Faulty Processors Research Articles

The t/k-diagnosability of m-ary n-cube networks

A crucial index used to evaluate the reliability of a multiprocessor system is diagnosability. The t/k-diagnosis strategy greatly improves the diagnosability of a system by allowing up to k fault-free processors to be misdiagnosed as faulty processors. Compared with the t-diagnosability and t1/t1-diagnosability, the t/k-diagnosability can better reflect the fault modes of a practical system. The m-ary n-cube Qnm (in order not to be confused with the “k” of the t/k-diagnosability, we replace k-ary n-cube Qnk with m-ary n-cube Qnm) is a significant and common network topology, which is used as the underlying network in the construction of many distributed memory multiprocessors. In this paper, we explore several useful lemmas and the t/k-diagnosability of Qnm. According to the sufficient condition for determining that a system is t/k-diagnosable, we show that for n≥2, m≥4, and 0≤k≤2n, Qnm is tk,n/k-diagnosable, where tk,n=2(k+1)n−12(k+1)(k+2)+1. Based on this, we provide a comparison among the t-diagnosability, the t1/t1-diagnosability, and the t/k-diagnosability of Qnm.

Embedding spanning disjoint cycles in augmented cube networks with prescribed vertices in each cycle

One of the important issues in evaluating an interconnection network is to study the Hamiltonian cycle embedding problems. For a positive integer k, a graph G is said to be spanning k-cyclable if for k prescribed vertices , there exist k disjoint cycles such that the union of spans G, and each contains exactly one vertex of . According to the definition, the problem of finding hamiltonian cycle focuses on k = 1. The notion of spanning cyclability can be applied to the problem of identifying faulty processors and other related issues in interconnection networks. The n-dimensional augmented cube is an important node-symmetric variant of the n-dimensional hypercube . In this paper, we prove that with is spanning k-cyclable for .

The g-Good-Neighbor Conditional Diagnosabilities of Hypermesh Optical Interconnection Networks Under the PMC and Comparison Models

Processor fault diagnosis is intended to identify faulty processors of a multicomputer system, guaranteeing its high reliability and availability. The [Formula: see text]-good-neighbor conditional diagnosability is a novel fault diagnosis method for various networks. Under the PMC model and comparison model, the [Formula: see text]-good-neighbor conditional diagnosabilities of hypermesh optical interconnection networks are determined, respectively. Directly applying the results, the [Formula: see text]-good-neighbor conditional diagnosabilities of hypercubes are derived under the PMC and comparison models.

The partial diagnosability of interconnection networks under the Hybrid PMC model

System level diagnosis is a primary method to identify the faulty elements in a multiprocessor system. Different from the traditional system level fault diagnosis models, the new fault diagnosis model, hybrid PMC model (short for HPMC model), is suitable to the circumstances that the processor and link faults occur simultaneously. Under the HPMC model, the q-edge restricted diagnosability of a system H, denoted by tqe(H), is the maximum number p such that H can correctly identify all the faulty elements containing p faulty processors and at most q faulty links. Similarly, the p-vertex restricted edge-diagnosability of a system H, denoted by spv(H), is the maximum number q such that H can correctly identify all the faulty elements containing q faulty links and at most p faulty processors. These two diagnosability are collectively referred to as partial diagnosability. We investigate and determine the q-edge restricted diagnosability of a system H under the HPMC model for q≥η(H), where η(H)=max⁡{|N(u)∩N(v)||uv∈E(H)}. In addition, we investigate the relationship between tqe(H) and spv(H) under the HPMC model, and show that under some conditions, spv(H)=q if tqe(H)=p. Applying our results, the two kinds of partial diagnosability of k-ary n-cubes, bijective connection networks and undirected Kautz graphs are established.

