<inline-formula> <tex-math notation="LaTeX">$t/t$ </tex-math> </inline-formula>-Diagnosability and <inline-formula> <tex-math notation="LaTeX">$t/k$ </tex-math> </inline-formula>-Diagnosability for Augmented Cube Networks

Abstract

Diagnosability is an important parameter for evaluating the reliability of multiprocessor systems. $t/t$ -diagnosability and $t/k$ -diagnosability are both new indexes for measuring the reliability of a system. An $n$ -dimensional augmented cube network ( $AQ_{n}$ ) is a variant of an $n$ -dimensional hypercube network. In this paper, we first prove that an $n$ -dimensional augmented cube network is $(4n-8)/ (4n-8)$ -diagnosable, which implies that the $t/t$ -diagnosability of $AQ_{n}$ is approximately two times larger than its classical $t$ -diagnosability. Some useful properties of $AQ_{n}$ not reported by previous studies are proposed. By employing these new properties, we prove that $AQ_{n}$ is $t/k$ -diagnosable, which implies that the $t/k$ -diagnosability is approximately $(k+1)$ times larger than $2n-1$ , i.e., the $t$ -diagnosability of $AQ_{n}$ , where $t=2(k+1)n-(({3(k+1)(k+2)})/{2})+1$ , $k\leqslant ({4n}/{9})-({13}/{9})$ , and $n> 5$ .