  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.