Simulation-based fault-tolerant multiprocessors system

System reliability is an important issue in designing modern multiprocessor systems. This paper proposes a fault-tolerant, scalable, multiprocessor system architecture that adopts a pipeline scheme. To verify the performance of the proposed system, the SimEvent/Stateflow tool of the MATLAB program was used to simulate the system. The proposed system uses twelve processors (P), connected in a linear array, to build a ten-stage system with two backup processors (BP). However, the system can be expanded by adding more processors to increase pipeline stages and performance, and more backup processors to increase system reliability. The system can automatically reorganize itself in the event of a failure of one or two processors and execution continues without interruption. Each processor communicates with its neighboring processors through input/output (I/O) ports which are used as bypass links between the processors. In the event of a processor failure, the function of the faulty processor is assigned to the next processor that is free from faults. The fast Fourier transform (FFT) algorithm is implemented on the simulated circuit to evaluate the performance of the proposed system. The results showed that the system can continue to execute even if one or two processors fail without a noticeable decrease in performance.

Characterization of component diagnosability of regular networks

Multiprocessor systems, which usually take interconnection network (or graph) as underlying topologies, are commonly deployed for big data analysis because of evolution in technologies such as cloud computing, IoT, social network and so on. Reliability evaluation is of significant importance to characterize fault tolerability for the topologies of multiprocessor systems, and system-level diagnosis is a primary strategy to identify the faulty processors in the systems. The g-component connectivity cκg(G) of a graph G is the size of a minimal vertex-set, whose removal will disconnect G to possess at least g components, which reflects the invulnerability of the topology graph. Based on g-component connectivity of graph G, the g-component diagnosability ctg(G) of regular networks has been proposed as a parameter to measure network fault-tolerability. The g-component diagnosability ctg(G) is the maximum t such that the graph G is g-component t-diagnosable. We first propose some general characterizations of the component diagnosability of regular networks under the classic PMC model and MM∗ model based on the component connectivity. And then we present some empirical analysis for some kinds of well-known regular networks, such as BC networks, star graphs, bubble-sort star graphs, alternating group graphs.

Faulty processor identification for a multiprocessor system under the Malek model using an improved binary bat algorithm

Faulty processor identification for a multiprocessor system under the Malek model using an improved binary bat algorithm

Vulnerability analysis of multiprocessor system based on burnt pancake networks

With the large scale network’s popularity in different fields, such as cloud computing, privacy protection, social networks and so on, the robustness analysis of networks against faulty processors has recently attracted more attention. When some processors are attacked and lose function, how to characterize the communication ability and efficiency of the surviving network is very crucial. From the viewpoint of graph theory, the size of the maximal component after removing certain faulty vertices can be viewed as a metric of fault tolerability, which is reckoned as an important dimension for the robustness analysis of a network. The h-component connectivity of a network G, denoted by cκh(G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least h components. In this paper, we show that when removing any subset X with order of at most 5n−9 in the burnt pancake network BPn(n≥5), BPn∖X has the largest component C with |C|≥|V(BPn)|−|X|−4 vertices, and all small components present the following structures: (a) an empty graph; (b) one 3-path, one K1,3, one K1,2, or an edge, or an isolated vertex; (c) one K1,2 and an edge, one K1,2 and an isolated vertex, two edges, one edge and an isolated vertex, or two isolated vertices; (d) an edge and two isolated vertices, or three isolated vertices; (e) four isolated vertices. Besides, we determine that the 6-component connectivity of BPn is cκ6(BPn)=5n−6 for n≥5.

Reliability Evaluation of Bicube-Based Multiprocessor System under the g-Good-Neighbor Restriction

Multiprocessor systems are commonly deployed for big data analysis because of evolution in technologies such as cloud computing, IoT, social network and so on. Reliability evaluation is of significant importance for maintenance and improvement of fault tolerance for multiprocessor systems, and system-level diagnosis is a primary strategy to identify the faulty processors in the systems. In this paper, we first determine the [Formula: see text]-good-neighbor connectivity of the [Formula: see text]-dimensional Bicube-based multiprocessor system [Formula: see text], a novel variant of hypercube. Besides, we establish the [Formula: see text]-good-neighbor diagnosability of the Bicube-based multiprocessor system [Formula: see text] under the PMC and MM* models.

