Multicast Tree Rearrangement to Recover Node Failures. in Overlay Multicast Networks

Transcription

1 Multcast Tree Rearrangeent to Recover Node Falures n Overlay Multcast Networks Hee K. Cho and Chae Y. Lee Dept. of Industral Engneerng, KAIST, Kusung Dong, Taejon, Korea Abstract Overlay ultcast akes use of the Internet as a low level nfrastructure to provde ultcast servce to end hosts. The strategy of overlay ultcast sldes over ost of the basc deployent ssues assocated wth IP ultcast, such as end-to-end relablty, flow and congeston control, and assgnent of an unque address for each ultcastng group. Snce each ultcast eber s responsble for forwardng ultcast packets, overlay ultcast protocols suffer fro ultcast node falures. To cope wth node falures n the overlay ultcast networks, the eployent of ultcast servce nodes (MSNs) s consdered whch allows relatvely hgh processng perforance to cover the dsconnected nodes. We are nterested n nzng the cost of both the MSNs and addtonal lnks when a node falure occurs. Overlay ultcast tree rearrangeent to connect ultcast ebers s dscussed and forulated as a bnary nteger prograng proble. The tree rearrangeent proble s solved by a heurstc based on the Lagrangean relaxaton. The perforance of the proposed algorth s nvestgated by carryng out experents n 50 and 100 node probles. The eployent of MSNs s llustrated to be dependent on the end-to-end delay bound n overlay networks and the degree constrant of eber nodes. 1

2 1. Introducton Multcastng provdes an effcent way of transttng data fro a sender to a group of recevers. Instead of sendng a separate copy of the data to each ndvdual group eber, a source node sends dentcal essages sultaneously to ultple destnaton nodes. An underlyng ultcast routng algorth deternes a ultcast tree connectng the source and group ebers. Data generated by the source flows through the ultcast tree, traversng each tree edge exactly once. As a result, ultcast s ore resource-effcent and s well suted to applcatons such as teleconferencng, vdeo-on-deand (VOD) servce, electronc newspapers, cyber educaton and edcal ages. However, despte the conceptual splcty of IP ultcast and ts obvous benefts, ts deployent s dffcult due to the coplexty of IP ultcast technology and lack of applcatons. To cope wth the ncreasng traffc of ulteda contents, the study on ultcastng wll becoe ore actve. Recent efforts to provde ultcast delvery have thus shfted to overlay ultcast that bulds a transport-layer overlay network aong ebers of a ultcast group. Recent topcs related to overlay ultcast ncludes [1, 2, 3, 4, 5, 6, 7, 8, 11] as an alternatve to IP ultcast. Overlay ultcast uses the Internet as a low level nfrastructure to provde ultcast servce to end hosts. Many basc deployent ssues such as end-to-end relablty, congeston control, and assgnent of an unque address for each ultcastng group that are assocated wth IP ultcast can be solved wth overlay ultcast. Current overlay ultcast projects can be classfed nto two catalogs accordng to the structure: end-to-end overlay [1, 3, 5] and proxy-based overlay [2, 4]. In end-to-end overlay, every eber n the ultcastng group shares the responsblty to forward data to other ebers. End hosts organze theselves nto a ultcastng tree. We call these end hosts ultcast nodes n the end-to-end overlay case. Scattercast [2] and Overcast [4] are typcal 2

