MPATH: A Loop-free Multipath Routing Algorithm

Transcription

1 1 MPATH: A Loop-free Multpath Routng Algorthm SRINIVAS VUTUKURY vutukury@cse.ucsc.edu Computer Scences Department Unversty of Calforna Santa Cruz, CA 9564 J.J. GARCIA-LUNA-ACEVES jj@cse.ucsc.edu Computer Engneerng Department Unversty of Calforna Santa Cruz, Calforna 9564 Networkng and Securty Center Sun Mcrosystems Laboratores Palo Alto, Calforna 9433 Abstract We present a dstrbuted routng algorthm for computng multple paths between each source-destnaton par n a computer network, such that the paths are loop-free at all tmes and are not necessarly of equal length. In ths algorthm, routers exchange second-to-last hop on the shortest path to destnatons n addton to shortest dstances, whch are used to prevent the well-know count-to-nfnty problem. The safety and lveness propertes of the algorthm are proved and ts performance s analyzed. I. INTRODUCTION RIP[8] and many other routng protocols based on the dstrbuted Bellman-Ford algorthm (DBF) for shortest-path computaton suffer from the bouncng effect and the countng-to-nfnty problems, whch lmt ther applcablty to small networks usng hop count as the measure of dstance. In the past several years, much research has been devoted to fxng these problems. In one approach, routers exchange query and reply messages to synchronze dstance updates, a technque that s sometmes called dffusng computatons [1]. Theloop-free routng algorthm DUAL[3], whch s used n EIGRP [2], and several algorthms based on dstance vectors have been proposed that use dffusng computatons to overcome the countng-to-nfnty problem of DBF [15], [11], [1], [2]. In another approach, routers exchange second-to-last hop to a destnaton n addton to dstance nformaton so that they can determne complete paths and prevent the count-to-nfnty problem. These algorthms are often called path-fndng algorthms or source-tracng algorthms [9], [14], [5]. All these algorthms elmnate DBF s countng to nfnty problem, and some of them (LPA [5]) are more effcent than any of the routng algorthms based on lnk-state nformaton proposed to date. Furthermore, LPA [5] s loop-free at every nstant. The MPATH routng algorthm presented n ths paper s a path-fndng algorthm [17], [18]. MPATH dffers from pror path-fndng algorthms n that t uses the nvarants, ntroduced n [19], to ensure multple loopfree paths of unequal cost. Another famly of routng algorthms exchange lnk nformaton to compute routng paths. These algorthms were frst proposed and wdely used because they do not suffer from the count-to-nfnty problem of the dstance vector algorthms. OSPF [12] and algorthms n [16], [13] are some that belong to ths famly, whch exchange complete topology nformaton. A couple of routng algorthms have been proposed that operate usng partal topology nformaton [4], [6], [7], [19] to elmnate the man lmtaton of topology-broadcast algorthms. Except DASM[2] and MPDA[19], all of the above routng algorthms focus on the provson of a sngle path to each destnaton. A drawback of DASM s that t uses mult-hop synchronzaton, whch can lmt ts scalablty. In ths paper, we present the frst path-fndng routng algorthm that (a) provdes multple paths of unequal cost to each destnaton that are free of loops at every nstant and (b) uses a Ths work was supported n part by the Defense Advanced Research Projects Agency (DARPA) under grants F and F C-38. synchronzaton mechansm that spans only one hop, whch makes t more scalable than routng algorthms based on dffusng computatons spannng multple hops. The paper s organzed as follows. Secton II descrbes MPATH. Secton III presents the correctness proofs showng that MPATH s loopfree at every nstant, safe, and lve. Secton IV analyzes the complexty of MPATH. Secton V provdes concludng remarks. II. DISTRIBUTED MULTIPATH ROUTING ALGORITHM A. Problem Formulaton A computer network s represented as a graph G =(N; L) where N s set of nodes (routers) and L s the set