The component connectivity, component diagnosability, and t/k-diagnosability of Bicube networks

With the rapid expansion of the scale of multiprocessor systems, the importance of fault tolerance and fault diagnosis is increasingly concerned. For an interconnection network G, the h-component connectivity is an important indicator for evaluating the fault tolerance of G, which is defined as the minimum number of vertices whose deletion will disconnect G such that the remaining has at least h components. The h-component diagnosability, a newly precise diagnosis strategy to analyze the reliability of G, is the diagnosability under the condition that the number of components is at least h in the resulting graph after removing the faulty processor set. The t/k-diagnosability is a classic imprecise diagnosis strategy, which can identify up to t faulty processors by sacrificing accuracy to a certain extent, namely misdiagnosing at most k fault-free processors. In this paper, we investigate some combinatorial properties and the fault tolerance ability of the n-dimensional Bicube network, denoted by BQn. Then we first prove that the (h+1)-component connectivity of BQn is h(n−1)−h(h−1)2+1 (n≥6, 1≤h≤n−1). Moreover, we derive that the (h+1)-component diagnosability of BQn is (h+1)(n−1)−h(h+1)2+1 under the PMC model and MM⁎ model (n≥7, 1≤h≤n−3). Furthermore, under the PMC model, we propose the t/k-diagnosis algorithm of BQn and then derive that BQn is [(k+1)n−k(k+3)2]/k-diagnosable (n≥6, 0≤k≤n−2).

The Reliability ofk-Aryn-Cube Based on Component Connectivity

AbstractConnectivity and diagnosability are two crucial subjects for a network’s ability to tolerate and diagnose faulty processors. The $r$-component connectivity $c\kappa _{r}(G)$ of a network $G$ is the minimum number of vertices whose deletion results in a graph with at least $r$ components. The $r$-component diagnosability $ct_{r}(G)$ of a network $G$ is the maximum number of faulty vertices that the system can guarantee to identify under the condition that there exist at least $r$ fault-free components. This paper first establishes that the $(r+1)$-component connectivity of $k$-ary $n$-cube $Q^{k}_{n}$ is $c\kappa _{r+1}(Q^{k}_{n})=-\frac{1}{2}r^{2}+\Big(2n-\frac{1}{2}\Big)r+1$ for $n\geq 2$, $k\geq 4$ and $1\leq r\leq n$. In view of $c\kappa _{r+1}(Q^{k}_{n})$, we prove that the $(r+1)$-component diagnosabilities of $k$-ary $n$-cube $Q^{k}_{n}$ under the PMC model and MM* model are $ct_{r+1}(Q^{k}_{n})=-\frac{1}{2}r^{2}+\Big(2n-\frac{3}{2}\Big)r+2n$ for $n\geq 4$, $k\geq 4$ and $1\leq r\leq n-1$.

Optimal Identifying Codes in Oriented Paths and Circuits

Identifying codes in graphs are related to the classical notion of dominating sets [1]. Since there first introduction in 1998 [2], they have been widely studied and extended to several application, such as: detection of faulty processor in multiprocessor systems, locating danger or threats in sensor networks. Let G=(V,E) an unoriented connected graph. The minimum identifying code in graphs is the smallest subset of vertices C, such that every vertex in V have a unique set of neighbors in C. In our work, we focus on finding minimum cardinality of an identifying code in oriented paths and circuits

A Formal Method for Determining the State of Processors in a Multiprocessor System under Testing

The paper deals with the problem of self-testing of multiprocessor systems within the framework of the Preparata–Metze–Chien model. A method is proposed to formalize the procedure for establishing the state of all processors of the $$M$$ -diagnosable system based on the analysis of the results of a certain set of mutual test checks by processors. The analysis is reduced to solving a Boolean equation based on the results of these checks and permits one to establish either the state (operational–faulty) of all processors in the system or the fact that the number of checks performed is not sufficient to do so (i.e., the state of some specific processors is undetermined). This remains valid in cases where the number of faulty processors in the system does not exceed $$M$$ .

