New Measure For Fault Diagnosis Research Articles

The r-Extra Diagnosability of Hyper Petersen Graphs

The diagnosability of a multiprocessor system or an interconnection network plays an important role in measuring the fault tolerance of the network. In 2016, Zhang et al. proposed a new measure for fault diagnosis of the system, namely, the [Formula: see text]-extra diagnosability, which restrains that every fault-free component has at least [Formula: see text] fault-free nodes. As a famous topology structure of interconnection networks, the hyper Petersen graph [Formula: see text] has many good properties. It is difficult to prove the [Formula: see text]-extra diagnosability of an interconnection network. In this paper, we show that the [Formula: see text]-extra diagnosability of [Formula: see text] is [Formula: see text] for [Formula: see text] and [Formula: see text] in the PMC model and for [Formula: see text] and [Formula: see text] in the MM[Formula: see text] model.

Hybrid fault diagnosis capability analysis of regular graphs

Fault diagnosis of systems is an important area of study in the design and maintenance of multiprocessor systems. A new measure for fault diagnosis of systems, namely, h-edge tolerable diagnosability, is the maximum number of faulty nodes that a system G can guarantee to locate when the number of faulty links does not exceed h, denoted by the(G). Particularly, t0e(G) is the traditional diagnosability of G. Chang et al. (2005) [5] determined the traditional diagnosabilities of regular graphs. In this paper, we establish the h-edge tolerable diagnosabilities of regular graphs under the PMC model and the MM⁎ model, which extends their results and provides a more precise characterization for the fault diagnosis capability of regular graphs. As applications, the h-edge tolerable diagnosabilities of many networks are determined under the PMC model and the MM⁎ model.

The 2-Extra Diagnosability of Alternating Group Graphs under the PMC Model and MM* Model

Diagnosability of a multiprocessor system is one important study topic. In 2015, Zhang et al. proposed a new measure for fault diagnosis of the system, namely, g-extra diagnosability, which restrains that every fault-free component has at least (g + 1) fault-free nodes. As a favorable topology structure of interconnection networks, the n-dimensional alternating group graph AGn has many good properties. In this paper, we give that the 2-extra diagnosability of AGn is 6n - 17 for n≥ 5 under the PMC model and MM* model.

On g-extra conditional diagnosability of hypercubes and folded hypercubes

Diagnosability of a multiprocessor system is one important study topic, which plays an important role in measuring of the reliability of multiprocessor systems. In the work of Zhang et al. in 2016, they proposed a new measure for fault diagnosis of systems, namely, g-extra conditional diagnosability. It is defined as the diagnosability of a multiprocessor system under the assumption that every fault-free component contains more than g vertices, which can measure the reliability of interconnection networks in heterogeneous environments more accurately than traditional diagnosability. As two kind of favorable topology structures of interconnection networks, the n-dimensional hypercubes Qn and folded hypercubes FQn have many good properties. In this paper, we investigate their g-extra conditional diagnosability and show that (a) the g-extra conditional diagnosability of Qn is (g+1)n−g−Cg2 for n≥5 and 1≤g≤n−14 under the MM* model; (b) the g-extra conditional diagnosability of FQn is (g+1)n−Cg2+1 for n≥9 and 1≤g≤n4 under the MM* model.

G-good-neighbor conditional diagnosability of star graph networks under PMC model and MM* model

Diagnosability of a multiprocessor system is an important study topic. S. L. Peng, C. K. Lin, J. J. M. Tan, and L. H. Hsu [Appl. Math. Comput., 2012, 218(21): 10406-10412] proposed a new measure for fault diagnosis of the system, which is called the g-good-neighbor conditional diagnosability that restrains every fault-free node containing at least g fault-free neighbors. As a famous topological structure of interconnection networks, the n-dimensional star graph S n has many good properties. In this paper, we establish the g-good-neighbor conditional diagnosability of S n under the PMC model and MM* model.

On g-good-neighbor conditional diagnosability of (n,k)-star networks

The g-good-neighbor conditional diagnosability is a new measure for fault diagnosis of systems. Xu et al. (2017) [27] determined the g-good-neighbor conditional diagnosability of (n,k)-star networks Sn,k (i.e., tg(Sn,k)) with 1≤k≤n−1 for 1≤g≤n−k under the PMC model and the MM⁎ model. In this paper, we determine tg(Sn,k) for all the remaining cases with 1≤k≤n−1 for 1≤g≤n−1 under the two models, from which we can obtain the g-good-neighbor conditional diagnosability of the star graph obtained by Li et al. (2017) [16] for 1≤g≤n−2.