of edges (lnks) connectng the nodes. A cost s assocated wth each lnk and can change over tme, but s always postve. Two nodes connected by a lnk are called adjacent nodes or neghbors. The set of all neghbors of a gven node s denoted by N. Adjacent nodes communcate wth each other usng messages and messages transmtted over an operatonal lnk are receved wth no errors, n the proper sequence, and wthn a fnte tme. Furthermore, such messages are processed by the recevng node one at a tme n the order receved. A node detects the falure, recovery and lnk cost changes of each adjacent lnk wthn a fnte tme. The goal of our dstrbuted routng algorthm s to determne at each node the successor set of for destnaton j, whch we denote by Sj (t) N, such that the routng graph SG j(t) consstng of lnk set f(m; n)jn 2 Sj m (t); m 2 Ng s free of loops at every nstant t, even when lnk costs are changng wth tme. The routng graph SG j (t) for sngle-path routng s a snk-tree rooted at j, because the successor sets Sj(t) have at most one member. In multpath routng, there can be more than one member n Sj (t); therefore, SGj (t) s a drected acyclc graph wth j as the snk node. There are potentally several SG j (t) for each destnaton j; however, the routng graph we wll construct s defned by the successor sets Sj (t) =fkjdj k (t) <Dj (t);k 2 N g,wheredj s the shortest dstance of node to destnaton j. We call such a routng graph the shortest multpath fordestnaton j. Aftera seresoflnk cost changes whch leave the network topology n arbtrary confguraton, the dstrbuted routng algorthm should work to modfy SG j n such a way that t eventually converges to the shortest multpath of the new confguraton, wthout ever creatng a loop n SG j durng the process. Therefore, our soluton to the routng problem conssts of frst computng Dj usng a shortest-path routng algorthm, and usng t to compute Sj. Because Dj k s node k s local varable, ts value has to be explctly or mplctly communcated to. If Djk s the value of Dj k as known to node, the problem now becomes one of computng Sj (t) =fkjd jk (t) <D j (t)g. However, because of non-zero propagaton delays, durng network transtons there can be dscrepances n the value of Dj k and ts copy Djk at, whch may cause loops to form n SG j. To prevent loops, therefore, addtonal constrants must be mposed when computng Sj. We show later that f the successor set

2 Report Documentaton Page Form Approved OMB No Publc reportng burden for the collecton of nformaton s estmated to average 1 hour per response, ncludng the tme for revewng nstructons, searchng exstng data sources, gatherng and mantanng the data needed, and completng and revewng the collecton of nformaton. Send comments regardng ths burden estmate or any other aspect of ths collecton of nformaton, ncludng suggestons for reducng ths burden, to Washngton Headquarters Servces, Drectorate for Informaton Operatons and Reports, 1215 Jefferson Davs Hghway, Sute 124, Arlngton VA Respondents should be aware that notwthstandng any other provson of law, no person shall be subject to a penalty for falng to comply wth a collecton of nformaton f t does not dsplay a currently vald OMB control number. 1. REPORT DATE REPORT TYPE 3. DATES COVERED --26 to TITLE AND SUBTITLE MPATH: A Loop-free Multpath Routng Algorthm 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Unversty of Calforna at Santa Cruz,Department of Computer Engneerng,Santa Cruz,CA, PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 1. SPONSOR/MONITOR S ACRONYM(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for publc release; dstrbuton unlmted 13. SUPPLEMENTARY NOTES 14. ABSTRACT 15. SUBJECT TERMS 11. SPONSOR/MONITOR S REPORT NUMBER(S) 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. REPORT unclassfed b. ABSTRACT unclassfed c. THIS PAGE unclassfed 18. NUMBER OF PAGES 6 19a. NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev. 8-98) Prescrbed by ANSI Std Z39-18

