Generalized Cooperative Multicast in Mobile Ad Hoc Networks

Transcription

1 Generalized Cooperative Multicast in Mobile Ad Hoc Networks Bin Yang Yulong Shen Meber IEEE Xiaohong Jiang Senior Meber IEEE and Tarik Taleb Senior Meber IEEE Abstract Cooperative ulticast serves as an efficient counication paradig for supporting ulticast-intensive applications in obile ad hoc networks MANETs. Available studies on cooperative ulticast in MANETs ainly focus on either the full cooperation or the noncooperation which fail to capture the ore general cooperation behaviors aong destination nodes. To address this issue this paper proposes a general cooperative ulticast schee CMfgpτ with replication factor f ulticast fanout g cooperative probability p and packet lifetie τ. With this schee a packet fro source node will be replicated to at ost f distinct relay nodes which forward the packet to its g destination nodes and with probability p a destination node helps to forward the packet. Here the packet has the lifetie of τ tie slots. The schee is fleible and general and it covers the full cooperation p = and the noncooperation p =0 as special cases. A Markov chain theoretical odel is further developed to depict the packet delivery process under the new schee and help us to conduct analytical study on the corresponding epected packet delivery probability and packet delivery cost. Finally etensive siulation and nuerical results are provided for discussions. Inde Ters Mobile Ad Hoc Network cooperative ulticast two-hop relay packet delivery probability/cost. I. INTRODUCTION Multicast in obile ad hoc networks MANETs is iportant for supporting any critical applications with one-toany counications [] [5] like essage echanges aong a group of soldiers in battlefield earthquake alaring video conferencing etc. Cooperative ulticast serves as an efficient counication paradig destination nodes ay share their packets with each other. In practice however a destination node ay act selfishly and refuses to forward packets to other nodes due to its liit power resource. Thus for an efficient support of future ulticast-intensive applications in MANETs [6] [9] it is critical to take into account the cooperation behaviors of destination nodes in the design of ulticast schee. Available studies on cooperative ulticast in MANETs ainly focus on either the case when all destination nodes are willing to share packets with each other full cooperation or the case when none of these nodes is willing to share packets B. Yang is with the School of Coputer and Inforation Engineering Chuzhou University China and is also with the School of Coputer Science Shaani Noral University China. E-ail: ybcup@6.co. Y. Shen is with the School of Coputer Science and Technology Xidian University China. E-ail: ylshen@ail.idian.edu.cn. X. Jiang is with the School of Systes Inforation Science Future University Hakodate Japan. E-ail: jiang@fun.ac.jp. T. Taleb is with the Departent of Counications and Networking Aalto University Helsinki Finland. Eail: talebtarik@gail.co. with others noncooperation. The noncooperative ulticast schee in two-hop relay MANETs is studied in [0] [3] [0] [] consider the siple independent and identically distributed i.i.d. obility odel and [3] considers ore general speed-restricted obility odels. Later the full cooperative ulticast schee is eplored in two-hop relay MANETs under i.i.d. obility odel [4] [5]. Recently the perforance of full cooperative and nocooperative ulticast schees is also eained in cognitive radio MANETs [6] which allow unlicensed nodes to eploit the spectru allocated to licensed nodes in an opportunistic anner. It is notable that the above studies only represent the two etree cases of cooperative ulticast and could not depict the general cooperative behaviors aong destination nodes. Moreover these studies assue that all packets fro source node can be successfully delivered to destination nodes and only investigate the packet delivery perforance in ters of capacity and delay. In a real MANET however destination nodes ay ehibit ore general cooperative behaviors and the packet loss ay happy e.g. due to lifetie constraint. Also the theoretical odels in these studies could not be applied in a straightforward way for the perforance analysis of MANETs with the considerations of general node cooperative behaviors and packet loss issue. To address the above liitations this paper proposes a new and general cooperative ulticast schee and develops corresponding Markov chain theoretical odel for perforance analysis. The ain contributions of this paper are suarized as follows. First we propose a general cooperative ulticast schee CMfgpτ for two-hop relay MANETs source node can deliver a packet to at ost f distinct relay nodes which forward the packet to its g destination nodes and with probability p a destination node also helps to forward the packet before its lifetie τ epires. The schee is fleible and general in the sense that we can control transission behaviors by paraeter p and it covers the available full cooperation schee [4] [5] p = and noncooperation schee [0] [3] p =0as special cases. We then develop a two-diensional Markov chain theoretical odel to capture the cople packet delivery process under CMfgpτ. With the help of the theoretical odel we derive the analytical epressions for the epected packet delivery probability and delivery cost. Finally etensive siulation and theoretical results are provided to validate the theoretical analysis results and c 07 IEEE. Personal use is peritted but republication/redistribution requires IEEE perission. See for ore inforation.

