# An Efficient Primary-Segmented Backup Scheme for Dependable Real-Time Communication in Multihop Networks

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5 GUMMADI et al.: EFFICIENT PRIMARY-SEGMENTED BACKUP SCHEME FOR DEPENDABLE REAL-TIME COMMUNICATION 85 Fig. 4. Modeling a network topology as a graph. (a) mesh topology with link weights. (b) Weighted directed graph G representing the network topology with a chosen primary path between two vertices S and D. than end-to-end backups and we have to find intermediate nodes where the backup segments meet the primary. Below, we provide an algorithm called Min_SegBak that solves this problem. Algorithm Min_SegBak: We model the network topology as a weighted directed graph. Every node in the network is represented by a unique vertex in the vertex set, while every link from node to with cost is represented by a directed edge from to with weight. (A duplex link is replaced with two links in either direction with the same cost.) Let and denote the source and destination nodes, respectively, for a dependable connection and,,, a sequence of vertices along some primary path between them. We define ( ) to denote that vertex that occurs after (before) vertex in sequence. In Fig. 4(a) and (b), we show a 3 4 mesh topology with links of unit cost and the corresponding weighted directed graph. Our algorithm to find the shortest segmented backup for in consists of three steps. In step 1, we generate a modified graph from on the same set of vertices. In the step 2, we find the shortest path between and in graph and use it in step3toobtain the segments ofthe minimum-cost segmented backup in. We elaborate on this below and illustrate it in Fig. 5. 1) Generate modified graph from. The following types of edges in graph are modified in the order given below: a) edges between successive vertices in the sequence ; the edges pointing in the direction from to are removed while those pointing in the reverse direction are assigned zero weight. b) edges, ; replace every such edge with where is the immediate predecessor of in. In other words, redirect every edge pointing to a vertex,to point to its immediate predecessor vertex. 2) Find the shortest path between and in. Run any least cost path algorithm for directed graphs (e.g., Dijkstra s algorithm) between and in. Let the path obtained be denoted by the vertex sequence,,. 3) Use the shortest path found in step 2 to find backup segments for in. Suppose the segmented backup consists of backup segments, ordered by the increasing position of their starting nodes in the sequence. We describe a method to generate the vertex sequences for these backup segments one after the other(i.e., first is generated, then, and so on until ) as we traverse the sequence,, (found in step 2). Initialize all vertex sequences to empty. At any stage of the traversal of, we use to denote the current backup segment being generated and to denote the current vertex. Initialize to. For every perform the first applicable action indicated in steps a d in that order. This process terminates on reaching destination. We use the terms and to denote the immediate predecessor and immediate successor vertices of in. Similarly, denotes the immediate successor of in. a) If then and. b) If then i) If then append to and stop. ii) If then, append to and stop. c) If for any, (i.e., ) then i) If then append to and. ii) If then, append to and. d) If for some, (i.e., ) then i) If then append to and. ii) If and then, append to and. iii) If and then. The resulting vertex sequences define backup segments which form the shortest segmented backup for the primary path in. We now illustrate the three steps above through an example in Fig. 5. The weighted directed graph obtained from the 3 4 mesh topology is shown in Fig. 5(a), while the graph obtained by modifying the graph using step 1 is shown in Fig. 5(b). Edges along the primary path in the direction from to (for example, the edge from to ) are removed, while those in the reverse direction (for example, the edge from to ) are assigned zero weight. Also, edges pointing to the nodes on the primary path (for example, the edge from to ) are redirected to point to preceding vertices on the primary path (the edge from to is redirected to ). The shortest path in found in step 2 of the algorithm is also shown in Fig. 5(b) with a dotted line. Finally, a segmented backup consisting of two backup segments ( and ) obtained from the shortest path using step 3 of the algorithm is shown in Fig. 5(c). Note that in step 1b we redirect the edges to ensure that successive backup segments overlap on at least one link of the primary path. This overlap is necessary to recover from node fail-

