  A classical queuing theory result states that in a parallel-queue
single-server model, the maximum stability region does not depend on the
scheduling decision epochs, and in particular is the same for preemptive and
non-preemptive systems. We consider here the case in which each of the queues
may be connected to the server or not, depending on an exogenous process. In
our main result, we show that the maximum stability region now does strongly
depend on how the decision epochs are defined. We compare the setting where
decisions can be made at any moment in time (the unconstrained setting), to two
other settings: decisions are taken either (i) at moments of a departure
(non-preemptive scheduling), or (ii) when an exponentially clock rings with
rate $\gamma$. We characterise the maximum stability region for the two
constrained configurations, allowing us to observe a reduction compared to the
unconstrained configuration. In the non-preemptive setting, the maximum
stability region is drastically reduced compared to the unconstrained setting
and we conclude that a non-preemptive scheduler cannot take opportunistically
advantage (in terms of stability) of the random varying connectivity. Instead,
for the $\gamma$ decision epochs, we observe that the maximum stability region
is monotone in the rate of the decision moments $\gamma$, and that one can be
arbitrarily close to the maximum stability region in the unconstrained setting
if we choose $\gamma$ large enough. We further show that Serve Longest
Connected (SLC) queue is maximum stable in both constrained settings, within
the set of policies that select a queue among the connected ones. From a
methodological viewpoint, we introduce a novel theoretical tool termed a ``test
for fluid limits'' (TFL) that might be of independent interest. TFL is a simple
test that, if satisfied by the fluid limit, allows us to conclude for
stability.
