Online Learning and Optimization for Queues with Unknown Demand Curve and Service Distribution (2303.03399v1)

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.