On [formula omitted]-good-neighbor conditional connectivity and diagnosability of hierarchical star networks

In this paper, we study the connectivity and diagonosability of n-dimensional hierarchical star network HSn based on the concept of forbidden faulty sets. In a forbidden faulty set, certain nodes cannot be faulty at the same time and this model can better reflect fault patterns in a real system than the existing ones. Under the condition every fault-free node in a network contains at least g fault-free neighbors, the g-good neighbor conditional connectivity is defined as the minimum number of faulty processors whose deletion makes the network disconnected, the g-good neighbor conditional diagnosability tg(G) is defined as the maximum number of faulty processors that the network can guarantee to identify solely by performing mutual tests among the processors. We investigate the g-good neighbor conditional connectivity and the g-good-neighbor conditional diagnosability of n-dimensional hierarchical star network HSn. Our results show that the g-good-neighbor conditional connectivity of HSn is (n−g)(g+1)!−1 and the g-good-neighbor conditional diagnosability of HSn under the PMC and MM* models is (n−g+1)(g+1)!−1 when 0≤g≤n−3,g≠2 and n≥4. In addition, we show the 2-good-neighbor conditional connectivity of HSn is 4n−8 and the 2-good-neighbor conditional diagnosability of HSn under the PMC and MM* models is 4n−5 when n≥4.

Theg-Extra Conditional Diagnosability of Graphs in Terms ofg-Extra Connectivity

The [Formula: see text]-extra conditional diagnosability and [Formula: see text]-extra connectivity are two important parameters to measure ability of diagnosing faulty processors and fault tolerance in a multiprocessor system. The [Formula: see text]-extra conditional diagnosability [Formula: see text] of graph [Formula: see text] is defined as the diagnosability of a multiprocessor system under the assumption that every fault-free component contains more than [Formula: see text] vertices. While the [Formula: see text]-extra connectivity [Formula: see text] of graph [Formula: see text] is the minimum number [Formula: see text] for which there is a vertex cut [Formula: see text] with [Formula: see text] such that every component of [Formula: see text] has more than [Formula: see text] vertices. In this paper, we study the [Formula: see text]-extra conditional diagnosability of graph [Formula: see text] in terms of its [Formula: see text]-extra connectivity, and show that [Formula: see text] under the MM* model with some acceptable conditions. As applications, the [Formula: see text]-extra conditional diagnosability is determined for some BC networks such as hypercubes, varietal hypercubes, and [Formula: see text]-ary [Formula: see text]-cubes under the MM* model.

Reliability Evaluation of Generalized ExchangedX-Cubes Based on the Condition ofg-Good-Neighbor

In the cloud computing environment with massive information services and decision-making resources, the accuracy and reliability of information are more important than previous single closed systems. Therefore, ensuring the reliability of information and the stable operation of the system are the core problems in the research fields such as the Internet Plus and the Internet of Things. The connectivity and diagnosability are two important measures for the fault tolerance of multiprocessor systems. Theg-good-neighbor conditional connectivity (Rg-connectivity) is the minimum number of nodes that make the graph disconnected, and each node has at leastgneighbors in every remaining component. Theg-good-neighbor conditional diagnosability (g-GNCD) is the maximum number of faulty processors that has been correctly identified in a system, and any fault-free processor has no less thangfault-free neighbors. ExchangedX-cubes are a class of irregular networks, obtained by deleting links from hypercubes and some variant networks of hypercubes (X-cubes). They not only combine the advantages ofX-cubes but also reduce the interconnection complexity. ExchangedX-cubes classify its nodes into two different classes clusters with a unique connecting rule. In this paper, we propose the generalized exchangedX-cubes framework so that architecture can be constructed by different connecting rules. Furthermore, we study theRg-connectivity andg-GNCD of generalized exchangedX-cubes under the PMC and MM∗models. As applications, theRg-connectivity andg-GNCD of generalized exchanged hypercubes, dual-cube-like networks, generalized exchanged crossed cubes, and locally generalized exchanged twisted cubes are determined, respectively.

