  We consider a simple discrete-time controlled queueing system, where the
controller has a choice of which server to use at each time slot and server
performance varies according to a Markov modulated random environment. We
explore the role of information in the system stability region. At the extreme
cases of information availability, that is when there is either full
information or no information, stability regions and maximally stabilizing
policies are trivial. But in the more realistic cases where only the
environment state of the selected server is observed, only the service
successes are observed or only queue length is observed, finding throughput
maximizing control laws is a challenge. To handle these situations, we devise a
Partially Observable Markov Decision Process (POMDP) formulation of the problem
and illustrate properties of its solution. We further model the system under
given decision rules, using Quasi-Birth-and-Death (QBD) structure to find a
matrix analytic expression for the stability bound. We use this formulation to
illustrate how the stability region grows as the number of controller belief
states increases.
  Our focus in this paper is on the simple case of two servers where the
environment of each is modulated according to a two-state Markov chain. As
simple as this case seems, there appear to be no closed form descriptions of
the stability region under the various regimes considered. Our numerical
approximations to the POMDP Bellman equations and the numerical solutions of
the QBDs hint at a variety of structural results.
