Sgnal Processng: Image Communcaton 23 (2008) 754 768 Contents lsts avalable at ScenceDrect Sgnal Processng: Image Communcaton journal homepage: www.elsever.com/locate/mage Dstrbuted meda rate allocaton n multpath networks $ Dan Jurca, Pascal Frossard Ecole Polytechnque Fédérale de Lausanne (EPFL), Sgnal Processng Insttute, CH-1015 Lausanne, Swtzerland artcle nfo Artcle hstory: Receved 28 March 2008 Receved n revsed form 9 September 2008 Accepted 10 September 2008 Keywords: Vdeo streamng Dstrbuted rate allocaton Optmal path selecton Dstorton optmzed vdeo streamng Multpath meda transmsson abstract The paper addresses the dstrbuted path computaton and rate allocaton problems for vdeo delvery over multpath networks. The streamng rate on each path s determned such that the end-to-end meda dstorton s mnmzed, when a meda clent aggregates packets receved va multple network channels to the streamng server. In common practcal scenaros, t s, however, dffcult for the server to have the full knowledge about the network status. Therefore, we propose here a dstrbuted path selecton and rate allocaton algorthm, where the network nodes partcpate to the optmzed path selecton and rate allocaton based on ther local vew of the network. Ths elmnates the need for end-to-end network montorng, and permts the deployment of large scale rate allocaton solutons. We desgn a dstrbuted algorthm for optmzed rate allocaton, where the meda clent teratvely determnes the best set of streamng paths, based on nformaton gathered by network nodes. Each ntermedate node then forwards ncomng meda flows on the outgong paths, n a dstrbuted manner. The proposed algorthm s shown to quckly converge to the rate allocaton that provdes a maxmal qualty to the vdeo clent. We also propose a dstrbuted greedy algorthm that acheves close-to-optmal end-to-end dstorton performance n a sngle pass. Both algorthms are shown to outperform smple heurstc-based rate allocaton approaches for numerous random network topologes. They offer an nterestng soluton for medaspecfc rate allocaton over large scale multpath networks. & 2008 Publshed by Elsever B.V. 1. Introducton As the nternet s far from provdng any wdely deployed guarantee of servce, effcent meda streamng strateges have to be devsed to cope wth the weaknesses of the network nfrastructure, and provde an acceptable qualty to multmeda applcatons. Multpath streamng emerged lately as an effectve soluton to overcome some lmtatons of lossy nternet paths [6,15]. It offers an ncrease n streamng bandwdth by balancng the load over multple network paths between the meda server $ Ths work has been supported by the Swss Natonal Scence Foundaton, under Grant PP-002-68737. Correspondng author. E-mal addresses: dan.jurca@gmal.com, jurca@docomolab-euro.com (D. Jurca), pascal.frossard@epfl.ch (P. Frossard). and the clent. It also provdes means to lmt packet loss effects, when combned wth error reslent streamng strateges and scalable encodng capabltes. Multpath streamng can be deployed n content delvery networks, overlay networks or wreless and peer-to-peer scenaros, where a clent has access to the meda sources smultaneously through multple network paths. Ths paper addresses the problem of meda-specfc rate allocaton for streamng applcatons n multpath networks. For gven network parameters, an optmal set of transmsson paths are selected, along wth ther respectve transmsson rate, such that the decoded meda qualty s maxmzed. However, path selecton performed exclusvely at the server or at the clent generally requres end-to-end network montorng, as well as the knowledge of the complete network status at one pont of the 0923-5965/$ - see front matter & 2008 Publshed by Elsever B.V. do:10.1016/j.mage.2008.09.002
D. Jurca, P. Frossard / Sgnal Processng: Image Communcaton 23 (2008) 754 768 755 topology. Ths unfortunately lmts the mplementaton of such algorthms to small-scale network scenaros. Therefore, we propose n ths paper, a dstrbuted soluton where ntermedate network nodes that are capable of handlng applcaton-level nformaton, partcpate to the path selecton and rate allocaton algorthm based on ther local vew of the network. Network nodes become actvely nvolved n the meda delvery between a streamng server and a meda clent; t permts to deploy effcent vdeo dstrbuton applcatons n large scale dstrbuted archtectures such as mesh or peer-to-peer networks. We buld on our pror work [10] that descrbes a server-drven framework for the analyss of jont path and rate allocaton n multpath vdeo streamng. We, however, bypass the lmtatons of centralzed approaches, and we propose here novel algorthms for computng the streamng paths and the streamng rates n a dstrbuted manner. We consder a network model composed of multple flows between the clent and the streamng server, e.g., overlay networks or networks wth path dversty. The ntermedate network nodes together report the resources avalable for the streamng sesson. Based on ths nformaton, the clent determnes the best path selecton and rate allocaton, and generates flow reservaton requests to the ntermedate network nodes and the streamng server. The clent-based flow reservaton s then accommodated wthn the network on a node-by-node bass. The jont path selecton and rate allocaton performs teratvely, untl all ntermedate nodes converge to a (unque) optmal soluton. The server fnally adapts the meda source rate accordngly by scalable codng, or packet flterng for example. The new dstrbuted path selecton strategy permts to relax the assumpton of full network status knowledge at the server and to elmnate the need for complex end-to-end network montorng strateges. We desgn a path selecton algorthm that quckly converges to the optmal vdeo rate allocaton soluton. In parallel, we propose a fast, one-step algorthm, whch provdes a close-to-optmal soluton. The performance of both algorthms are analyzed n detals and compared to smple heurstc-based approaches. Thanks to the optmal meda-specfc allocaton, the proposed algorthms clearly outperform other strateges based on pure network metrcs. Thanks to ther dstrbuted nature and to the meda-specfc consderatons, these algorthms offer effectve solutons to the meda streamng rate allocaton problem n medum to large scale multpath networks. The rest of ths paper s organzed as follows. Secton 2 dscusses the related work and motvates the need for dstrbuted rate allocaton solutons. Secton 3 descrbes n detal the streamng scenaro consdered n ths paper, and presents the rate allocaton optmzaton problem. We present our dstrbuted solutons n Secton 4 and we analyze the characterstcs of the proposed algorthms n Secton 5. Extensve smulaton results are fnally presented n Secton 6 for numerous network topologes. 