A Probabilistic Predictive Multicast Algorithm in Ad Hoc Networks (PPMA)

Transcription

1 A Probablstc Predctve Multcast Algorthm n Ad Hoc Networks (PPMA) Daro Pompl Marco Vttucc Broadband and Wreless Networkng Laboratory School of Electrcal and Computer Engneerng Georga Insttute of Technology, Atlanta, GA 333 daro@ece.gatech.edu Dpartmento d Informatca e Sstemstca Unversty La Sapenza of Rome, Italy, 84 ccuttvocram57@tscal.t Abstract Ad hoc networks are collectons of moble nodes communcatng usng wreless meda, wthout any fxed nfrastructure. Exstng multcast protocols fall short n a harsh ad hoc moble envronment because node moblty causes conventonal multcast trees to rapdly become outdated. The amount of bandwdth resources requred for buldng up a multcast tree s commonly less than that requred for other delvery structures, snce a tree avods unnecessary duplcaton of data. However, a tree structure s more subject to dsrupton due to lnk/node falure and node moblty than more meshed structures. Ths paper explores these contrastng ssues, and proposes PPMA, a new probablstc predctve multcast algorthm n ad hoc networks, whch leverages the tree delvery structure for multcastng, solvng ts drawbacks n terms of lack of robustness and relablty n hghly moble envronment. PPMA overcomes the exstng tradeoff between the bandwdth effcency to set up a multcast tree, and the tree robustness to node energy consumpton and moblty, by decouplng tree effcency from moblty robustness. By explotng the non-determnstc nature of ad hoc networks, the proposed algorthm takes nto account the estmated network state evoluton n terms of node resdual energy, lnk avalablty, dsplacement of set up multcast trees, and node moblty forecast, n order to mze the multcast tree lfetme. The algorthm statstcally tracks the relatve movements among nodes to capture the dynamcs n the ad hoc network. Ths way, PPMA estmates the nodes future relatve postons, n order to calculate a long lastng multcast tree. To do so, t explots those more stable lnks n the network, whle mnmzng the total network energy consumpton. We propose PPMA n both ts centralzed and dstrbuted verson, provdng performance evaluaton through extensve smulaton campagn run on an ad hoc C++ based smulator. I. INTRODUCTION Ad hoc networks are collectons of moble nodes communcatng usng wreless meda, wthout any fxed nfrastructure. Conventonal multcast routngs are nadequate n these scenaros, as moblty can cause rapd and frequent changes n the network topology. Exstng multcast protocols fall short because node moblty causes conventonal multcast trees to rapdly become outdated. Frequent state changes requre constant updates, reducng the already lmted bandwdth avalable for data, and possbly never convergng to accurately portray the current topology. Moblty represents the most challengng ssue for multcast routng protocols to address. In fact, f an algorthm shows robustness to moblty, often ncurs n other shortcomngs, as protocol overhead or loop formaton. Conversely, f a protocol s prmarly desgned to lmt or to optmze the network wth respect to sgnalng overhead and power consumpton, commonly ts performance degrades wth the ncrease of moblty. Multcast communcatons can be classfed nto source specfc and group shared. In source specfc multcast communcaton, only one node n the multcast group sends data whle all the other member nodes receve data. In group shared multcast communcaton, each node n the multcast group wants to send/receve data to/from member nodes. A tree that spans all member nodes s called multcast tree. Multcast trees can be classfed nto source rooted and shared trees, based on the communcaton strategy. A source rooted tree has the source node as root and s optmzed for source specfc multcast communcatons. A shared tree, on the other hand, s optmzed for group shared communcatons, and connects each group member wth all the other group 8

2 members. Tree based multcast s a very well establshed concept n wred networks, both for source specfc and group shared applcaton support []. In the tree based approach, multcast routng uses a source based or group shared tree among sources and recevers, dependng on the applcaton requrements. Ths approach s characterzed by hgh bandwdth effcency, snce only one path exsts between any par of nodes. The amount of bandwdth resources requred for buldng up a tree s commonly less than that requred for other multcast delvery structures, snce a multcast tree avods unnecessary duplcaton of data []. Ths way, the optmzaton routng problem n tree-based multcast s to fnd the mnmum-weght tree that spans all the nodes n the multcast group [3][4][5]. However, a multcast tree s more subject to dsrupton due to lnk/node falure and node moblty than more branched structures. Ths paper explores these contrastng ssues, and proposes PPMA, a new Probablstc Predctve Multcast Algorthm n ad hoc networks, whch leverages the tree delvery structure for multcastng, solvng ts drawbacks n terms of lack of robustness and relablty n hghly moble envronment. There s, n fact, a tradeoff between the bandwdth effcency to set up a multcast tree, and the tree robustness to energy node consumpton and moblty. The prmary objectve of the proposed algorthm s to address ths tradeoff, by decouplng tree effcency from moblty robustness. The ntuton ths paper s based on s that the determnstc nature whch characterzes tradtonal multcast protocols tends to become ther lmtng factor when amng at robustness and scalablty, partcularly n hghly dynamc ad hoc networks. By explotng the non-determnstc nature of ad hoc networks, PPMA takes nto account the estmated network state evoluton n terms of node resdual energy, lnk avalablty, dsplacement of set up multcast trees, and node moblty forecast, n order to mze the multcast tree lfetme. The algorthm statstcally tracks the relatve movements among nodes to capture the dynamcs n the ad hoc network. PPMA estmates the nodes future relatve postons, n order to calculate a long lastng multcast tree. To do so, t explots those more stable lnks n the network, whle mnmzng the total network energy consumpton. We propose PPMA n both ts centralzed and