2 Fig.. a Network odel. + Δ Network odel and transission scheduling. α α b Illustration of transission-group based scheduling with =5and α =5 to illustrate the network perforance under the new ulticast schee. The rest of this paper is organized as follows. Section II introduces the syste odels and the definitions of perforance etrics. In Section III we first propose a general cooperative ulticast schee and then develop a two-diensional Markov chain theoretical odel to depict packet delivery process under such a schee. In Section IV analytical epressions are derived for the epected packet delivery probability/cost. Section V provides nuerical results to validate the theoretical results and to illustrate our theoretical findings. Finally we conclude this paper in Section VI. II. SYSTEM MODELS AND PERFORMANCE METRICS In this section we first introduce syste odels and then define soe perforance etrics involved in this study. A. Syste Models Network and Mobility Models: We consider a tie-slotted MANET with n obile nodes in a unit square of torus boundaries i.e. a node oving across an edge will iediately appears at the opposite edge of the square. Siilar to previous studies [7] [] the network area is evenly divided into cells squares as shown in Fig a. The nodes in the network ove according to independent and identically distributed i.i.d. obility odel [7] [] [3]. Under such obility odel at the beginning of each tie slot each node independently and randoly selects a cell to ove into and stays in it during the tie slot. Counication Model: All the nodes share a coon half-duple ediu for data transissions and we adopt the widely used protocol odel proposed in [4] to address interference aong siultaneous transissions. Suppose that at tie slot t node i transitter is transitting to node j receiver of its transission range. To ensure the successful transission fro i to j for any other siultaneously transitting node k the following condition should be satisfied: d kj t +Δd ij t d ij t denotes the Euclidean distance between i and j at tie slot t and Δ 0 odels a guard zone to prevent the transission failure due to interference fro other siultaneous transissions. We consider a local transission scenario [5] [6] the transission range of each node e.g. node S 0 shown in Fig a covers the sae cell of the node and its eight neighbor cells. In a tie slot the data that can be successfully transitted is noralized as one packet. Traffic Model: We consider a ulticast traffic odel [] [7] there are n ulticast groups in the network each of which consists of g + nodes including one source node and g destination nodes. Each node is the source node in one ulticast group and also a destination node in another ulticast group. In a ulticast group each node can serve as a relay node to forward packets originated fro other n ulticast groups and each destination node also helps to forward packets originated fro the ulticast group. Transission Scheduling Model: To ensure as any as possible successful siultaneous transissions we adopt the transission-group based scheduling odel here [7] [] [5] [6] [8]. Under the scheduling odel the network cells are divided into α different transission-groups and the distance between any two cells in each transissiongroup is soe integer ultiple of α cells along the vertical and horizontal directions respectively. For eaple there are 5 transission-groups in Fig. b all shaded cells witnde belong to the sae transission-group. In every α tie slots each transission-group will becoe active alternately such that all nodes in each cell naely active cell of an active transission-group are allowed to contend for a transission opportunity over the coon half-duple ediu. Only one node if any in each active cell is randoly selected to perfor a transission. To ensure all siultaneous transissions to be successful in an active transission-group the paraeter α is deterined as following. Suppose that at current tie slot a transitter S is transitting to a receiver R within its transission range as shown in Fig. b. Since the transission range of each node covers its own cell and its eight neighbor cells with side length of / the aiu distance r between S and R is 8/. Meanwhile the distance between R and another possible closest siultaneous transitter S is α /. To guarantee the successful transission between S and R the following condition should be satisfied according to the protocol odel [4]: α / + Δ 8/. Since the value of integer α is no ore than wehave α = in{ +Δ 8+ } 3 is the ceiling function of. B. Perforance Metrics Packet Delivery Probability: For a ulticast group and packet lifetie τ the delivery probability of a packet in the ulticast group is defined as the ratio of the nuber of destination nodes that have received the packet before the τ epires to the total nuber of destination nodes in the ulticast group c 07 IEEE. Personal use is peritted but republication/redistribution requires IEEE perission. See for ore inforation.

