pbl

Document Sample
pbl Powered By Docstoc
					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
                      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
Processors
                  memory          memory      memory     memory

Links
                  l1    l2   l3   l4    l5   l6    l7   l8   l9
             b1
Buses        b2
             b3
     Processor-Bus-Link (PBL) Graph

P1         P2                 P3         P4    P-nodes   P1         P2         P4
          l3              l6              l9                       l3              l9
l1                                                       l1
               l4
                                    l7

          l8        l2         l5                                        l5
     b1                  b2          b3        B-nodes        b1              b3


When there is no fault                             when processor p3, bus b2,
PBL graph G’                                        and link l8 are faulty
      Component Adjacency Graph

processor adjacency graph                              b1
  (PAG) Gp’

 p1        p2
                              l1                 b3         b2
                       l8              l2
                                                      BAG Gb’
 p4        p3
                  l9                        l6

                  l7                        l4

                        l5             l3
                             LAG Gl’
P1         P2         P3              P4          P4                    l1
                                                                 l2           l3
          l3    l4                                 l12
l1                               l8                      l10                           l4
                          l6

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




 b1     b2      b3         b3

 The complete graph K3,4                 LAG of K3,4
           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
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.

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:10
posted:12/14/2011
language:
pages:16