  This paper studies a service system in which arriving customers are provided
with information about the delay they will experience. Based on this
information they decide to wait for service or to leave the system.
Specifically, every customer has a patience threshold and they balk if the
observed delay is above the threshold. The main objective is to estimate the
parameters of the customers' patience-level distribution and the corresponding
potential arrival rate, using knowledge of the actual queue-length process
only. The main complication, and distinguishing feature of our setup, lies in
the fact that customers who decide not to join are not observed, remarkably, we
manage to devise a procedure to estimate the underlying patience and arrival
rate parameters. The model is a multi-server queue with a Poisson stream of
customers, enabling evaluation of the corresponding likelihood function of the
state-dependent effective arrival process. We establish strong consistency of
the MLE and derive the asymptotic distribution of the estimation error. Several
applications and extensions of the method are discussed. The performance is
further assessed through a series of numerical experiments. By fitting
parameters of hyperexponential and generalized-hyperexponential distributions
our method provides a robust estimation framework for any continuous
patience-level distribution.
