Ergodic Risk Sensitive Control of Markovian Multiclass Many-Server Queues with Abandonment

TL;DR

This paper investigates the asymptotic optimality of ergodic risk-sensitive control policies for multiclass queueing networks with abandonment, using novel variational and empirical measure techniques in the Halfin-Whitt regime.

Contribution

It introduces a new approach to analyze ergodic risk-sensitive control in queueing networks by leveraging variational representations and auxiliary controls, establishing asymptotic optimality.

Findings

Proves convergence of ERSC values to the limiting diffusion's ERSC value.

Develops a novel variational approach for ERSC costs involving auxiliary controls.

Establishes tightness of mean empirical measures for extended processes.

Abstract

We study the optimal scheduling problem for a Markovian multiclass queueing network with abandonment in the Halfin--Whitt regime, under the long run average (ergodic) risk sensitive cost criterion. The objective is to prove asymptotic optimality for the optimal control arising from the corresponding ergodic risk sensitive control (ERSC) problem for the limiting diffusion. In particular, we show that the optimal ERSC value associated with the diffusion-scaled queueing process converges to that of the limiting diffusion in the asymptotic regime. The challenge that ERSC poses is that one cannot express the ERSC cost as an expectation over the mean empirical measure associated with the queueing process, unlike in the usual case of a long run average (ergodic) cost. We develop a novel approach by exploiting the variational representations of the limiting diffusion and the Poisson-driven queueing dynamics, which both involve certain auxiliary controls. The ERSC costs for both the diffusion-scaled queueing process and the limiting diffusion can be represented as the integrals of an extended running cost over a mean empirical measure associated with the corresponding extended processes using these auxiliary controls. For the lower bound proof, we exploit the connections of the ERSC problem for the limiting diffusion with a two-person zero-sum stochastic differential game. We also make use of the mean empirical measures associated with the extended limiting diffusion and diffusion-scaled processes with the auxiliary controls. One major technical challenge in both lower and upper bound proofs, is to establish the tightness of the aforementioned mean empirical measures for the extended processes. We identify nearly optimal controls appropriately in both cases so that the existing ergodicity properties of the limiting diffusion and diffusion-scaled queueing processes can be used.