Robustness of subsystem reliability of $k$-ary $n$-cube networks under probabilistic fault model

Abstract

With the emergence of the Big Data era, as multiprocessor systems consisting of multiple processors play a vital role in big data analytics, we are prompted to explore the qualitative and quantitative metric to characterize the reliability of the systems. As the size of the multiprocessor systems grows, the probability of the occurrence of failing processors increases. One metric of the macroscopic reliability of a system is the measure of the collective effect when its subsystems are out of function. The subsystem reliability of a system is the quantitative metric that a fault-free subsystem of specific size is operational as before with the occurrence of individual faults. Although some networks have the same order and similar topologies, there are differences in their subsystem reliabilities. In this work, we focus on the comparison of two distinct topologies of <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-ary <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula>-cube networks with the same order and calculate the robustness of reliability bounds of <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-ary <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula>-cube networks. We analytically show that the subsystem reliability is negatively correlated with the dimension <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula>, even if two subsystems of <inline-formula><tex-math notation="LaTeX">$Q_{n}^{k}$</tex-math></inline-formula> are of the same order. That is, the smaller <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is, the larger subsystem reliability of <inline-formula><tex-math notation="LaTeX">$Q_{n}^{k}$</tex-math></inline-formula> will be. This work provides a theoretical methodology to choose the more dependable topology of <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-ary <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula>-cube networks with the same order. Finally, we apply some numerical simulations to validate the results we established.