3 2 Procedure INIT-PATH finvoked when the node comes up.g 1. Intalze all tables. 2. Run PATH algorthm. End INIT-PATH Algorthm PATH finvoked when a message M s receved from neghbor k, or an adjacent lnk to k has changed cost or when a node s ntalzed.g 1. Run NTU to update neghbor tables. 2. Run MTU to update man tables. 3. For each destnaton j marked as changed, Add update entry [j, D j, p j] to the new message M. 4. Wthn fnte amount of tme, send message M to each neghbor. End PATH Fg. 1. The PATH Algorthm at each node for each destnaton j satsfy certan condtons called loop-free nvarant condtons, then the snapshot at tme t ofthe routng graph SG j (t) mpled by S j (t) s free of loops. B. Node Tables and Message Structures As n DBF, nodes executng MPATH exchange messages contanng dstances to destnatons. In addton to the dstance to a destnaton, nodes also exchange the dentty of the second-to-last node, also called predecessor node, whch s the node just before the destnaton node on the shortest path. In ths respect MPATH s akn to several pror algorthms [5], [14], [9], but dffers n ts specfcaton, verfcaton and analyss and, more mportantly, n the multpath operaton descrbed n the next secton. The followng nformaton s mantaned at each node : 1. The Man Dstance Table contans Dj and p j,wheredj s the dstance of node to destnaton j and p j s the predecessor to destnaton j on the shortest path from to j. The table also stores for each destnaton j, the successor set Sj, feasble dstance FDj, reported dstance RDj and two flags changed and report-t. 2. The Man Lnk Table T s the node s vew of the network and contans lnks represented by (m; n; d) where (m; n) s a lnk wth cost d. 3. The Neghbor Dstance Table for neghbor k contans Djk and p jk where Djk s the dstance of neghbor k to j as communcated by k and p jk s the predecessor to j on the shortest path from k to j as notfed by k. 4. The Neghbor Lnk Table Tk s the neghbor k s vew of the network as known to and contans lnk nformaton derved from the dstance and predecessor nformaton n the neghbor dstance table. 5. Adjacent Lnk Table stores the cost lk of adjacent lnk to each neghbor k. If a lnk s down ts cost s nfnty. Nodes exchange nformaton usng update messages whch have the followng format. 1. An update message can one or more update entres. An update entry s a trplet [j, d, p], where d s the dstance of the node sendng the message to destnaton j and p s the predecessor on the path to j. 2. Each message carres two flags used for synchronzaton: query and reply. Procedure NTU fcalled by PATH to process an event.g 1. If event s a message M from neghbor k, a. For each entry [j, d, p] n M //Note d = D k j, p = p k j.) Set D jk d and p jk p. b. For each destnaton j wth an entry n M, Remove exstng lnks (n, j) nt k and add new lnk (m, j, d) tot k,whered = D jk, D mk and m = p jk. 2. If lnk-down event, clear the neghbor tables of k; 3. If lnk-up or lnk-cost change event,update l k; End NTU C. Computng D j Fg. 2. Neghbor Table Update Algorthm Ths subsecton descrbes the shortest-path algorthm PATH (1) and the next subsecton descrbes how PATH s extended to compute multple next-hops are determned based on D j. INIT-PATH s called at node startup to ntalze the tables; dstances are ntalzed to nfnty and node denttes to a null value. PATH s executed n response to an event: ether a recept of an update message from a neghbor or detecton of an adjacent lnk cost or lnk status (up/down) change. PATH nvokes procedure NTU (Fg. 2), whch frst updates the neghbor dstance tables and then updates T k wth lnks (m; n; d), where d = D nk, D mk and m = p nk. PATH then nvokes procedure MTU (Fg. 3), whch constructs T by mergng the topologes T k and the adjacent lnks l k. PATH s vrtually dentcal to PDA [19], n the sense that though they dffer n the nformaton the routers exchange, nternally they construct the partal topologes. The mergng process s straghtforward f all neghbor topologes (T k) contan consstent lnk