dstrbuted verson, provdng performance evaluaton through extensve smulaton campagn run on an ad hoc C++ based smulator. The remander of the paper s organzed as follows. In Secton II we revew the man wreless ad hoc network multcast routng protocols whch the present work s related to. In Secton III we descrbe the motvatons and goals of ths paper, ntroducng our novel probablstc cost functon. In Secton IV we explore the terms our cost functon s based on, and pont out ther need to reach the descrbed goals. In Secton V we present PPMA, a Probablstc Predctve Multcast Algorthm n ad hoc networks, n both ts centralzed and dstrbuted verson. In Secton VI we show numercal results through extensve smulatons run on an ad-hoc C++ based smulator. Fnally, n Secton VII we conclude the paper. II. RELATED WORK There have been several multcast routng protocols proposed for wreless ad hoc networks n lterature. In the followng we present those whch the present work s related to, focusng on some tree based algorthms whch face the problem of determnng a robust and relable multcast tree n moble ad hoc networks. We wll pont out whch are ther strengths and weaknesses. In partcular n II-A, II-B, and II-C we present pros and cons of PAST-DM (Progressvely Adapted Sub-Tree n Dynamc Mesh) [6], ITAMAR (Independent-tree ad hoc multcast routng) [7], and AODV (Ad Hoc On-Demand Dstance Vector Protocol) [8], respectvely. In Secton III we descrbe how our proposed probablstc multcast algorthm effectvely addresses most of ther drawbacks. A. PAST-DM: Progressvely Adapted Sub-Tree n Dynamc Mesh PAST-DM [6] tres to optmze multcast trees n terms of ther total lnk cost and data delvery delay. It utlzes a vrtual mesh topology that gradually adapts to the changes n underlyng network topology n a fully dstrbuted manner, wth mnmum control cost. The multcast tree for packet delvery s progressvely adjusted accordng to the latest local topology nformaton. A multcast sesson begns wth the constructon of a vrtual mesh connectng all group members. Each member node starts a neghbor dscovery process. PAST- DM [6] makes each member node mantan the topology map of the vrtual mesh, represented as a lnk state table. The lnk state of a node wll eventually be carred to the faraway nodes after several exchanges. Through the lnk state tables, each node has a local vew of the whole vrtual topology. Thus, each source constructs ts own data delvery tree based on ts local lnk state table, wth no extra overhead of control message. Ths s the key dfference between PAST-DM method and other source-based tree protocols. The topology 9

3 nformaton s more up-to-date and accurate close to the source and progressvely less accurate as hop dstance ncreases. Thus, between two vrtual lnks wth the same cost durng the tree constructon, the one that s closer to the source node s favored. To address ths property, PAST-DM explots a Source-Based Stener tree algorthm [3], that bulds up a spannng tree by usng an adapted cost,.e. a lnk cost weghted by the dstance to the source. Usng adapted cost s a good way to optmze some parameter such as power consume or delvery delay. A lnk adjacent to the source has dstance and hence adapted cost equal to. The source makes all ts neghbors as ts chldren n the multcast tree and dvdes the remanng nodes nto subgroups. Each subgroup forms a subtree rooted at one of the frst-level chldren. The source node does not need to compute the whole multcast tree. It puts each subgroup nto a packet header, combnes the header wth a copy of the data packet, and uncasts the packet to the correspondng chld. Each chld s then responsble of further delverng the data packet to all nodes n ts subgroup and t does so by repeatng the Source-Based Stener tree algorthm [3][6]. Man dsadvantages: Vrtual mesh as well as any herarchcal structure takes the advantage of scalng very well, snce vrtual topology can hde the real one regardless of network dmenson. However, PAST-DM does not take nto account any predcton on nodes moblty n the adapted cost, and thus t could be useless weghtng the lnk cost by a dstance that has rapdly changng. B. ITAMAR: Independent-Tree Ad hoc MultcAst Routng ITAMAR [7] fnds a set of pre-calculated alternate trees to promptly react to lnk breaks. Ths way delay could be reduced whenever a vable backup tree s avalable at the tme of falure of the current tree. Backup trees have the advantage of makng the tree based scheme more robust to node moblty. Specfcally, velocty value, whch leads to nconvenent route dscovery and mantenance overhead, s much hgher, when compared to other tree based schemes. Cost of the multcast tree s optmzed along wth mnmzng the mutual correlaton of falure tmes of each par of trees under the constrants of partal knowledge of the network. The basc dea s that backup multcast trees wth mnmal overlap could be used, one after another, to reduce the number of servce nterruptons. Ths would also mprove the mean tme between route dscovery cycles for a gven nterrupton rate and hence reduce the control overhead and the rate of data loss. At the same tme, ITAMAR ams to keep the cost of transmsson low. Ths method s effectve only f the falure tmes of the trees are ndependent of one another [7]. Thus, under constrant of nodes movng ndependently of one another, trees must have no common nodes and hence no common edges. Snce totally ndependent trees could not be found n many cases, ITAMAR concentrates on mnmzng the dependence between the falure tmes,.e. the correlaton of the falure tmes of the two trees. Man dsadvantages: The man dsadvantage f ITA- MAR [7] s that t s not convenent n all stuatons to choose ndependent trees. In fact, f only some lnks fal, a tree wth a hgh cost can be chosen. Also, preventng a lnk-falure by pre-calculatng a tree as much ndependent of the prevous tree as possble, could result n a greater control messagng overhead (requred for route establshment) than a method capable of reparng only faled lnks, leavng dentcal all other tree nodes. Another dsadvantage of ITAMAR mplementaton s that Djkstra algorthm does not allow to compute the trees n a dstrbuted manner. Consequently, the protocol does not scale