On Reliability of Multiprocessor System Based on Star Graph

Abstract

As a critical parameter in evaluating the reliability of a multiprocessor system when processors malfunction, the \boldsymbol $h$ -extra connectivity ( $h$ -EC) of a multiprocessor system modeled by a graph $G$ , denoted by $\kappa ^{(h)}_{o}(G)$ , is an $h$ -extra vertex-cut with minimum cardinality. Both of the $h$ -extra conditional diagnosability ( $h$ -ECD) and the $t/h$ -diagnosability of the multiprocessor system are vital to tolerate and diagnose faulty processors. These two parameters rely on the resolving of $h$ -EC. For the multiprocessor system based on star graph $S_{n}$ , we show that the 5-EC $\kappa ^{(5)}_{o}(S_{n})$ of $S_{n}$ ( $n\geq 5$ ) is $6n-18$ . As a by-product, we present a novel proof of $\kappa ^{(2)}_{o}(S_{n})=3n-7$ (resp., $\kappa ^{(4)}_{o}(S_{n})=5n-14$ ) by relaxing the restriction $n\geq 10$ (resp., $n\geq 7$ ) to $n\geq 5$ (resp., $n\geq 5$ ). Furthermore, we determine that the $h$ -ECD of $S_{n}$ $(n\geq 5)$ under the preparata, metze, and chien (PMC) model is $(h+1)n-2h-1$ for $1\leq h\leq 3$ and $(h+1)n-3h+2$ for $4\leq h\leq 5$ . In addition, we show that $S_{n}$ is $[(h+1)n-4h+2]/h$ -diagnosable for $4\leq h\leq 5$ , which extends the result that $S_{n}$ is $[(h+1)n-3h-1]/h$ -diagnosable for $1\leq h\leq 3$ by [Zhou et al. “The t/k-diagnosability of star graph networks,” IEEE Trans. Comput. , vol. 64, no. 2, pp. 547–555, Feb. 2015].