A stochastic optimization algorithm for revenue maximization in a service system with balking customers

TL;DR

This paper develops a stochastic gradient descent algorithm for revenue maximization in a queueing system with balking customers, using only observable effective arrivals to adapt prices dynamically.

Contribution

It introduces a novel IPA-based method to estimate the effective arrival rate and proves convergence of the pricing algorithm under mild conditions.

Findings

The algorithm effectively maximizes expected revenue per unit time.

It relies solely on observable customer behavior, not full system state.

Convergence to the optimal price is theoretically guaranteed.

Abstract

This paper analyzes a service system modeled as a single-server queue, in which the service provider aims to dynamically maximize the expected revenue per unit of time. This is achieved by constructing a stochastic gradient descent algorithm that dynamically adjusts the price. A key feature of our modeling framework is that customers may choose to balk - that is, decide not to join - when facing high congestion. A notable strength of our approach is that the revenue-maximizing algorithm relies solely on information about effective arrivals, meaning that only the behavior of customers who choose not to balk is observable and used in decision-making. This results in an elaborate interplay between the pricing policy and the effective arrival process, yielding a non-standard state dependent queueing process. An important contribution of our work concerns a novel Infinitesimal Perturbation Analysis (IPA) procedure that is able to consistently estimate the stationary effective arrival rate. This is further leveraged to construct an iterative algorithm that converges, under mild regularity conditions, to the optimal price with provable asymptotic guarantees.