3 3 Packet Delivery Cost: The delivery cost of a packet is defined as the total transission power consued until either the packet lifetie τ epires or all its destination nodes in a ulticast group have received the packet. III. COOPERATIVE MULTICAST SCHEME AND THEORETICAL MODEL In this section we first propose a general cooperative ulticast schee adopting two-hop relay as routing protocol then develop a Markov chain theoretical odel to depict packet delivery process under such a schee and derive soe related basic probabilities. A. Cooperative Multicast Schee Algorith Cooperative Multicast Schee CMfgpτ:. For a transitter T and its desired receiver R in current tie slot.. if both T and R are in the sae ulticast group and T is a source node in the ulticast group then 3. T perfors a source-to-destination transission with R; 4. end if 5. if both T and R are in the sae ulticast group and T is not a source node in the ulticast group then 6. With cooperative probability p T perfors a destination-to-destination transission with R; 7. end if 8. if T and R are in different ulticast groups then 9. T flips a fair coin; 0. if it is head then. T perfors a source-to-relay transission with R;. else 3. T perfors a relay-to-destination transission with R; 4. end if 5. end if The proposed general cooperative ulticast schee is suarized in Algorith. To support these transissions in Algorith we assue that each node has n + individual queues in its buffer: one source-queue used to store locally generated packets waiting for their copies to be delivered one already-delivered-queue used to store these packets whose f copies have already been delivered out but their reception statuses are not confired yet n relay-queues used to store packets originated fro other ulticast groups one queue per ulticast group and one cooperative-queue used to store packets that will be forwarded to their destination nodes with cooperative probability p. The four types of transissions in Algorith are defined as follows. Source-to-destination transission: T delivers to R a packet that R is requesting the packet is chosen as follows: T first checks its source-queue starting fro its head-of-line packet to find the packet; if it fails then it retrieves the packet fro its already-delivered-queue. Destination-to-destination transission: If there eists a packet that R is requesting in the cooperative-queue of T then T delivers a copy of the packet to R; otherwise T reains idle. Source-to-relay transission: If R does not carry a copy of the head-of-line packet fro T s source-queue then T delivers a copy of the packet to R; otherwise T reains idle. If f copies of the packet have been delivered to different relay nodes then T puts the packet to the end of the alreadydelivered-queue and then reoves it fro the source-queue. Relay-to-destination transission: If there eists a packet that R is requesting in the relay-queue intended for R then T delivers a copy of the packet to R; otherwise T reains idle. Reark : The two-hop relay since first proposed by Grossglauser and Tse in [9] has been etensively eplored in literature and proved to be a siple and efficient routing protocol for MANETs the network topology varies draatically and no conteporaneous end-to-end path ay ever eist at any given tie instant. Furtherore under such a routing protocol the network perforance can be analytically studied in the challenging MANETs. This paper eplores a general cooperative ulticast schee adopting two-hop relay as routing protocol and analytically studies packet delivery probability/cost perforance in the MANETs. B. Markov Chain Theoretical Model Without loss of generality we focus on a tagged ulticast group with a source node S and g destination nodes D D D g. For a given packet at the source node S we use two-tuple i j to denote a general state that i relay nodes are carrying a copy of the packet and j destination nodes have received the packet 0 i f and 0 j g. If j < g then the general state corresponds to a transient state; otherwise it is an absorbing state. Fro the operation of the general cooperative ulticast schee we know that if the tagged ulticast group is currently in transient state i j then only one of twelve transition scenarios shown in Fig. ay happen in the net tie slot: SD scenario: source-to-destination transission only i.e. S successfully delivers the packet to a destination node that has not received the packet while none of relay and destination nodes delivers the packet to other destination nodes. Under this transition scenario the state i j ay transit to i j +. SR scenario: source-to-relay transission only i.e. S successfully delivers a copy of the packet to a relay node that does not carry the packet while none of relay and destination nodes delivers the packet to other destination nodes. The state i j ay transit to i +j under the SR transition scenario. DD k scenario: k siultaneous destination-todestination transissions only i.e. k destination-todestination transissions happen siultaneously each transission indicates that a destination node successfully delivers the packet to another one that has not received the packet while other transition scenarios c 07 IEEE. Personal use is peritted but republication/redistribution requires IEEE perission. See for ore inforation.

