The Reliability of Subgraphs in the Arrangement Graph

Abstract

As the size of a multiprocessor computer system grows, the probability of having faulty (i.e., malfunctioning or failing) processors in the system increases. It is then important to quantify how the faults collectively affect the entire system. The reliability of subsystems in a system, defined as the probability that a fault-free subsystem of a certain size still exists when the system has faults, is a measure for the faults' effect on the whole system. It can be used as an indicator of system health. In this paper, we will present two schemes to calculate the reliability of an -subgraph in the -Arrangement Graph , an extensively studied interconnection network proposed for multiprocessor computers. The first scheme will use a probability fault model and the Principle of Inclusion-Exclusion to establish an upper-bound of the reliability, by taking into account the intersection of not more than three subgraphs. The second scheme uses basically the same idea, but completely neglects the intersection among subgraphs to calculate an approximate reliability. The results of the two schemes are compared, and are shown to be in good agreement, especially as the single-node reliability goes low.

The Extra Connectivity, Extra Conditional Diagnosability, and <inline-formula> <tex-math notation="LaTeX"> $t/m$</tex-math> </inline-formula>-Diagnosability of Arrangement Graphs

“…. Fault tolerant routing and conditional diagnosability of arrangement graphs have been widely investigated [5], [6], [8], [22]- [24], [38], [39].…”

Reliability Analysis of Subsystem in Balanced Hypercubes

“…Because the number of faulty nodes increases over time, in [15], [17], [21], the expected number of faulty nodes at moment t, denoted by f (t), is determined using f (t) = N (1 − e −at ), where N is the total number of nodes in a network. Accordingly, the node reliability function at moment t, denoted by p(t), can be estimated by the following formula:…”

“…Two schemes were proposed by Wu and Latifi [17] to analyze the sub-star reliability in the star-graph network under the probability fault model: one scheme is to establish an upper bound of the substar reliability by considering the intersection of no more than a certain number of subgraphs, and another one is to derive an approximation by completely neglecting any intersection among the sub-stars. Recently, Li et al [8], Huang et al [18], Kung and Hung [19], Kung et al [20], and Lin et al [21] also applied these two schemes to analyze the subgraph reliability of the (n, k)-star graph, the arrangement graph, the bubblesort graph, the split-star graph and the alternating group graph under the probability fault model, respectively.…”

The influence of datacenter usage on symmetry in datacenter network design

“…As regards recursively defined interconnection networks, the existence of a sub-network X (h − 1) within a host network X (h) has been considered via studies on reliability: each node is apportioned some failure probability and an analysis of X (h) is undertaken as to the likelihood of there existing a healthy copy of X (h − 1) within X (h). This model for reliability was first proposed in [8] (see, for example, [50] for some more recent developments). Of course, an analogous analysis of reliability in DCNs would be directly relevant to embedding within recursively defined DCNs, but, as far as we are aware, no such reliability analysis exists (the only consideration of notions of reliability in DCNs that we know of can be found in [13]).…”