Diagnosability for two families of composition networks

Diagnosability is an important parameter to assess the reliability of multiprocessor systems. A multiprocessor system is called t-diagnosable if all faulty processors can be identified without replacement as long as the number of faulty processors does not exceed t. The diagnosability is the maximum value of t such that multiprocessors system is t-diagnosable. Matching composition networks (MCNs) and cycle composition networks (CCNs) are two typical network structures to construct multiprocessor systems. Let G be a MCN which is obtained by adding a perfect matching between two graph G0 and G1, and let H be a CCN obtained by adding a perfect matching between Gi and Gi+1(modm) for each i∈{0,1,…,m−1} with m≥3. In this paper, we show that the diagnosability of G under the PMC (δ(G)≥2) model and MM⁎ model (δ(G)≥5) is n−1 if G0≅G1≅Kn; otherwise, δ(G). Furthermore, we show that the diagnosability of H under the PMC model (δ(H)≥4) and MM⁎ model (δ(H)≥5) is δ(H).

System management recovery in NoC-based many-core systems

The technology nodes reduction enabled the emergence of NoC-based many-cores with dozens to hundreds of processing elements (PEs). Despite the processing power offered by a large number of processors and communication flexibility due to the adoption of NoCs, it is necessary to manage the many-core resources to ensure scalability. The execution of the management tasks requires a PE reserved exclusively to execute such actions. These processors are named managers PE–MPE. A centralized approach would induce a significant load to the MPE in large-scale systems, and a permanent fault in the MPE would compromise the entire system. The adoption of a distributed approach, organization adopted in this work, with MPEs hierarchically organized, reduces the management load, and a fault in an MPE would compromise only the PEs managed by the faulty MPE. The literature presents several fault-tolerant proposals targeting the NoC or the processors. However, there is a significant gap related to fault-tolerant methods at the system level, i.e., related to fault-tolerant techniques regarding the MPEs. The goal of this paper is to present a recovery method when an MPE became faulty, and propose a protocol to migrate the management software safely to a new PE. The method adopts task migration to release a processor if there is no processor to receive the kernel that was executing in a faulty processor. The proposal is transparent to the applications running in the many-core, with an overhead in the execution time varying between 1.5 and 1.65 ms during the management and task migration.

The Relationship Between Extra Connectivity and t/k-Diagnosability of Regular Networks

The g-extra connectivity of a multiprocessor system modeled by a graph G, denoted by [Formula: see text] (G), is the minimum number of removed vertices such that the network is disconnected and each residual component has no less than g + 1 vertices. The t/k-diagnosis strategy can detect up to t faulty processors which might include at most k misdiagnosed processors. These two parameters are important to measure the fault tolerant ability of a multiprocessor system. The extra connectivity and t/k-diagnosability of many well-known networks have been investigated extensively and independently. However, the general relationship between the extra connectivity and the t/k-diagnosability of general regular networks has not been established. In this paper, we explore the relationship between the k-extra connectivity and t/k-diagnosability for regular networks under the classic PMC diagnostic model. More specifically, we derive the relationship between 1-extra connectivity and pessimistic diagnosability for regular networks. Furthermore, the t/k-diagnosability and pessimistic diagnosability of some networks, including star network, BC networks, Cayley graphs generated by transposition trees etc., are determined.

The 1-good-neighbor diagnosability of unidirectional hypercubes under the PMC model

The hypercubes are a famous class of networks for multiprocessor systems and the unidirectional hypercubes are hypercube interconnection topologies with simplex unidirectional links. Under the classic PMC model, each processor in a multiprocessor system tests a subset of its neighbors. The collection of tests in this system can be modeled by a directed graph. The diagnosability of a system is the maximum number of faulty processors that the system may identify according to the outcomes of the tests, and the g-good-neighbor diagnosability is a more accurate indicator than the diagnosability. In this paper, we first determine the 1-good-neighbor connectivity of unidirectional hypercubes and then determine the diagnosability and 1-good-neighbor diagnosability of hypercube networks when unidirectional hypercubes are used as the collection of tests under the PMC model.