3 exaples of proxy-based overlay structure that for a herarchcal structure copared to endto-end overlay. The ultcastng servce s perfored wth the help of proxy nodes, whch can duplcate data and forward the to end hosts wth predefned routng algorth. In proxy-based overlay, proxy s the ultcast node and end hosts just receve the ultcast data fro the correspondng proxes. Representatve research [7] on overlay ultcast protocol ncludes Scattercast, Overcast, Narda [3] and ALMI [5]. Each protocol has dfferent desgn objectves, whch leads to dfferent propertes. Narda and Scattercast ntend to nze the delay fro a ultcast source to each eber. ALMI strves to nze the ultcast tree cost, where the cost of each lnk s defned as the round-trp delay between group ebers. Overcast, on the other hand, axzes avalable bandwdth for each eber. In the tree constructon process, Narda and Scattercast use a esh-frst approach. That s, group ebers are frst connected nto a esh and then the ultcast tree s bult on top of t. On the other hand, Overcast and ALMI use a drect approach. The step to buld the esh s bypassed and the ultcast tree s fored drectly. In overlay ultcast, snce ultcast ebers are responsble for forwardng ultcast packets, they suffer fro ultcast node falures. When a ultcast node fals, rapd ultcast tree recovery s essental to dstrbute ultcast data to dsconnected nodes. However, any researchers have focused ther attenton anly on constructng the ntal overlay ultcast tree [1, 2, 3, 4, 5, 6, 8, 11]. In ths paper, we are nterested n solvng the tree rearrangeent proble by establshng new connectons for the ultcast ebers when a ultcast node falure occurs. 2. Packet Delvery and Node Falures n Overlay Multcast Networks Overlay ultcast avods the deployent hurdles of IP ultcast at the cost of data delvery latency due to ts neffcent ultcast data dstrbuton. Ths neffcency s llustrated n the 3

4 exaple network of Fgure 1. In Fgure 1, nodes A, B, C, D, and E are ebers of a ultcast group, nodes R1, R2, and R3 are routers n the network, and the dashed lnes connectng the nodes are physcal lnks. The flows n Fgure 1 (a) llustrate how IP ultcast forwards data sent by A to the other ebers of the group. Fgure 1 (b) shows a possble overlay ultcast traffc flow. Fgure 1 (c) shows the overlay wthout the underlyng physcal network. Coparng Fgure 1 (a) and (b), t s clear that data delvery usng overlay ultcast experences longer delay than tradtonal IP ultcast delvery. Ths exaple shows that latency between the source and the ultcast ebers n an overlay ultcast largely depends on the desgn of the overlay route. Thus, n ths paper, we assue that end-to-end delay constrant s requred to avod excessve delay. Note n Fgure 1 (c) that each lnk n the overlay ultcast tree represents an ndependent uncast sesson. Node C of Fgure 1 (c) handles three uncast sessons. However, each ultcast node has a lt n the nuber of uncast sessons that can be handled sultaneously. It s due to the CPU perforance, network nterface card capacty, buffer sze and others. Thus we consder the degree constrant of a ultcast node n the overlay ultcast tree. The degree constrant represents the axu nuber of uncast sessons that a ultcast node can handle dependng on the capacty. B A B A A R1 R2 R1 R2 B C R3 E D C R3 E D D C E (a) Exstng Multcast (b) Overlay Multcast (c) Overlay wthout the underlyng physcal network Fgure 1. Coparson of packet delvery 4

5 B A Sngle doan R1 R2 C R3 D E Fgure 2. Overlay network for nter doan ultcast Overlay ultcast can also be used n nter-doan ultcast. Only a host fro each doan receves the ultcast data strea fro the reote sender va the overlay ultcast. The other recevers n the doan receve the applcaton data strea fro the host as llustrated n Fgure 2. The doan n ths paper represents a doan that s anaged by a coon ultcast protocol. Now, consder a tree rearrangeent n case of a node falure. In Fgure 3, suppose that node A has faled n the overlay ultcast tree. Thus, the chldren of node A have to rejon the ultcast sesson. The overlay ultcast tree s dvded nto several fragents as shown n Fgure 3 (b). Fgure 3 (c) shows the case where the end-to-end delay constrant s not satsfed by soe nodes after the ultcast tree rearrangeent, due to the degree bound of the node. The end-to-end delay can be reduced by the eployent of a node wth a hgher degree n the overlay ultcast tree [8]. Thus, to satsfy the end-to-end delay bound, we consder the eployent of the ultcast servce nodes (MSNs) whch allow relatvely hgh processng perforance by coverng all the dsconnected nodes. The eployent of an MSN requres addtonal lnks to connect the dsconnected ultcast nodes. The overlay ultcast tree rearrangeent, therefore, causes expenses for the MSNs and the lnks. For effcent tree rearrangeent, specal consderaton s requred n the selecton of the MSNs and addtonal lnks. In ths paper, we are nterested n nzng the cost of both 5

