[Paper Review] Stability of parallel queueing systems with coupled service rates

Sem Borst, Matthieu Jonckheere+1 more|Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands|Jan 10, 2010

Advanced Queuing Theory Analysis3 citations

TL;DR

This paper establishes necessary and sufficient conditions for the stability of parallel queueing systems with coupled service rates, where each queue's service rate depends on the total workload across all queues. Using stochastic monotonicity and marginal drift analysis of multiclass birth-death processes, it provides a sharp characterization of stability, particularly for systems with decreasing service rates and uniform limits, and demonstrates that stability regions can be nonconvex.

ABSTRACT

This paper considers a parallel system of queues fed by independent arrival streams, where the service rate of each queue depends on the number of customers in all of the queues. Necessary and sufficient conditions for the stability of the system are derived, based on stochastic monotonicity and marginal drift properties of multiclass birth and death processes. These conditions yield a sharp characterization of stability for systems, where the service rate of each queue is decreasing in the number of customers in other queues, and has uniform limits as the queue lengths tend to infinity. The results are illustrated with applications where the stability region may be nonconvex.

Motivation & Objective

• To derive necessary and sufficient conditions for positive recurrence in parallel queueing systems with state-dependent service rates.
• To extend Foster–Lyapunov and fluid limit techniques to systems where service rates depend on the aggregate queue lengths across all queues.
• To characterize stability in systems where each queue’s service rate decreases with the number of customers in other queues.
• To analyze partial stability, where only a subset of queues remain stable.
• To demonstrate that stability regions in such systems can be nonconvex, challenging classical intuition.

Proposed method

• The authors model the system as a continuous-time multiclass birth-death process with state-dependent transition rates.
• They apply stochastic coupling techniques to compare the original system with simpler, tractable systems for stability analysis.
• Marginal drift criteria are derived based on the mean drift of individual queue lengths, enabling stability assessment without full Lyapunov function construction.
• The analysis leverages uniform limits of service rate functions as queue lengths tend to infinity, particularly for decreasing functions.
• Fluid limit approximations and strong law of large numbers for time-changed Poisson processes are used to support asymptotic stability arguments.
• Theoretical results are validated through two illustrative applications, including systems with nonconvex stability regions.

Experimental results

Research questions

• RQ1 Under what conditions is a parallel queueing system with coupled service rates positive recurrent?
• RQ2 How do state-dependent service rates—especially those decreasing in other queues’ lengths—affect system stability?
• RQ3 Can stability regions for such systems be nonconvex, and if so, how can this be characterized?
• RQ4 What role do uniform limits of service rates at infinity play in determining system stability?
• RQ5 How can marginal drift analysis be used to derive stability conditions without constructing full Lyapunov functions?

Key findings

• Necessary and sufficient conditions for stability are derived using marginal drift properties and stochastic monotonicity in multiclass birth-death processes.
• For systems with service rates decreasing in other queues’ lengths and possessing uniform limits at infinity, the stability condition is sharp and fully characterizable.
• The stability region may be nonconvex, which contradicts classical convexity assumptions in traditional queueing models.
• The method applies to systems with at most three queues where service rates depend only on whether queues are empty or not, recovering known results from transform methods and ergodic theory.
• Partial stability—where only a subset of queues remain stable—can be analyzed using the proposed drift criteria.
• The proof of Proposition 6 was corrected in the revised version, ensuring the validity of the stability characterization.