# Brownian control problems for a multiclass M/M/1 queueing problem with model uncertainty

arXiv: https://arxiv.org/abs/1702.06479 · HTML: https://mlsanity.com/paper/1702.06479/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Feb 2017.

## Built on by (most cited first, 1 of 1)

- [Learning-based Optimal Admission Control in a Single Server Queuing System](https://mlsanity.com/paper/2212.11316/) (2022)

## Abstract (arXiv)

We consider a multidimentional Brownian control problem (BCP) with model uncertainty that formally emerges from a multiclass M/M/1 queueing control problem under heavy-traffic with model uncertainty. The BCP is formulated as a multidimensional stochastic differential game with two players: a minimizer that has an equivalent role to the decision maker in the queueing control problem and a maximizer whose role is to set up the uncertainty of the model. The dynamics are driven by a Brownian motion. We show that a state-space collapse propery holds. That is, the multidimensional BCP can be reduced to a one-dimensional BCP with model uncertainty that also takes the form of a two-player stochastic differential game. Then, the value function of both of the games is characterized as the unique solution to a free-boundary problem from which we extract equilibria for both games. Finally, we analyze the dependence of the value function and the equilibria on the ambiguity parameters.