Stability of Fluid Queuing Systems with Parallel Servers and Stochastic Capacities

TL;DR

This paper models the stability of traffic queues in parallel systems with stochastic capacities using a PDQ framework, providing conditions for stability based on capacity modes and control policies.

Contribution

It introduces a PDQ model for stochastic capacity fluctuations and derives new stability conditions involving BMI feasibility and mode analysis.

Findings

Stability depends on average inflows not exceeding average capacities.

Feasibility of a bilinear matrix inequality ensures queue stability.

Conditions are strengthened for systems with two capacity modes.

Abstract

This note introduces a piecewise-deterministic queueing (PDQ) model to study the stability of traffic queues in parallel-link transportation systems facing stochastic capacity fluctuations. The saturation rate (capacity) of the PDQ model switches between a finite set of modes according to a Markov chain, and link inflows are controlled by a state-feedback policy. A PDQ system is stable only if a lower bound on the time-average link inflows does not exceed the corresponding time-average saturation rate. Furthermore, a PDQ system is stable if the following two conditions hold: the nominal mode's saturation rate is high enough that all queues vanish in this mode, and a bilinear matrix inequality (BMI) involving an underestimate of the discharge rates of the PDQ in individual modes is feasible. The stability conditions can be strengthened for two-mode PDQs. These results can be used for design of routing policies that guarantee stability of traffic queues under stochastic capacity fluctuations.