2. Related work Ths paper addresses the multpath routng problem from a meda applcaton perspectve. The process of selectng the paths for transmsson and ther respectve rate allocaton, targets an mproved streamng experence measured n terms of vdeo dstorton. Several pror works have dscussed the path selecton problem based on pure network metrcs. These works generally lead to suboptmal solutons n terms of meda qualty. For example, numerous routng algorthms have been proposed n order to optmze a gven network QoS metrc [28], to mprove the performance of TCP over wreless ad-hoc networks [20], to dscover multple avalable network paths to one source [32], or to optmze the network resource allocaton n overlay multcasts [3,25]. In addton, the authors of [33] adapt the DSR protocol for ad-hoc networks to provde multple vable paths for multmeda transmssons. However, none of these works specfcally consders the multmeda applcaton characterstcs n the routng decsons. They rather rely on routng algorthms that fnd the best path (or set of paths), gven some establshed network metrcs. Whle ths may be optmal n terms of network utlzaton, t s, however, suboptmal from the pont of vew of the qualty of servce for the meda streamng applcaton. In 30 80% of the cases, the best paths found by classc routng algorthms are suboptmal from a meda perspectve [26]. At the same tme, several works have nvestgated the problem of multpath streamng, as a way to ncrease the multmeda qualty of servce on lossy network nfrastructures. Nevertheless, most of the research work dedcated to multpath streamng focuses on the streamng process tself (meda schedulng aspects), but generally not towards fndng whch paths should deally be used for the streamng applcaton, for a gven network topology. More specfcally, the multpath problem s addressed n the case of meda streamng n [21] for a multcast scenaro. The authors present a FEC scheme combned wth server dversty and a packet schedulng mechansm, whch ntends to mnmze the cumulatve dstorton of ndvdual erroneous vdeo packets. Multstream codng, combned wth multpath transmsson, has been presented n [17] as a soluton to fght aganst network errors n an ad-hoc network envronment. The authors of [1] multple path streamng scenaro for the transmsson of vdeo sequences encoded n multple descrptons. Other works n dstrbuted vdeo streamng [13,35,22] deal wth resource allocaton and schedulng on multple, a pror chosen streamng paths, wth the fnal goal of mnmzng the overall dstorton perceved by the meda clents. All these works rely on a gven set of transmsson paths, and try to optmally explot these network resources. However, none of these specfcally targets the optmal choce of the streamng paths or the rate allocaton problem for mprovng the multmeda qualty of servce. The work presented n [30] addresses the problem of choosng the best paths from a meda perspectve. However, t only nvestgates the effcency of path swtchng schemes from the meda applcaton pont
756 D. Jurca, P. Frossard / Sgnal Processng: Image Communcaton 23 (2008) 754 768 of vew, wthout analyzng the benefts of multpath streamng. In ths work, we address the problem of jont selecton of network paths and rate allocaton, such that the end-toend meda dstorton s mnmzed. In addton to consderng multpath strateges that have been shown to mprove streamng performances, our work also nnovates by proposng a dstrbuted soluton for path computaton. It allevates the need for expensve end-toend path montorng systems, and thus can be deployed n large scale network scenaros. Our streamng framework s qute generc and apples to any streamng system that obeys an addtve rule for the aggregated transmtted rate and packet loss probabltes. The optmal routng and rate allocaton decson determnes the best usage of end-toend transmsson paths, so that the meda dstorton s mnmzed when network flows are aggregated at the decoder. 3. The dstrbuted multpath rate allocaton problem 3.1. Network and vdeo model We consder that the meda streamng applcaton s deployed on a large scale network, e.g., overlay networks or networks wth path dversty. The network s modelled as a fully connected, drected acyclc graph G(V,E), between the streamng server S and the clent C (Fg. 1). V s the set of nodes n the network, and E s the set of lnks. Each node N AV has a local vew N ¼fI ; O g of the network topology, where I DE and O DE represent the sets of ncomng and respectvely outgong network lnks for the node N. Each lnk L u AE has two assocated postve metrcs: the avalable bandwdth r u 40 and the average packet loss probablty p u A[0,1], assumed to be ndependent of the streamng rate. We defne P C,1ppn, as an end-to-end path between S and C n G, wth parameters b C and p C beng the end-toend bandwdth and the loss probablty, respectvely, and Server S Clent C I = {L u } O = {L u } L u (b u, p u ) Node Local Network Vew At Node Fg. 1. Multpath network scenaro and network vew at Node N. n the total number of dstnct paths. A flow 1 transmtted on path P C has a streamng rate r C pb C ¼ mn (r u ), and s affected by the loss probablty p C ¼ 1 PLu2P Cð1 p Lu2P u Þ. C We consder applcatons wth strngent delay constrants and no possblty for packet retransmsson. Typcally, we use UDP streamng strateges that are generally preferred to TCP solutons n many streamng scenaros. The vdeo qualty depends n ths case on the actual streamng/encodng rate and transmsson loss probabltes. We consder that the end-to-end meda dstorton can be computed as the sum of the source dstorton and the channel dstorton. It s commonly admtted that the qualty experenced at the clent, depends on both the dstorton due to a lossy encodng of the meda nformaton and the dstorton due to losses experenced n the network (see for example [5,29]). For a gven vdeo encoder, the source dstorton D S s mostly drven by the encodng rate R (also called streamng rate n ths paper), and the meda sequence content, whose characterstcs nfluence the rate-dstorton characterstcs of the encoder. The channel dstorton D L s dependent on the average loss probablty e sustaned by vdeo nformaton, and the sequence characterstcs. It s roughly proportonal to the number of vdeo enttes (e.g., frames) that cannot be decoded, and the loss probablty e corresponds to the actual vdeo packet loss rato when vdeo frames are encapsulated nto dstnct network packets. The average end-to-end dstorton can thus be wrtten as D ¼ D S þ D L ¼ ar x þ b 2 (1) where a, bar + and xa[ 1,0] are parameters that depend on the vdeo sequence. In the above multpath streamng scenaro, the streamng rate can smply be wrtten as the sum of the rates of the dfferent flows R ¼ Xn ¼1 r C. At the same tme, when the loss processes on dfferent paths are ndependent, the overall loss probablty becomes P n ¼1 2¼ p C r C P n. ¼1 r C The average end-to-end dstorton model s a smple and general approxmaton, sutable for most common streamng strateges where the number of packets per frame s ndependent of the encodng rate, whch can be adjusted to the network offered streamng rate, e.g., through scalable encodng. Note that the actual vdeo loss process s lkely to present a low correlaton, due to the usage of multple paths. Under the gven network assumptons, the vdeo dstorton metrc becomes qute nsenstve to the actual lnk error model, and s mostly nfluenced by the average loss probabltes. A valdaton of ths model through vdeo experments can be found n Ref. [10]. 1 Throughout ths paper, the terms flow and end-to-end network path are used nterchangeably.
