An accurate performance model for network-on-chip and multicomputer interconnection networks

In this paper, we present a mathematical background for a new approach for performances modeling of interconnection networks, based on analyzing the packet blocking and waiting time spent in each channel passing through all possible paths in the channel dependency graph. We have proposed a new, simple and very accurate analytical model for deterministic routing in wormhole networks, which is general in terms of the network topology and traffic distribution. An accurate calculation of the variance of the service time has been developed, which overcomes the rough approximation used, as a rule, in the existing models. The model supports two-dimensional mesh topologies, widely used in network-on-chip architectures, and multidimensional topologies, popular in multicomputer architectures. It is applicable even for irregular topologies and arbitrary application-specific traffic. Results obtained through simulation show that the model achieves a high degree of accuracy.     

A fault-tolerant wormhole routing scheme for torus networks with nonconvex faults

In this paper, we present a fault-tolerant routing algorithm for torus networks by using only 4 virtual channels. The proposed algorithm is based on the solid fault model, which includes rectangular faults and many practical nonconvex faults. Previous works need at least 6 virtual channels to achieve the same fault-tolerant ability.     

Performance evaluation of deterministic wormhole routing in k-ary n-cubes

We present a new analytical approach for the performance evaluation of deterministic wormhole routing in k-ary n-cubes. Our methodology achieves closed formulas for average time values through the analysis of network flows. The comparison with simulation models demonstrates that our methodology gives accurate results for both low and high traffic conditions. Another important quality is the flexibility of our approach. We demonstrate that it can be used to model dimension-ordered-routing in several k-ary n-cubes such as hypercubes, 3D symmetric and asymmetric tori, architectures with uni- and bi-directional channels.     

A new performance measure for characterizing fault rings in interconnection networks

One of the fundamental issues in parallel computers is how to efficiently perform routing in a faulty network where each component fails with some probability. Adaptive fault-tolerant routing algorithms in such systems have been frequently suggested as a means of providing continuous operations in the presence of one or more failures by allowing the graceful system degradation. Many algorithms involve adding buffer space and complex control logic to the routing nodes. However, the addition of extra logic circuits and buffer space makes nodes more liable to failure and less reliable. Further, if the shape of fault pattern is confined, then many non-faulty nodes will be sacrificed and hence their resources are wasted. This is clearly an undesirable solution and motivates solutions that provoke efficient use of non-faulty nodes. One such approach to reducing the number of functional nodes that must be marked as faulty is based on the concept of fault rings to support more flexible routing around rectangular fault regions. Before such schemes can be successfully incorporated in networks, it is necessary to have a clear understanding of the factors that affect their performance potential. In this paper, we propose the first general solution for computing the probability of message facing the fault rings with and without overlapping in the well-known torus networks. We also conduct extensive simulation experiments using various fault patterns, the results of which are used to confirm the good accuracy of the proposed analytical models.     

The impact of virtual channel allocation on the performance of deterministic wormhole-routed k-ary n-cubes

Virtual channels yield significant improvement in the performance of wormhole-routed networks as they can greatly reduce message blocking over network resources. K-ary n-cubes with deterministic routing have been widely analysed using analytical modelling tools. Most existing models, however, have either entirely ignored the effects of virtual channel multiplexing or have not considered the impact of virtual channels allocation on message latency. This paper discusses two different organisations of virtual channels in k-ary n-cubes, resulting in two deterministic routing algorithms. It then proposes an analytical model to compute message latency for the two routing algorithms. The proposed model is used in a case study to demonstrate the sensitivity of network latency to the way virtual channels are allocated to messages.     

An accurate mathematical performance model of adaptive routing in the star graph

Analytical modelling is indeed the most cost-effective method to evaluate the performance of a system. Several analytical models have been proposed in the literature for different interconnection network systems. This paper proposes an accurate analytical model to predict message latency in wormhole-switched star graphs with fully adaptive routing. Although the focus of this research is on the star graph but the approach used for modelling can be, however, used for modelling some other regular and irregular interconnection networks. The results obtained from simulation experiments confirm that the proposed model exhibits a good accuracy for various network sizes and under different operating conditions.     

Adaptive routing in wormhole-switched necklace-cubes: Analytical modelling and performance comparison

The necklace hypercube has recently been introduced as an attractive alternative to the well-known hypercube. Previous research on this network topology has mainly focused on topological properties, VLSI and algorithmic aspects of this network. Several analytical models have been proposed in the literature for different interconnection networks, as the most cost-effective tools to evaluate the performance merits of such systems. This paper proposes an analytical performance model to predict message latency in wormhole-switched necklace hypercube interconnection networks with fully adaptive routing. The analysis focuses on a fully adaptive routing algorithm which has been shown to be the most effective for necklace hypercube networks. The results obtained from simulation experiments confirm that the proposed model exhibits a good accuracy under different operating conditions.     

