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 P1P2P3P4 memory l1 l2l3l4l5l6l9l8l7 b1 b3 b2 Processors Links Buses

4. Processor-Bus-Link (PBL) Graph When there is no fault PBL graph G’ P1P4P3P2 b1b2b3 l1 l2 l3 l4 l5 l6l9 l8 l7 P1P4P2 b1b3 l1 l3 l5 l9 P-nodes B-nodes when processor p3, bus b2, and link l8 are faulty

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

6. A PBL Graph G’’ P1P4P3P2P4 b1b2b3 l1 l2 l3 l4 l5 l6 l9 l8 l7 l10 l11 l12 p1 p4 p2 p3 P5 BAG Gb’’ b1 b3 b2 l1 l2l3 l4 l5 l6 l9 l8 l7 l11 l10 l12 LAG Gl’’ PAG Gp’’ b4

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. 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

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. 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

11. 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. P1P3P2 b1b2b3 LAG of K3,4 The complete graph K3,4

13. 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

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.