The Role of Information in System Stability with Partially Observable Servers

Abstract

We present a methodology for analyzing the role of information on system stability. For this 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. 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. The example that we consider in this paper is a case of two servers where the environment of each is modulated like a Gilbert-Elliot channel. As simple as this case seems, there appear to be no closed form descriptions of the stability region under the various regimes considered. However, the numerical approximations to the POMDP Bellman equations together with the numerical solutions of the QBDs, both of which are in agreement, hint at a variety of structural results.