nformaton, but when two or more neghbors lnk tables contan conflctng nformaton regardng a partcular lnk, the conflct must be resolved. Two neghbor tables are sad to contan conflctng nformaton regardng a lnk, f ether both report the lnk wth dfferent cost or one reports the lnk and the other does not. Conflcts are resolved as follows: f two or more neghbor tables contan conflctng nformaton of lnk (m; n), then T s updated wth lnk nformaton reported by the neghbor k that offers the shortest dstance from the node to the head node m of the lnk,.e., l k + D mk = mnfl k + D mkjk 2 N g. Tes are broken n a consstent manner; one way s to break tes always n favor of lower address neghbor. Because tself s the head of the lnk for adjacent lnks, any nformaton about an adjacent lnk suppled by neghbors wll be overrdden by the most current nformaton about the lnk avalable to node. After mergng the topologes, MTU runs Djkstra s shortest path algorthm to fnd the shortest path tree and deletes all lnks from T that are not n the tree. Because there can be more than one shortest-path tree, whle runnng Djkstra s algorthm tes are agan broken n a consstent manner. The dstances D j and predecessors p j are then obtaned from T. The tree s compared wth the prevous shortest path tree and only the dfferences are then reported to the neghbors. If there are no dfferences, no updates are reported. Eventually all tables converge such that D j gve the shortest dstances and all message actvty wll cease. D. Computng S j MPATH descrbed n ths secton determnes the successor sets S j, by enforcng the Loop-free Invarant condtons (descrbed below) usng a neghbor-to-neghbor synchronzaton. Let FD j, called the feasble dstance, be an estmate of the ds-

4 3 Procedure MTU 1. Clear lnk table T. 2. For each node j 6= occurrng n at least one T k, a. Fnd MIN mnfd jk + l kjk 2 N g. b. Let n be such that MIN =(D jn + l n). (Tes are broken consstently. Neghbor n s the preferred neghbor for destnaton j). For each lnk (j, v, d)nt n, Add lnk (j; v; d) to T. 3. Update T wth each lnk l k. 4. Run Djkstra s shortest path algorthm on T to fnd new D j,andp j. 5. For each destnaton j, fd j or p j changed from prevous value, set changed and report-t flags for j. End MTU Fg. 3. Man Table Update Algorthm tance of node to node j n the sense that FD j s equal to D j when the network s n stable state, but to prevent loops durng perods of network transtons, t s allowed to be temporarly dffer from D j.the key to loop-free routng s n mantanng FD j such that the followng condtons are satsfed. Loop-free Invarant Condtons(LFI)[19]: FD j(t) D k j(t) k 2 N (1) S j(t) = f k j D jk(t) <FD j (t)g (2) The nvarant condtons (1) and (2) state that, for each destnaton j, a node can choose a successor whose dstance to j, as known to, s less than the dstance of node to j that s known to ts neghbors. Theorem 1: [19] If the LFI condtons are satsfed at any tme t, the SG j (t) mpled by the successor sets S j (t) s loop-free. Proof: Let k 2 S j(t) then from (2) we have D jk(t) < FD j(t) (3) At node k, because node s a neghbor, from (1) we have Combnng (3) and (4) we get FD k j (t) D jk(t) (4) FD k j (t) < FD j (t) (5) Eq.(5) states that, f k s a successor of node n a path to destnaton j, thenk s feasble dstance to j s strctly less than the feasble dstance of node to j. Now, f the successor sets defne a loop at tme t wth respect to j, then for some node p on the loop, we arrve at the absurd relaton FD p j (t) <FDp j (t). Therefore the LFI condtons are suffcent for loop-freedom. The nvarants used n LFI are ndependent of whether the algorthm uses lnk states or dstance vectors; n lnk-state algorthms, such as MPDA, the Djk are computed locally from the lnk-states communcated by the neghbors whle n dstance-vector algorthms, lke the MPATH presented here, the Djk are drectly communcated. The nvarants (1) and (2) suggest a technque for computng Sj (t) such that the successor graph SG j(t) for destnaton j s loop-free at every nstant. The key s determnng FDj (t) n Eq. (1), whch requres Procedure INIT-MPATH finvoked when the node comes up.g 1. Intalze tables and run MPATH. End INIT-MPATH Algorthm MP AT H finvoked when a message M s receved from neghbor k, or an adjacent lnk to k has changed.g 1. Run NTU to update neghbor tables. 