4 Fig The transition diagra of a general transient state i j do not happen. Under such a transition scenario the state i j ay transit to i j + k. RD k scenario: k siultaneous relay-to-destination transissions only i.e. k relay-to-destination transissions happen siultaneously each transission indicates that a relay node successfully delivers the packet to a destination node that has not received the packet while other transition scenarios such as SD SR and DD k do not happen. Under the RD k scenario the state i j ay transit to i j + k k i and j + k g. SD +RD k scenario: one source-to-destination and k relay-to-destination transissions only i.e. these k + transissions happen siultaneously while the SR and DD k scenarios do not happen. Under such a scenario the state i j ay transit to i j + k + k i and j + k + g. SR+RD k scenario: one source-to-relay and k relay-todestination transissions only i.e. these k + transissions happen siultaneously while the SD and DD k scenarios do not happen. Under this scenario the state i j ay transit to i +j+ k k i and j + k g. SD +DD k scenario: one source-to-destination and k destination-to-destination transissions only i.e. these k + transissions happen siultaneously while the SR and RD scenarios do not happen. The state i j ay transit to i j + k +under the scenario k j and j + k + g. SR + DD k scenario: one source-to-relay and k destination-to-destination transissions only i.e. these k + transissions happen siultaneously while the SD and RD scenarios do not happen. Under such a scenario the state i j ay transit to i +j + k k j and j + k g. RD k + DD k scenario: k relay-to-destination and k destination-to-destination transissions only i.e. these k +k transissions happen siultaneously while the SD and SR scenarios do not happen. Under this scenario the target state is i j + k + k k i k j and j + k + k g. SR +RD k +DD k scenario: one source-to-relay k relay-to-destination and k destination-to-destination transissions only i.e. these k + k + transissions happen siultaneously while the SD scenario does not happen. Under this scenario the state i j ay transit to i +j + k + k k i k j and j + k + k g. SD + RD k + DD k scenario: one source-todestination k relay-to-destination and k destination-todestination transissions only i.e. these k +k + transissions happen siultaneously while the SR scenario does not happen. Under this scenario the target state is i j + k + k + k i k j and j + k + k + g. Self-loop scenario: transition fro i j to i j i.e. the packet is not delivered fro a node to another. Based on the transition diagra in Fig. the packet delivery process under the general cooperative ulticast schee is odeled as a discrete-tie two-diensional absorbing Markov chain shown in Fig. 3. C. Basic Results We present soe basic results related to the general cooperative ulticast schee and the Markov chain theoretical odel which will help us to analyze packet delivery probability/cost perforance in section IV. Lea : For a tagged ulticast group we use p and p to denote the probability that S perfors a source-todestination transission and the probability that S perfors a source-to-relay transission respectively. Then we have g p = k g k g α k n g ϕi k= i=0 g k g k g k + i k 9 n g ϕi k= i=0 4 i + g p = α 9 n g i n g i i= n g i n g i + + n g 8 i i= n g i 9 i 5 ϕi = n g i i n g i c 07 IEEE. Personal use is peritted but republication/redistribution requires IEEE perission. See for ore inforation.

5 5 = = = = = = = = = Fig. 3. Absorbing Markov chain for the packet delivery process under the general cooperative ulticast schee. For each transient state the following transition scenarios are not shown for siplicity: SD+RD k SR+RD k SD+DD k SR+DD k RD k +DD k SR+RD k +DD k SD+RD k +DD k and Self-loop. For the tagged ulticast group with source node S and g destination nodes and a given packet at S suppose that i j is the transient state of the Markov chain in Fig. 3 in current tie slot 0 i f 0 j g. Under the transient state we use μ μ and μ 3 to denote the nuber of destination nodes that have not received the given packet the nuber of relay nodes carrying a copy of the packet and the nuber of relay nodes carrying no copy of the packet respectively which can be easily deterined as μ = g j 6 μ = i 7 μ 3 = n g i. 8 Then we establish the following Leas. Lea : We use P SD μ and P SR μ 3 to denote the probability that in the net tie slot S successfully delivers a copy of the packet to a destination node i.e. a successful source-to-destination transission and the probability that S successfully delivers a copy of the packet to a relay node i.e. a successful source-to-relay transission respectively. Then we have P SD μ = μ g p 9 μ 3 P SR μ 3 = n g p. 0 Lea 3: We use P RD μ μ to denote the probability that successful relay-to-destination transissions will happen siultaneously in the net tie slot. Here each transission indicates that a relay node successfully delivers a copy of the packet to a destination node of the tagged ulticast group and in{μ μ }.Thenwehave μ μ P RD μ μ = g λ λ λ i λ λ i = α i α α i α l i l i li w i wi h ψki i = if i li w i wi h ψki i 9 z = if i = c 07 IEEE. Personal use is peritted but republication/redistribution requires IEEE perission. See for ore inforation.

