Estimating Customer Impatience in a Service System With Unobserved Balking

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 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, and remarkably, we manage to devise a procedure to estimate the underlying patience and arrival rate parameters. The model is a multiserver 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. Funding: The research of Yoshiaki Inoue is supported in part by JSPS KAKENHI [Grant JP18K18007]. The research of Liron Ravner and Michael Mandjes is partly funded by NWO Gravitation Project Networks [Grant 024.002.003]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/stsy.2022.0101 .