The Tightly Super 2-extra Connectivity and 2-extra Diagnosability of Locally Twisted Cubes

Connectivity plays an important role in measuring the fault tolerance of an interconnection network [Formula: see text]. A faulty set [Formula: see text] is called a g-extra faulty set if every component of G − F has more than g nodes. A g-extra cut of G is a g-extra faulty set F such that G − F is disconnected. The minimum cardinality of g-extra cuts is said to be the g-extra connectivity of G. G is super g-extra connected if every minimum g-extra cut F of G isolates one connected subgraph of order g + 1. If, in addition, G − F has two components, one of which is the connected subgraph of order g + 1, then G is tightly [Formula: see text] super g-extra connected. Diagnosability is an important metric for measuring the reliability of G. A new measure for fault diagnosis of G restrains that every fault-free component has at least (g + 1) fault-free nodes, which is called the g-extra diagnosability of G. The locally twisted cube LTQn is applied widely. In this paper, it is proved that LTQn is tightly (3n − 5) super 2-extra connected for [Formula: see text], and the 2-extra diagnosability of LTQn is 3n − 3 under the PMC model ([Formula: see text]) and MM* model ([Formula: see text]).

The 2-good-neighbor (2-extra) diagnosability of alternating group graph networks under the PMC model and MM* model

Diagnosability of a multiprocessor system is one important study topic. In 2012, Peng et al. proposed a measure for fault tolerance of the system, which is called the g-good-neighbor diagnosability that restrains every fault-free node containing at least g fault-free neighbors. In 2016, Zhang et al. proposed a new measure for fault diagnosis of the system, namely, the g-extra diagnosability, which restrains that every fault-free component has at least (g+1) fault-free nodes. As a favorable topology structure of interconnection networks, the n-dimensional alternating group graph network ANn has many good properties. In this paper, we obtain that (a) the 2-good-neighbor diagnosability of ANn is 3n−7 for n ≥ 4 under the PMC model and MM* model; (b) the 2-extra diagnosability of ANn is 3n−7 for n ≥ 4 under the PMC model, and the 2-extra diagnosability of ANn is 3n−7 for n ≥ 5 under the MM* model. These results are optimal with respect to 2-extra diagnosability of ANn.

The 2-good-neighbor connectivity and 2-good-neighbor diagnosability of bubble-sort star graph networks

Connectivity plays an important role in measuring the fault tolerance of interconnection networks. The g-good-neighbor connectivity of an interconnection network G is the minimum cardinality of g-good-neighbor cuts. Diagnosability of a multiprocessor system is one important study topic. A new measure for fault diagnosis of the system restrains that every fault-free node has at least g fault-free neighbor vertices, which is called the g-good-neighbor diagnosability of the system. As a famous topology structure of interconnection networks, the n-dimensional bubble-sort star graph BSn has many good properties. In this paper, we prove that 2-good-neighbor connectivity of BSn is 8n−22 for n≥5 and the 2-good-neighbor connectivity of BS4 is 8; the 2-good-neighbor diagnosability of BSn is 8n−19 under the PMC model and MM∗ model for n≥5.

The g-good-neighbor conditional diagnosability of n-dimensional hypercubes under the MM⁎ model

Diagnosability of a multiprocessor system is one important study topic. In 2012, Peng et al. proposed a new measure for fault diagnosis of the system, which is called g-good-neighbor conditional diagnosability that restrains every fault-free node containing at least g fault-free neighbors. As a famous topology structure of interconnection networks, the n-dimensional hypercube has many good properties. In this paper, we give the g-good-neighbor conditional diagnosability of the n-dimensional hypercube under the MM⁎ model.

The 2-good-neighbor diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM⁎ model

Diagnosability is an important metric for measuring the reliability of multiprocessor systems. In 2012, Peng et al. proposed a new measure for fault diagnosis of the system, which is called g-good-neighbor diagnosability that restrains every fault-free node containing at least g fault-free neighbors. As a favorable topology structure of interconnection networks, the Cayley graph CΓn generated by the transposition tree Γn has many good properties. In this paper, we give that the 2-good-neighbor diagnosability of CΓn under the PMC model and MM⁎ model is g(n−2)−1, where n≥4 and g is the girth of CΓn.