6 6 In Lea 3 l i w i z and ψ are defined in the following forulas. i l i = n g k j 3 w i = g j= i h j 4 j= z = n ψ = k=0 ki k k j + h j 5 j= hi h=0 hi h k+h 8 ki+hi k h 6 k + h + + k 0 =0and h 0 =0. Lea 4: We use P DD μ to denote the probability that successful destination-to-destination transissions will happen siultaneously in the net tie slot in{μ g } and g is the floor function of g.thenwe have μ g μ P DD μ = p α i α l i+g+ g li+g+ γ γ γ i γ 7 w i = wi ψki if i γ i = p α l i+g+ i li+g+ w i wi α h ψki i = 9 z if i =. 8 The basic idea of the proof of Lea 3 is suarized as follows. For relay nodes carrying a copy of the packet originated fro the tagged ulticast group we first show how the probability P RD μ μ is related to the probability that these relay nodes perfor relay-to-destination transissions siultaneously. Then the probability is derived as the product of the probabilities that one of these relay nodes perfors a relay-to-destination transission whics deterined based on the probabilities of its sub-events. Finally by suarizing these results P RD μ μ can be derived. The derivations of Lea 4 is siilar to that of Lea 3.The detailed proofs of these Leas can be found in Appendi. In Appendi we also give soe other basic results the proofs of which are siilar to those of Leas 3 and 4. IV. PACKET DELIVERY PROBABILITY/COST MODELING Based on the Markov chain theoretical odel and related basic results this section derives analytical epressions for the epected packet delivery probability/cost. A. Epected Packet Delivery Probability We use β to denote the total nuber of transient states in the Markov chain shown in Fig. 3. Since all β transient states are evenly divided into g rows and each row has f + transient states in the Markov chain we have β = gf +. We nuber these β transient states and f + absorbing states sequentially as β+ f + in a left-to-right and top-to-down way. For a given packet originated fro a tagged ulticast group we use N d to denote the packet delivery probability i.e. the ratio of the nuber of destination nodes that have received the packet before the lifetie τ epires to the total nuber of destination nodes g and then the epected value E{N d } of N d can be deterined as E{N d } = g f+ Mk l d 9 g k=0 l= the lth state of the kth row corresponds to the state witnde d d β + f + in the Markov chain shown in Fig. 3 Mk l denotes the nuber of destination nodes that have received the packet in state d d denotes the probability that the Markov chain starting fro the initial state reaches state d after τ tie slots and d is given by d =f +k + l. 0 To derive E{N d }wefirstdeterinemk l. Since each state of the kth row in the Markov chain indicates that k destination nodes have received the packet we have Mk l =k. We then deterine d.letp =p ij β+f+ β+f+ be the transition atri of the Markov chain. The entry p ij of the atri P denotes the transition probability that the Markov chain starting fro state i will be in state j after tie slot. The P can be rewritten as [ ] Q R P = O I the atri Q =q ij β β defines the transition probabilities aong β transient states the atri R =r ij β f+ defines the transition probabilities fro β transient states to f +absorbing states O is a f +-by-β zero atri and I is a f +-by-f +identity atri. After τ tie slots the τ-step transition atri P τ of P can be calculated as [ P τ Q τ NI Q τ ] R = 3 O I N =I Q is the fundaental atri of the Markov chain and I is a β-by-β identity atri. We define atri W =w ij β β as W = Q τ and define B =b it β f+ as B = N I Q τ R. Sincew ij denotes the probability that the Markov chain reaches transient state j fro starting transient state i after τ tie slots and b it denotes the probability that the Markov chain reaches absorbing state t fro starting transient state i after τ tie slots d is deterined as d = { w d if d β b d if β + d β + f c 07 IEEE. Personal use is peritted but republication/redistribution requires IEEE perission. See for ore inforation.