6 86 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 11, NO. 1, FEBRUARY 2003 Fig. 5. Illustration of Min_SegBak algorithm. (a) Weighted directed graph G with a chosen primary path between two vertices S and D. (b) Modified graph G obtained from G along with a shortest path between S and D; these are obtained using steps 1 and 2 of the algorithm. (c) Directed graph G with the primary and a segmented backup; the backup is obtained from the shortest path using step 3 of the algorithm. ures. If, however, it is sufficient to recover from link failures and not necessarily node failures, step 1b can be omitted from the algorithm. We now state an important theorem of optimality of the segmented backups generated. This theorem, coupled with the fact that every end-to-end backup is also a segmented backup, guarantees a more efficient resource reservation using our algorithm. Theorem 2: The segmented backup generated by the Min_SegBak algorithm is a minimum cost segmented backup for the chosen primary path (proof given in Appendix II). Complexity of the Min_SegBak algorithm: The complexity of step 1 of the algorithm is at most the number of edges in, which is. Step 2 involves finding the least weight path in ; assuming one uses Dijkstra s algorithm, its complexity is. The complexity of step 3 is the cost of traversal of the shortest path in which is. Since, the overall complexity of the algorithm is which is the complexity of the least weight path algorithm over. This is same as the complexity of end-to-end backup. Thus, our algorithm offers improved resource utilization without additional complexity. V. IMPLEMENTATION OF PRIMARY-SEGMENTED BACKUP SCHEME In this section, we deal with the protocols required to implement our scheme, their complexity, and their applicability in the Internet as it exists today. We discuss three protocols: 1) routing protocol, to determine the primary as well as the backup paths; 2) reservation protocol, to reserve resources efficiently along these paths; and 3) recovery protocol, to recover from any failure. We present the recovery protocol as a simple extension to the Internet Engineering Task Force (IETF)-recommended resource reservation protocol, RSVP [15], and show that its complexity is comparable to that of RSVP. Finally, we reflect on the relevance of these schemes in the rapidly evolving Internet. A. Design and Analysis of Implementation Protocols In order to implement our scheme we need two protocols, namely, routing and reservation-recovery protocols. Design of the Routing Protocol: The routing protocol determines the primary and segmented backup paths using the Min_SegBak algorithm described in Section IV. To compute the backup path, our algorithm assumes that every router has the global knowledge of the network topology. To obtain information about all routers and links in the network, our routing protocol can use any variant of link state protocol such as the Open Shortest Path First (OSPF) [17] protocol (a popular intra-domain routing protocol). With this global knowledge of the network, the primary path can be computed as the least weight path using Dijkstra s shortest path algorithm, while the backup path could be decided later by running the Min_SegBak algorithm at the source node. Analysis of the Routing Protocol: We showed in Section IV that the complexity of the Min_SegBak algorithm is same as that of the complexity of the least weight path algorithm such as Dijkstra s algorithm, which is also the complexity of the widely used OSPF routing protocol [16]. Thus, while our protocol does require extra computations at routers for the backup paths compared with OSPF, the increased computation is bounded by a small constant factor. We feel that this should not be a problem in these days when Moore s law ensures that the computation power of the router processors doubles every two years. Further, note that using link state protocols could potentially limit the scalability of our scheme to within a single autonomous system. While we believe that our subsequent work [21] on a distributed version of the Min_SegBak algorithm could eliminate this limitation (as it does not need global knowledge of any kind), we do not explore it here as it is outside of the scope of this paper. However, later in this section, we show that there are interesting applications even with this scalability bottleneck.

