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.