Multi-Processors by the Numbers: Mathematical Foundations of Spaceflight Grid Computing

LaForge, L.A.; Turner, Jacob

doi:10.1109/aero.2006.1655990

Cited by 2 publications

(11 citation statements)

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“…-Addressing a general audience with an overview of our work, the diagrams and informal narrative of "Multi-Processors by the Numbers" [44] unfold how feasible regions govern multi-processor design and operation, and elaborate grid computing as mentioned in the Abstract above. -Technologists and engineers may also be interested in "Vertical Cavity Surface Emitting Lasers for Spaceflight Multi-Processors" [43], our broadly-scoped report on VCSELs as enablers for tunable architectures.…”

Section: +17757846951mentioning

confidence: 99%

“…To put a face on this formula, let the worst-case fault tolerance f = f (n) scale as a function of the total number of nodes n (cf. [44] Fig. 13), and substitute f (n) into the righthand side of (3).…”

Section: Tunable Multi-processor Topologiesmentioning

confidence: 99%

“…Provably bounded, minimum or near-minimum latency is particularly crucial to avionics multi-processors and realtime networks. For any count n of nodes, Harary-Hayes graphs optimize the tradeoff between overall channel network cost and [44] worst-case fault tolerance f. If our objective is to minimize latency, on the other hand, then Harary-Hayes graphs are about as far from optimal (cf. Section 2.2) as we can get: the radius and diameter are very nearly the worst possible!…”

Section: )mentioning

confidence: 99%

“…Section 2.2) as we can get: the radius and diameter are very nearly the worst possible! Additional illustrations in [44] Additional illustrations in [44] In a recent request for proposals, for example, the United States Navy [57] issued a call for W Band avionics based on A complete mathematical foundation … for end-to-end operation that provides predictable, stable performance. The performance characteristics of the whole network must be guaranteed and stable within the constraints of temporal accuracy, stability and high utilization.…”

Section: )mentioning

confidence: 99%

“…To this end, Figure 9 of [44] depicts how we would benefit from an automated knowledge catalog of theorems and algorithms. Recognizing the interdisciplinary nature of this knowledge, we will henceforth refer to it as the mathematics of connectivity.…”

Section: )mentioning

confidence: 99%

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Spaceflight Multi-Processors with Fault Tolerance and Connectivity Tuned from Sparse to Dense

LaForge¹,

Moreland²,

Fadali³

2006 IEEE Aerospace Conference

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A fault is a condition which renders the node or channel untrustworthy, hence (in this paper) unusable. A network instantaneously subjected to faulty nodes and channels tolerates them if the remaining healthy nodes and channels form a quorum. Also in this paper: we assume that faulty nodes have been diagnosed, and that hardware and software can deny their connection to healthy nodes. The fault tolerance is the maximum number of faults that an n-node network can tolerate. Under a graph model, the tolerance f to faulty nodes is at most the tolerance to faults in channels, or any combination of faults in nodes and channels, for n > f+1 > 1. Throughput this paper we therefore employ a conservative model of faults in nodes. The worst-case (nodal) network fault tolerance f equals one less than the (vertex) connectivity f+1 of the corresponding graph. The fractional fault tolerance f / n equals the fault tolerance divided by the number of nodes. The fractional connectivity ( f + 1) / n is only slightly less than the fractional fault tolerance, and is more convenient for comparing the cost of worst-case fault tolerance with the probabilistic cost of tolerating a constant fraction of faults that are distributed, say, with Bernoulli probability pdi)graph: directed, weighted Vertex, edge, arc, hyperedge: pp. 4, 5, 13. Planar: pp. 10, 14. Cf. adjacency. Strict graph: each edge's endpoints are distinct (no loops) and appear in but one edge. Cf. multi-graph p. 9 Hamiltonian; p. 8 Path or cycle spanning all vertices Hamming distance, graph Definitions: p. 13. Also see Gray code, p. 13 (hyper)cube; mesh Definitions: pp. 13, 20. Qualifiers: dimension, (majorized ) radix. Cf. hyperedge, j-edge. (hyper)edge, separator Definitions: pp. 13, 14. Also see: hyperedge matching, hyperfactorization, p. 14 interior-disjoint, IDJ; p. 5 Two paths are interior-disjoint ( IDJ ) if, apart from their endpoints, they do not intersect metric space, distance; distance space; Table 1 Real-valued distance |⋅, ⋅| ≥ 0 mapping all pairs from a given set, such that: a) |x, y| = 0 if and only if x = y; b) symmetry: |x, y| = |x, y|; c) triangle inequality: |x, y| ≤ |x, y| + |x, y| order, size; p. 10 Vertex resp. edge cardinality path, endpoint, (path)length; pp. 4, 17Endpoint vertex x, or union P(x … z) of path P(x … y) with edge ( y, z ), such that y and z ∉ P(x … y) are endpoints of P(x … y) resp. P(x … z). The (path)length of P is its size quorum; p. 4 A configuration of processing nodes capable of accomplishing the mission. A quorum is achieved if sufficient healthy nodes (i.e., nodes that have not failed) remain capable of communicating among themselves. In graph-theoretic terms, a quorum is equivalent to a connected subgraph (or component) induced by deleting, from the graph corresponding to the original network, vertices and edges corresponding to faulty nodes and channels. The basic idea of a quorum as a connected induced subgraph may be varied by imposing constraints (e.g., we could insist the quorum be a tree [30] or an array [36]), or by relaxing...

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Section: +17757846951mentioning

confidence: 99%

Section: Tunable Multi-processor Topologiesmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

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