Connective Fault Tolerance in Multiple-Bus System Hung-Kuei Ku and John P. Hayes IEEE Transactions...
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Transcript of Connective Fault Tolerance in Multiple-Bus System Hung-Kuei Ku and John P. Hayes IEEE Transactions...
Connective Fault Tolerance in Multiple-Bus System
Hung-Kuei Ku and John P. HayesIEEE Transactions on parallel and distributed System, VOL.
8, NO. 6, June 1997
元智大學 資訊工程所 陳桂慧
1999.05.19
Outline
• PBL Graph model
• Connectivity and faults– Processor fault tolerance
– Bus fault tolerance
– Link fault tolerance
• Application– M-LANs
– Spanning Bus Hypercubes
• Conclusion
A Representative multiple-bus system
P1 P2 P3 P4
memory memory memory memory
l1 l2 l3 l4 l5 l6 l9l8l7
b1
b3b2
Processors
Links
Buses
Processor-Bus-Link (PBL) Graph
When there is no fault
PBL graph G’
P1 P4P3P2
b1 b2 b3
l1
l2
l3
l4
l5
l6 l9
l8
l7
P1 P4P2
b1 b3
l1l3
l5
l9
P-nodes
B-nodes
when processor p3, bus b2, and link l8 are faulty
Component Adjacency Graph
processor adjacency graph (PAG) Gp’
BAG Gb’
LAG Gl’
l1l2
l3
l4
l5
l6l9
l8
l7
b1
b3 b2p1
p4
p2
p3
A PBL Graph G’’
P1 P4P3P2 P4
b1 b2 b3 b3
l1
l2
l3 l4
l5
l6
l9
l8
l7l10 l11
l12
p1
p4
p2
p3
P5
BAG Gb’’
b1
b3
b2
l1l2 l3
l4
l5
l6
l9l8
l7
l11
l10
l12
LAG Gl’’
PAG Gp’’
b4
Processor Fault Tolerance
• P-nodes are connected in a PBL graph if and only if its PAG is connected.
• THEOREM 1.
A PBL graph G is (K(Gp)-1)-PFT where Gp is the PAG of G.– Kp-PFT, Kp is the degrees of processor fault tolerances of G.– The (node) connectivity K(G) of G is the cardinality of a smallest node
cut of G, a node cut of G is a set of nodes whose result in a disconnected or trivial graph.
• COROLLARY 1.
The minimum critical processor-fault sets of a PBL graph G are the minimum node cuts of its PAG Gp.
• LEMMA 1.
Let G be a PBL graph with no isolated P-node or B-node, and let G contain at least two P-nodes and B-nodes. The P-nodes of G are connected if and only if its B-nodes are connected.
• THEOREM 2.
Let G be a PBL graph that contain b B-nodes. If ßp(G)<b, then G is (min{K(Gb),ßp(G)}-1)-BFT, where Gb is the BAG of G; otherwise, G is (ßp(G)-1)-BFT.
Bus Fault Tolerance
Bus Fault Tolerance (2)
• COROLLARY 2.
Let G be the BAG of a PBL graph G. Then the minimum critical bus-fault set of G are given by the following three cases:– the minimum node cuts of Gb, if ßp(G)<b and
K(Gb)<ßp(G);
– the minimum neighborhoods of the P-nodes of G, if ßp(G)=b, K(Gp)>ßp(G), or K(Gb)=ßp(G)=b-1;
– all the minimum critical bus-fault sets described in the precious two cases, if K(Gb)=ßp(G)<b-1.
• LEMMA 2.
The P-nodes of a PBL graph G with no isolate P-node are connected if and only if the edges of G are connected.
• THEOREM 3.
A PBL Graph G is (min{K(Gl), ðp(G)}-1)-LFT where Gl is the LAG of G
Link Fault Tolerance
Link Fault Tolerance
• COROLLARY 3.
Let Gl be the LAG of a PBL graph G. Then the minimum critical link-fault sets of G are given by the following three cases:– the minimum node cuts of Gl, if K(Gl)<ðp(G);
– edge sets, each of which contains all the edges incident with a P-node of degree ðp(G), if K(Gl)>ðp(G);
– all the minimum critical link-fault sets described in the previous two cases, if K(Gl)=ðp(G).
Application - M-LAN
• M-LAN, multi-channel local area network, every processor is connected to all buses.
P1 P3P2
b1 b2 b3 b3
LAG of K3,4The complete graph K3,4
• PAG of Kp,b is the complete graph Kb of p nodes, => Kp,b is (p-2)-PFT, – ßp(Kp,b) = b, => (b-1)-BFT.
• LEMMA 3.
The LAG Gl of the complete bipartite graph Kp,b is (p+b-2)-connected.
• THEOREM 4.
An M-LAN with p>=2 processors and b buses is (p-2,b-1,b-1)-FT
Application - M-LAN
Application - Spanning Bus Hypercubes
A w-wide d-dimensional spanning-bus hypercube is (d(w-1), d-1, d-1)-FT
Conclusion
• The PBL model is straightforward but very general.– The B-node can represent any communication medium that
transfers signals/data to all the components connect to it.
• The PBL graph model can efficiently model a wide variety of faults such as processor, bus, and link faults.– The component adjacency graphs derived from the PBL
graph are particularly useful for identifying a network’s “weak” point, such as its minimum critical fault sets.