  We study a multi-server queueing system with a periodic arrival rate and
customers whose joining decision is based on their patience and a delay proxy.
Specifically, each customer has a patience level sampled from a common
distribution. Upon arrival, they receive an estimate of their delay before
joining service and then join the system only if this delay is not more than
their patience, otherwise they balk. The main objective is to estimate the
parameters pertaining to the arrival rate and patience distribution. Here the
complication factor is that this inference should be performed based on the
observed process only, i.e., balking customers remain unobserved. We set up a
likelihood function of the state dependent effective arrival process (i.e.,
corresponding to the customers who join), establish strong consistency of the
MLE, and derive the asymptotic distribution of the estimation error. Due to the
intrinsic non-stationarity of the Poisson arrival process, the proof techniques
used in previous work become inapplicable. The novelty of the proving mechanism
in this paper lies in the procedure of constructing i.i.d. objects from
dependent samples by decomposing the sample path into i.i.d. regeneration
cycles. The feasibility of the MLE-approach is discussed via a sequence of
numerical experiments, for multiple choices of functions which provide delay
estimates. In particular, it is observed that the arrival rate is best
estimated at high service capacities, and the patience distribution is best
estimated at lower service capacities.
