residual allocation model proposed by
Engen in the context of species diversity, and rediscovered by Perman and the
authors in the study of excursions of Brownian motion and Bessel processes. For
$0 < \alpha < 1, \mathsf{PD}(\alpha, 0)$ is the asymptotic distribution
of ranked lengths of excursions of a Markov chain away from a state whose
recurrence time distribution is in the domain of attraction of a stable law of
index $\alpha$. Formulae in this case trace back to work of Darling, Lamperti
and Wendel in the 1950s and 1960s. The distribution of ranked lengths of
excursions of a one-dimensional Brownian motion is $\mathsf{PD}(1/2, 0)$, and
the corresponding distribution for a Brownian bredge is $\mathsf{PD}(1/2,
1/2)$. The $\mathsf{PD}(\alpha, 0)$ and $\mathsf{PD}(\alpha, \alpha)$
distributions admit a similar interpretation in terms of the ranked lengths of
excursions of a semistable Markov process whose zero set is the range of a
stable subordinator of index $\alpha$.