6 the MSNs and addtonal lnks when a node falure occurs. Faled node A (a) Sngle ultcast node falure (b) Separaton of ultcast tree End-to-end delay constrant s not satsfed. (c) Tree rearrangeent wthout the delay constrant Multcast Servce Node (MSN) (d) Tree rearrangeent wth the delay constrant Fgure 3. Node falure and tree rearrangeent 6

7 3. Multcast Tree Rearrangeent n Overlay Multcast Networks Gven a graph G = (V, E) where V s the set of nodes and E s the set of lnks, consder an overlay ultcast tree, T = (N, E1) where N V s the set of ultcast nodes and E1 E s the set of lnks n T. Let R N be the set of ultcast nodes to rejon the ultcast tree due to a node falure. The set of the black rectangle nodes n Fgure 3 corresponds to R. For a rejon node R, a sngle path s requred to connect the node to the ultcast source. If no path exsts between the ultcast source and node, the tree rearrangeent s possble. However, by assung at least one path between the ultcast source and node, the faled tree s rearranged to connect every ultcast node. Let x be a bnary varable to represent a lnk (,j) that s eployed for a path between the source and rejon ultcast eber. If lnk (,j) s eployed to connect the source and node, then x = 1. Otherwse, x = 0. The dotted lnes of Fgure 3 (c) and (d) represent lnks wth x = 1. Then the followng relaton holds for every node ncludng the source node s. x x = 1 k k for = s and all x x = 0 for V /{, s} and x x = 1 k for = and all R all In the above constrants, notce that several paths between the source and rejon nodes ay possbly traverse through lnk (,j). Snce the ultcast tree rearrangeent proble consders the cost of each addtonal lnk, we ntroduce a constant M to reflect the ultple new paths that share the sae lnk. Let y represent the adopton of lnk (,j) for the new paths, then the followng equaton holds. R R x + x j My for < j Note that the above equaton holds for any lnk n the overlay ultcast tree. 7

8 Now, as dscussed n the prevous secton, data delvery n the overlay ultcast tree s perfored by ndependent uncast sessons. Because each ultcast node ncludng the MSN has a capacty lt n the nuber of sultaneous uncast sessons, we consder the degree constrant of a ultcast node. The constrant represents the nuber of uncast sessons a ultcast node can handle sultaneously. By lettng w be the degree constrant of node, we have y + k y w z for all In addton to the above constrants, we need to consder the end-to-end delay between the source node and ultcast ebers. In ths proble, we assue that every ultcast eber has a delay bound. In other words, every path between the source and a rejon node has to satsfy the delay bound constrant. By lettng d be the lnk delay and D be the end-to-end delay bound, we have d (, j) E x D for all R Now, we need to nze the cost of the newly nstalled MSNs and that of the newly ncluded lnks. Let u and c respectvely be the cost of the newly nstalled nodes and lnks to rearrange the ultcast tree. Then the total cost s gven by L u z + (, j) E < j c 2 y where L V s a set of canddate MSNs and E2 E s a set of lnks newly ncluded nto the overlay ultcast tree. Fro the above dscusson, the ultcast tree rearrangeent proble can now be forulated as follows. Proble P: Mnze c y + subject to: (, j) E 2 < j L u z 8

9 x x = 1 for = s and all R (1) k k x x = 0 for V /{, s} and all R (2) x x = 1 for = and all R (3) k + x j My for < x j (4) y + y w z for all (5) d (, j) E x k D x, y, z are bnary varables for all R Consderng the NP-hardness of the overlay ultcast tree proble wthout the delay constrant [11], the proposed bnary nteger prograng proble s NP-hard. In the above forulaton constrants (1), (2) and (3) are well known constrants for the shortest path proble. Thus, relaxaton of constrants (4), (5) and (6) ay lead the proble to a less coplcated verson. Lagrangean relaxaton s a general soluton strategy for solvng such a relaxed proble. The procedure perts us to decopose a proble nto several easy subprobles by explotng ts specal structure. Ths soluton approach s used for solvng any odels wth ebedded network structure, such as the one consdered n [10, 15]. In fact, the Lagrangean relaxaton leads the overlay ultcast tree reconstructon proble nto three decoposed subprobles. The frst subproble s reduced nto a sple shortest path proble where any sophstcated algorths are avalable. The other two subprobles are reduced nto unconstraned nzaton probles that are easy to handle. Therefore, n the next secton we thus develop a Lagrangean relaxaton algorth as a prosng soluton procedure for the overlay tree rearrangeent proble. (6) 9

