  Given a growth rule which sequentially constructs random permutations of
increasing degree, the stochastic process version of the rencontre problem asks
what is the limiting proportion of time that the permutation has no fixed
points (singleton cycles). We show that the discrete-time Chinese Restaurant
Process (CRP) does not exhibit this limit. We then consider the related
embedding of the CRP in continuous time and thereby show that it does have this
and other limits of the time averages. By this embedding the cycle structure of
the permutation can be represented as a tandem of infinite-server queues. We
use this connection to show how results from the queuing theory can be
interpreted in terms of the evolution of the cycle counts of permutations.
