Online Learning and Optimization for Queues with Unknown Demand Curve and Service Distribution

TL;DR

This paper introduces an online learning framework for queue management that optimizes service fees and capacity without relying on separate parameter estimation, improving robustness and solution quality.

Contribution

It develops an integrated online learning approach that directly learns optimal policies, overcoming the sensitivity of traditional predict-then-optimize methods to estimation errors.

Findings

The online learning method converges and has bounded regret.

Simulation experiments confirm the effectiveness of the approach.

Compared to PTO, the online method is more robust to parameter uncertainties.

Abstract

We investigate an optimization problem in a queueing system where the service provider selects the optimal service fee p and service capacity \mu to maximize the cumulative expected profit (the service revenue minus the capacity cost and delay penalty). The conventional predict-then-optimize (PTO) approach takes two steps: first, it estimates the model parameters (e.g., arrival rate and service-time distribution) from data; second, it optimizes a model based on the estimated parameters. A major drawback of PTO is that its solution accuracy can often be highly sensitive to the parameter estimation errors because PTO is unable to properly link these errors (step 1) to the quality of the optimized solutions (step 2). To remedy this issue, we develop an online learning framework that automatically incorporates the aforementioned parameter estimation errors in the solution prescription process; it is an integrated method that can "learn" the optimal solution without needing to set up the parameter estimation as a separate step as in PTO. Effectiveness of our online learning approach is substantiated by (i) theoretical results including the algorithm convergence and analysis of the regret ("cost" to pay over time for the algorithm to learn the optimal policy), and (ii) engineering confirmation via simulation experiments of a variety of representative examples. We also provide careful comparisons for PTO and the online learning method.