D. Jurca, P. Frossard / Sgnal Processng: Image Communcaton 23 (2008) 754 768 757 In the remander of ths secton, we frst revew brefly the results for the multpath meda rate allocaton problem n centralzed systems. The relaxaton of the assumpton about full knowledge of the network then leads to a novel problem formulaton for dstrbuted path selecton n multpath vdeo streamng. The soluton of ths problem leads to the rate allocaton algorthms descrbed n ths paper. 3.2. Optmal rate allocaton Ths secton brefly overvews the soluton to the optmal rate allocaton, when the server has full knowledge about the status of the network. In our prevous work [10], we have derved the analytcal rules that allow dervng the optmal soluton to the jont path selecton and rate allocaton problem for a gven vdeo stream, wth a smple algorthm whose complexty s lnear n the number of avalable end-to-end network paths. Namely, once the parameters of all paths P C are known by S, the three followng theorems lead to the optmal greedy rate allocaton soluton n any loop-free flow-equvalent network graph. A flow-equvalent graph G denotes a network topology where the maxmum bandwdth offered to the applcaton (e.g., the maxflow of G) does not depend on the choce of transmsson paths. Flowequvalent networks represent most of the typcal topologes encountered n practce, where bottleneck lnks typcally le on the edge of the network, or between domans. Interested readers are referred to Ref. [9] for ample dscussons and proofs of the followng theorems. They are gven here for the sake of completeness, as they are used n the dstrbuted algorthm proposed n the next secton. Theorem 1 (on off flows). Gven a flow-equvalent network graph G wth ndependent flows F C havng rates r C A[0, b C ] and a dstorton metrc as defned n Eq. (1), the optmal soluton of the rate allocaton problem when all the paths are dsjont, les at the margns of the value ntervals for all r C,.e., the optmal value of r C s ether 0 or b C, 8:1ppn. Theorem 2 (parameter decouplng). Gven a flow-equvalent network graph G wth ndependent flows F C havng rates r C A[0, b C ] and a dstorton metrc as defned n Eq. (1), the structure of the optmal rate allocaton s F* ¼ [b 1 C, b 2 C, y, b C,0,0,y0]. Theorem 3 (bottleneck bandwdth sharng). Let L u be a bottleneck lnk for the set of paths B u ¼fP k Cg n G. The k bottleneck lnk bandwdth r u shall be shared among paths P C n a greedy way, startng wth the path affected by the lowest loss probablty. When the characterstcs of all paths P C are known by the server S, Theorems 1 3 offer the gudelnes for a complete optmal path selecton and rate allocaton soluton. They show that the optmal rate allocaton can be acheved by a greedy path selecton algorthm that starts wth the paths affected by the smallest end-to-end loss probablty. At the same tme, the rate of bottleneck lnks that are shared by multple network paths should also be splt n a greedy manner among meda flows. Once a path P C s chosen for transmsson, t s optmal to stream at rate r C ¼ b C, from the meda applcaton perspectve. 2 Based on these rules, the optmal path selecton and rate allocaton can be acheved by a greedy algorthm, whch provdes a low complexty soluton to meda-specfc resources optmzaton n flow-equvalent networks. We present now novel dstrbuted mechansms for computng the avalable end-to-end paths on the network graph, so that the strong assumpton of full network knowledge at the streamng server can be relaxed. We eventually buld on Theorems 1 3 n order to compute the optmal rate allocaton on these paths. 3.3. Dstrbuted optmzaton problem We now formalze the dstrbuted path selecton and rate allocaton problem addressed n ths paper. When no sngle node N AV (ncludng S), s aware of the entre network topology G, we want to fnd the optmal path selecton and flow rate allocaton that mnmzes the overall dstorton D at the clent. Under the assumptons that the streamng rate can be controlled and that the packet loss rate s ndependent of the streamng rate, the server S eventually adapts the vdeo encodng rate to the aggregate rate of the avalable network paths used for streamng, and to the loss processes experenced on these paths. 3 The optmzaton problem can be formulated as follows: 3.3.1. Dstrbuted multmeda rate allocaton problem (DMMR) Gven the flow-equvalent network graph G(V, E) whose lnks L u have a maxmal bandwdth r u and an average loss rato p u, gven the node local vews N ; 8N 2 V and gven the vdeo sequence characterstcs G ¼ (a, b, x), fnd the complete set of end-to-end paths P C,1ppn and the optmal rate allocaton R ~ ¼½r 1 C ;...; rn C Š that mnmzes the dstorton metrc D ~R ¼ argmn ~R under the constrants D ¼ argmnðar x þ b 2Þ (2) ~R r C pb C X ; 8P C ; 1ppn r pr C u ; 8u such that L u 2 E P C :Lu2P C where R ~ represents the set of possble rate allocatons on G(V, E), R ¼ Pn r C and ¼ Sn ¼1 p C r C : S n ¼1 r C ¼1 2 Note that n our developments, we assume that b C s the total far share of bandwdth allocated by the network for the streamng applcaton on path P C, and that the streamng flows do not suffer from self-congeston. 3 For a dscusson of a more general transmsson error process, we refer the nterested reader to [12]; [27].