2. Run MTU to obtan new D j and p j. 3. If node s PASSIVE or node s ACTIVE ^ last reply arrved, Reset goactve flag. For each destnaton j marked as report-t, a. FD j mnfd j ;RD jg b. If D j >RD j,setgoactve flag. c. RD j D j d. Add [j, RD j, p j] to message M. e. Clear report-t flag for j. Otherwse, the node s ACTIVE and watng for more reples, For each destnaton j marked as changed, f. FD j mnfd j ;FD jg 4. For each destnaton j marked as changed, a. Clear changed flag for j b. S j fkjd jk <FD jg 5. For each neghbor k, a. M M. b. If event s a query from k,setreply flag n M. c. If goactve set, Set query flag n M. d. If M non-empty, send M to k. 6. If goactve set, become ACT IV E, otherwse become PASSIV E. End MP AT H Fg. 4. Mult-path Loop-free Routng Algorthm node to know Dj(t), k the dstance from to node j n the topology table T k that node communcated to neghbor k. Because of non-zero propagaton delay, T k s a tme-delayed verson of T. We observe that, f node delays updatng of FDj wth Dj untl k ncorporates the dstance Dj n ts tables, then FDj satsfes the LFI condton. MPATH (Fg. 4) enforces the LFI condtons by synchronzng the exchange of update messages among neghbors usng query and reply flags. If a node sends a message wth a query bt set, then the node must wat untl a reply s receved from all ts neghbors before the node s allowed to send the next update message. The node s sad to be n ACTIVE state durng ths perod. The nter-neghbor synchronzaton used n MPATH spans only one hop, unlke algorthms that use dffusng computaton that potentally span the whole network(e.g., DASM [2]). Assume that all nodes are n PASSIVE state ntally wth correct dstances to all other nodes and that no messages are n transt or pendng to be processed. The behavor of the network, where every node runs MPATH, s such that when a fnte sequence of lnk cost changes occurs n the network wthn a fnte tme nterval, some or all nodes go through a seres of PASSIVE-to-ACTIVE and ACTIVE-to-PASSIVE state transtons, untl eventually all nodes become PASSIVE wth correct dstances to all destnatons. Let a node n PASSIVE state receve an event that results n changes n ts dstances to some destnatons. Before the node sends an update message to report new dstances, t checks f the dstance Dj to any destnaton j has ncreased above the prevously reported dstance RDj. If none of the dstances ncreased, then the node remans n PASSIVE state. Otherwse, the node sets the query flag n the update

5 4 message, sends t to each neghbor, and goes nto ACTIVE state. When n ACTIVE state, a node cannot send any update messages or ncrease FD j. After recevng reples from all ts neghbors the node s allowed to ncrease FD j and report any changes that may have occurred snce the tme t has transtoned to ACTIVE state, and f none of the dstances ncreased beyond the reported dstance, the node transtons to PASSIVE state. Otherwse, the node sends the next update message wth the query bt set and becomes ACTIVE agan, and the whole cycle repeats. If a node receves a message wth the query bt set when n PASSIVE state, t frst modfes ts tables and then sends back an update message wth the reply flag set. Otherwse, f the node happens to be n ACTIVE state, t modfes the tables but because the node s not allowed to send updates when n ACTIVE state, the node sends back an empty message wth no update nformaton and the reply bt set. If a reply from a neghbor s pendng when the lnk to the neghbor fals then an mplct reply wth nfnte dstance s assumed, Because reples are gven mmedately to queres and reples are assumed to be gven upon lnk falure, deadlocks