well snce t requres every nodes to have a global knowledge of the network topology. C. AODV: Ad hoc On Demand Dstance Vector protocol AODV [8] routng protocol s capable of uncast, broadcast, and multcast communcaton. One of the man advantage of combnng uncast and multcast communcaton ablty n the same protocol s that route nformaton obtaned when searchng for a multcast route can also ncrease uncast routng knowledge, and vce versa. Uncast and multcast routes are dscovered on demand as n DSR [9], along wth hop-based routng as n DSDV [], usng a broadcast route dscovery mechansm. In order to reduce communcaton overheads, updates are propagated only along actve routes,.e. routes that have montored some track n the recent past. Broadcast data delvery s provded by AODV [8] by usng the Source IP Address and Identfcaton felds of the IP header as a unque dentfer of the packet. As nodes jon the multcast group, a multcast tree composed of group members and nodes connectng the group members s created. A multcast group leader mantans the multcast group sequence number. Multcast group members must also agree to be routers n the multcast tree. Man dsadvantages: There are some dsadvantages n AODV protocol concernng latency, utlzaton ef-

4 fcency, low moblty robustness, and poor scalablty property under partcular condtons. As far as concerns latency, snce n AODV routes are not always the shortest ones, data delvery latency s expected to be worse than a shortest path algorthm. In terms of resource utlzaton effcency, source routng utlzes a lot of bandwdth due to the use of lsts of addresses whch ncreases the sze of the header of data packets. Moreover, snce AODV keeps hard-state n ts routng table, the protocol has to actvely track and react to changes n the tree. AODV uses perodc beaconng to keep routng tables updated, thus addng a sgnfcant overhead to the protocol. Moreover, snce each node mantans a routng table entry for each multcast group for whch the node s a member or a router, t suffers hgh-rate moblty due to transmsson of many routng packets. Another dsadvantage of AODV s that snce nodes make use of ther routng caches to reply to route queres, a storm of reples and repettve updates n hosts caches may occur, leadng to poor scalablty performance. A. Motvatons and Goals III. PROBLEM SETUP The man motvatons whch convnced us to propose PPMA, a new probablstc predctve algorthm for multcastng n ad hoc networks, are summarzed hereafter. The determnstc nature whch characterzes tradtonal multcast protocols tends to become ther lmtng factor when amng at robustness and scalablty, partcularly n hghly dynamc ad hoc networks. The amount of bandwdth resources requred for buldng up a tree s less than that requred for other more meshed multcast delvery structures. The tradeoff between the bandwdth effcency to set up a multcast tree, and the tree robustness to energy node consumpton and moblty has never been evaluated through extensve smulaton campagn. By explotng the non-determnstc nature of ad hoc networks, PPMA takes nto account the estmated network state evoluton n terms of node resdual energy, lnk avalablty, dsplacement of set up multcast trees, and node moblty forecast, to mze the multcast tree lfetme. The algorthm statstcally tracks the relatve movements among nodes to capture the dynamcs n the ad hoc network. Ths way, PPMA estmates the nodes future relatve postons, and computes a long lastng multcast tree. To do so, t explots those more stable lnks n the network, whle mnmzng the total network energy consumpton. To acheve these goals, we ndvduate a set of general rules that ams to acheve such objectves: ) The more battery charge a node avals, the more ts avalablty to take part n the tree should be; ) The hgher the number of multcast trees a node belongs to, the less the node avalablty should be; 3) If the avalable battery charge goes under a predetermned threshold E mn, then a node should no more be consdered avalable to take part to multcast communcatons; 4) The larger s the dstance between two nodes, the smaller ther avalablty to establsh a communcaton should be. Obvously, f the dstance s larger than a lmt range D, no lnk between the nodes should be consdered; 5) The more prone a lnk s to fal or break, the smaller ts probablty of beng ncluded n a branch of a multcast tree should be. Such a property clearly depends on postons, speeds, and drectons of nodes. B. Transmsson Energy Model An accurate model for node energy consumpton per bt at the physcal layer s n []: E = E trans elec + βd γ + E rec elec () where: Eelec trans s the energy utlzed by transmtter electroncs (PLLs, VCOs, bas currents, etc) and dgtal processng. Ths energy s ndependent of dstance. Eelec rec s the energy utlzed by recever electroncs, and βd γ accounts for the radated power necessary to transmt over a dstance d between source and destnaton (β and γ are parameters whch depend on the envronment, and ther values are reported n Table I). As n [], we assume E trans elec = E rec elec = E elec () Thus, the overall expresson for E n eq. smplfes to C. Probablstc Lnk Cost E = E elec + βd γ (3) In order to synthesze the propertes presented n Subsecton III-A, we dentfy a probablstc lnk cost functon, composed of four multplcatve terms, two Energy Terms, a Dstance Term, and a Lfetme Term, whch wll be extensvely explaned n Subsectons IV-A, IV-B, and IV-C, respectvely. For two generc nodes and

