This work concerns the asymptotic behavior of critical fluid model solutions for a multiclass processor sharing queue under general distributional assumptions. Such critical fluid model solutions are measure-valued functions of time. We prove that critical fluid model solutions converge to the set of invariant states as time goes to infinity, uniformly for all initial conditions lying in certain relatively compact sets. This generalizes an earlier single-class result of Puha and Williams to the more complex multiclass setting. In particular, several new challenges are overcome, including formulation of a suitable relative entropy functional and identifying a convenient form of the time derivative of the relative entropy applied to trajectories of critical fluid model solutions.

中文翻译：

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基于相对熵的多类处理器共享队列临界流体模型的渐近行为

这项工作涉及在一般分布假设下多类处理器共享队列的临界流体模型解决方案的渐近行为。这种临界流体模型解决方案是时间的测量值函数。我们证明了临界流体模型解随着时间趋于无穷大而收敛到一组不变状态，对于位于某些相对紧凑集合中的所有初始条件是一致的。这将 Puha 和 Williams 的早期单类结果推广到更复杂的多类设置。特别是，克服了几个新的挑战，包括制定合适的相对熵泛函和识别应用于临界流体模型解决方案轨迹的相对熵的时间导数的方便形式。