due to nter-neghbor synchronzaton cannot occur. Eventually, all nodes become PASSIVE wth correct dstances to destnatons, whch we prove n the next secton. III. CORRECTNESS OF MPATH To show the correctness of MPATH, we prove the followng: (1) MPATH eventually converges wth Dj gvng the shortest dstances and (2) the successor graph SG j s loop-free at every nstant and eventually converges to the shortest multpath. PATH works essentally lke PDA[19] except that the knd of update nformaton exchanged s dfferent; PDA exchanges lnk-state whle PATH exchanges dstance-vectors wth predecessor nformaton. Internally both represent ths nformaton as partal topologes communcated by the neghbors. So, the correctness proof of PATH s dentcal to PDA. The convergence of MPATH drectly follows from the convergence of PATH because extensons to MPATH are such that update messages n MPATH are only delayed a fnte amount of tme. A node generates update messages only to report changes n dstances and predecessor, so after convergence no messages wll be generated. The followng theorems show that MPATH provdes nstantaneous loop-freedom. Theorem 2: For the algorthm MPATH executed at node,lett n be the tme when RD j s updated and reported for the n-th tme. Then, the followng condtons always hold. FD j(t n) mnfrd j (t n,1);rd j(t n)g (6) FD j (t) FD j (t n) t 2 [tn;t n+1) (7) Proof: From the workng of MPATH n Fg. 4, we observe that RDj s updated at lne 3c when (a) the node goes from PASSIVEto-ACTIVE because of one or more dstance ncreases (b) the node receves the last reply and goes from ACTIVE-to-PASSIVE state (c) the node s n PASSIVE state and remans n PASSIVE state because the dstance dd not ncrease for any destnaton (d) the node receves the last reply but mmedately goes nto ACTIVE state. The reported dstance RDj remans unchanged durng the ACTIVE phase. Because FDj s updated at lne 3a each tme RDj s updated at lne 3c, Eq. (6) follows. When the node s n ACTIVE phase, FDj may also be modfed by the statement on lne 3f, whch mples Eq. (7). Theorem 3: (Safety property) At any tme t, the successor sets S j (t) computed by MPATH are loop-free. Proof: The proof s based on showng that the FD j and S j computed by MPATH satsfy the LFI condtons. Let t n be the tme when RD j s updated and reported for the n-th tme. The proof s by nducton on the nterval [t n;t n+1]. Let the LFI condton be true up to tme t n, we show that From Theorem 2 we have FD j(t) D k j(t) t 2 [t n;t n+1] (8) FD j (t n) mnfrd j (t n,1);rd j (t n)g (9) FD j (t n+1) mnfrd j (t n);rd j (t n+1)g (1) FD j(t) FD j (t n) t 2 [tn;t n+1) (11) Combnng the above equatons we get FD j(t) mnfrd j (t n,1);rd j(t n)g t 2 [tn;t n+1](12) Let t be the tme when message sent by at t n s receved and processed by neghbor k. Because of the non-zero propagaton delay across any lnk, t s such that t n <t <t n+1 and because RD j s modfed at t n and remans unchanged n (t n;t n+1) we get RD j(t n,1) D k j(t) t 2 [t n;t ) (13) From Eq. (13) and (14) we get RD j (t n) D k j (t) t 2 [t ;t n+1] (14) mnfrd j (t n,1);rd j(t n)g D k j (t) t 2 [t n;t n+1] (15) From (12) and (15) the nductve step (8) follows. Because FDj (t) Dj(t k ) at ntalzaton, from nducton we have that FDj (t) Dj(t) k for all t. Gven that the successor sets are computed based on FDj, t follows that the LFI condtons are always satsfed. Accordng to the Theorem 1 ths mples that the successor graph SG j s always loop-free. The followng theorem shows that MPATH correctly computes the shortest multpath. Theorem 4: (Lveness property) A fnte tme after the last change n the network, the D j gve the correct shortest dstances and S j = fkjd k j <D j;k 2 N g. Proof: The proof s smlar to the proof of Theorem 4 n [19]. The convergence of MPATH follows drectly from the convergence of PATH because the update messages n MPATH are only delayed a fnte tme as allowed at lne 4 n algorthm PATH. Therefore, the dstances D j n MPATH also converge to