Reliability
4.5 The Octagonal Mesh Network
4.5.1 The Routing Relation
Under the L00-metric, nodes lying along the same row or the same column of the mesh are connected by a multiple number of routes, provided that they are not adjacent to each other. In other words, using the new octagonal mesh and the L00-metric based routing relations, we have succeeded in removing the bottlenecks in our original 2D rectilinear mesh. Unfortunately, in so doing, we have created a different set of bottlenecks: Nodes lying along the same diagonals are now joined only by single routes. Clearly, we need to introduce extra alternate routes in addition to those allowed under the L00-metric, provided that, collectively all the generated routes remain acyclic.
To proceed, we observe that the two Li- and L00-metrics defined on a 2D mesh are compatible with each other over the octagonal mesh. They are compatible in the sense that moving in a direction that decreases one metric will never result in an increase in the other metric. This is summarized in the following lemma:
Lemma 4.3 The Li- and L00-metrics are compatible with each other over the 2D octagonal mesh.
Proof. Let 8X and 8Y denote the corresponding change in absolute distances from the destination in the X and Y directions when a message is forwarded across a channel.
We observe that during every move, 8X and 8Y can only assume the discrete values from the set { -1, 0,
+
1}. To decrease the Li-metric, we must have 8 X+
8Y < 0 ⇒8X:::; 0 /\ 8Y :::; 0, which can never increase the L00-metric. Similarly, to decrease the L00-metric, we must have 8X
<
0 V 8Y<
0 ⇒ 8X+
8Y :::; 0. In other words, it can neverincrease the Li-metric. ■
Compatibility of the Li- and L00 - metric allows us to define a new metric M as the sum of the two old metrics:
It is straightforward to see that M, defined above, is a genuine metric over the octagonal- mesh network. We shall now define an alternative set of routing relations, R*
=
{R!j}, based on the reduction of this new metric, M:which guarantees the acyclicity of routes generated collectively by the R*s. Notice that another possible way to generate a combined metric for the octagonal mesh is to define
Figure 4.14: A Worst Possible Route in the Octagonal Mesh
it lexicographically as (dL00, dL1). We favor the summation-generated metric because its values are more closely related to actual distances. Under this combined metric, the routes generated are no longer shortest graph-theoretic distance paths. However, since the routes are generated so as to reduce at least one of the component metrics of M, and dL1 can be twice as much as dL00, it is clear that the route lengths cannot be more than three times those of the shortest distance paths. In fact, on the octagonal mesh, the worst-case route length is at most 3dL1 - 1, since the last move on any route must simultaneously reduce both dL1 and dL00• That such a worst case does exist is exemplified in Figure 4.14. The advantage we gain from paying such a price is that between every pair of nonadjacent nodes, there now exists at least two node- disjoint routes joining them. Hence, with this new set of routing relations, we succeed in removing the bottlenecks that have caused the low yield results in the two-dimensional rectilinear mesh.
4.5.2 Reliability Assessment
In this section, we present the simulation and computation results for the octagonally connected mesh network discussed in the previous section. Again, we have attempted to
differentiate the effects of node failure and channel failure by simulating them separately.
The results are plotted in Figures 4.15-4.18. The followings observations are made regarding these figures:
• The average yield statistics of the 2D octagonal mesh are far better than the corre- sponding statistics of the 2D rectilinear mesh under identical failure probabilities.
• The statistical patterns are comparatively more dispersive than those obtained from all the previous simulations.
• The statistical patterns for the node faults are much more dispersive than those obtained from the channel faults, with a number of occasional outlying points.
The relatively dispersive nature of the yield statistics and, in particular, the existence of these outlying points require some further explanation; this also helps us to gam additional insights into how the octagonal mesh achieves the much better yield over that of the rectilinear mesh. In Figures 4.21 and 4.22, we have shown the computed kernel configuration of a few scattered isolated faults, and that of a single cluster of faults. Notice that as long as the fault cluster sizes are small, they can be readily routed around by the redundant paths generated under the routing relations, R*s. However, as the fault cluster sizes increase further, they can no longer be accommodated, and a phenomenon so familiar in the original rectilinear mesh commences: Entire columns, rows, or diagonals are restrained to operate as pure switches. For randomly generated faults, such large fault clusters remain relatively rare until the failure probabilities be- come relatively high. On the other hand, the occasional occurrences of the few large fault clusters result in the existence of the observed outlying points. In particular, the formation of clusters are much more likely if the failure probabilities of neighboring resources are highly correlated. Since node faults are in effect highly correlated channel faults, this explains why many more outlying points are observed under the indepen- dent node faults assumption than are observed under the independent channel faults assumption.
Figures 4.19 and 4.20 compare the yield statistics of the octagonal mesh, the rectilin- ear mesh, and the binary-n-cube. It is interesting to observe that the yield statistics for the 2D octagonal mesh is consistently better than those obtained for the binary-n-cube
of the same size. However, it should be pointed out that the significant gain in yield is achieved at the expense of reducing the effective available network bandwidth, as we shall find out in the following subsection.
256 Nodes, 930 Channels 1.0
0.9 +++ + +
y
t
+*+I +
E 0.8 + +
D L + +
+
0.7 +
0.6 +
0 2 4 6 8 10 12 14 16
Percentage of Node Faults
Figure 4.15: 16 X 16 Octagonal Mesh with Node Faults
256 Nodes, 930 Channels 1.00
0.95 y 0.90 E I L D 0.85
0.80
+ + + + 0.75
0 2 4 6 8 10 12
Percentage of Channel Faults
Figure 4.16: 16 x 16 Octagonal Mesh with Channel Faults
1024 Nodes, 3906 Channels 1.00
0.95 0.90 y I 0.85 E L 0.80 D
0.75
+ + + + + + 0.70
0.65 + +
0 2 4 6 8 10 12 14
Percentage of Node Faults
Figure 4.17: 32 X 32 Octagonal Mesh with Node Faults
1024 Nodes, 3906 Channels 1.00
0.95 y 0.90 E I D L 0.85
0.80 0.75
0 2 4 6 8 10 12
Percentage of Channel Faults
Figure 4.18: 32 x 32 Octagonal Mesh with Channel Faults
1024 Nodes 1.0
0.8 y 0.6 E I D L 0.4
0.2 32 X 32 Rectilinear Mesh
0.0
0 2 4 6 8 10 12 14
Percentage of Node Faults
Figure 4.19: Another Comparison of Yield with Node Faults
1024 Nodes 1.0
32 x 32 Octagonal Mesh
0.8 y 0.6
I E L D 0.4
0.2 32 x 32 Rectilinear Mesh
0.0
0 2 4 6 8 10 12
Percentage of Channel Faults
Figure 4.20: Another Comparison of Yield with Channel Faults
Figure 4.21: A Kernel Configuration Induced by Isolated Faults
Figure 4.22: A Kernel Configuration Induced by a Cluster of Faults