5 j, we consder as ther lnk cost C j the followng: where: P E C j = P E = P E (E, E mn P D j, E P L j (4), W out, W ) (5) s the Energy Term for node and t weghts how much resdual energy node avals for communcatons, P D j = Pj D (d j, D [r req, ɛ mod b, ηb mod ], γ) (6) s the Dstance Term between node and node j, and t takes nto consderaton the transmsson power needed by node to communcate to node j, and P L j = Pj L (d j, D [r req, ɛ mod b, ηb mod ], σ, t) (7) s the Lfetme Term between node and node j, and t helps statstcally evaluate the probablty that the dstance between these two nodes remans bounded by the mum transmsson range of node, D, gven the current dstance. The explanaton of the parameters and varables each term depends on are reported hereafter. E [J] s the battery state of node (resdual charge); E mn [J] s a threshold under whch the node s no more avalable, and E [J] s the node mum charge; mn and operators are extended to all nodes n the network; W out [W ] s the nstantaneous power spent for the multcast communcatons by node, and W [W ] s the mum nstantaneous power node can consume for communcatons; d j [m] s the dstance between the two nodes and j at tme t [s], and D [m] s the mum rado range node can reach at ts requested btrate r req [bt/s], gven the energy-per-bt ɛ mod b [J/bt] used n rado transmsson, and the spectral effcency ηb mod [bt/s/hz] of the adopted modulaton system; β and γ are parameters whch depend on the envronment; t [s] s the tme nterval used for moblty predcton, and σ [m] s the standard devaton of the gaussan process used for nter-node dstance predcton. All terms are normalzed and range n [, ], so that they can be vewed as pseudo-probablty terms. Thus, C j n eq. 4 can be stll vewed as a pseudo-probablty. Actually, only the Lfetme Term Pj L s a correctly defned probablty, whle the other terms are normalzed so as to mantan values of C j n the same varaton range. Ths way, because of the statstcal meanng of the lnk cost C j, we can assocate a probablty meanng to the multcast trees computed by the proposed probablstc predctve algorthm. The tree probablty wll be the product of all the costs of those lnks ncluded n the tree. Ths tree probablty s obvously expected to decrease over tme, f nodes are not statonary and f ther future movements are unknown. The meanng of the tree probablty can be nterpreted as the probablty that all the lnks n a multcast tree wll survve at least for a tme perod t, crtcal parameter n eq. 7. In Secton IV we descrbe n detal all the terms of the lnk cost functon, why each of them s necessary, and ther synergc effect to meet the goals descrbed n Subsecton III-A. IV. PROBABILISTIC LINK COST FUNCTION TERMS A. Energy Term The man purpose of the Energy Term s to keep all the nodes of the network alve as long as possble, respectng the general rules ), ), 3), presented n Secton III-A. Also, we want to nterpret the energy term assocated to node as the probablty of choosng node durng the multcast tree constructon. So, the energy term vares n the range [, ] and s a functon of the battery state of the node (resdual charge). Specfcally, t assumes greater values for those nodes that have a greater resdual charge. Hence, a frst attemp for the energy term assocated to node could be: ( P (E ) = E mn {E mn } } mn {E mn {E u - (E mn {E mn }) where the unt step u - s defned as: u - (E mn {E mn }) = } ) { E > mn {E mn } elsewhere Roughly, P (E ) can be vewed as the percentage charge of node, wth respect to the mum possble charge of nodes ( {E }). The reason for the presence of mn and operators n eq. 8 s that we want to assocate to the energy term of a node ts physcal resdual charge. In fact, f we desgned P (E ) as the percentage charge of the mum charge of node, we would mss the objectve of mzng the lfetme of all nodes n the network. Ths can be better understood through an example. Fg. shows P (E ) and P j (E j ) assocated to nodes and j, respectvely, accordng to the followng defnton, where no mn and operators are used: ( Ek E mn ) k P k (E k ) = Ek Ek mn u - (E k Ek mn ) () (8) (9)

6 probablty F (E ) F j (E j ) battery state E * Fg. EXAMPLE OF THE wrong ENERGY TERM EXPRESSION IN EQ. where we chose E =3 [J], Ej =5 [J], and E mn = Ej mn = [J], for sake of clearness. We note that eq. concdes wth the percentage charge of the total charge of the node k. Also, we note that node j has a larger total charge Ej and that, consequentally, P j (E j ) has a smaller slope. Now, we see that probabltes of choosng node and j that have equal charge E * are dfferent, so that the charges of the two nodes are not equally weghted. In partcular, node wll be more lkely to jon multcast trees, although equpped wth a smaller energy supply. Ths smple example prove the real need for the mn and operators n eq. 8. Let us pont out a drawback n eq. 8: f a new node h, equpped wth an energy supply Eh larger than any other node n the network, joned the network, we should redefne eq. 8 by replacng {E } wth Eh. Such an obstacle can be avoded ether by knowng a pror {E }, or by choosng a {E } large enough to hold for most of the nodes n the network. If the node battery charge becomes lower than ts E mn, such a node cannot jon anymore multcast trees; consequently, the term must be equal to zero. Otherwse, f a node s not currently nvolved n any multcast communcaton, ts avalablty lnearly depends on charge E. Thus, all other parameters beng equal, we choose the most charged nodes among the possble ones. Conversely, f a node s nvolved n some multcast communcatons, ts avalablty should be less than the prevous case. Moreover, the more multcast paths a node shares, the less ts avalablty should be. Ths behavor can be ensured by an exponent added to eq. 8. Therefor, E E j the fnal Energy Term has the followng formula: ( P E = E mn {E mn } mn {E mn {E W ) } W +δ out } u - ( E mn {E mn } ) () where: δ s a postve constant near to whose objectve s to avod a possble dvson by zero. The exponent n eq. s an a-dmensonal term, and t vares from to, whle W out ncreases from to W. Thus, as W out grows to W, the energy term should deally tend to a two slope broken lne. Precsely, t should have slope equal to zero n E [E mn, E ), and slope equal to n E. In ths lmt case the term s always equal to zero but n E B. Dstance Term. Ths term syntheszes the general rule 4) n Secton III-A. It takes nto account how much power wll be spent by nodes and j to mantan the multcast tree. It can be expressed as follows: P D j = ( D D d j ) γ u -(D d j ) () We hde n D the dependence of the cost functon on the modulaton type: D = D (r req, ɛ mod b, ηb mod ), V (3) Clearly, f d j > D, the communcaton lnk between node and j does not respect the QoS requrements n terms of mnmum avalable rate r req, and so Pj D must be. Pj D should decrease from to as d j ncreases from to D, at least wth a quadratc trend. In fact, receved power decreases over dstance wth an exponent equal to γ, accordng to our propagaton model n eq. 3. In free space γ s equal to, whle n a real ndoor or outdoor envronment, t can range n the nterval [3,4.5]. The unt step functon u n eq. s needed to ensure Pj D not to be negatve for dstance d j greater than D. C. Lfetme Term Ths term, whch syntheszes the general rule 5) n Secton III-A, s a predctve one and t s defned as a correct probablty functon. P L j = P rob{d j (t + t) D (r req, ɛ mod b, ηb mod )} (4) It represents the probablty that the dstance d j (t + t) between nodes and j, after that t seconds elapsed, s 3