Strongly Hamiltonian laceability of the even k-ary n-cube

The interconnection network considered in this paper is the k-ary n-cube that is an attractive variance of the well-known hypercube. Many interconnection networks can be viewed as the subclasses of the k-ary n-cubes include the cycle, the torus and the hypercube. A bipartite graph is Hamiltonian laceable if there exists a Hamiltonian path joining every two vertices which are in distinct partite sets. A bipartite graph G is strongly Hamiltonian laceable if it is Hamiltonian laceable and there exists a path of length N − 2 joining each pair of vertices in the same partite set, where N = |V(G)|. We prove that the k-ary n-cube is strongly Hamiltonian laceable for k is even and n  2.     

Embedding paths and cycles in 3-ary n-cubes with faulty nodes and links

The k-ary n-cube, denoted by , is one of the most important interconnection networks for parallel computing. In this paper, we consider the problem of embedding cycles and paths into faulty 3-ary n-cubes. Let F be a set of faulty nodes and/or edges, and n?2. We show that when |F|?2n-2, there exists a cycle of any length from 3 to in . We also prove that when |F|?2n-3, there exists a path of any length from 2n-1 to between two arbitrary nodes in . Since the k-ary n-cube is regular of degree 2n, the fault-tolerant degrees 2n-2 and 2n-3 are optimal.     

Augmented k-ary n-cubes

We define an interconnection network AQ_n,k which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube AQ_n,k has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube AQ_n,k: is a Cayley graph, and so is vertex-symmetric, but not edge-symmetric unless n = 2; has connectivity 4n − 2 and wide-diameter at most max{(n − 1)k − (n − 2), k + 7}; has diameter , when n = 2; and has diameter at most , for n ? 3 and k even, and at most , for n ? 3 and k odd.     

Performance comparison of different multicast routing strategies in disruption tolerant networks

Disruption Tolerant Networks (DTNs) technologies are emerging solutions to networks that experience frequent partitions. As a result, multicast design in DTNs is a considerably more difficult problem compared to that in Internet and mobile ad hoc networks. In this paper, we first investigate three basic DTN multicast strategies, including unicast-based multicast (U-Multicast), static-tree-based multicast (ST-Multicast) and dynamic-tree-based multicast (DT-Multicast) strategies. Then we focus on studying two DT-Multicast routing schemes: Dynamic Tree Based Routing (DTBR) and On-demand Situation-aware Multicast (OS-Multicast), which address the challenges of utilizing opportunistic links to conduct dynamic multicast structures in DTNs. Performances of different strategies are then evaluated by simulations, including applying the real-world DTN traces. Our results show that OS-Multicast and DTBR can achieve higher message delivery ratio than that of using U-Multicast and ST-Multicast strategies. Also, to get better performance, we recommend that system designers select OS-Multicast when the source traffic rate is low.     

Modeling and analyzing the performance of adaptive hierarchical networks

As networks become larger, scalability and QoS-awareness become important issues that have to be resolved. A large network can be effectively formed as a hierarchical structure, such as the inter/intra-domain routing hierarchy in the Internet and the Private Network-to-Network Interface (PNNI) standard, to resolve these critical issues. Methods of modeling and analyzing the performance of QoS-capable hierarchical networks become an open issue. Although the reduced load approximation technique has been extensively applied to flat networks, the feasibility of applying it to the hierarchical network model has seldom been investigated. Furthermore, most of the research in this area has focused on the performance evaluation with fixed routing. This work proposes an analytical model for evaluating the performance of adaptive hierarchical networks with multiple classes of traffic. We first study the reduced load approximation model for multirate loss networks, and then propose a novel performance evaluation model for networks with hierarchical routing. This model is based on a decomposition of a hierarchical route into several analytic hierarchical segments; therefore the blocking probability of the hierarchical path can be determined from the blocking probabilities of these segments. Numerical results demonstrate that the proposed model for adaptive hierarchical routing yields accurate blocking probabilities. We also investigate the convergence of the analysis model in both the originating-destination (O-D) pair and the alternative hierarchical path. Finally, the blocking probability of the adaptive hierarchical O-D pair is demonstrated to depend on the blocking of all hierarchical paths but not on the order of the hierarchical path of the same O-D pair.     

Edge-bipancyclicity of the k-ary n-cubes with faulty nodes and edges

The k-ary n-cube has been one of the most popular interconnection networks for massively parallel systems. In this paper, we investigate the edge-bipancyclicity of k-ary n-cubes with faulty nodes and edges. It is proved that every healthy edge of the faulty k-ary n-cube with f_v faulty nodes and f_e faulty edges lies in a fault-free cycle of every even length from 4 to kⁿ − 2f_v (resp. kⁿ − f_v) if k ? 4 is even (resp. k ? 3 is odd) and f_v + f_e ? 2n − 3. The results are optimal with respect to the number of node and edge faults tolerated.     

