标题： 马尔可夫网络过程在随机环境中的相关公式 显示英文标题

标题： Correlation formulas for Markovian network processes in a random environment

摘要： 我们考虑马尔可夫过程，例如排队网络过程，在一个随机环境中，该环境通过确定节点的随机故障来影响网络，并随后需要进行维修。 从文献中可用的显式乘积形式稳态分布出发，我们注意到这个稳态分布不提供关于时间和空间（在节点上）相关结构的信息。 我们通过排队-环境过程的一步相关性来研究这种相关结构。 尽管这些相关性的绝对值的公式很复杂，但相关网络的相关性差异简单且具有良好的结构。 因此，我们在具有相同不变分布的随机环境中比较两个网络，并关注当环境变化或旅行规则被扰动时，此类网络中过程的时间行为。 评估比较公式，我们比较了相关过程的谱间隙和渐近方差。 显示英文摘要

摘要： We consider Markov processes, which describe e.g. queueing network processes, in a random environment which influences the network by determining random breakdown of nodes, and the necessity of repair thereafter. Starting from an explicit steady state distribution of product form available in the literature, we notice that this steady state distribution does not provide information about the correlation structure in time and space (over nodes). We study this correlation structure via one step correlations for the queueing-environment process. Although formulas for absolute values of these correlations are complicated, the differences of correlations of related networks are simple and have a nice structure. We therefore compare two networks in a random environment having the same invariant distribution, and focus on the time behaviour of the processes when in such a network the environment changes or the rules for traveling are perturbed. Evaluating the comparison formulas we compare spectral gaps and asymptotic variances of related processes.