shortest dstances. Because changes to D j are always reported to the neghbors and are ncorporated by the neghbors n ther tables n fnte tme D jk = Dk j for k 2 N after convergence. From lne 3a n MPATH, we observe that when node becomes passve FD j = D j holds true. Because all nodes are passve at convergence t follows that S j = fkjd jk < FD j;k 2 N g = fkjd k j <D j;k 2 N g. IV. SIMULATION RESULTS The smulatons compare the control overhead and convergence tmes of MPATH, TOPB and DASM. TOPB s a lnk-state algorthm that closely approxmates OSPF, whch s a lnk-state algorthm for whch commercal mplementatons exst and whose convergence tme s farly constant and depends on the dameter of the network. Ideally, MPATH should approach the convergence tmes of TOPB, that s the extra tme needed to enforce loop-freedom should be neglgble. We expect MPATH to have far less message overhead because of ts relance on only partal topology. On the other hand DASM s the only dstance-vector routng algorthm to date that provdes loop-free multpaths to each destnaton. DASM acheves loop-freedom through dffusng computatons that span the whole network. In contrast, MPATH

6 MESSAGE LOAD IN BYTES Fg. 5. CAIRN Topology used n smulatons 35 3 Fg. 8. Lnk recoveres. Message overhead MESSAGE LOAD IN BYTES TIME IN MILLISECONDS Fg. 6. Lnk falures. Message overhead Fg. 9. Lnk recoveres. Convergence tmes TIME IN MILLISECONDS Fg. 7. Lnk falures. Convergence tmes uses only neghbor-to-neghbor synchronzaton. It s nterestng to see how convergence tmes are effected by the synchronzaton mechansms. Also, t s not obvous how the control message overheads of DASM and MPATH compare. The performance metrcs used for comparson are the control message overhead and the convergence tmes. We use the event-drven real-tme smulator CPT from Noka and perform smulatons on the CAIRN topology shown n Fg. 5 ( For smplcty, we use a flat topology wthout area aggregaton; there s no reason to beleve area aggregaton would favor one routng algorthm over others. Two types of events are trggered n the network: lnk-status changes (lnk falures and lnk recovery) and lnk-cost changes. In practce lnks and nodes are hghly relable and change status much less frequently than lnk costs whch are a functon of the traffc on the lnk. For smplcty, We do not smulate node falures because of the problems resultng due to loss of sequence numbers by the nodes, whch only effect the functonng of TOPB here. We also restrct lnk-status changes to a sngle change; that s, only one lnk falure or lnk recovery can occur at any tme durng the measurement nterval. Because n backbone networks the lnks and nodes n the network are hghly relable, smultaneous multple topologcal changes are much less lkely to occur and t s reasonable to assume that tables converge between topologcal changes. However, lnk costs of multple lnks can change smultaneously and repeatedly before the tables converge to the latest costs. Ths s the case when near-optmal delay routng of [19] s used, n whch the lnk costs are perodcally measured and reported. For these reasons, we smulate only sngle lnkstatus changes and multple lnk-cost changes. Lnk-status changes: Each lnk s made to fal and recover n turn, and the control message overhead and convergence tmes are measured n each case. The worst-case and the averages of control message overhead and convergence tmes are gven n Table 1. Fgs. (6)-(9) show the overheads assocated wth each event. For lnk falures and recoveres MPATH has lower average message overhead than TOPB, whch s due to use of partal topologes. However, due to synchronzaton used for provdng loop-freedom, the worst-case message overhead s hgher for MPATH.. MPATH has larger overhead than DASM under lnk recoveres because, though nether nvokes synchronzaton, MPATH exchanges predecessor nformaton n addton to dstances. Under lnkfalures, DASM requres more messages than MPATH because of the multhop synchronzaton that DASM