7 GUMMADI et al.: EFFICIENT PRIMARY-SEGMENTED BACKUP SCHEME FOR DEPENDABLE REAL-TIME COMMUNICATION 87 Fig. 6. Illustration of resource reservation using setup messages. (a) Source initiates the process by sending a PrimPath message. (b) Destination responds with a PrimResv message. (c) Source computes backup segments and sends SegPath message. As it traverses, the start nodes of backup segments initiate BackSegPath messages. (d) End nodes of backup reply with BackSegResv messages. Design of the Reservation Protocol: We propose simple extensions to RSVP [15] to design the reservation protocol. These extensions are along the lines of extensions discussed in [4]. We begin with a brief overview of RSVP. RSVP is a protocol to reserve resources along the network paths for unicast and multicast data flows in an integrated services network. The protocol primarily consists of the following three types of messages: 1) setup messages to reserve resources along paths; 2) maintainance messages to notify the end nodes of any failures along the path; and 3) teardown messages to free the resources reserved. We will discuss a few important messages of each type. 1) Setup messages. The source router initiates the reservation by sending a Path message to the destination along a route selected by the routing protocol. On receiving the message, the destination router sends a Resv message back to source in the reverse direction using the path state set up by the Path message. Resources are reserved at the routers along the path by the Resv message. 2) Maintenance messages. Error messages such as PathErr and ResvErr are used to inform the end nodes of a failure in establishing the path. 3) Teardown messages. Messages like PathTear and ResvTear are propagated much like their setup message counterparts to reclaim the resources reserved for the data flow. In our reservation protocol, we have additional messages to reserve resources along the backup paths. We also add a few objects to the messages discussed above. 1) Setup messages. We illustrate the connection setup process in Fig. 6. The reservation is initiated by the source with a PrimPath message to the destination as shown in Fig. 6(a). The destination responds with a Prim- Resv message back to the source. The PrimResv message contains an object called PrimaryPath, which is filled up with the routers along the primary path as the message traverses it, as shown in Fig. 6(b). This PrimaryPath object is used by the source to compute the primary and backup segments using Min_SegBak algorithm. Each pair of primary and backup segments is copied into a separate SegmentPath object. All the SegmentPath objects are copied into a SegmentPathList object, which is sent in a SegPath message from the source to the destination. As the SegPath message traverses the primary path, the SegmentPathList is used by routers to identify the start nodes of backup segments. These nodes initiate a BackSegPath message along the backup segment. This message is tagged with the SegmentPath object (contains the corresponding primary and backup segments) to help in backup multiplexing. This is illustrated in Fig. 6(c). On receiving the BackSegPath message, the last node of a backup segment replies with a BackSegResv message, which traverses the backup segments in reverse direction as shown in Fig. 6(d). Resources are reserved along the backup segments by the BackSegResv message using backup multiplexing. 2) Maintenance messages. Upon failure during the set up of either primary path or any backup segment, error messages like PrimPathErr, PrimResvErr, BackSegPathErr, and BackSegResvErr are generated. These messages also initiate the corresponding reservation teardown messages to free any resource reservations made prior to the routing failure. 3) Teardown messages. The messages PrimPathTear, Prim- ResvTear, BackSegPathTear, and BackSegResvTear are used to free the resources reserved. They are propagated along the primary and backup paths. Analysis of the Reservation Protocol: The reservation state maintained by the routers is soft state and must be updated periodically through refresh messages. If a periodic refresh message is not received, a router can reallocate the reserved resources for some other flow. Thus, the resources can be recovered even when the source or destination nodes fail to generate explicit Teardown messages. The amount of state maintained at the routers by the reservation protocol is a very important concern for integrated services. Below, we argue that the state stored in our scheme is only marginally greater than the state maintained in the

10 90 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 11, NO. 1, FEBRUARY 2003 Fig. 8. Comparing the amount of spare resources reserved by segmented backup and end-to-end backup schemes over mesh topologies of various sizes. (a) mesh. (b) mesh. (c) mesh. (d) mesh. Fig. 9. Comparing the ACAR for segmented backup and end-to-end backup schemes over mesh topologies of various sizes. (a) 525 mesh. (b) mesh. (c) mesh. (d) mesh. our experiments, the average hop counts of primary and backup paths are directly proportional to the amount of resources reserved for the paths as every connection has the same one unit of bandwidth requirement. For example, suppose that at 40% network load, the average hop counts of primary and segmented backup are 12 and 6, respectively. This implies that while 40% of the total network resources are reserved, two-thirds ( ) of the reserved resources are allocated for primary paths (i.e., 27% of the total network resources) while one-third ( ) of the reserved