7 less or equal than D. Dfferently from [3][4], we consder the followng probablty: P rob{d j (t + t) D d j (t) D } (5) where D s the mum dstance a transmtter can reach. It means that we assume that a tme t node j s n transmsson range of node (d j (t) D ). The determnaton of ths lfetme term s a challengng ssue, because t generally depends on several factors. It could be well determned only f the current postons and the future destnatons of nodes are exactly known. Furthermore, a closed form expresson of Pj L s hard to fnd when consderng all possble statstcal parameters, even under rough approxmatons of node moblty. Consequently, some assumptons about future movements of nodes are necessary n order to obtan a useful expresson of Pj L. An essental nformaton to be known s an esteem of the current dstance d j (t). It s worth pontng out that such a pece of data has to come from some knd of measurements, and so t s prone to errors. In most cases, that error s accurately descrbed by a Gaussan Probablty Densty Functon (PDF) N (d j (t), σ d ), centered at the measured value d j (t) and wth varance σ d, weghted by a unt step functon. So, t can be expressed as follows: P dj(t)(x) == π σd e x d j (t) σ d u - (x) (6) where the unt step functon u - (x) s needed because dstances are postve defned. Varance σ d can be set ether to a fxed value dependng on the measurement system, as n the case of nodes equpped wth GPS for locaton trackng, or to a value proportonal to d j (t), as n the case of nodes capable of measurng relatve dstances by message exchange. If the only avalable pece of data s d j (t), we propose to consder d j (t+ t) as a stochastc varable, dstrbuted wth a gaussan probablty densty functon. We suppose ts mean value to be d j (t),.e. the measured value of d j (t), and ts varance to be the sum of two dstnct terms. Before explorng these terms, let us note that such measurement s avalable to be processed after a tme nterval t d, that s not necessarly well-known. If we rely on a esteem of t d or we consder t reasonably constant, then we can ncorporate ths factor nto t. If not, we have to take a larger σ d. Thus, actually, we should deal wth a gaussan stochastc varable d j (t + t), whose mean value s another gaussan stochastc varable. Ths causes the varance of d j (t + t) to be the sum of the varance of d j (t) (.e. σd ) and of a term dependng on the node relatve speeds and the tme perod t (σv ). So, we have: P dj(t+ t)(x) = π σ e x d j (t) σ u - (x) (7) where σ = σd + σ v s derved n appendx. Intutvely we can say that σv depends on the relatve speed v j between node and node j. In fact, the greater v j, the hgher the possble values of d j (t + t). To derve an expresson for σv, we can start from the PDF of the node speed. We assume a zero mean gaussan PDF for each x- and y-component of the velocty vector. Ths s equvalent to assume a Brownan moton for each node around ts current poston. So, for the generc node we have: P VX V Y (v x, v y ) = e vx +v y σ (8) π σ A proftable way to estmate σv s consderng the PDF of the ampltude of the velocty vector (whch we wll refer to as speed),.e. P V (x) = x σ e x σ u - (x) (9) where V = VX + VY. Eq. 9 s a Raylegh PDF. Under these assumptons, t s possble to tune σ n order to properly ft the model. The x- and y-component of the relatve speed between two nodes are: { VXj = V X = V X V Xj () V Yj = V Y = V Y V Yj Because V X and V Y are dfference of gaussan stochastc varables, they stll are gaussan stochastc varables. Furthermore, by supposng statstcally ndependent the x- and y-components of the node speed, t results that V X and V Y are statstcally ndependent too. V X has mean equal to the dfference of the means of V X and V Xj, and varance equal to the sum of the varances of V X and V Xj. The same holds for V Y. So, P VX V Y (x, y) = π (σ + σj ) e x +y σ +σ j () Now, we need to relate the V j ampltude to σv. A reasonable assumpton s that σv s proportonal to the product of the expected ampltude value V j and to the tme nterval t. Thus, σ v t E{ V j } = k v t E{ V j } () 4

8 where E{ V j } s the expected value of V j, and k v s a constant. Because V X and V Y are zero mean gaussan varables, t results that V j has a Raylegh PDF too. So, takng σv j = σ + σj, t yelds: P Vj (x) = x σ V j e π E{ V j } = σ Vj = x σ V j u - (x) (3) π (σ + σ j ) (4) π σv = k v t (σ + σj ) = ki v t (σ + σj ) (5) On the other hand, f an esteem of V j s avalable, then σv t V j = kv II t V j V. CENTRALIZED AND DISTRIBUTED PPMA Algorthm represents the pseudo-code for PPMA, the proposed probablstc predctve multcast algorthm, n ts centralzed verson. It works lke the centralzed Bellman-Ford algorthm, wth the excepton of the choce of the father of a node n the computaton of the multcast tree. Centralzed Bellman-Ford fnds the shortest path from node x to the source s, wth respect to a certan metrc, for every possble number of hops and for every node x. In Algorthm we do not show the for loop assocated wth the number of hops, for sake of smplcty. For a gven number of hops, a node x wll have a set of potental fathers F(x). The Centralzed Belman- Ford Algorthm chooses the father f x by mnmzng the cost of the assocated path. Centralzed PPMA, nstead, gves hgher prorty to those potental fathers that have other chldren, n order to explot the pecularty of multcastng. Among these fathers, hgher prorty s gven to those that have a current rado transmttng range whch allows node x to receve packets from them. If any so defned fathers exst, node x wll choose the closest one. If all potental fathers have to ncrease