  The large-time behavior of a nonlinearly coupled pair of measure-valued
transport equations with discontinuous boundary conditions, parameterized by a
positive real-valued parameter $\lambda$, is considered. These equations
describe the hydrodynamic or fluid limit of many-server queues with reneging
(with traffic intensity $\lambda$), which model phenomena in diverse
disciplines, including biology and operations research. For a broad class of
reneging distributions with finite mean and service distributions with finite
mean and hazard rate function that is either decreasing or bounded away from
zero and infinity, it is shown that if the fluid equations have a unique
invariant state, then the Dirac measure at this state is the unique random
fixed point of the fluid equations, which implies that the stationary
distributions of scaled $N$-server systems converge to the unique invariant
state of the corresponding fluid equations. Moreover, when $\lambda\ne 1$, it
is shown that the solution to the fluid equation starting from any initial
condition converges to this unique invariant state in the large time limit. The
proof techniques are different under the two sets of assumptions on the service
distribution. When the hazard rate function is decreasing, a reformulation of
the dynamics in terms of a certain renewal equation is used, in conjunction
with recursive asymptotic estimates. When the hazard rate function is bounded
away from zero and infinity, the proof uses an extended relative entropy
functional as a Lyapunov function. Analogous large-time convergence results are
also established for a system of coupled measure-valued equations modeling a
multiclass queue.