resources are allocated for backup paths (i.e., 13% of the total network resources). We expect the advantages of using our scheme over end-to-end backup schemes to increase as the length of the primary increases. This is because longer primary paths have more backup segments and all of the advantages that go with them. To capture this effect in our study, we introduced a parameter called Min_Path_Len and requested only connections between nodes that have the length of the shortest path between them greater than Min_Path_Len. We chose Min_Path_Len depending on the size and diameter of the network topology. For square meshes of sizes 5, 7, 9, and 12, we chose Min_Path_Len to be 3, 4, 6, and 8, respectively, while for USANET we chose it to be 0. These values are reasonable as a significant percentage (USANET 100%, 5 5 mesh 66%, %) of node pairs in these network topologies have the length of the shortest path between them greater than the chosen value of Min_Path_Len. In Fig. 8, we compare the resource reservation requirements of our scheme with that of the end-to-end backup scheme for meshes of varying sizes. Similarly, we compare the ACAR of the two schemes in Fig. 9. Finally, in Fig. 11, we compare the performance of the schemes over USANET. We now go into a detailed analysis of the results in each of these figures. Comparing the Amount of Spare Resources Reserved: The graphs in Fig. 8(a) (d) show the average hop counts of primary path, end-to-end backup and segmented backup at various network loads for meshes of sizes 5 5, 7 7, 9 9, and 12 12, respectively. Note that for the average hop count of primary paths, we plot a single curve rather than separate curves for the two schemes. In our simulations, we found that the hop counts

12 92 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 11, NO. 1, FEBRUARY 2003 have also given an efficient backup route selection algorithm. We evaluated the proposed scheme through extensive simulations and demonstrated the superior performance of our method compared to earlier schemes. In order to realize the full potential of the method of segmented backups, routing strategies must be developed for backup segments to achieve better QoS guarantees on delay and bounded time failure recovery. Fig. 10. The 28-node topology of the USANET. Fig. 11. Comparing the performance of segmented backup and end-to-end backup schemes over USANET topology. both schemes is similar as pointed in observation 2. Finally, as observation 3 points out, about 6% of the requested calls are rejected because of lack of an end-to-end backup in scenarios where a segmented backup can be found. To summarize, our simulations demonstrate that our scheme is capable of delivering better resource efficiency as well as better call acceptance rate compared with existing end-to-end backup schemes. However, the benefits of our scheme vary with the network topology and load. Our scheme performs significantly better for larger networks with low connectivity (more nodes and less edges) at low and moderate loads. VII. CONCLUSION In this paper, we have proposed segmented backups: a failurerecovery scheme for dependable real-time communication in multihop networks. This mechanism not only improves resource utilization and call acceptance rate but also can provide faster failure recovery and better QoS guarantees on end-to-end delays without compromising the level of fault tolerance provided. We APPENDIX I PROOF OF THEOREM 1 Theorem 1 Whenever two disjoint paths exist between a source and a destination in a network, segmented backups are guaranteed to exist for any choice of primary path between the end nodes. However, there are no such guarantees for end-to-end backups. Proof: In Fig. 2(a), we demonstrated a network topology where disjoint paths exist between a pair of end nodes but end-to-end backups do not exist for a chosen primary path. Here, we prove that segmented backups exist whenever disjoint paths exist. We start with a simple observation. We refer to the backup segment spanning a primary segment that contains an intermediate node as a backup segment covering. For example, in Fig. 1, the three backup segments over links A C, D G, and H K cover the nodes,, and, respectively. Observe that a segmented backup for a primary path can be constructed by taking the set of all backup segments covering each of the intermediate nodes in, as in Fig. 1. Below, we prove our claim of existence of a segmented backup by showing the existence of backup segments that cover every intermediate node. In our graph, let the two disjoint paths between source and destination be denoted by and, respectively. Let be any chosen primary path between them and let denote its length. Two cases arise: Case 1) (i.e., has only one edge ). This is the special case with no intermediate nodes. One of or is a segmented backup for, as edge cannot be in both and. Case 2) (i.e., has at least 1 intermediate node). Let denote any intermediate node on. We need to show the existence of a backup segment that covers. Since and are disjoint, at least one of them does not contain. Without loss of generality let us assume that does not lie on. We claim that since 1) and have the same end points and, and 2) and, a segment (a contiguous subpath) of acts as a backup segment covering. We prove it using recursion. Base Case for Recursion: and are disjoint. Clearly, is a suitable backup segment covering the primary segment containing. Recursive Step: We apply it when and are not disjoint. We show the existence of sub paths and for paths and (i.e., and ) such that 1) and have the same end points and 2) and.