10 4. Lagrangean Relaxaton Based Heurstc for Tree Rearrangeent In ths secton, we consder the Lagrangean relaxaton [9] to solve the proble forulated n Secton 3. The relaxed proble s decoposed nto three subprobles whch can easly be solved by eployng the known algorths for the underlyng network structures [10]. Consder the tree rearrangeent proble P forulated n the prevous secton. The odel s an nteger prograng proble whch s, n general, dffcult to solve. Rather than solvng the dffcult optzaton proble drectly, we cobne the cubersoe constrants wth the orgnal objectve functon, where soe ultpler values act as penalty factors. Then the proble as a whole s transfored to a ore tractable for. The otvaton for adoptng ths approach s based on the fact that the orgnal proble P has an attractve substructure, the shortest path proble, whch we would lke to explot algorthcally. Let us dscuss the Lagrangean relaxaton procedure n ore detal. By relaxng constrants (4), (5), and (6), the relaxed proble P L s obtaned as follows. Proble P L Mnze Z L (λ) = ( R + L (, j) E ( u ( λ (, j) + λ ( ) d 1 w λ ( )) z 2 3 ) x λ ( ) D) + 3 ( c (, j) E 2 < j λ (, j) M 1 + λ ( ) + λ ( j)) y 2 2 Subject to x x = 1 k k for = s and all x x = 0 for V /{, s} and x x = 1 k x, y, z are bnary varables for = and all R all R R 10

11 The Lagrangean ultplers λ 1, λ 2 and λ 3 correspond to the constrant sets (4), (5) and (6) respectvely. Snce λ 1 (,j) s defned only for <j, we let λ 1 (,j) = λ 1 (j,) for coputatonal convenence. Here, note that proble P L can be decoposed nto followng three ndependent subprobles: Subproble P L1 Mnze Subject to ( 1 3 λ3 R (, j) E x x = 1 k k ( λ (, j) + λ ( ) d ) x ( ) D) for = s and all x x = 0 for V /{, s} and x x = 1 k x are bnary varables for = and all R all R R Subproble P L2 Mnze ( c λ1 (, j) M + λ ( ) + λ ( j)) 2 2 y (, j) E 2 < j Subproble P L3 Mnze ( u wλ2 ( )) z L For a specfc rejon node, subproble P L1 s a shortest path proble. Subproble P L2 and P L3 are unconstraned nzaton probles. Now we present a systeatc soluton procedure for the decoposed subprobles. () Soluton to subproble P L1 11

12 The well-known Dkstra s algorth [9, 14] s used for fxed n subproble P L1. The algorth fnds the shortest path fro the source node to the rejon ultcast node n a network. Dkstra s algorth solves the shortest path proble n O(N 2 ) te. Snce totally shortest path probles have to be solved at every teraton, the coputatonal coplexty of subproble P L1 s O(N 2 ). () Soluton to subproble P L2 Subproble P L2 s an unconstraned nzaton proble. If the coeffcent of y s negatve n the objectve functon, then y = 1 nzes the objectve functon value of subproble P L2. Otherwse, y = 0. Thus, the decson crteron of y s as follows. Let l = λ (, j) M + λ ( ) + λ ( )) ( c j y = 0 f l 0 y = 1 otherwse () Soluton to subproble P L3 Subproble P L3 s also an unconstraned nzaton proble. Thus, the decson crteron of z s as follows. Let v = u w λ ( ) 2 Z = 0 f v 0 z = 1 otherwse A soluton to the Lagrangean relaxaton proble P L s obtaned by solvng the above subprobles. However, the soluton to P L s usually not a feasble soluton to the orgnal proble P. Thus, a feasble soluton s obtaned at each teraton as follows. The soluton acqured n Lagrangean relaxaton proble P L s transfored nto tree for through Step 1 and Step 2. Step 3 akes the above soluton satsfy the delay and degree constrants by 12