ther rado range to reach node x, then x wll choose that one that has to ncrease less that range, for power effcent reasons. If all potental fathers have not any chldren, node x wll choose that father f x that mnmzes the LINK COST(x, f x ). Algorthm represents the pseudo-code for PPMA, the proposed probablstc predctve multcast algorthm, n ts dstrbuted verson. Dstrbuted PPMA uses two dfferent costs: a prvate cost (C prv ) and a publc cost (C pub ). The frst s needed to fnd a mnmum cost path toward the source, wthout optmzng the multcast tree. The second s used to enable a node to jon an exstent multcast tree, tryng to reduce the number of nodes that belong to the tree. If a node s a recever for a gven multcast group m M, t may have just joned the tree or not. If the recever has joned the tree, t goes on fndng better paths (P new ) than the current one (P current ). If a smaller cost path s found, the recever wll change path only f new cost < current cost. Ths condton ensures that path change s convenent, meanng that the new path should have a smaller cost than the current one, so that resources spent for changng path wll be repad by the resource savng nduced by the use of the new path. If the recever has not joned the tree, t fnds the best publc path. If t s found, the recever jons the tree. It must send to ts new father a JOIN ACK for establshng the lnk, t must publcze to ts neghbor the publc cost of the new path (C Pnew ) and, fnally, t must set the publc cost of the current path to the prvate one (C Pcurrent C Pcurrent prv ). If a node s not a recever, t pub fnds the most convenent path between publc path and prvate path, and stores the related fathers. Whenever some other node asks for the cost of path to the source, the node wll gve the lowest-cost path and the related father. Fnally, f a node s not a recever and does not receve any query for lnks, t must replace the publc cost wth the prvate one. pub VI. PERFORMANCE EVALUATION A. Network Moblty Model The network of nodes s represented as (V, D), where V = {v,.., v N } s a fnte set of nodes n a fntedmenson terran, wth N = V, and D s the matrx whose element (, j) contans the value of the dstance between nodes v and v j. As far as concerns the moblty models n ad hoc networks, the most commonly used models wll be brefly descrbed hereafter. ) Random WayPont Model (RWPM): n ths model, nodes n a large room choose some destnaton, and move there at a random speed unformly chosen n (, V ]. Once a node has reached ts destnaton pont, t pauses for a tme P unformly chosen n [, P ]. If the border effects are modeled consderng a wrap around envronment, n the steady state a unform dstrbuton of nodes n the whole regon s generated, whereas f a bounced back s consdered, a lower node densty n proxmty of the borders can be notced. ) Random Walk Model (RWM): ths model has been proposed by Ensten n 96 to mmc partcle s Brownan movement. A node starts movng by pckng up a random drecton unformly n [, π]. It contnues 5

9 TABLE I SIMULATION PARAMETERS Parameter E elec Value 5 [nj/bt] β [pj/bt m γ ] γ 3 [E mn, E ] [., ] [J] W.55 [mw ] T erran 3 3 [m ] Nodes 65 t mob P acket sze r req TABLE II.5 [s] 8 [bt] 3 [Kbps] NETWORK NODE MOBILITY Velocty and Acceleraton constrants Moblty * V [m/s] A mn [m/s ] A [m/s ] Low 5 3 Medum 5 3 Hgh * Velocty s unformly dstrbuted wth V mn = for each scenaro the movement for a certan tme, and then repeats the drecton selecton agan. If durng ths perod t hts the regon border, dfferent polces can be mplemented to take nto account the border effect. The node, n fact, can ether be bounced back, or contnue ts movement as f t were n a wrap around envronment, or be deleted and replaced n a random poston nsde the regon. The frst two solutons are n ths model equvalent, gvng both n the steady state a unform dstrbuton of nodes, whle the latter soluton concentrates nodes n the center of the regon. The choce of speeds s the same as that n the Random WayPont Model. 3) Random Drecton Model (RDM): ths model s smlar to the Random Walk Model, dfferng from that only as far as concerns the node behavor n the border of the regon. In ths model a node httng the border wll choose ts next drecton towards the nsde half plane. The angle formed by node movement drecton and the tangent lne of the border s chosen unformly n [, π]. 4) Extended-Random Walk Model (E-RWM): n ths paper we extended the Random Walk Model to nclude acceleratons and deceleratons, the man dfference beng that the speed of a node can change accordng to an nstantaneous acceleraton and deceleraton, as shown n the followng formula whch uses parameters n Table II. v(t + t mob ) = (V mn, mn(v(t) + a t mob, V )) { a [Amn, ], p a >.5 a [, A ], p a.5; { x(t + tmob ) = x(t) + v(t + t mob ) cos(φ) y(t + t mob ) = y(t) + v(t + t mob ) sn(φ) (6) (7) where p a and φ are stochastc varables unformly dstrbuted n (, ] and [, π], respectvely. Moreover, nodes belongng to the same multcast group are grouped nto a cluster. Clusters are modeled as a set of nodes deployed n a crcular area wth a gven radus. The source s n the center of ths area. Insde a cluster, all nodes move wth a speed coherent to the speed of the source. However, not all nodes n a cluster are recevers or senders. B. Smulaton Results In ths work, a random ad hoc network has been generated, and parameters reported n Table I have been used to run smulatons. The trees bult up by PPMA are compared to the trees bult up by the Stener algorthm [3], whch bulds multcast trees by mnmzng ther total cost. Each algorthm bulds a tree wth two dfferent lnk cost functons: the frst ncludes the Dstance Term, whereas the second ncludes all terms (the Dstance, the Energy, and the Lfetme) terms. For each smulaton several experments have been run to ensure 95% relatve confdence ntervals smaller that 5%. Startng from a completely unloaded randomly generated ad hoc network, source rooted multcast trees are bult accordng to the two competng algorthms. Multcast groups are sequentally randomly generated. Multcast group members (source and recevers) are randomly chosen among ad hoc network nodes. In partcular, we consdered multcast requests from Small Groups (5 recevers) and Large Groups ( recevers), n order to test the network under dfferent load condtons. In Fg. the average Tree Lfetme for PPMA and Stener algorthm n a medum and hgh moblty envronment for small multcast group sze s shown, whle 6

