In this paper we study the uniform stability properties of two classes of parallel server networks with multiple classes of jobs and multiple server pools of a tree topology. These include a class of networks with a single nonleaf server pool, such as the ‘N’ and ‘M’ models, and networks of any tree topology with class-dependent service rates. We show that with \(\sqrt{n}\) safety staffing, and no abandonment, in the Halfin–Whitt regime, the diffusion-scaled controlled queueing processes are exponentially ergodic and their invariant probability distributions are tight, uniformly over all stationary Markov controls. We use a unified approach in which the same Lyapunov function is used in the study of the prelimit and diffusion limit. A parameter called the spare capacity (safety staffing) of the network plays a central role in characterizing the stability results: the parameter being positive is necessary and sufficient that the limiting diffusion is uniformly exponentially ergodic over all stationary Markov controls. We introduce the concept of “system-wide work conserving policies," which are defined as policies that minimize the number of idle servers at all times. This is stronger than the so-called joint work conservation. We show that, provided the spare capacity parameter is positive, the diffusion-scaled processes are geometrically ergodic and the invariant distributions are tight, uniformly over all “system-wide work conserving policies." In addition, when the spare capacity is negative we show that the diffusion-scaled processes are transient under any stationary Markov control, and when it is zero, they cannot be positive recurrent.

在本文中，我们研究了具有多类作业和树形拓扑的多个服务器池的两类并行服务器网络的统一稳定性属性。其中包括具有单个非叶服务器池的一类网络，例如“N”和“M”模型，以及具有依赖于类的服务速率的任何树形拓扑的网络。我们表明，在 Halfin-Whitt 机制中，使用\(\sqrt{n}\)安全人员配置并且没有放弃，扩散尺度的受控排队过程是指数遍历的，并且它们的不变概率分布是紧密的，在所有静止的马尔可夫上均匀分布控制。我们使用统一的方法，其中相同的 Lyapunov 函数用于研究预限制和扩散限制。一个名为的参数网络的备用容量（安全人员配备）在表征稳定性结果中起着核心作用：参数为正是必要且充分的，以使限制扩散在所有静止马尔可夫控制上呈均匀指数遍历。我们引入了“全系统工作节约政策”的概念”，它们被定义为始终最小化空闲服务器数量的策略。这比所谓的联合工作守恒更强。我们表明，如果备用容量参数为正，则扩散尺度过程是几何遍历的并且不变分布是紧密的，在所有“系统范围的工作保存策略”中是一致的。此外，当备用容量为负时，我们证明扩散尺度过程在任何平稳马尔可夫控制下都是瞬态的，当它为零时，它们不能是正循环的。