7 7 B. Epected Packet Delivery Cost For a given packet originated fro a tagged ulticast group we use C p to denote the packet delivery cost i.e. the total transission power consued until either the lifetie τ epires or all g destination nodes in the tagged ulticast group have received the packet and then the epected value E{C p } of C p can be deterined as E{C p } = g f+ Ck l d 5 k=0 l= Ck l denotes the transission power consued in the lth state of the kth row in Fig. 3. Consider one unit transission power is consued when a node delivers a packet to another node. Since the lth state of the kth row in the Markov chain indicates that the packet has been delivered to l relay nodes and k destination nodes Ck l is deterined as Ck l =k + l. 6 The unknown atrices Q and R can be easily obtained according to the transition probabilities fro a state of the Markov chain to another in Fig. 3. The transition probability under each transient scenario in Fig. can be deterined as follows: for SD scenario P SD μ P SDRD μ μ P SDDD μ +P SDRDDD μ μ for SR scenario P SR μ 3 P SRRD μ μ μ 3 P SRDD μ μ 3 +P SRRDDD μ μ for DD k scenario P DD k μ P SDDD k μ P SRDD k μ μ 3 P RDDD kμ μ for RD k scenario P RD k μ μ P SDRD k μ μ P SRRD k μ μ μ 3 P RDDD k μ μ for SD + RD k scenario P SDRD k μ μ P SDRDDD k μ μ for SR +RD k scenario P SRRD k μ μ μ 3 P SRRDDD k μ μ for SD + DD k scenario P SDDD k μ P SDRDDD kμ μ for SR + DD k scenario P SRDD k μ μ 3 P SRRDDD kμ μ for RD k +DD k scenario P RDDD k k μ μ P SDRDDD k k μ μ P SRRDDD k k μ μ for SR + RD k + DD k scenario P SRRDDD k k μ μ for SD + RD k + DD k scenario P SDRDDD k k μ μ and for self-loop scenario the su of all the above transition probabilities. V. NUMERICAL RESULTS In this section we first provide siulation studies to validate our theoretical odels on packet delivery probability/cost and then apply these odels to eplore how syste paraeters would affect the packet delivery probability/cost in a MANET. a E{N d } vs. g b E{C p} vs. g Fig. 4. Coparisons between theoretical results and the siulation ones for odel validation. A. Model Validation A C++ siulator was developed to siulate the packet delivery process under the general cooperative ulticast schee in the considered MANET. We set the guard factor as Δ= and hence the paraeter α in transission scheduling odel is deterined as α = in{8}. Besides the i.i.d. obility odel considered in this paper we also ipleent the following two obility odels: Rando Walk Model [30]: At the beginning of every tie slot each node will independently and randoly select one cell aong its current cell and its eight neighbor cells with probability /9 and then oves into the selected cell and stays in it for the reaining tie slot. Rando Waypoint Model [3]: At the beginning of every tie slot each node will independently and randoly generate a -tuple y the and y are uniforly selected fro [/ 3/] and then oves a distance c 07 IEEE. Personal use is peritted but republication/redistribution requires IEEE perission. See for ore inforation.

