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.

中文翻译：

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信息在部分可观察的服务器的系统稳定性中的作用

我们提出了一种方法，用于分析信息对系统稳定性的作用。为此，我们考虑一个简单的离散时间控制排队系统，其中控制器可以选择在每个时隙使用哪个服务器，并且服务器性能根据马尔可夫调制随机环境而有所不同。在信息可用性的极端情况下，即在没有完整信息或没有信息的情况下，稳定区域和最大稳定策略是微不足道的。但是，在更实际的情况下，仅观察到所选服务器的环境状态，仅观察到服务成功或仅观察到队列长度，找到使控制律最大化的吞吐量是一个挑战。为了处理这些情况，我们设计了该问题的部分可观察的马尔可夫决策过程（POMDP）公式，并说明了其解决方案的性质。我们进一步在给定的决策规则下对系统进行建模，使用准生死（QBD）结构来找到稳定边界的矩阵解析表达式。我们使用此公式来说明稳定区域如何随着控制器置信状态数量的增加而增长。我们在本文中考虑的示例是两个服务器的情况，其中每个服务器的环境都像Gilbert-Elliot通道那样进行调制。就象这种情况一样简单，在所考虑的各种情况下，似乎没有对稳定区域的封闭形式描述。但是，对POMDP Bellman方程的数值近似以及QBD的数值解，两者都一致，