10 n Fg. 3 the same metrc for large multcast group sze s shown. The curves related to the dstance term show performance that gets worse wth a larger multcast group sze. Instead, curves related to all terms are less correlated to the multcast group sze. Moreover, the ncrease of moblty affects the dstance term based trees, shortenng ther lfetme, whle trees bult by also consderng the lfetme term mprove ther lfetme as network moblty grows. In Fg. 4 the number of Connected Recevers for PPMA and Stener algorthm n a medum and hgh moblty envronment for small multcast group sze s depcted, whle n Fg. 5 the same metrc for large multcast group sze s shown. The number of connected recevers ncreases at the ncrease of both group sze and moblty. Analogously, dfferences among dstance term based trees and all term based trees get more evdent at the growng of both group sze and moblty. In Fg. 6 the average Battery Charge of the nodes for PPMA and Stener algorthm n a medum and hgh moblty envronment for small multcast group sze s shown, whle n Fg. 7 the same metrc for small multcast group sze s shown. The energy term affects the slopes of the curves startng at about 5 seconds. In Fg. 8 the Tree Lfetme for PPMA and Stener algorthm for large multcast groups n a hgh moblty for three dfferent values of the sgnalng energy cost for a tree swtchng s shown. C swtch = % corresponds to the energy requred to the network to send one 8 bt packet along the multcast tree. Although the tree lfetme decreases at the ncrease of the consdered swtchng cost, PPMA shows a less senstve behavor, resultng n a greater robustness to the varyng of C swtch. In all these bunches of smulatons t s consstently shown that our predctve PPMA algorthm outperforms the Stener algorthm, both n the Small and Large Group smulatons. Moreover, PPMA manages to leverage better the avalable network resources than the competng algorthm, by explotng the moblty predctve features t s endowed wth. PPMA shows also good robustness propertes to moblty, as can be ponted out from Fg. -5. VII. CONCLUSIONS Ths paper proposed PPMA, a new probablstc predctve multcast algorthm n ad hoc networks, whch leverages the tree delvery structure for multcastng, overcomng ts lmtatons n terms of lack of robustness and relablty n hghly moble envronment. PPMA INIT : Algorthm CENTRALIZED PPMA : nodes set of the network V : multcast groups set M 3: m M : 4: source s; current node x; father of x f x 5: F(x)={f V f x f}{potental father set} 6: H(x)={h V f h x}{chldren set} 7: N (x)={n V LINK COST(n, x)< }{neghbor set} 8: K current (x)=k curr (x)={k V d(x, k) < D current 9: K (x)={k V d(x, k) < D x } x } CENTRALIZED PPMA: : for all m M do : for all x V do 3: for all V H(x) do 4: F Dcurrent x FIND SET(N (x) K curr () F(x)) 5: f F Dcurrent x then 6: F D x FIND SET(N (x) K () F(x)) 7: f F D x then 8: F new FIND SET(N (x) K ()) 9: f F new then : f x NULL : else {F new } : f x MIN j (LINK COST(x, j)), j F new 3: end f 4: else {F D x } 5: f x MIN z (d(x, z) Dz current 6: end f 7: else {F Dcurrent x } 8: f x MIN w (d(x, w)), w F Dcurrent x 9: end f : end for : end for : end for ), z F D x explots the non-determnstc nature of ad hoc networks, by takng nto account the estmated network state evoluton n terms of node resdual energy, lnk avalablty, dsplacement of set up multcast trees, and node moblty forecast, to mze the multcast tree lfetme. The algorthm statstcally tracks the relatve movements among nodes to capture the dynamcs n the ad hoc network. Ths way t estmates the nodes future relatve postons, n order to calculate a long lastng multcast tree. To do so, t explots those more stable 7

11 INIT : Algorthm DISTRIBUTED PPMA : nodes set of the network V : multcast groups set M 3: set of recevers R m, m M 4: multcast tree T m 5: source s m ; current node x 6: path P T m ;{path from x to s m } 7: publc cost of path P C P pub ; 8: prvate cost of path P C P prv 9: father of x va P f P x : H(x) = {h V P T, f P h DISTRIBUTED PPMA: : for all m M do : f (x s m ) then 3: f (x R m ) then 4: f (x T m ) then 5: P prv mn MIN P(Cprv P ) 6: P new FIND PATH(MIN(P prv x}{chldren set} mn, CPcurrent prv )) 7: f P new NULL then 8: COMPUTE ĊOSTS(P new, P current ) 9: f (new cost < current cost) then : SWITCH FATHER(fx Pcurrent : UPDATE COST(C Pcurrent, f Pnew x ) pub, P current ) : UPDATE COST(C Pnew pub, P new) 3: end f 4: end f 5: else {x T m } 6: P FIND PATH(MIN P (Cpub P )) 7: JOIN(f P x ) 8: UPDATE COST(C P pub, P ) 9: end f : else {x R m } : P prv mn MIN P(Cprv P ) : P + FIND PATH(MIN(P prv mn, MIN P(Cpub P ))) 3: f P + NULL then 4: STORE(P +, C P + prv, CP + pub ) 5: end f 6: end f 7: end f 8: end for {Daemon runnng x V} Average tree lfetme [s] Average tree lfetme [s] COMPUTE COSTS: : f f Pcurrent x : MAX y (LINK COST(x, y)), y H(x) 3: MAX z (LINK COST(f, z)), z H(f) {x} 4: mn MIN(, +C Pcurrent prv ) 5: 3 MAX w (LINK COST(f, w)), w H(f) 6: new cost mn + C Pnew prv 7: current cost C Pcurrent prv Stener tree: (Lfetme Term)+(Energy Term)+(Dstance Term) Stener tree: (Dstance Term) PPMA: (Lfetme Term)+(Energy Term)+(Dstance Term) PPMA: (Dstance Term) Smulaton tme [s] (a) Average tree lfetme [s] 6 4 Fg Stener tree: (Lfetme Term)+(Energy Term)+(Dstance Term) Stener tree: (Dstance Term) PPMA: (Lfetme Term)+(Energy Term)+(Dstance Term) PPMA: (Dstance Term) Smulaton tme [s] TREE LIFETIME FOR PPMA AND STEINER ALGORITHM FOR SMALL MULTICAST GROUPS IN A medum MOBILITY (FIG. (A)) (b) AND hgh MOBILITY ENVIRONMENT (FIG. (B)) Stener tree: (Lfetme Term)+(Energy Term)+(Dstance Term) Stener tree: (Dstance Term) PPMA: (Lfetme Term)+(Energy Term)+(Dstance Term) PPMA: (Dstance Term) Smulaton tme [s] (a) Average tree lfetme [s] 5 5 Fg. 3 5 Stener tree: (Lfetme Term)+(Energy Term)+(Dstance Term) Stener tree: (Dstance Term) PPMA: (Lfetme Term)+(Energy Term)+(Dstance Term) PPMA: (Dstance Term) Smulaton tme [s] TREE LIFETIME FOR PPMA AND STEINER ALGORITHM FOR large MULTICAST GROUPS IN A medum MOBILITY (FIG. 3(A)) AND hgh (b) MOBILITY ENVIRONMENT (FIG. 3(B)) 8

