Connective Fault Tolerance in Multiple-Bus System

1. Connective Fault Tolerance in Multiple-Bus System Hung-Kuei Ku and John P. Hayes IEEE 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

2. Outline • PBL Graph model • Connectivity and faults • Processor fault tolerance • Bus fault tolerance • Link fault tolerance • Application • M-LANs • Spanning Bus Hypercubes • Conclusion

3. A Representative multiple-bus system P1 P2 P3 P4 Processors memory memory memory memory Links l2 l3 l4 l5 l6 l7 l8 l9 l1 b1 Buses b2 b3

4. Processor-Bus-Link (PBL) Graph When there is no fault PBL graph G’ P1 P2 P3 P4 P1 P2 P4 P-nodes l3 l6 l9 l3 l9 l1 l1 l4 l7 l5 l5 l8 l2 b1 b2 b3 b1 b3 B-nodes when processor p3, bus b2, and link l8 are faulty

5. Component Adjacency Graph processor adjacency graph (PAG) Gp’ b1 p2 p1 b2 l1 b3 l2 l8 BAG Gb’ p4 p3 l9 l6 l7 l4 l3 l5 LAG Gl’

6. P1 P2 P3 P4 P4 l1 l2 l3 l3 l4 l12 l8 l4 l1 l10 l6 A PBL Graph G’’ l6 l9 l11 l7 l5 l2 l11 l10 b1 b2 b3 b3 l5 l12 l9 l7 l8 p1 b1 b2 LAG Gl’’ p2 P5 b3 b4 p4 p3 BAG Gb’’ PAG Gp’’

7. 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.

8. Bus Fault Tolerance • 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 connectedif and only ifits 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.

9. 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.

10. Link Fault Tolerance • 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

11. 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).

12. Application - M-LAN • M-LAN, multi-channel local area network, every processor is connected to all buses. P1 P2 P3 b1 b2 b3 b3 The complete graph K3,4 LAG of K3,4

13. Application - M-LAN • 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

14. Application - Spanning Bus Hypercubes

15. A w-wide d-dimensional spanning-bus hypercube is (d(w-1), d-1, d-1)-FT

16. 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.