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

## TL;DR

This paper studies a multidimensional Brownian control problem with model uncertainty derived from a heavy-traffic multiclass M/M/1 queue, reducing it to a one-dimensional problem and characterizing its solution via a free-boundary problem.

## Contribution

It introduces a novel approach to handle model uncertainty in multiclass queueing systems by formulating a stochastic differential game and establishing a state-space collapse.

## Key findings

- State-space collapse reduces the multidimensional problem to one dimension.
- Unique solution to a free-boundary problem characterizes the value function.
- Analysis of how ambiguity parameters affect the solution.

## Abstract

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.