# Application of Adaptive Probing for Fault Diagnosis in Computer Networks 1

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4 B. Probe station selection in a non-deterministic environment While selecting probe stations in a non-deterministic environment, because of the involved probabilities, the independence of the two paths p and q cannot be declared with absolute certainty and hence needs to be represented with a certain belief value. This value represents the confidence in our belief that the two paths are independent. The belief value can be computed as follows: B(I (p,q) ) = 1 P ( R n(p, q)) (1) n Nodes(p) Nodes(q) where the term R n (p, q) represents the event that probes p and q both pass through node n, and P (R n (p, q)) = P (p, n) P (q, n). The higher the value of B(I (p,q) ), the stronger is the belief that probes p and q are independent. For a candidate probe station c, the probability that c provides a path to a shadow node s that is independent of the paths to s from already selected probe stations in S can be obtained by computing B(I (P ath(s,n),p ath(c,n)) ) = 1 P ( Q m(p ath(s, n), P ath(c, n))) (2) where m Nodes(Path(S, n)) Nodes(Path(c, n)). The term Q m (P ath(s, n), P ath(c, n)) represents the event that Path(S, n) and Path(c, n) both pass through the node m and P (Q m (P ath(s, n), P ath(c, n))) = P (P ath(s, n), m) P (P ath(c, n), m). If this value is greater than a threshold, then the path is considered to be independent. A candidate node that provides maximum number of independent paths is selected as the next probe station. Once a probe station c is selected, it is added to the set S and the probability P(Path(S, n), m) for each node n to which node c provides an independent path and for each node m used by these paths, is updated as follows: P (P ath(s c, n), m) = P ( (Q m(p ath(s, n), P ath(c, n)))) (3) VI. PROBE SELECTION We define the problems of probe selection for failure detection and fault localization and prove them to be NP-Complete. We then present heuristic-based algorithms to select probes to perform failure detection and adaptive probe selection algorithms to perform fault localization. We first assume a deterministic environment representing the dependencies between the probes and the nodes using a deterministic dependency model. We later relax the assumptions on the available dependency information and consider a non-deterministic environment. We also present a preplanned probing algorithm where a set of probes is selected to localize all possible faults in the network and is sent periodically over the network. Preplanned probing involves a high computational complexity and we show through simulation results that preplanned probing is less accurate and requires much larger number of probes as compared to adaptive probing. We present the following algorithms: 1) Algorithm GFD: Algorithm for probe selection for failure detection. We compare this algorithm with the Additive algorithm presented by Brodie et. al. [1] through experimental evaluation. 2) Algorithm GFL: Algorithm for adaptive-probing-based probe selection for fault localization using two approaches: Min Search and Max Search. 3) Algorithm BSFL: Algorithm for adaptive-probing-based probe selection for fault localization, where probes are selected to test failed probe paths in a binary search fashion. 4) Algorithm PPFL: Probabilistic preplanned-probingbased probe selection algorithm. 5) Algorithm PGFD: Probabilistic probe selection algorithm for failure detection. 6) Algorithm PGFL: Probabilistic adaptive-probing-based probe selection algorithm for fault localization using two approaches: Min Search and Max Search. A. Failure detection Probes for failure detection should be selected such that, in the presence of a fault in the network, some of the selected probes should fail, causing the detection of the failure by the network manager. As the probes for failure detection are sent at periodic intervals, they should be optimized to prevent overwhelming the network resources and affecting the performance of other applications using the network. The probe set selection problem for failure detection can then be defined as: Given a set of probes and given the dependency information between the probes and the network nodes, select the smallest set of probes such that every node in the network is covered by some probe. In the dissertation, we first prove that the probe set selection for failure detection is NP-Complete by reducing Minimum Set Cover problem to the Minimum Failure-Detection-Probe- Set Selection problem. We then present an approximation algorithm for the selection of such a probe set. We present a greedy approximation algorithm (Algorithm GFD) that explores the information contained in the dependencies between probes and network components. The algorithm selects the network element n which is probed by the least number of probes, using the dependency information between probes and probed elements. Out of all the probes probing element n, the algorithm selects the probe which goes through a maximum number of nodes that are not yet probed. Probe selection is done till all nodes are covered by the selected probes. 1) Failure detection in a non-deterministic environment: The probe selection criteria for failure detection is based on identifying the nodes where the uncertainty in selection is minimum, and then applying the Greedy approach of selecting a probe that gives maximum coverage of nodes among all the probes that pass through this node. With a probabilistic dependency model, we identify the node with minimum probe