13 GUMMADI et al.: EFFICIENT PRIMARY-SEGMENTED BACKUP SCHEME FOR DEPENDABLE REAL-TIME COMMUNICATION 93 Let, and,, denote the nodes along the paths with and representing the th and th vertices along the paths and respectively. is the th intermediate node on the path. As and are not disjoint, they must have some common node such that for some,. As node either or. In either case, we define and as follows: If, and.if then,, and,. Clearly, 1) paths and have same end points and 2) and. Further, and. If and are disjoint the base case assures us of the existence of a backup segment covering. If not, this step is applied recursively. Since the paths are of finite lengths and decrease in each iteration, this process always terminates in the base case. Thus, every node along the primary path is guaranteed a backup segment that covers it and a segmented backup can be generated by taking all the backup segments together. APPENDIX II PROOF OF THEOREM 2 First, we state and prove two lemmas. We use them later to prove the theorem. Lemma 1 The weight (cost) of the segmented backup, i.e., the sum of weights of all backup segments generated, is equal to the weight of the shortest path B found in step 2 of the algorithm. Proof: Every directed edge in which does not point to a vertex in is included in one of the backup segments by steps 3b and 3c. We replaced every edge which starts from a vertex not in but points to a vertex in with an edge of equal weight in step 3d(i). This leaves us with the case of edges between vertices along primary path. Edges from a vertex in to its successor vertex are included in backup segments by step 3d(ii), while edges from a vertex in to its predecessor vertices are excluded. However, these excluded edges are of zero weight and do not contribute to the weight of the path. Finally, there are no extra edges added to the backup segments. This is illustrated in Fig. 5(b) and (c), where both the shortest path and segmented backup weigh six units. Thus, we conclude that the weight of the segmented backup generated is equal to the weight of path B. Lemma 2 Every segmented backup for primary path between and in can be mapped to a path between the same nodes in that is of equal weight. Proof: For any chosen segmented backup for in,we show the existence of a path in between the same end nodes that is of equal weight. Suppose the chosen backup consists of backup segments. Denote the corresponding primary segments as. Let, and denote the first, last, and penultimate nodes of the th primary segment, respectively. Similarly, let, and denote the first, last, and the penultimate nodes of the th backup segment, respectively. Clearly,,. Also, note that the first and the penultimate nodes of a segment with only a single link are the same. To show the existence of a path from to in that is of equal weight, we claim that: 1) there exists a path from to that is of same weight as ; 2) there exists a path from to that is of same weight as ; and 3) there exists a path from to that is of zero weight. All edges of the backup segments that do not point to any vertex in are left unchanged while modifying into by step 1 of the algorithm. Thus only the last edge in is changed and no edge is changed in. For each this last edge from to is redirected to point to. Thus, in the modified graph the edges in form a path from to and the edges in form a path from to. This proves our claims 1 and 2 above. As successive primary segments overlap over at least one edge of the primary path, either or. From step 1a of the algorithm, we know that there is zero weight path from any node on primary path to its predecessors. This proves our claim 3 above. By taking the edges of the paths in claims, and above, we obtain a single path from to in that is of same weight as the segmented backup. We now use the Lemmas 1 and 2 to prove Theorem 2. Theorem 2 The segmented backup generated by the Min_SegBak algorithm is a minimum cost segmented backup for any chosen primary path. Proof: To prove the theorem we need to show that: 1) the backup generated is a valid segmented backup; and 2) it is a minimum-cost segmented backup. While we avoid formal arguments for 1) here, one can easily prove it by establishing the converse of Lemma 2, namely, every path between and in can be mapped to a valid segmented backup in by following step 3 of the algorithm. Instead, we assume 1) holds and prove 2). It follows from Lemma 2 that the weight of the shortest path in between and is at most the weight of the minimum-cost segmented backup. However, Lemma 1 states that the weight of the segmented backup generated by our algorithm equals the weight of the shortest path between and in. Thus, we conclude that the weight of the segmented backup generated by Min_SegBak is at most the weight of minimum-cost segmented backup. REFERENCES [1] D. G. Anderson et al., Resilient overlay networks, in Proc. 18th ACM Symp. Operating System Principles, Banff, Canada, Oct. 2001, pp [2] J. Anderson, B. Doshi, S. Dravida, and P. Harshavardhana, Fast restoration of ATM networks, IEEE J. Select. Areas Commun., vol. 12, pp , Jan [3] A. Banerjea, Simulation study of the capacity effects of dispersity routing for fault-tolerant real-time channels, in Proc. ACM SIGCOMM, Aug. 1996, pp

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