Estimation of service value parameters for a queue with unobserved balking

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.

💡 Research Summary

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This paper studies a variant of Naor’s classic M/M/1 admission‑pricing model in which customers differ in their intrinsic service value R. The values are assumed to be independent draws from a parametric distribution Fθ, while all customers share a common, known waiting‑cost rate C per unit of time. Upon arrival a customer observes the current queue length q and a fixed admission price p; she joins the system if the expected net benefit R − p − C·(q+1)/μ is non‑negative, otherwise she balks and disappears without leaving any trace. Consequently the system operator can only observe the effective queue‑length process, not the balking customers.

The authors model the observable queue length as a continuous‑time Markov chain (a birth‑death process) with state‑dependent arrival rates
λq = λ·