  We consider a model for a queue in which only a fixed number $N$ of customers
can join. Each customer joins the queue independently at an exponentially
distributed time. Assuming further that the service times are independent and
follow an exponential distribution, this system can be described as a
two-dimensional Markov process on a finite triangular region $\mathfrak S$ of
the square lattice. We interpret the resulting random walk on $\mathfrak S$ as
a Dyck path that is weighted according to some state-dependent transition
probabilities that are constant along one axis, but are rather general
otherwise. We untangle the resulting intricate combinatorial structure by
introducing appropriate generating functions that exploit the recursive
structure of the model. This allows us to derive a fully explicit expression
for the probability density function of the number of customers served in any
busy period (equivalently, of the length of any excursion of the Dyck path
above the diagonal) as a weighted sum with alternating sign over a certain
subclass of Dyck paths, whose study is of independent interest.
