[Paper Review] Stability and Control of Piecewise-Deterministic Queueing Systems.

Li Jin, Saurabh Amin|arXiv (Cornell University)|Apr 7, 2016

Advanced Queuing Theory Analysis31 references3 citations

TL;DR

This paper studies piecewise-deterministic queueing systems (PDQs) with Markov-modulated capacity and state-feedback control, establishing necessary and sufficient stability conditions. It derives a bilinear matrix inequality condition for stability and proves unique ergodicity, with explicit results for two-mode systems and applications to single and parallel networked links.

ABSTRACT

We consider a piecewise-deterministic queueing (PDQ) model to study traffic queues due to stochastic capacity fluctuations in transportation facilities. The saturation rate (capacity) of the PDQ model switches between a finite set of values (modes) according to a Markov chain. The inflow to the PDQ is controlled by a state-feedback policy. The main results of this article are stability conditions of PDQs, i.e. conditions under which the distribution of the queue length converges to a unique invariant probability measure. On one hand, a necessary condition for stability is that the average inflow does not exceed the average saturation rate. On the other hand, based on the Foster-Lyapunov criteria, we derive a sufficient condition that requires a bilinear matrix inequality to admit positive solutions and the invariant probability measure to be unique. We also study the rate of convergence for stable PDQs. Furthermore, for PDQs with two modes, a necessary and sufficient condition for stability is available. In addition, we present examples for the stability analysis of feedback control policies for single PDQs as well as a network of two PDQ links in parallel.

Motivation & Objective

• To analyze the stability of piecewise-deterministic queueing systems (PDQs) under stochastic capacity fluctuations modeled by a Markov chain.
• To derive necessary and sufficient conditions for the existence of a unique invariant probability measure governing queue length distribution.
• To evaluate the rate of convergence to the invariant measure in stable PDQ systems.
• To develop feedback control policies that ensure system stability under dynamic capacity regimes.
• To apply the theoretical framework to single PDQs and parallel networks of two PDQ links.

Proposed method

• Model the PDQ as a piecewise-deterministic Markov process where capacity switches between discrete modes according to a continuous-time Markov chain.
• Use state-feedback control to regulate inflow based on current queue length and system mode.
• Apply Foster-Lyapunov criteria to derive a sufficient stability condition involving a bilinear matrix inequality with positive definite solutions.
• Establish a necessary condition for stability: average inflow must not exceed average saturation rate across all modes.
• For two-mode PDQs, derive a necessary and sufficient stability condition using explicit analytical solutions.
• Analyze convergence rates using Lyapunov function techniques and properties of the invariant measure.

Experimental results

Research questions

• RQ1 Under what conditions does the queue length distribution of a PDQ converge to a unique invariant probability measure?
• RQ2 What is the relationship between average inflow and average capacity that ensures system stability?
• RQ3 How can feedback control policies be designed to maintain stability under Markov-modulated capacity?
• RQ4 What is the rate of convergence to the invariant distribution in stable PDQ systems?
• RQ5 Can necessary and sufficient stability conditions be derived for two-mode PDQs?

Key findings

• A necessary condition for stability is that the average inflow does not exceed the average saturation rate across all capacity modes.
• A sufficient condition for stability is that a bilinear matrix inequality admits a positive definite solution and the invariant measure is unique.
• For PDQs with exactly two capacity modes, a necessary and sufficient condition for stability is derived explicitly.
• The rate of convergence to the invariant distribution is quantified using Lyapunov function methods and system parameters.
• Feedback control policies are validated through examples on single PDQs and parallel two-link networks, demonstrating stability under the derived conditions.
• The invariant probability measure of the queue length process is unique under the proposed sufficient stability condition.