Estimation of service value parameters for a queue with unobserved balking

TL;DR

This paper develops a maximum likelihood estimation method for service value parameters in queues with unobserved balking, enabling dynamic pricing and revenue optimization based on queue length data.

Contribution

It introduces a novel estimation approach for customer service values in queues with unobserved balking, including consistency and asymptotic normality analysis.

Findings

Estimator is consistent and asymptotically normal under certain conditions.

The proposed pricing scheme effectively estimates revenue-maximizing prices.

Simulation results demonstrate the estimator and pricing algorithm perform well.

Abstract

In Naor's model [17], customers decide whether or not to join a queue after observing its length. This work considers a variation in which customers are heterogeneous in their service value (reward) $R$ from completed service and homogeneous in the cost of staying in the system per unit of time. It is assumed that the values of customers are independent random variables generated from a common parametric distribution. The manager observes the queue length process, but not the balking customers. Assuming that the distribution of $R$ admits a known parametric form, a Maximum Likelihood Estimator based on the queue length data is constructed for the underlying parameters of $R$. We provide verifiable conditions for which the estimator is consistent and asymptotically normal. The estimation procedure is further leveraged to construct a dynamic pricing scheme that estimates the revenue maximizing admission price by iteratively updating the price using the estimated parameters. The performance of the estimator and the pricing algorithm are studied through a series of simulation experiments.