13 reconstructng the feasble path between the source and rejon node. The soluton n Step 3 s further proved n Step 4. Let p() be an exstng path between the ultcast source and rejon node, eployed for the current ultcast tree, and let p () be a new path between the ultcast source and rejon node. In Step 4, f p () has lower cost than p(), p() s reoved fro the ultcast tree and p () s added to the ultcast tree. The heurstc for generatng feasble solutons s as follows. Step 1. y = 0 for all, j. z = 0 for all. Step 2. If Σ x 1, y = 1. If (y = 1) and ( P), z = 1. If (y = 1) and (j P), z j = 1. Step 3. Check feasblty for each node. If the soluton s feasble for all node, then Goto Step 4. Otherwse, update the network for feasble node and copute Dkstra s algorth to recopute x for nfeasble node. Goto Step 1. Step 4. If ternaton crteron s satsfed, Whle all rejon nodes are not searched, Select a rejon node that s not searched. Fnd a p (). If p () has lower cost than p(), p() s reoved fro the current soluton. p () s added to the current soluton. Stop. 13

14 Otherwse, Stop. Accordng to the Lagrangean boundng prncple [9] we can calculate the axu allowable error range for each of the feasble solutons obtaned. For any vector λ of the Lagrangean ultplers, the objectve functon value Z L (λ) of the Lagrangean dual functon s a lower bound to the optal objectve functon value Z P * of the orgnal optzaton proble P. Thus, we have Z L (λ) Z P *. The boundng prncple edately ples one advantage of the Lagrangean relaxaton approach. The ethod gves us a useful ternaton crteron nvolvng axu allowable error range. At each teraton, Lagrangean relaxaton proble P L s solved by usng updated ultpler value λ. A feasble soluton to the proble P s then generated va the proposed heurstc fro the soluton of the relaxed proble. Let Z P be the objectve functon value of proble P by the feasble soluton. Then the axu allowable error range ε s defned as Z ε = P Z L ( λ) 100% Z ( λ) L Start Copute Lower bound and feasble soluton Solve sub-proble 1 Solve sub-proble 2 Is ternaton crtera satsfed? Yes End Solve sub-proble 3 No Update ultplers wth subgradent ethod Fgure 4. Procedure of the proposed heurstc 14

15 Lagrangean relaxaton proble P L s repeatedly solved by usng updated ultpler values untl the axu allowable error range ε satsfes a ternaton threshold or the nuber of teratons exceeds the threshold lt. We use the subgradent ethod [10, 13] for updatng Lagrangean ultplers. Fgure 4 shows the overall procedure of the proposed algorth. 5. Coputatonal Results In ths secton, we dscuss the coputatonal experent of the Lagrangean relaxaton based heurstc for the tree rearrangeent proble. Multcast networks wth 50 and 100 nodes are generated. In each network, lnks are randoly generated such that the total nuber of lnks becoes E = 5 V. The average lnk cost s assued to be 10. Each lnk has average delay value of three. The end-to-end delay bound has the value of 10 and 20 n the network wth 50 nodes and 15 and 25 n the network of 100 nodes. The average cost of MSN s assued to be 20 and degree constrant has the value of 3 and 8 n the network wth 50 nodes. These values are 5 and 15 n the network of 100 nodes. All soluton procedures are run on a Pentu III-660MHz PC. Table 1 and 2 respectvely show the result of the proposed algorth n 50 and 100 nodes ultcast networks. The CPLEX [12] s eployed to have optal solutons. The proposed Lagrangean heurstc shows good perforance n every proble category of delay bound and degree constrants. The soluton gap fro the optal soluton s slghtly hgher n probles wth lower delay bound and degree constrant. As shown n the table, the average soluton gap fro the optal soluton s 5% and 5.6% respectvely n the 50 and 100 node probles. The CPU seconds are also copared as shown n the tables. The coputatonal effort requred by the proposed algorth s draatcally reduced when copared to the exact soluton procedure. The coputatonal result s further nvestgated by exanng the requred nuber of MSNs, 15

