  We consider sojourn (or response) times in processor-shared queues that have
a finite customer capacity. Computing the response time of a tagged customer
involves solving a finite system of linear ODEs. Writing the system in matrix
form, we study the eigenvectors and eigenvalues in the limit as the size of the
matrix becomes large. This corresponds to finite capacity models where the
system can only hold a large number $K$ of customers. Using asymptotic methods
we reduce the eigenvalue problem to that of a standard differential equation,
such as the Airy equation. The dominant eigenvalue leads to the tail of a
customer's sojourn time distribution. Some numerical results are given to
assess the accuracy of the asymptotic results.