uses. Same argument can be appled for the convergence tmes. Multple lnk-cost changes: When near-optmal routng framework s mplemented as n [19], multple lnks change cost smultaneously. To study the protocol behavor under such scenaros, costs of multple lnks s changed smultaneously and the performance s measured. The average message overhead and convergence tmes are shown n the Table 1. MPATH has lower worst-case and average message overhead than TOPB and DASM. MPATH has lower worst-case and average convergence tmes than DASM. The average convergence tme for MPATH s also lower than TOPB. Only n the worst-case, MPATH showed hgher convergence tmes than TOPB, whch s agan due to synchronzaton used n MPATH. V. CONCLUSIONS We have presented the frst path-fndng routng algorthm that provdes multple paths between each source-destnaton par that need not necessarly have equal costs and that are loop-free at every nstant. The

7 6 TABLE 1 Control messages (bytes) Worst-case Avg Std-dev Lnk falures TOPB DASM MPATH Lnk recoveres TOPB DASM MPATH Lnk-cost changes TOPB DASM MPATH Convergence tmes (ms) Worst-case Avg Std-dev Lnk falures TOPB DASM MPATH Lnk recoveres TOPB DASM MPATH Lnk-cost changes TOPB DASM MPATH [19] S. Vutukury and J.J. Garca-Luna-Aceves. A Smple Approxmaton to Mnmum Delay Routng. Proc. of ACM SIGCOMM, Sept [2] W. T. Zaumen and J.J. Garca-Luna-Aceves. Loop-Free Multpath Routng Usng Generalzed Dffusng Computatons. Proc. IEEE INFOCOM, March routng algorthm s desgned around a set of loop-free nvarant condtons and uses nter-nodal synchronzaton that spans no more than one hop. Usng smulatons, the performance of the routng algorthm, n terms of control message overhead and convergence tmes, s compared wth other algorthms. The multple next-hop choces that MPATH makes avalable at each node can be used for traffc load-balancng and mnmzng delays n the network [19]. REFERENCES [1] E.W.Djkstra and C.S.Scholten. Termnaton Detecton for Dffusng Computatons. Informaton Processng Letters, 11:1 4, August 198. [2] D. Farnach. Introducton to enhanced IGRP(EIGRP). Csco Systems Inc., July [3] J.J. Garca-Luna-Aceves. Loop-Free Routng Usng Dffusng Computatons. IEEE/ACM Trans. Networkng, 1:13 141, February [4] J.J. Garca-Luna-Aceves and J. Behrens. Dstrbuted, scalable routng based on vectors of lnk states. IEEE Journal on Selected Areas n Communcatons, October [5] J.J. Garca-Luna-Aceves and S. Murthy. A path-fndng algorthm for loop-free routng. IEEE/ACM Trans. Networkng, February [6] J.J. Garca-Luna-Aceves and M. Spohn. Scalable lnk-state nternet routng. Proc. Internatonal Conference on Network Protocols, October [7] J.J. Garca-Luna-Aceves and M. Spohn. Source tree adaptve routng. Proc. Internatonal Conference on Network Protocols, October [8] C. Hendrck. Routng Informaton Protocol. RFC, 158, june [9] P. A. Humblet. Another Adaptve Dstrbuted Shortest Path Algorthm. IEEE Trans. Commun., 39:995 13, June 91. [1] J. M. Jaffe and F. H. Moss. A Responsve Dstrbuted Routng Algorthm for Computer Networks. IEEE Trans. Commun., 3: , July [11] P. M. Merln and A. Segall. A Falsafe Dstrbuted Routng Protocol. IEEE Trans. Commun., 27: , September [12] J. Moy. OSPF Verson 2. RFC, 1247, August [13] R. Perlman. Fault-tolerant broadcast of routng nformaton. Computer Networks and ISDN, 7, [14] B. Rajagopalan and M. Faman. A Responsve Dstrbuted Shortest-Path Routng Algorthm wth Autonomous Systems. Internetworkng: Research and Experence, 2:51 69, March [15] A. Segall. Optmal dstrbuted routng for vrtual lne-swtched data networks. IEEE Trans. Commun., 27:21 29, January [16] J. Spnell and R. Gallager. Event Drven Topology Broadcast wthout Sequence Numbers. IEEE Trans. Commun., 37: , [17] S. Vutukury and J.J. Garca-Luna-Aceves. An algorthm for multpath computaton usng dstancevectors wth predecessor nformaton. Proc. of ICCCN, Oct [18] S. Vutukury and J.J. Garca-Luna-Aceves. A Dstrbuted Algorthm for Multpath Computaton. GLOBECOM 99, 1999.