12 Number of connected recevers Stener tree: (Lfetme Term)+(Energy Term)+(Dstance Term) Stener tree: (Dstance Term) 3 PPMA: (Lfetme Term)+(Energy Term)+(Dstance Term) PPMA: (Dstance Term) Smulaton tme [s] Number of connected recevers Stener tree: (Lfetme Term)+(Energy Term)+(Dstance Term) Stener tree: (Dstance Term) 3 PPMA: (Lfetme Term)+(Energy Term)+(Dstance Term) PPMA: (Dstance Term) Smulaton tme [s] Average avalable node energy [J] Stener tree: (Lfetme Term)+(Energy Term)+(Dstance Term) Stener tree: (Dstance Term) PPMA: (Lfetme Term)+(Energy Term)+(Dstance Term) PPMA: (Dstance Term) Smulaton tme [s] Average avalable node energy [J] Stener tree: (Lfetme Term)+(Energy Term)+(Dstance Term) Stener tree: (Dstance Term) PPMA: (Lfetme Term)+(Energy Term)+(Dstance Term) PPMA: (Dstance Term) Smulaton tme [s] (a) (b) (a) (b) Fg. 4 NUMBER OF CONNECTED RECEIVERS FOR PPMA AND STEINER ALGORITHM FOR small MULTICAST GROUPS IN A medum MOBILITY (FIG. 4(A)) AND hgh MOBILITY ENVIRONMENT (FIG. 4(B)) Fg. 6 AVERAGE AVAILABLE NODE ENERGY FOR PPMA AND STEINER ALGORITHM FOR small MULTICAST GROUPS IN A medum MOBILITY (FIG. 6(A)) AND hgh MOBILITY ENVIRONMENT (FIG. 6(B)) Number of connected recevers Stener tree: (Lfetme Term)+(Energy Term)+(Dstance Term) Stener tree: (Dstance Term) PPMA: (Lfetme Term)+(Energy Term)+(Dstance Term) PPMA: (Dstance Term) Smulaton tme [s] Number of connected recevers Stener tree: (Lfetme Term)+(Energy Term)+(Dstance Term) Stener tree: (Dstance Term) PPMA: (Lfetme Term)+(Energy Term)+(Dstance Term) PPMA: (Dstance Term) Smulaton tme [s] Average avalable node energy [J] Stener tree: (Lfetme Term)+(Energy Term)+(Dstance Term) Stener tree: (Dstance Term) PPMA: (Lfetme Term)+(Energy Term)+(Dstance Term) PPMA: (Dstance Term) Smulaton tme [s] Average avalable node energy [J] Stener tree: (Lfetme Term)+(Energy Term)+(Dstance Term) Stener tree: (Dstance Term) PPMA: (Lfetme Term)+(Energy Term)+(Dstance Term) PPMA: (Dstance Term) Smulaton tme [s] (a) (b) (a) (b) Fg. 5 NUMBER OF CONNECTED RECEIVERS FOR PPMA AND STEINER ALGORITHM FOR large MULTICAST GROUPS IN A medum MOBILITY (FIG. 5(A)) AND hgh MOBILITY ENVIRONMENT (FIG. 5(B)) Fg. 7 AVERAGE AVAILABLE NODE ENERGY FOR PPMA AND STEINER ALGORITHM FOR large MULTICAST GROUPS IN A medum MOBILITY (FIG. 7(A)) AND hgh MOBILITY ENVIRONMENT (FIG. 7(B)) lnks n the network, whle mnmzng the total network energy consumpton. We proposed PPMA n both ts centralzed and dstrbuted verson, provdng performance evaluaton through extensve smulaton campagn run on an ad hoc C++ based smulator. REFERENCES [] L. H. Sahasrabuddhe and B. Mukherjee, Multcast Routng Algorthms and Protocols: A Tutoral, IEEE Network, Jan/Feb, pp. 9-. [] R. T. Wong, A dual ascent approach for Stener tree problems on a drected graph, Mathematcal Programmng, Vol. 8, pp. 7-87, 984. [3] R. K. Ahuja, T. L. Magnant, and J. B. Orln, Network Flows: Theory, Algorthms, and Applcatons, Prentce Hall, Feb [4] D. Pompl, L. Lopez, and C. Scoglo, DIMRO, a DffServ- Integrated Multcast algorthm for Internet Resource Optmzaton n source specfc multcast applcatons, IEEE ICC 4. [5] H. Takahash and A. Matsuyama, An approxmate soluton for the Stener problem n graphs, Math. Japonca Vol. 6, pp , 98. [6] C. Gu and P. Mohapatra, Effcent Overlay Multcast for Moble Ad Hoc Networks, Proc. of IEEE WCNC 3, Mar. 3. [7] Sajama and Z. J. Haas, Independent-tree ad hoc multcast routng (ITAMAR), Moble Networks and Applcatons, Vol. 8, No. 5, Oct. 3. [8] E. M. Royer and C. E. Perkns, Multcast Operaton of the 9

13 Average tree lfetme [s] Stener tree: C swtch =% Stener tree: C swtch =5% Stener tree: C swtch =% PPMA: C swtch =% PPMA: C swtch =5% PPMA: C swtch =% Smulaton tme [s] P dj(t+ t)(z) = π σ e z d j (t) σ Our objectve s the followng probablty: u - (Z) (9) P rob{d j (t + t) D } (3) From the Theorem of Total Probablty, t yelds: P rob{d j (t + t) D } = Fg. 8 TREE LIFETIME FOR PPMA AND STEINER ALGORITHM FOR large MULTICAST GROUPS IN A hgh MOBILITY, FOR THREE VALUES OF C swtch Ad Hoc On-Demand Dstance Vector Routng Protocol, ACM MobCom, Aug. 999, pp. 78. [9] D. B. Johnson and D. A. Maltz, Dynamc source routng n ad hoc wreless networks, n Moble Computng, Imelnsk and Korth, Eds. Kluwer Academc Publshers, 996, Vol [] C. E. Perkns and P. Bhagwat, Hghly Dynamc Destnaton Sequenced Dstance Vector Routng (DSDV) for Moble Computers, n Proceedngs of ACM Sgcomm Conference on Communcaton Archtectures, Protocols and Applcatons, pp , Aug [] W. B. Henzelman, A. P. Chandrakasan, and H. Balakrshnan, An Applcaton-Specfc Protocol Archtecture for Wreless Mcrosensor Networks, IEEE Transactons on Wreless Communcatons, Vol., No. 4, October. [] T. Meloda, D. Pompl, and Ian F. Akyldz, Optmal Local Topology Knowledge for Energy Effcent Geographcal Routng n Sensor Networks, Proc. of IEEE Infocom 4, Hong Kong SAR, PRC [3] A. B. McDonald and T. F. Znat, A Path Avalablty Model for Wreless Ad-Hoc Networks, IEEE Wrreless Communcatons and Networkng Conference (WCNC99), New Orleans, LA, Sep. -4, 999. [4] A. B. McDonald and T. F. Znat, Predctng Node Proxmty n Ad-Hoc Networks: A Least Overhead Adaptve Model for Selectng Stable Routes, IEEE APPENDIX We assume the followng PDF for d j (t) to be: P dj(t)(x) = π σd e x d j (t) σ d u - (x) (8) where d j (t) s the measured value. Also, we assume the condtonal probablty of d j (t + t) gven d j (t) to be: d j(t) P rob{d j (t + t) D d j (t)} P rob{d j (t)} Thus, n the contnuous case: D e z x σv π σv x d j (t) σ d dz e π σd (3) (3) We suppose σ d s small wth respect to d j (t), so that we can extend to the lmt of the external ntegral n eq. 33. Ths s a reasonable assumpton f the measurement system has good accuracy, and the measurement s updated frequently enough. By exchangng the order of ntegraton, we have: π σ v σ d we ntegrate: π σ v σ d D D ( and, fnally, we smplfy: π (σd + σ v ) e z x σv σ v σ d π σd + σ v D z d e x d j (t) z d e j (t) σ d v +σ σ d dx dx j (t) σ d v +σ ) dz (33) (34) dz (35) Thus, t has been proved that, under the gven assumptons, the varance of d j (t + t) s equal to σ d + σ v. dz 3