758 D. Jurca, P. Frossard / Sgnal Processng: Image Communcaton 23 (2008) 754 768 4. Dstrbuted rate allocaton 4.1. Dstrbuted path computaton We present n ths secton two algorthms for dstrbuted path selecton and rate allocaton. The algorthms dffer n the computaton of the paths between the server S and the clent C. Before descrbng n detal the dstrbuted path computaton and rate allocaton strateges, we brefly ntroduce the notaton and assumptons necessary to ther presentaton. Recall that every node N AV has only a local vew of the network topology, denoted by N ¼fI ; O g. I and O are the sets of ncomng and respectvely outgong lnks to/from N. We assume that N possesses an estmate of the bandwdth r u and loss probablty p u on all the outgong lnks (.e., 8L u AO ). Let P k denote a path connectng the node N to the server. In addton to maxmal bandwdth b k and loss probablty p k, a path s characterzed by two decsons flags that are used by the dstrbuted rate allocaton algorthms. The flag f k s a path reservaton flag that can only be set or reset by the clent C, respectvely the server S, and the flag d k s a decson flag that can be updated by any ntermedate node on the path P k. Whle f k s used to advertse the network flows requested by the clent C, d k s used to sgnal the feasblty of a requested flow at an ntermedate node. We denote by P ¼ {P k } the set of all dstnct paths between the server S and the node N. Note that two dstnct paths P k l and P may not necessarly be fully dsjont, as they may share one or more network lnks. Wthout loss of generalty, we assume that the paths n P are ordered accordng to the ncreasng value of the path loss probabltes p k. Let fnally P u DP be the set of dstnct paths between the server S and the node N, whch share the ncomng lnk L u AI. End-to-end paths between the server and the clent are then bult n a dstrbuted manner, snce no node has the full knowledge of the network status. These paths are computed by path extenson, whch s performed ndependently at each network node. We choose the notaton - to represent the path extenson operator that adds a lnk L u AO leavng node N, to an ncomng path P k AP.In other words, f lnk L u connects nodes N and N j, we can wrte P l j ¼ P k -L u, wth P l j AP u j and P k AP. We can compute the bandwdth and loss probablty parameters for the extended path P l j ¼ P k -L u, respectvely, as b l j ¼ mn(b k,r u ) and p l j ¼ 1 (1 p k )(1 p u ). We propose two dfferent methods for dstrbuted path computaton (employed by the two proposed algorthms), whch respectvely constructs all the possble paths, or bulds them n a greedy manner. Formally, the two path extenson rules can be stated as follows. Rule 1: Each ncomng path P k AP at node N s extended towards all the outgong lnks L u AO. If the set of outgong lnks drectly connect N to several nodes N j, the set of extended paths at node N can be wrtten as O ¼ {P l j ¼ P k -L u P k AP, L u AO }. The subset of the extended paths that borrow the partcular outgong lnk L u s wrtten as O u ¼ {P l j ¼ P k -L u P k AP }. All paths wth null bandwdth are obvously omtted. It s easy to see n ths case that O u ¼ P and that O ¼ P O. The sze of the set s multplcatve n the number of ncomng flows and n the number of outgong lnks [16]. It has to be noted that resource allocaton for flows n O s constraned by the avalable bandwdth on jont bottleneck lnks, and that all the paths may not be used smultaneously at ther full transmsson bandwdth. Rule 2: The ncomng paths P k AP at node N, taken n order of ncreasng loss probablty p k are extended towards the outgong lnks L u AO, taken n decreasng order of relablty. Smlarly to a water-fllng algorthm, the total outgong bandwdth s greedly allocated to the set of ncomng paths, untl all the ncomng paths are extended, or untl no more bandwdth s avalable. When the sets of outgong lnks and the ncomng paths are both ordered along the ncreasng values of loss probablty, the set of extended paths at node N can be wrtten as 8 < G ¼ P l j ¼ X u Pk! L u r m 4 Xk 1 b n and Xu 1 r m o Xk : m¼1 n¼1 m¼1 b n n¼1 9 = The subset of the paths n G that borrow the outgong lnk L u s denoted G u. Note that n ths case, smultaneous resource allocaton for all flows n G, s feasble on G. Based on the dstrbuted path computaton that follows ether Rule 1 or 2, we now descrbe two rate allocaton strateges called, respectvely, Algorthms 1 and 2. Then we present both an optmal and a greedy rate allocaton algorthm for dstrbuted multpath meda streamng. 4.2. Dstrbuted path selecton and rate allocaton The dstrbuted path computaton and rate allocaton algorthms proceed frst by determnng the paths avalable between the server and the clent. Then they reserve paths accordng to the optmal allocaton computed by the clent. They proceed n two phases, the path dscovery and the path reservaton phases, respectvely. To ths am, control messages are exchanged at the applcaton layer between the server S and the clent C, and forwarded by the ntermedate nodes, as llustrated n Fg. 2. We assume the exstence of a bdrectonal control channel between any two nodes n G that are connected by a network segment L u. The server ntates the frst phase of the algorthms by sendng path dscovery messages Path u on all outgong lnks. The messages are forwarded by the ntermedate nodes n the consdered network overlay, on the control channel assocated wth lnk L u. At each ntermedate node, the Path messages contan the nformaton (b k and p k ) related to every possble flow between the server and node N, along wth nformaton related to prevously successfully reserved flows. The node then extends the path accordng to Rule 1 or 2 (n the case of Algorthm 1 or 2, respectvely), and forwards path dscovery message Path u that contans nformaton about the paths that borrow the lnk L u. Dependng on the path extenson strategy that uses ether Rule 1 or 2, the clent eventually receve nformaton about all possble paths, or ;