Multi-Radio Multi-Channel无线传感器网络中信道分配和路由策略

针对无线传感器网络中传输时延长、传输冲突大和吞吐量低等问题,提出了一种在Multi-Radio Multi-Channel无线传感器网络中信道分配和路由策略.该策略动态地建立k元n立方体拓扑结构,使用优化的静态信道分配算法提高节点的吞吐量,使用维序寻径的路由算法减少传输冲突.该方法适用于网络节点稠密、节点相互之间通信冲突大的情况,并且在单跳和多跳的网络环境下均适用.实验结果表明,基于k元n立方体这一拓扑结构的信道分配和路由策略与传统方法相比,有效地减少了端到端时延,降低了网络冲突,减少了节点能量消耗,延长了网络寿命,提高了网络吞吐量.     

Upper bounds on the queuenumber of k-ary n-cubes

A queue layout of a graph consists of a linear order of its vertices, and a partition of its edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph G, denoted by qn(G), is called the queuenumber of G. Heath and Rosenberg [SIAM J. Comput. 21 (1992) 927-958] showed that boolean n-cube (i.e., the n-dimensional hypercube) can be laid out using at most n−1 queues. Heath et al. [SIAM J. Discrete Math. 5 (1992) 398-412] showed that the ternary n-cube can be laid out using at most 2n−2 queues. Recently, Hasunuma and Hirota [Inform. Process. Lett. 104 (2007) 41-44] improved the upper bound on queuenumber to n−2 for hypercubes. In this paper, we deal with the upper bound on queuenumber of a wider class of graphs called k-ary n-cubes, which contains hypercubes and ternary n-cubes as subclasses. Our result improves the previous bound in the case of ternary n-cubes. Let denote the n-dimensional k-ary cube. This paper contributes three main results as follows:

(1)

if n?3.

(2)

if n?2 and 4?k?8.

(3)

if n?1 and k?9.

     

Pancyclicity of ternary n-cube networks under the conditional fault model

A graph G is said to be conditional k-edge-fault pancyclic if after removing k faulty edges from G, under the assumption that each vertex is incident to at least two fault-free edges, the resulting graph contains a cycle of every length from its girth to |V(G)|. In this paper, we consider ternary n-cube networks and show that they are conditional (4n−5)-edge-fault pancyclic.     

Parallel Lagrange interpolation on <Emphasis Type="Italic">k</Emphasis>-ary <Emphasis Type="Italic">n</Emphasis>-cubes with maximum channel utilization

This paper proposes an efficient parallel algorithm for computing Lagrange interpolation on k-ary n-cube networks. This is done using the fact that a k-ary n-cube can be decomposed into n link-disjoint Hamiltonian cycles. Using these n link-disjoint cycles, we interpolate Lagrange polynomial using full bandwidth of the employed network. Communication in the main phase of the algorithm is based on an all-to-all broadcast algorithm on the n link-disjoint Hamiltonian cycles exploiting all network channels, and thus, resulting in high-efficiency in using network resources. A performance evaluation of the proposed algorithm reveals an optimum speedup for a typical range of system parameters used in current state-of-the-art implementations.

A Performance Model of Adaptive Wormhole Routing in k-Ary n-Cubes in the Presence of Digit-Reversal Traffic

Several analytical models of fully adaptive routing in wormhole-routed k-ary n-cubes under the uniform traffic pattern have recently been proposed in the literature. Although the uniform reference model has been widely used by researchers, it is not always true in practice as there are many applications that exhibit traffic nonuniformity. There has hardly been any study that describes an analytical model of fully adaptive routing under nonuniform traffic conditions. This paper describes the first analytical model of fully adaptive routing in k-ary n-cubes in the presence of nonuniform traffic generated by the digit-reversal permutation, which is an important communication operation found in many matrix computation problems. Results obtained through simulation experiments confirm that the model predicts message latency with a good degree of accuracy under different working conditions.     

On the k-tuple domination of de Bruijn and Kautz digraphs

In a digraph G, a vertex u is said to dominate itself and vertices v such that (u,v) is an arc of G. For a positive integer k, a k-tuple dominating set D of a digraph is a subset of vertices such that every vertex is dominated by at least k vertices in D. The k-tuple domination number of a given digraph is the minimum cardinality of a k-tuple dominating set of the digraph. In this letter, we give the exact values of the k-tuple domination number of de Bruijn and Kautz digraphs.     

Panconnectivity and edge-pancyclicity of 3-ary <Emphasis Type="Italic">N</Emphasis>-cubes

We study two topological properties of the 3-ary n-cube Q _n ³. Given two arbitrary distinct nodes x and y in Q _n ³, we prove that there exists an x–y path of every length ranging from d(x,y) to 3 ⁿ −1, where d(x,y) is the length of a shortest path between x and y. Based on this result, we prove that Q _n ³ is edge-pancyclic by showing that every edge in Q _n ³ lies on a cycle of every length ranging from 3 to 3 ⁿ .