交叉立方体的1好邻连通度和诊断度

连通度和诊断度是度量多处理器系统故障诊断能力的重要参数。2012年，Peng等提出了一个新的系统故障诊断方法，称为g好邻诊断度，它限制每个非故障顶点至少有g个非故障邻点。n维交叉立方体是超立方体的一个重要变形。本文证明了交叉立方体的1好邻连通度是2n – 2 (n ≥ 4)，又证明了交叉立方体在PMC模型下的1好邻诊断度是2n – 1 (n ≥ 4)和在MM*模型下的1好邻诊断度是2n – 1 (n ≥ 5)。 Connectivity and diagnosability are important parameters in measuring the fault diagnosis of multiprocessor systems. In 2012, Peng et al. proposed a new measure for fault diagnosis of the system, which is called g-good-neighbor diagnosability that restrains every fault-free node con-taining at least g fault-free neighbors. The n-dimensional crossed cube is an important variant of the hypercube. In this paper, we prove that the 1-good-neighbor connectivity of crossed cube is 2n − 2 for n ≥ 4, and the 1-good-neighbor diagnosability of crossed cube is 2n − 1 under the PMC model for n ≥ 4 and the MM* model for n ≥ 5.

The ${\schmi g}$-Good-Neighbor Conditional Diagnosability of ${\schmi k}$-Ary ${\schmi n}$-Cubes under the PMC Modeland MM* Model

The diagnosability of a system is defined as the maximum number of faulty processors that the system can guarantee to identify, which plays an important role in measuring of the reliability of multiprocessor systems. In the work of Peng et al. in 2012, they proposed a new measure for fault diagnosis of systems, namely, $g$ -good-neighbor conditional diagnosability. It is defined as the diagnosability of a multiprocessor system under the assumption that every fault-free node contains at least $g$ fault-free neighbors, which can measure the reliability of interconnection networks in heterogeneous environments more accurately than traditional diagnosability. The $k$ -ary $n$ -cube is a family of popular networks. In this study, we first investigate and determine the $R_g$ -connectivity of $k$ -ary $n$ -cube for $0\le g\le n.$ Based on this, we determine the $g$ -good-neighbor conditional diagnosability of $k$ -ary $n$ -cube under the PMC model and MM* model for $k\ge 4, n\ge 3$ and $0\le g\le n.$ Our study shows the $g$ -good-neighbor conditional diagnosability of $k$ -ary $n$ -cube is several times larger than the classical diagnosability of $k$ -ary $n$ -cube.

Conditional diagnosability of balanced hypercubes under the PMC model

Diagnosability is a critical metric for determining the reliability of a multiprocessor system. In 2005, a new measure for fault diagnosis of a system, namely conditional diagnosability, was proposed to improve the number of faulty processors identified. In this paper, we study the conditional diagnosability of balanced hypercubes under the PMC model and show that the conditional diagnosability of the n-dimensional balanced hypercube is 4n−3 for n⩾1.

Conditional Diagnosability of Matching Composition Networks Under the PMC Model

In the work of Lai in 2005, they proposed a new measure for fault diagnosis of systems, namely, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">conditional diagnosability</i> . It assumes that no fault set can contain all the neighbors of any vertex in the system. In the same paper, they showed that the conditional diagnosability of hypercube <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Qn</i> is 4( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> - 2) + 1 for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ges 5. In this brief, we generalize this result by considering a family of more popular networks, namely, matching composition networks (MCNs), which are a class of networks composed of two components of the same order linked by a perfect matching under PMC (Preparata, Metze and Chien) model. We determine in Theorem 7 the conditional diagnosability for some MCNs, from which we deduce that the hypercube <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Qn</i> , the crossed cube CQ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , the twisted cube TQ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , and the MOumlbius cube MQ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> all have the same conditional diagnosability of 4( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> - 2) + 1 for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ges 5. We show that the bijective connection (BC) networks in the work of Fan and He in 2003 and the work of Zhu in 2008 satisfy the conditions of Theorem 7, and thus, our conditional diagnosability result also applies to BC networks. Finally, we show that the MCNs satisfying the conditions of Theorem 7 are more general than the BC networks.