**4. Results**

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// *k* and *l* are the center of the second region with the radius *r*

*twoRegionsFailureAlgorithm* (*r*, *¨r*, *nt, nfl*, *nfw*, *nsth*)

2 **for** *i ĸ r* to *nfl*, *i* incremented by *¨r*  3 **for** *j ĸ r* to *nfw*, *j* incremented by *¨r* 

1 *regions ĸ* { }

// *i* and *j* are the center of one region with the radius *r*

 *idr1 ĸ i ||* : || *j // idr1 is the ID of the first region swr1 ĸ* { } *// the set of sensors within the region idr1 swr1 ĸ findSensorsWithinRegionAlgorithm* (*i, j, r, nt*) *nt1 ĸ removeFailedSensorsAlgorithm* (*nt, swr1*)

10 *idr2 ĸ k ||* : || *l // idr2 is the ID of the second region // twoRegid represents an ID for the underlying two regions*

 *// the two regions identified by the ID twoRegid*

13 *swr2 ĸ* { } *// the set of sensors within the region idr2* 14 swr*<sup>2</sup> ĸ findSensorsWithinRegionAlgorithm* (*k, l, r, nt1*) 15 *nt2 ĸ removeFailedSensorsAlgorithm* (*nt1, swr2*)

17 *paths ĸ findAllPaths* ({ }, {*s*}, *nt2, s, sink*) 18 **if** *paths is* empty *// no path exists*

23 *regions ĸ regions* U {*twoRegid*, *dis*}

 *// return the set of dual-failed regions with their associated disconnected nodes*

**Figure 9.** Algorithm for Finding the number of disconnected nodes under dual region-failures

find all sensors located within the region by the algorithm shown in Fig. 9. Sensors located within the failure region are then removed from the routing tables of all nodes in the network topology by executing the algorithm shown in Fig. 6. The above mentioned steps are repeated for the second region-failure. Now, we have come up with a network topology without the failed nodes due to the dual region-failure. Then, we examine the path availability from each node in the network topology to the sink node by executing the algorithm depicted in Fig. 7. If no path is available from a node to the sink node the node is marked as a disconnected node. On the other hand, if a path is found the node is marked as connected node. then we

19 *dis ĸ dis* U *s // the node s is disconnected*

*// number of disconnected nodes are above certain threshold nsth*

 *// the dis is the set of the disconnected nodes associated with*

8 **for** *k ĸ i* to *nfl*, *k* incremented by *¨r*  9 **for** *l ĸ r* to *nfw*, *l* incremented by *¨r* 

11 *twoRegid ĸ idr1 || idr2* 

16 **foreach** sensor *s* ࣅ *nt2*

22 **if** *length* (*dis*) *> nsth*

12 *dis ĸ* { }

20 **end** 21 **end**

24 **end** 25 **end** 26 **end** 27 **end**

28 return *regions*

**end**

In this section, we present our simulation results. In our simulations we consider the network topology shown in Fig. 10 in which node 0 is the sink node. The failure information of different region failures generated during our simulations results are shown in Table 1 and Table 2, respectively.

**Figure 10.** Network Topology

Note that, due to the large number of region-failures generated, we present only region failures that lead to disconnecting more than two and eight nodes for the single and dual failure scenarios, respectively.

Hereafter, we investigate the location of the worst-case region-failure under Link-based, Node-based (without a mission critical node) and Node-based with a mission critical node in Fig. 12, Fig. 14 and Fig. 16, respectively. In the Node-based with a mission critical node, the node 28 is chosen as a Mission-Critical(MC) node.

The results shown in Fig. 11 show that, based on the traditional definition of the worst-case region cut, the failure region with id 5 is considered the worst-region cut as its failure leads to having the maximum number of failed links. The location of this failure-region is depicted

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**Table 1.** Region-failure information


in Fig 12. It is also notable that, according to Table 1 the failure of this region results in disconnecting 3 sensor nodes namely, node 33, 34, and 35 as depicted in Fig. 12.