16 due to the dfferent values of delay bounds and degree constrants. Fgure 5 shows that ore MSNs are eployed as the proble becoes ore dffcult. As the delay bound and degree constrant becoes tghter, the eployed nuber of MSNs ncreases for the tree rearrangeent. The ncrease sees to be ore senstve to the delay bound when the degree constrant s tght. Further experents are perfored to exane the cost senstvty of the overlay tree rearrangeent. Fgure 6 shows the nuber of eployed MSNs when dfferent unt MSN cost s appleds. It s clear that ore MSNs are eployed as the nuber of rejonng ultcast nodes ncreases. The ncrease of the requred MSNs s also senstve to the unt cost. Hgher unt cost constrants the nuber of MSNs to cover the rejon hosts. 6. Concluson To overcoe node falures n overlay ultcast networks, ultcast tree rearrangeent s consdered by eployng the ultcast servce nodes (MSNs). The servce nodes generate new paths fro a source to ultcast ebers such that the end-to-end delay bound s satsfed. Snce each ultcast eber s responsble for forwardng the packets n the overlay network, the degree constrant of a ultcast node s also consdered. The tree rearrangeent proble s forulated as a bnary nteger lnear progra that nzes the cost of MSNs and newly adopted lnks. Lagrangean relaxaton based heurstc s developed to solve the proble. Soe ntractable constrants are relaxed and added nto the objectve as a penalty. Then the proble s decoposed nto three subprobles that are easy to handle. Networks wth 50 and 100 nodes are generated for coputatonal experents. In each network, four categores of probles are tested wth dfferent delay bounds and degree constrants. The average soluton gap fro the optal soluton s 5% and 5.6% respectvely n 16

17 the 50 and 100 node probles. The coputatonal te requred by the proposed algorth s draatcally reduced copared to the exact soluton procedure. The requred nuber of MSNs due to the dfferent values of delay bound and degree constrant s also nvestgated. As the proble becoes tghter, the eployed nuber of MSNs ncreases to recover the node falure. The ncrease s ore senstve to the delay bound when the degree constrant s tghter. The cost of MSN also affects the eployent of the nodes. 17

18 Heurstc Optal Optal Proble (Delay bound, CPU Nuber Soluton value CPU GAP* Degree constrant) seconds (HS) (OPT) seconds 1 (10,3) (10,3) (10,3) (10,3) (10,3) (10,8) (10,8) (10,8) (10,8) (10,8) (20,3) (20,3) (20,3) (20,3) (20,3) (20,8) (20,8) (20,8) (20,8) (20,8) *GAP = (HS-OPT)/OPT Table 1. Perforance of the proposed heurstc wth 50 nodes 18

19 Heurstc Optal Optal Proble (Delay bound, CPU Nuber Soluton value CPU GAP* Degree constrant) seconds (HS) (OPT) seconds 1 (15,5) (15,5) (15,5) (15,5) (15,5) (15,15) (15,15) (15,15) (15,15) (15,15) (25,5) (25,5) (25,5) (25,5) (25,5) (25,15) (25,15) (25,15) (25,15) (25,15) *GAP = (HS-OPT)/OPT Table 2. Perforance of the proposed heurstc wth 100 nodes 19

20 3 2.5 (15,5) 100 node probles 50 node probles The nuber of MSNs (10,3) (10,8) (15,15) (25,5) (20,3) (25,15) (20,8) (Delay bound, Degree constrant) Fgure 5. The nuber of MSNs eployed for the tree rearrangeent The nuber of MSNs Average MSN cost = 10 Average MSN cost = 20 Average MSN cost = Rejon ultcast nodes Fgure 6. The nuber of MSNs for the tree rearrangeent 20

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