We use Hamilton equations to identify most likely scenarios of long queues being formed in ergodic Jackson networks. Since the associated Hamiltonians are discontinuous and piecewise Lipschitz, one has to invoke methods of nonsmooth analysis. Time reversal of the Hamilton equations yields fluid equations for the dual network. Accordingly, the optimal trajectories are time reversals of the fluid trajectories of the dual network. Those trajectories are shown to belong to domains that satisfy a certain condition of being “essential.” As an illustration, we consider a two-station Jackson network. In addition, we prove certain properties of substochastic matrices, which may be of interest in their own right.

中文翻译：

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大队列的几何

我们使用汉密尔顿方程来确定遍历杰克逊网络中形成长队的最可能情况。由于相关的哈密顿量是不连续的和分段的Lipschitz，因此必须调用非平滑分析方法。汉密尔顿方程的时间反转产生了对偶网络的流体方程。因此，最佳轨迹是对偶网络的流体轨迹的时间反转。这些轨迹显示为属于满足一定条件的域，这些条件是“必需的”。作为说明，我们考虑一个两站的Jackson网络。此外，我们证明了亚随机矩阵的某些属性，它们本身可能会引起人们的兴趣。