The results shown in Fig. 13 clearly indicate that, in absence of mission-critical nodes within the network topology the proposed model is very clever to find out the worst-case region-failure. The worst-case region-failure is the region-failure that has the maximum

**Figure 11.** Single Failure-region weights under link-based approach.

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Region id Failed Nodes IDs No. of Failed links No. of Disconnected Nodes

Region id Failed Nodes' IDs at Region 1 Failed Nodes IDs at Region 2

in Fig 12. It is also notable that, according to Table 1 the failure of this region results in

The results shown in Fig. 13 clearly indicate that, in absence of mission-critical nodes within the network topology the proposed model is very clever to find out the worst-case region-failure. The worst-case region-failure is the region-failure that has the maximum

disconnecting 3 sensor nodes namely, node 33, 34, and 35 as depicted in Fig. 12.

1 12 4 8 2 11, 12 6 8 3 12, 13 6 7 4 2, 3 8 5 5 31, 32 11 3 6 13 3 3 7 7, 8 6 3 8 7 3 3 9 21 3 2

**Table 1.** Region-failure information

**Table 2.** Failed Nodes due to dual region-failure

**Figure 12.** Worst-case region failure under Link-based approach.

impact on the network performance which is region 2 in this case as its failure results in disconnecting 8 sensor nodes, namely, 13, 14, 15, 16, 17, 18, 19 and 20. However, according to Fig. 11, the failure of this region leads to failure of 6 links only.

The results shown in Fig. 15 indicate clearly that, with the proposed model of a region-cut, introducing a mission-critical node into the network topology leads to a change in selecting the worst-case region cut which is region 4 as its failure results in disconnecting 5 sensor nodes, namely, 7, 8, 21, 23, and 28 as shown in Table 1. However, according to Fig. 11, the failure of this region leads to failure of 8 links.

**Figure 13.** Single Failure-region weights under node-based approach without MC-node.

**Figure 14.** Worst-case region cut using Node-based approach.

Therefore, we claim that, using the failed links as the only criteria for defining the worst-case region cut is impractical as it ignores the case that the failure of some nodes may lead to failure of few links however its impact on the network is more severe due to disconnecting larger number of nodes. Moreover, it disregards the fact that some network nodes have higher priority than others.

Hereafter, we investigate the location of the worst-case dual region-failures without including a MC-node and with a MC-node in Fig. 17 and Fig. 18, respectively.

The locations of the worst-case dual region-failures shown in Fig. 17 indicate that, using the link-based approach (blue regions including nodes 11,12 and 31, 32) lead to cutting of 17 links and disconnecting 12 nodes. On the other hand, under the dual region failure scenario,

**Figure 15.** Worst-case single region failure under Node-based approach with MC node

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**Figure 13.** Single Failure-region weights under node-based approach without MC-node.

Therefore, we claim that, using the failed links as the only criteria for defining the worst-case region cut is impractical as it ignores the case that the failure of some nodes may lead to failure of few links however its impact on the network is more severe due to disconnecting larger number of nodes. Moreover, it disregards the fact that some network nodes have higher

Hereafter, we investigate the location of the worst-case dual region-failures without including

The locations of the worst-case dual region-failures shown in Fig. 17 indicate that, using the link-based approach (blue regions including nodes 11,12 and 31, 32) lead to cutting of 17 links and disconnecting 12 nodes. On the other hand, under the dual region failure scenario,

**Figure 14.** Worst-case region cut using Node-based approach.

a MC-node and with a MC-node in Fig. 17 and Fig. 18, respectively.

priority than others.

**Figure 16.** Worst-case region failure underNode-based approach with MC node

using the proposed Node-based approach, the worst-case region-failure depicted in Fig. 17 (red regions including nodes 1,4 and 2) have only 8 link-cuts however, it results in a full disconnection of all the network nodes. This is due to the fact that, based on our approach, the worst-case region cuts are located near by the sink node and the failure of these nodes lead to isolating the sink node from the whole network. The results shown in Fig. 18 show that, by introducing the MC-node number 28, the worst-case dual region failure remains unchanged as the earlier case without MC-node. This is due to fact that , by introducing the MC-node, no further influence can affect the network as the maximum impact has been already happened by having a complete disconnected sensor network.

**Figure 17.** Worst-case Dual-Region Failures

**Figure 18.** Worst-case Dual-Region Failures with a mission node
