Evaluating Robustness of Subnetworks for the Split-Star Network

The robustness of subnetworks for the interconnection network of a computer system is an important consideration for the system performance. It can be measured by the extent to which subnetworks can stay fault-free when faults are present in the system. In this paper, we evaluate the subnetwork robustness for the n-dimensional split-star network $S_{n}^{2}$. Let $S_{n-m}^{2}$, $1\leq m\leq n-3$, be a subnetwork of $S_{n}^{2}$, and let p be the node reliability, the probability that a single node remains fault-free. We determine two values that reflect how robust $S_{n-m}^{2}$ subnetworks are, from two perspectives. We first establish the upper/lower bounds for $\mathcal{F}_{m}(S_{n}^{2})$, the minimum number of faulty nodes to make all $S_{n-m}^{2}$ subnetworks faulty. Then, we determine the subnetwork reliability, denoted by $\mathcal{R}_{m}(S_{n}^{2},p)$, which is the probability that at least one fault-free $S_{n-m}^{2}$ subnetwork exists in $S_{n}^{2}$, given the node reliability p. The upper/lower bounds and an approximation expression for $\mathcal{R}_{m}(S_{n}^{2},p)$ are obtained. We also propose a simulation method to estimate $\mathcal{R}_{m}(S_{n}^{2},p)$. The experimental results show that a) when p is relatively low, $\mathcal{R}_{m}(S_{n}^{2},p)$ can be approximated by the mean value of its upper and lower bounds, or the estimation value by the approximation expression; b) when p is high, $\mathcal{R}_{m}(S_{n}^{2},p)$ can be more accurately estimated by our simulation method.