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

TL;DR

This work tackles dynamic revenue optimization in a single-server queue with balking, modeling how price and congestion influence effective demand. It develops an online stochastic gradient method that updates price using an Infinitesimal Perturbation Analysis (IPA) gradient estimator based solely on effective arrivals, avoiding explicit dependence on unobservable balked customers. A novel recursive IPA gradient framework and coupling arguments enable convergence guarantees to the optimal price p*, with explicit bias and variance bounds and a sublinear regret bound. Numerical experiments demonstrate robustness across service-time distributions and joining rules, highlighting practical applicability to price-based congestion management in service systems.

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.

Figures (11)

• Figure 1: Workload dynamics of an M/G/1 + H queue with $\Lambda = 2$, $p = 5$, $H(p, V) = \exp(-\theta_1p -\theta_2V)$. The green dots represent those customers that join, while red dots represent those that balk.
• Figure 2: CDF of an effective interarrival time in the setup from Example \ref{['example: Psi and H']} for different combinations of $p, \overline{W}$.
• Figure 3: Inverse of the CDF of $A$ in the setup from Example \ref{['example: Psi and H']} for different combinations of $p, \overline{W}$
• Figure 4: Visualization of Lemma \ref{['lemma:example:gradient_bounds']}. We simulate an M/G/1 + $H(p, V)$ queue with price $p = 20$, and model parameters $\Lambda = 20$, $\theta_1 = 0.1$, and $\theta_2 = 0.2$. For each effective arrival $i$, we record the value of the price derivative of the waiting time, denoted $\nabla_p \overline{W}_i$. The plot shows that $\nabla_p \overline{W}_i$ consistently lies in the interval $\left[ -{\theta_1}{\theta_2}, 0 \right] = [-0.5, 0]$ for all $i$.
• Figure 5: Visualizations of Proposition \ref{['propn:model:theoretical_results']}. The top plot illustrates the observation from the proof of part (b) that the two queues with distinct initial workloads are eventually coupled with a time-lag. We observe that $\vert \widetilde{A}^1_n - \widetilde{A}^2_n \vert$ does not change from some point on. The second and third plots illustrate parts (a) and (b).

Theorems & Definitions (40)

• Example A
• Example A
• Proposition 2.2
• Proposition 3.1
• proof
• Remark 3.2
• Example A
• Example A
• Example A
• Lemma 3.5