D. Jurca, P. Frossard / Sgnal Processng: Image Communcaton 23 (2008) 754 768 759 A. Intate Path messages. B. 1. Aggregate ncomng Path messages; 2. Compute the rate allocaton accordng to Rule 1 or Rule 2; 3. Update and forward Path messages. C. 1. Compute flow reservaton; 2. Intate Resv messages. Server E. 1. Aggregate Resv messages; 2. Compute flows; 3. Start streamng/next teraton. Path Message Resv Message D. 1. Aggregate Resv messages; 2. Make flow reservaton; 3. Update and forward Resv messages. Clent Fg. 2. Dstrbuted path selecton and reservaton. respectvely about only a subset of them. In the latter case, the paths are computed n a greedy manner along the decreasng relablty of the end-to-end channels. Upon recepton of path dscovery messages, the clent C computes the optmal path selecton P* C usng the Theorems 1 3, and the nformaton t gets from the nodes about end-to-end paths. It should be noted that these theorems greatly smplfy the rate allocaton n flowequvalent networks. They ndeed state that paths should ether be used at ther full bandwdth, or smply dropped. The clent then ntates the second phase of our algorthms, by sendng path reservaton messages, Resv u, whch are forwarded by the network nodes N to the server, on the backward control channel assocated wth lnk L u. 4 A path reservaton message Resv u contans nformaton about the flow(s) that should be reserved on lnk L u for the streamng sesson (e.g., requested rate b k C, end-to-end loss probablty p k C and flags f k and d k, whch are both set to 1 by C). However, there s no guarantee that all paths n P* C, can be accommodated smultaneously. Once all Resv messages are receved at node N (one for each outgong lnk), the node N attempts to greedly allocate the bandwdth for the requested flows (d k ¼ f k ¼ 1) on the outgong lnks, followng the order of ncreasng loss probablty p k C. It eventually marks the flows that cannot be reserved at the requested rate b C,by settng the flag d k ¼ 0. Once a vald subset of paths P*DP* C s successfully reserved by S (.e., all d k flags are set to 1), the nodes update ther local vew of the network, N 0 ¼ N \P, and new path dscovery messages are ssued. The clent aggregates nformaton about the resdual network resources, and updates the path selecton P* C accordngly. The process s terated untl convergence to the fnal rate allocaton, whch s reached 4 Due to practcal mplementaton consderatons, an empty Resv message should be sent even on lnks that do not contan any reserved flow. Alternatvely, tmeouts should be mplemented at each ntermedate node. when all flows reserved by C can be accommodated by the network at the requested rate b k C. The dstrbuted path selecton and rate allocaton algorthms llustrated n Fg. 2 are fnally summarzed n Algorthms 1 and 2, where the left-hand sde and rghthand sde columns, respectvely, correspond to the path dscovery and path extensons phases. Intally, both algorthms start at the server sde, wth step 4. The algorthms dffer n the path extenson rule (step 3 n the bottom left block). The path extenson rule drectly controls the convergence to the stable rate allocaton, but also the qualty of the rate allocaton soluton. Comprehensve nformaton about end-to-end paths as created by Rule 1 permts to reach an optmal rate allocaton, but possbly at the expense of several teratons of the path reservaton schemes. The algorthm, however, converges n a small number of rounds to a feasble soluton, Algorthm 1. Dstrbuted path selecton and rate allocaton algorthm-optmal Server S: Node N : Upon receve Resv u, 8L u AO S : Upon receve Resv u, 8L u AO : 1. compute P* C based on flags f k ; 2. update P* based on flags d k ; 3. f P* ¼ øorp* ¼ P* C, return P*;. 4. else update network vew N 0 S send Path u, 8L u AO S. 1. 8 paths P k A{P k } P k - L u AResv u \P*: set d k ¼ 0f b C k 4r 0 u, where the avalable output bandwdth r 0 u s updated accordng to a greedy allocaton; 2. send Resv v, 8L v AI. Node N : Clent C: Upon receve Path u, 8L u AI : Upon receve Path u, 8L u AI C : 1. update network graph 1. compute the set of N 0 ; avalable paths P C ; 2. compute avalable paths 2. compute the optmal P accordng to N 0 ; allocaton P* C from P C ;
760 D. Jurca, P. Frossard / Sgnal Processng: Image Communcaton 23 (2008) 754 768 3. compute extended paths O resp. G, 8L v AO, accordng to Rule 1; 4. send dscovery messages Path u, 8L v AO. 3. 8P C k AP* C, set f k ¼ d k ¼ 1; 4. send reservaton messages Resv v, 8L v AI C. Algorthm 2. Dstrbuted path selecton and rate allocaton algorthm-greedy Server S: Node N : Upon receve Resv u, 8L u AO S : Upon receve Resv u, 8L u AO : 1. compute P* C based on flags f k ; 2. update P* based on flags d k ; 3. f P* ¼ øorp* ¼ P* C, return P*; 4. else update network vew N 0 S send Path u, 8L u AO S. 1. 8 paths P k A{P k } P k - L u AResv u \P*: set d k ¼ 0f b C k 4r 0 u, where the avalable output bandwdth r 0 u s updated accordng to a greedy allocaton; 2. send Resv v, 8L v AI. Node N : Clent C: Upon receve Path u, 8L u AI : Upon receve Path u, 8L u AI C : 1. update network graph 1. compute the set of N 0 ; avalable paths P C ; 2. compute avalable paths 2. compute the optmal P accordng to N 0 ; allocaton P* C from P C ; 3. compute extended paths 3. 8P k C AP* C, set f k ¼ d k ¼ 1; O, resp. G 8L v AO, 4. send reservaton accordng to Rule 2; messages Resv v, 8L v AI C. 4. send dscovery messages Path v, 8L v AO. gven the network graph G. The Rule 2 constructs only a lmted subset of end-to-end network paths, gven a greedy forwardng soluton at each ntermedate node N. It leads to a qucker computaton of the soluton, whch may however be suboptmal. Both algorthms are analyzed n Secton 5 and ther performance s compared n Secton 6. 5. Analyss and dscusson 5.1. Convergence propertes Ths secton proposes an analyss of the path selecton and rate allocaton algorthms ntroduced n the prevous secton. Under the assumpton that the network s stable durng the executon of our algorthms, we derve hard bounds on the convergence of the rate allocaton towards the optmzed soluton. Observe that one teraton of the algorthms requres one complete message exchange between S and C, on the avalable paths. Hence, the tme requred by one round s n the order of the round-trp tme (RTT) of the slowest paths n the network. The computatons at ntermedate nodes and at S and C are trval and ther duraton can be neglected. The rate allocaton algorthms generally converge n a very small number of steps (as shown n the next secton), so that the assumpton about the stablty of the network n terms of average bandwdth and loss probablty of the network lnks s usually vald. Snce the total number of paths s qute small n general [10], the algorthms reach a stable soluton n a convergence tme correspondng to only a small number of RTTs. Ths s equal to the number of rounds of the teratve path selecton that are needed for convergence, and the average lnk characterstcs are lkely to stay unchanged durng that perod. In order to derve exact bounds on the performance of our algorthms, we further assume that the control channel s relable, and that nodes are synchronzed,.e., any node receves all dedcated control packets n a bounded tme nterval. Note that perfect synchronzaton s not crucal for the desgn of the proposed algorthms, whch can work wth looser synchronzaton. Loose node synchronzaton can be acheved by employng separate synchronzaton protocols [18]. Most works addressng decentralzed systems [24] assume loose node synchronzaton n order to derve bounds on protocol performance. We consder frst the Algorthm 1, whch uses Rule 1 for path extenson: the clent has a complete vew of endto-end paths to compute the path selecton. We show that the Algorthm 1 converges to the optmal soluton n one round f paths are dsjont. Then, we show that n the worst case, one round of the algorthm reserves at least the path wth the lowest loss probablty. Consequently, the Algorthm 1 termnates n a fnte number of rounds. We now formally prove these three propertes. Property 1. If the paths requested by C do not share any bottleneck jont lnk L u, Algorthm 1 converges n one round. Proof. Let P C be the set of avalable paths between S and C dscovered by Algorthm 1, and let P* C ¼ {P 1 C, y, P m C }be the optmal set of paths chosen by C for transmsson, accordng to Theorems 1 3. If b k C represents the avalable rate of on requested path P k C AP* C,wehaveb k C pr u, 8L u AP k C. Snce, by hypothess, the chosen paths P k C do not contan any jont bottleneck lnk L u, we have r u X P k:lu2pcb k k C, 8L u AP k C and 8P k C AP* C. Ths means that any node N, upon the recepton of reservaton packets, Resv, can allocate the requested bandwdth on the outgong lnks for all requested flows. Therefore, no flow s marked wth d k ¼ 0, and the server S can compute the optmal allocaton P* ¼ P* C, after one round of the protocol. & Property 2. Let the network graph that corresponds to the avalable resources at one stage of the algorthm be denoted G 0 ¼ [ N 0 Durng each round, Algorthm 1 reserves n G 0 :N 2V at least the end-to-end flow P C between S and C whch s affected by the smallest loss probablty P C. Proof. Let P C AP* C \P* be the lowest loss probablty path requested by C but not yet reserved by our algorthm. Observe that P C s the lowest loss probablty path n the resdual graph G 0, and also n the local vew N 0 observed by each node N. Hence, at every node N traversed by P C, the flow P C wll have prorty durng the greedy reservaton phase of Algorthm 1. Indeed, from the path extenson operaton we have b C pr u, 8L u AP C. Hence P C s successfully reserved at each
D. Jurca, P. Frossard / Sgnal Processng: Image Communcaton 23 (2008) 754 768 761 ntermedate node N on the path. Fnally, the flow P C reaches S wth the Resv packets wth both flags d ¼ f ¼ 1, hence the server S ntegrates the flow to the set of successfully reserved paths:p* ¼ P* S P C. & Property 3. Algorthm 1 converges, and termnates n at most m rounds, where m s the number of allocated flows, whch s moreover not larger than the total number of avalable dstnct paths n G. Proof. Ths result s a drect consequence of Property 2. At each round, the algorthm reserves at least one flow, and the avalable rate of the lnks n the resdual network decreases. Hence, n subsequent rounds of the algorthm, the clent C wll not be able to request an nfnte number of flows. & The prevous propertes show that gven the avalable flow-equvalent network graph G, Algorthm 1 converges to the optmal path selecton n a lmted number of rounds, no more than the total number of avalable endto-end paths between S and C. Moreover, n the case of dsjont network paths, our protocol manages to reserve the optmal set of flows needed for transmsson n a sngle round. Fnally, for the more general case, the algorthm secures at least one transmsson flow from the optmal allocaton. We now concentrate on the second algorthm, and demonstrate that t converges n a sngle teraton. Moreover, we show that the soluton offered by Algorthm 2 s actually dentcal to the optmal soluton provded by Algorthm 1 f each network node has only one outgong lnk. Property 4. Algorthm 2 converges after one round of path dscovery and selecton phases. Proof. Let P C be the set of avalable paths between S and C, as dscovered n the path dscovery phase of Algorthm 2, based on path extenson Rule 2. Let further P* C ¼ {P 1 C, y, P m C } be the optmal set of paths chosen by C for transmsson accordng to Theorems 1 3, based on the nformaton receved from the network nodes. Let fnally b k C be the rate of the requested path P k C AP* C, wth b k C pr u, 8L u AP k C. The greedy rate allocaton n the path extenson gven by Rule 2 ensures that at any node N, and 8L u AO, we have P k:lu2pcb k pr k C u. Ths means that any node N, upon recepton of reservaton packets, can allocate the bandwdth on the outgong lnks for all requested flows. Therefore, no flow s marked wth d k ¼ 0, and the server S can compute the optmal allocaton P* ¼ P* C, after one round of the protocol. & Property 5. Algorthm 2 provdes the same soluton as Algorthm 1 f the outdegree of every ntermedate node N s equal to 1. Proof. In ths partcular type of networks, we observe that the rate allocaton operatons durng path extenson n the path dscovery phase becomes dentcal for both Algorthms 1 and 2. Snce the rest of the algorthms s totally dentcal, they wll provde the exact same soluton, whch s moreover optmal. & Algorthm 2 offers a fast way of computng avalable end-to-end network paths between S and C. Snce rate allocaton decsons are taken ndependently at each ntermedate node, the algorthm works for any type of loop-free DAGs. 5.2. Practcal mplementaton We dscuss here the practcal mplementaton of the proposed algorthms, and propose a few examples for deployment n real network scenaros. In large scale networks, montorng end-to-end paths between any two gven nodes becomes hghly complex and costly. Nor actve nether passve montorng solutons scale well n terms of executon tme, accuracy and complexty wth a growng number of ntermedate nodes and network segments [7]. Snce full knowledge about network status cannot be acheved n large scale networks, dstrbuted path computaton solutons are certanly advsable. They addtonally permt to release the computatonal burden of a sngle node/server, and dstrbute t among several ntermedate nodes [16]. Networkng protocols have been proposed to organze large scale random network graphs nto DAGs [34], or sets of multple end-to-end paths [32] and even to ensure specal network propertes lke path dsjontness and survvablty [14]. We address the decentralzed path computaton and rate allocaton problem, from the perspectve of a meda streamng applcaton. The forwardng decsons are taken n order to maxmze the qualty of servce of such specfc applcatons, n partcular to mnmze the loss probablty and aggregate enough transmsson bandwdth. Our algorthms run at the applcaton layer and present a low complexty n terms of message passng and executon tme. In varable network scenaros, where the lnk parameters change slowly over tme, our algorthms can be run perodcally n order to adapt the rate