[Paper Review] An ergodic control problem for many-server multi-class queueing systems with help

Anup Biswas|arXiv (Cornell University)|Feb 10, 2015

Advanced Queuing Theory Analysis13 references3 citations

TL;DR

This paper studies an ergodic control problem in a many-server multi-class queueing system with cross-trained servers (M/M/N+M) in the Halfin-Whitt regime. It establishes asymptotic convergence of value functions and characterizes the limiting controlled diffusion with a state-dependent control set, proving the limiting ergodic control problem is fully solvable under convex polynomial growth costs.

ABSTRACT

A $M/M/N+M$ queueing network is considered with $d$ independent customer classes and $d$ server pools in Halfin-Whitt regime. Class $i$ customers have priority for service in pool $i$ for $i=1, ..., d$, and may access other pool if the pool has an idle server. We formulate an ergodic control problem where the running cost is given by a non-negative convex function with polynomial growth. We show that the limiting controlled diffusion is governed by a control set that depends on the state. We provide a complete analysis for the limiting ergodic control problem and establish asymptotic convergence of the value functions for the queueing model.

Motivation & Objective

• To model and analyze a many-server multi-class queueing system with flexible staffing and priority service.
• To formulate an ergodic control problem with non-negative convex running cost having polynomial growth.
• To derive the limiting controlled diffusion process under the Halfin-Whitt regime.
• To characterize the state-dependent control set governing the limiting diffusion.
• To establish asymptotic convergence of value functions from the finite queueing model to the diffusion limit.

Proposed method

• Formulates a queueing model with d customer classes and d server pools, where each class has priority in its dedicated pool and can use idle servers in other pools.
• Analyzes the system in the Halfin-Whitt heavy-traffic regime, scaling the number of servers N to infinity.
• Derives the limiting controlled diffusion process via functional central limit theorem arguments.
• Identifies that the control set in the limiting diffusion depends on the current state, reflecting available server pools.
• Uses convex analysis and viscosity solution techniques to solve the limiting ergodic control problem.
• Proves asymptotic convergence of the value functions of the finite queueing system to the value function of the limiting diffusion control problem.

Experimental results

Research questions

• RQ1 How does the state-dependent availability of server pools affect the ergodic control of multi-class queueing systems?
• RQ2 What is the structure of the limiting controlled diffusion process in a many-server multi-class system with cross-service flexibility?
• RQ3 Can the value function of the finite queueing system be asymptotically approximated by a diffusion control problem under polynomial growth costs?
• RQ4 What conditions ensure the solvability of the limiting ergodic control problem with state-dependent controls?
• RQ5 How does the priority structure and server flexibility influence the long-run average cost minimization?

Key findings

• The limiting controlled diffusion process is governed by a control set that depends on the current state, reflecting the availability of idle servers across pools.
• The limiting ergodic control problem is completely solvable, meaning an optimal control exists and the value function is well-defined.
• The value functions of the finite queueing systems converge asymptotically to the value function of the limiting diffusion control problem.
• The running cost is a non-negative convex function with polynomial growth, ensuring regularity and integrability in the limit.
• The analysis confirms that the Halfin-Whitt regime yields a tractable diffusion approximation for the ergodic control of flexible multi-class systems.
• The state-dependent control set captures the realistic feature that only idle servers can be assigned, leading to a non-constant control constraint.