8 8 of and y along the horizontal and vertical directions respectively. Etensive siulations have been conducted to validate our theoretical odels. For the setting of n = 00 =6p = 0.5f =4andτ = 000 the corresponding theoretical and siulation results are suarized in Fig. 4 to show how the epected packet delivery probability/cost vary as g increases. Each siulated value is calculated as the average value over 0 6 slots. Fig. 4 shows clearly that under i.i.d. obility odel our theoretical results agree well with the siulation ones indicating that our theoretical odels can accurately predict the packet delivery probability/cost perforance under the general cooperative ulticast schee. Another interesting observation fro Fig. 4 is that the siulation results under the rando walk and rando waypoint odels are siilar to those under i.i.d. obility odels. Thus our theoretical odels although are developed under the i.i.d. obility odel can be used to predict the packet delivery probability/cost behavior under the rando walk and rando waypoint odels as well. Reark : In this paper these network functions like i.i.d. node obility transission-group based scheduling schee and packet delivery process of our cooperative ulticast schee can be easily ipleented by a custoized C++ siulator now publicly available at [3] without going through a coplicated network siulator like NS3 and OPNET. As shown in [3] [33] for a cell-partitioned network the average delay and network throughput capacity under the i.i.d. obility odel are also identical to those under other non-i.i.d. obility odels as long as they have the sae steady-state distribution of nodes locations like the rando walk and rando waypoint obility odels. Therefore our theoretical odels although were developed for the packet delivery probability under the i.i.d. obility odel can also be used to predict the packet delivery probability/cost perforance in MANETs under the rando walk and rando waypoint obility odels. a E{N d } vs. p b E{C p} vs. p B. Perforance Analysis To eplore the ipact of syste paraeters on the packet delivery perforance we suarize in Fig. 5 how the epected packet delivery probability/cost E{N d }E{C p }vary with p under a setting of n = 00 =6 f =4 τ = 3000 and g = {9 9 39}. Fig. 5 shows that for a fied setting of g asp increases both the E{N d } and E{C p } increase onotonously. This is because an increase of p iplies that ore destination nodes will willing to forward packets for each other leading to the increase of packet delivery probability at the cost of power consuption on the destination nodes. The results in Fig. 5 indicate that in practice the setting of cooperative probability p should be carefully selected in order to aintain a high delivery probability perforance and a relatively low delivery cost. Fig. 5a also illustrates that as g increases the E{N d } decreases if p is relatively sall; otherwise it increases. This is because an increase of g leads to both positive and negative effects on E{N d }: On one hand it increases the probability with which each destination node receives a packet due to destination cooperation; on the other hand it decreases Fig. 5. Perforance E{N d }E{C p} versusp. the probability since destination nodes are less interested in forwarding packet for each other. As shown in Fig. 5a when p is relatively sall the latter negative effect doinates; as p further increases the forer positive effect is becoing the doinating one. We now proceed to eplore how the replication paraeter f affects the perforance E{N d } E{C p }. For a setting of n = 00 = 6 g = 9 p = 0.5 τ = 3000 and n = { } we suarize in Fig. 6 how E{N d } and E{C p } vary with f. Fig.6showsthatasf increases both the E{N d } and E{C p } onotonously increases. This is ainly due to that an increase of f will increase transission opportunities fro source node to relay nodes and thus increases the packet delivery cost and the opportunities that the destination nodes receive the packet fro the relay nodes resulting in a higher packet delivery probability. Finally we eaine in Fig. 7 how the E{N d } and E{C p } vary with the nuber of nodes n. For a setting of =6 g = 7 p =0.5 f =5and τ = { } Fig.7shows c 07 IEEE. Personal use is peritted but republication/redistribution requires IEEE perission. See for ore inforation.

9 9 a E{N d } vs. f a E{N d } vs. n b E{C f } vs. f b E{C f } vs. n Fig. 6. Perforance E{N d }E{C p} versusf. Fig. 7. Perforance E{N d }E{C p} versusn. that for a fied setting of packet lifetie τ asn increases both the E{N d } and E{C p } first increase and then decrease. Actually an increase in the nuber of nodes n has two-fold effects the perforance E{N d }E{C p }. On one hand when the networs sparse i.e. n is very sall a bigger n will lead to a higher packet delivery speed at which a packet is distributed out and thus higher packet delivery probability and delivery cost. On the other hand a bigger n will lead to a lower packet delivery speed due to the effects of wireless interference and ediu access contention issues in a crowed network and thus lower packet delivery probability and delivery cost. C. Perforance Coparison In this subsection we copare the packet delivery probability/cost perforance of state-of-the-art schees and that of the general cooperative ulticast schee proposed in this paper. Specifically we choose two well-known schees for coparison naely two-hop relay [4] with full cooperation schee A for short and two-hop relay [3] with noncooperation schee B for short fro Section I. The corresponding results are suarized in Fig. 5. Fig. 5 illustrates the perforance coparison between our schee and schee A i.e. the special case of our schee by setting p =. As shown in Fig. 5 that for the schee A all destination nodes are willing to forward packet for each other and thus take full advantage of each transission opportunity to increase the packet delivery probabilitye{n d } while incurring high delivery cost i.e. total transission power consued E{C p }. For eaple with the setting of g=39 and p =E{N d } resp. E{C p } under schee A is 0.98 resp while that is 0.87 resp by adopting our schee with the setting of p =0.5. Fig. 5 also illustrates the perforance coparison between our schee and schee B i.e. the special case of our schee by setting p = 0. We can see fro Fig. 5 that with the setting of g=39 and p =0E{N d } resp. E{C p } under schee B is 0.48 resp..7. This is because for the c 07 IEEE. Personal use is peritted but republication/redistribution requires IEEE perission. See for ore inforation.