allocaton to a dynamc network topology. Observe that the fastest network parameter estmaton algorthms offer good results on tmescales of a few seconds [23], whle the executon of our path computaton algorthms takes one, or a few RTTs. Hence, runnng our algorthm perodcally, on tmescales equal to the network estmaton ntervals ensures effcent routng and rate allocaton wth the latest estmaton about the network state. Fnally, the control overhead can be lmted to two packets on each lnk of the network, for each teraton of the dstrbuted algorthms. For most typcal scenaros, the overhead stays very low compared to the streamng rate. It typcally depends on the perodcty chosen for the computaton of the dstrbuted rate allocaton. We fnally dentfy a few typcal scenaros where effectve rate allocaton between multple stream paths can brng nterestng benefts n terms of meda qualty. The lst s certanly not exhaustve, and t rather descrbes a few practcal stuatons where the applcaton of the algorthms proposed above s straghtforward. Overlay network scenaros (e.g., structured peer-topeer networks or Content Dstrbuton Networks). The meda nformaton from a server s forwarded towards
762 D. Jurca, P. Frossard / Sgnal Processng: Image Communcaton 23 (2008) 754 768 the clent by multple edge servers or proxy, whch belong to the same overlay network. The clent consumes the aggregated meda stream from multple transmsson flows employed by the applcaton. Wreless network scenaros (e.g., WF networks). A wreless clent can aggregate the meda nformaton transmtted on multple wreless channels. Interference among transmsson channels can be mnmzed by choosng non-overlappng wreless channels (e.g., there are eght non-overlappng channels accordng to the IEEE 802.11a standard specfcatons), and by optmzng the transmsson schedule n the wreless network [31]. Hybrd network scenaros (e.g., UMTS/GPRS/WF networks). A moble clent can smultaneously beneft from multple wreless servces n order to retreve the meda nformaton from a server connected to the nternet backbone. Our framework for path selecton and rate allocaton can be appled to any of the above-mentoned scenaros n a straghtforward manner. For the case of shared network resources n many-to-many setups, smple modfcatons to our algorthms can yeld good resource allocatons among clents, gven an optmzaton metrc. Consder F as the resource sharng polcy mplemented at an ntermedate node. F s desgned accordng to the fnal optmzaton metrc of the overall system, e.g., maxmzng system qualty [4]. Farness and congeston control mechansms on the end-to-end dscovered paths can also be successfully appled [19], smple dstrbuted resource sharng and packet prortzaton schemes can be mplemented based on the dfferent mportance of the smultaneous sessons [2]. Based on F, each node can take an approprate decson on how to allocate ts resources, (namely the bandwdth of the outgong lnks) among the concurrent applcatons, based on predefned utlty functons for example. Whle the desgn of a truly far dstrbuton of resources between concurrent sessons s outsde the scope of ths paper, our generc framework permts to easly lmt the bandwdth offered to a sngle sesson. The mplementaton of ndependent congeston control solutons becomes, therefore, straghtforward. 6. Smulatons 6.1. Smulaton setup We analyze the performance of our path computaton algorthms n dfferent network scenaros, and we compare them to smple heurstc-based rate allocaton algorthms. Results are presented n terms of convergence tme, and vdeo qualty performance. We frst study the average behavor of the algorthms n random network graphs, and we eventually dscuss n detals a specfc scenaro, mplemented n ns2 [8] n the presence of cross traffc. In all smulatons, the test mage sequence s bult by concatenaton of the foreman sequence, n CIF format, n order to produce a 1500-frame vdeo stream, encoded n H.264 format at 30 frames per second (equvalent to 50 s of vdeo). The encoded btstream s packetzed nto a sequence of network packets, where each packet contans nformaton related to at most one vdeo frame. The sze of the packets s lmted by the sze of the maxmum transmsson unt (MTU) on the underlyng network. The packets are sent through the network on the chosen paths, n a FIFO order, followng a smple schedulng algorthm [11]. The vdeo decoder fnally mplements a smple frame repetton error concealment strategy n case of packet loss. A vdeo packet s correctly decoded at the clent, unless t s lost durng transmsson due to the errors on the network lnks, or unless t arrves at the clent past ts decodng deadlne. We consder typcal vdeo-on-demand (VoD) streamng scenaros, where the admssble playback delay s large enough,.e., larger than the tme needed to transmt the bggest packet on the lowest bandwdth path. 6.2. Random network graphs We generate two types of network topologes: () typcal Wreless network graphs, wth low bandwdth and hgh error probablty for the network lnks and () Hybrd network scenaros, where the server s connected to the wred nfrastructure (hgh rate, low loss probablty), and the clent can access the nternet va multple wreless lnks, that have a reduced bandwdth, and an ncreased loss probablty. For both scenaros, we generate 500 random graphs. Any two nodes n the graph are drectly connected wth a probablty g. The parameters for each lnk are randomly chosen accordng to a normal dstrbuton, n the nterval [R mn, R max ] for the bandwdth, and, respectvely, [p mn, p max ] for the loss probablty. The parameters for representatve wred and wreless lnks are presented n Table 1. We lmt the sze of the generated graphs to 15 nodes. In ths case, the number of avalable paths between the server and the clent s already much larger than the number of paths actually chosen for the meda transmsson (6.89 and 6.73 avalable paths n average for the Hybrd and Wreless case, respectvely). The results below can be easly extended to larger networks, where the graph processng algorthms, however, rapdly become the computatonal bottleneck. Note that these algorthms are ndependent of path selecton and dstrbuted rate allocaton strateges studed n ths paper. Frst, we analyze the number of rounds n whch Algorthm 1 converges to the optmal rate allocaton gven by a centralzed algorthm, as proposed n Ref. [10]. The results for both network scenaros are presented n Fg. 3. We observe that the large majorty of the cases requre less than three teratons n order to reach the optmal rate allocaton. Ths shows that our algorthm performs very fast and needs only a very small number of control messages to converge to the optmal rate allocaton. Next, we propose to examne n Fg. 4 the convergence of Algorthm 1, computed n terms of vdeo dstorton, as compared to the qualty of the stream acheved wth the optmal rate allocaton. We observe that the dstorton due to Algorthm 1 rapdly decreases, and that the partal
