As the size of a multiprocessor system grows, the probability that faults occur in this system increases. One measure of the reliability of a multiprocessor system is the probability that a fault-free subsystem of a certain size still exists with the presence of individual faults. In this paper, we use the probabilistic fault model to establish the subgraph reliability for |$AG_n$|⁠, the |$n$|-dimensional alternating group graph. More precisely, we first analyze the probability |$R_n^{n-1}(p)$| that at least one subgraph with dimension |$n-1$| is fault-free in |$AG_n$|⁠, when given a uniform probability of a single vertex being fault-free. Since subgraphs of |$AG_n$| intersect in rather complicated manners, we resort to the principle of inclusion–exclusion by considering intersections of up to five subgraphs and obtain an upper bound of the probability. Then we consider the probabilistic fault model when the probability of a single vertex being fault-free is nonuniform, and we show that the upper bound under these two models is very close to the lower bound obtained in a previous result, and it is better than the upper bound deduced from that of the arrangement graph, which means that the upper bound we obtained is very tight.

中文翻译：

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具有一致和非一致顶点无故障概率的交替群图的子图可靠性

随着多处理器系统大小的增加，该系统中发生故障的可能性也会增加。多处理器系统可靠性的一种度量是存在个别故障的情况下，仍然存在一定大小的无故障子系统的可能性。在本文中，我们使用概率故障模型为| $ AG_n $ |⁠，| $ n $ |建立子图可靠性。维交替组图。更准确地说，我们首先分析概率| $ R_n ^ {n-1}（p）$ | 至少一个尺寸为| $ n-1 $ |的子图 | $ AG_n $ |⁠中的|是无错的，如果给定单个顶点无错的统一概率。由于| $ AG_n $ |的子图在以相当复杂的方式相交时，我们通过考虑最多五个子图的相交并求出概率的上限来求助于包含-排除的原理。然后，当单个顶点无故障的概率不均匀时，我们考虑概率故障模型，并且表明这两个模型下的上限非常接近先前结果中获得的下限，并且优于从排列图的上限推导出上限，这意味着我们获得的上限非常紧密。