5 selection uncertainty by computing the entropy of the probabilities by which the node is probed by probes. The entropy for a node n is computed as follows: H(n) = P (p n)log(p (p n)) (4) p (p AvailableP robes)&(p (p n)>0) where P(p n) represents the probability that probe p passes through node n. The node with minimum entropy is chosen as the next node to cover. Once a node n is selected from the set of nodes N, of all the probes that probe this node, the probe that gives maximum coverage is selected. In a non-deterministic environment, two factors decide the probe selection for failure detection: (1) probability gain obtained in covering node n, and (2) probability gain obtained in covering other nodes. For each node m, we maintain a value Coverage(m) to represent the probability that the node m has been covered by the probes selected so far. We select a probe that maximizes the metric to evaluate the improvement that can be obtained in the probability of covering node n and the probability of covering other nodes by selecting a certain probe p. We represent this metric as follows: Gain(p) = (P (p n) P (p n) Coverage(n)) + (P (p m) P (p m) Coverage(m)) (5) B. Fault localization m N {n} We present algorithms for probe selection for fault localization. We first describe the preplanned approach for probe selection for fault localization and show that the problem of selecting such a probe set is NP-Complete by reducing the Minimum Test Collection problem to the Minimum Fault- Localization-Probe-Set Selection problem. We then present adaptive probing algorithms for fault localization based on different heuristics. We present a greedy algorithm (Algorithm GFL) to select probes for fault localization. The algorithm maintains sets of failed, passed, and suspected nodes. The set of suspected nodes contains the nodes whose health needs to be determined. This suspected node set is initialized to all nodes that are present on the failed probe paths. The success and failure of the probes sent affect the sets of failed, passed, and suspected nodes. The nodes lying on the paths of successful probes are added to the set of passed nodes and removed from the set of suspected nodes. A node n is declared as failed and added to the failed nodes set when a failed probe goes through a set of nodes such that all nodes other than node n on that path have already been found to have good health. In other words, no other node on that path is present in the suspected node set. In each iteration, the algorithm builds a probe set to be sent over the network to determine the health of the remaining suspected nodes. We present two approaches to select the probes for probing the nodes in the suspected node set. One approach is to iteratively select a probe that covers maximum number of suspected nodes. The success of such a probe gives a large Fig. 3. Network with nodes 1 and 7 as probe stations. (a., b.) Nodes probed by a small set of long probes and larger set of short probes respectively; (c.) Probes sent in a binary search fashion on a failed probe path. amount of information by removing all the nodes on that probe path from the suspected node set. However if the probe fails then the probe does not give much information to significantly narrow down the search space of a failed node. Hence another approach could be to select a probe for each suspected node such that it goes through the least number of other suspected nodes. The success of such a probe gives the information about good health of a small number of nodes, reducing the suspected node set only by a small amount. However, the failure of such a probe narrows down the search space significantly. Figures 3a and 3b show an example of how a network can be probed by a set of long and short probes respectively. Success of probe 1 8 gives information about good health of nodes 4, 5, 6, 7, and 8, while its failure narrows down the failure to a set of 5 nodes. This set would need more probes for further diagnosis. On the other hand, success of a smaller probe 1 2 gives little information indicating good health of the single node 2, but failure of this probe narrows the fault localization to a single node, node 2, requiring no more probes for further localization. The greedy approach does not diagnose each probe path independently. Instead, it builds a single set of suspected nodes consisting of nodes on all failed probe paths. We present a binary search approach, where we propose to diagnose each failed probe path independently. On each failed probe path, additional probes are sent in a binary search fashion till one failure on that path is diagnosed. On a failed probe path, a probe is first sent from the probe station half way on the probe path. If this probe fails, further diagnosis is done on the first half of the probe path. On the other hand, if this probe succeeds, then the later half of the probe path is diagnosed in similar fashion. Figure 3c shows an example of how probes are sent in a binary search fashion to identify a failed node on the probe path. Consider that node 5 has failed and that the failure was detected on observing a failure of probe 1 8. Binary search probe selection then sends probe 1 6. On observing a failure on this path, the first half of the probe path is analyzed focusing on nodes 4, 5, and 6. Continuing probe selection in the binary search fashion, probe 1 4 is sent. Success of this probe indicates good health of node 4, leaving nodes 5 and 6

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