D. Jurca, P. Frossard / Sgnal Processng: Image Communcaton 23 (2008) 754 768 763 Table 1 Parameters for random graph generaton Parameter Wred lnks Wreless lnks Connectvty probablty g 0.6 R mn 10 6 bps 10 5 bps R max 3 10 6 bps 7 10 5 bps p mn 10 4 10 3 p max 5 10 3 4 10 2 0.9 0.8 Number of rounds needed by Algorthm 1 Wreless Case Hybrd Case Average Devaton from Optmal Dstorton (MSE) 1.8 1.6 1.4 1.2 1 0.8 0.6 Dstorton mprovement per round Algorthm 1 Wreless Case Hybrd Case Fracton of Cases 0.7 0.6 0.5 0.3 0.1 0 1 2 3 4 Nr. of Rounds Fg. 3. Probablty densty functon of the number of rounds necessary for the teratve rate allocaton to converge to the optmal soluton of Algorthm 1. solutons are very close to the optmal one, even after the frst round of the teratve rate allocaton strategy. It clearly llustrates that the proposed dstrbuted algorthm converges very fast to the optmal soluton, and that the most crtcal paths n terms of vdeo qualty are already allocated by the very ntal rounds of the dstrbuted soluton. In both Fgs. 3 and 4, we can observe that Algorthm 1 performs better n the Hybrd network scenaro, than n the Wreless case. Ths s due to the fact that ths network scenaro has n average less bottleneck lnks. Please observe that n ths smulated scenaro, the bottleneck lnks are usually the wreless lnks, snce the rates of the wred lnks are much hgher. Therefore, Algorthm 1 s expected to converge faster to the optmal soluton n the Hybrd scenaro, where path are less lkely to share bottleneck lnks. Ths s n accordance wth the propertes of ths algorthm presented n the prevous secton. Then, we analyze the performance of the proposed algorthm, n terms of vdeo qualty obtaned wth the rate allocaton soluton. We compare the results obtaned wth Algorthm 1, to the ones obtaned by a smpler dstrbuted heurstc that forwards the ncomng network flow at each ntermedate node on the best outgong lnk n terms of loss probablty (e.g., sngle best-path streamng). We compute the dstrbuton of the penalty n qualty suffered by the heurstc scenaro, for 500 dfferent network graphs. Results are represented n Fg. 5. They llustrate 0 1 2 3 4 Nr. of Rounds Fg. 4. Convergence of Algorthm 1, measured n terms of vdeo dstorton (MSE), as compared to the optmal soluton. Fracton of Cases 1 0.9 0.8 0.7 0.6 0.5 0.3 0.1 Algorthm 1 compared to Heurstc Algorthm Wreless Case Hybrd Case 0 0 50 100 150 200 Mnmum Qualty Improvement (%) Fg. 5. Improvement n qualty offered by Algorthm 1 vs. a heurstc rate allocaton algorthm. the probablty for the mprovement n qualty to be above a predefned threshold x. We observe that, for both network scenaros, our algorthm obtans sgnfcantly better results n more than 80% of the cases. Ths motvates the extra control overhead ntroduced by Algorthm 1, whch s needed to reach the optmal rate allocaton. A smlar behavor s shown n Fg. 6, where we observe that Algorthm 2 also performs much better than the sngle best path strategy, n a large fracton of the cases consdered, and for both network scenaros. Algorthms 1 and 2 are compared n Fgs. 7 and 8. Fg. 7 represents the dfference n qualty ncurred by Algorthm 2, wth respect to the optmal allocaton offered by Algorthm 1. A smlar representaton s proposed n Fg. 8, where the qualty provded by Algorthm 1 s computed based on the rate allocaton obtaned after the frst round of the teratve algorthm, as opposed to the optmal allocaton that s used n Fg. 7. From both fgures,
764 D. Jurca, P. Frossard / Sgnal Processng: Image Communcaton 23 (2008) 754 768 Fracton of Cases 1 0.9 0.8 0.7 0.6 0.5 0.3 0.1 Algorthm 2 compared to Heurstc Algorthm 0 20 40 60 80 100 Mnmum Qualty Improvement (%) Wreless Case Hybrd Case Fg. 6. Improvement n qualty offered by Algorthm 2 vs. a heurstc rate allocaton algorthm. Fracton of Cases 0.9 0.8 0.7 0.6 0.5 0.3 0.1 Algorthm 2 compared to Algorthm 1 Wreless Case Hybrd Case 0 0 20 40 60 80 100 Mnmum Qualty Dfference (%) we see that, for the Wreless scenaro, the performance of the greedy scheme s equal to the optmal soluton n more than 50% of the cases. Algorthm 2 s even better, when compared to the executon of the optmal algorthm after the frst round (80% of the cases provdng smlar results). Ths s due to the very small number of paths chosen for transmsson, and to the fact that lnk parameters n the Wreless scenaro are qute homogeneous. In the pathologcal case where all network lnks would have the same parameters, the performance of the two algorthms would be dentcal. However, n the Hybrd network scenaro, we observe that the greedy algorthm offers bad results n a sgnfcant number of cases, snce qualty attans only 50% of the optmal soluton n almost 20% of the cases. Ths s manly due to the heterogenety of the network lnks parameters n hybrd scenaros. Fnally, we compare Algorthms 1 and 2 n terms of number of paths chosen for the streamng applcaton. The results for the Wreless and Hybrd network scenaros are presented n Fgs. 9 and 10, respectvely. We observe that n general Algorthm 2 uses a smaller number of flows for transmsson. Ths can be explaned by the greedy allocaton of paths, when Rule 2 s used for path extenson. Smlar results can be observed when the average streamng rate s computed for the solutons provded by both algorthms, for each type of networks. Table 2 shows that Algorthm 2 generally results n a smaller transmsson rate. However, the performance n terms of receved vdeo qualty s very close to the optmal one, snce the paths wth the lowest loss probablty are prortzed n both algorthms. In addton, the partcular network setup used n the smulaton allows for average streamng rates that already offer a good encodng qualty, where the rate-dstorton gradent s not very large. Overall, the prevous results show that Algorthm 1 represents a fast path computaton soluton n most types of networks that present a low number of bottleneck lnks. On the other sde, Algorthm 2 offers a vable, lower complexty alternatve for very large network scenaros Fg. 7. Improvement n qualty for Algorthm 1 vs. Algorthm 2. 1 0.9 Algorthm 2 compared to Algorthm 1 frst round Wreless Case Hybrd Case 0.35 Number of flows Wreless Case Algorthm 1 Algorthm 2 0.8 0.3 Fracton of Cases 0.7 0.6 0.5 0.3 0.1 Fracton of Cases 5 0.15 0.1 0.05 0 0 20 40 60 80 100 0 1 2 3 4 5 or more Mnmum Qualty Dfference (%) Number of Flows Fg. 8. Improvement n qualty for Algorthm 1 lmted to one teraton only, vs. Algorthm 2. Fg. 9. Probablty densty functon of the number of paths used by Algorthms 1 and 2 n the Wreless Network Case.