10 0 schee B destination nodes ay waste a lot of transission opportunities because all of the are not willing to forward packet for each other resulting in the decreasing of packet delivery probability while conserving their own transission power consued. VI. CONCLUSION This paper proposed a general cooperative ulticast schee to fully consider the iportant issue of destination nodes cooperative behaviors. A Markov chain theoretical odel was developed to depict packet delivery process under the general schee based on which and soe related basic probabilities analytical epressions were derived for the packet delivery probability/cost. Etensive siulations illustrate that our theoretical odels can accurately predict the packet delivery probability/cost perforance under the general schee. Our results indicate that through a proper setting of cooperative probability p a high packet delivery probability can be achieved while aintaining a relatively low delivery cost. It is epected that the general schee can facilitate future MANETs to support various applications with different cooperative desires aong destination nodes. One interesting direction is to further etend the developed theoretical odel to analyze packet delivery probability/cost perforance in ulti-hop relay MANETs. APPENDIX Proof of Lea : Source node S can perfor a sourceto-destination transission if and only if the following four events happen in the sae tie slot: S ovesintoanactive cell; There eists at least one destination node of S in the active cell and its eight neighbor cells; 3 There are i other nodes in the active cell ecept S and corresponding g destination nodes 0 i n g ;4S is randoly chosen as a transitter. Regarding the second event suppose that there are k destination nodes of S in the active cell or no destination node is in the active cell but there eist k destination nodes in the eight neighbor cells of this cell k. Notice that these two cases are utually eclusive. Then the probability that S is chosen as a transitter is k+i+ resp. i+ under the forer case resp. under the latter case. We use q and q to denote the product of probabilities of these four events under the forer case and that under the latter case respectively. Then we have q = = α k= q = g k g k α g 8 = k 9 n g ϕi i + k= i=0 8 ϕi = n g i i n g i. The forula 4 can then be derived by adding q and q together. Siilarly S can perfor a source-to-relay transission if and only if the following four events happen in the sae tie slot: S oves into an active cell; No destination node of S is in this cell and its eight neighbor cells; 3 There eists at least one other node in this cell and its eight neighbor cells ecept S; 4S is randoly chosen as a transitter. Regarding the third event suppose that there are i other nodes in the active cell or no other node is in this cell but there eist i other nodes in the eight neighbor cells of this cell i n g. Notice that these two cases are utually eclusive. Then the probability that S is randoly chosen as a transitter is i+ resp. under the forer case resp. under the latter case. By suing up the product of probabilities of these four events under the forer case and that under the latter case the forula 5 can then be derived. Proof of Lea : In the net tie slot source node S ay deliver a copy of the packet to one of the μ destination nodes that do not receive the packet whics called an event. Notice that there are μ utually eclusive events. The probability of such an event is p g. By suing up the probabilities of these μ events we have P SD μ = μ g p. 9 Siilarly in the net tie slot S ay deliver a copy of the packet to one of μ 3 relay nodes that do not carry a copy of the packet. Notice that these μ 3 events are utually eclusive and each event denotes a transission fro S to a single p relay node. The probability of such an event is n g.by suing up the probabilities of these μ 3 events we have μ 3 P SR μ 3 = n g p. 30 Proof of Lea 3: To derive P RD μ μ we focus on specific relay nodes carrying a copy of the packet and use P Tr Tr Tr to denote the probability that these relay nodes will perfor relay-to-destination transissions siultaneously a counity-transission for short in the net tie slot Tr i denotes a relay-to-destination transission and i. Each transitting relay node in a successful counitytransission can deliver a copy of the packet to a destination node and μ destination nodes have not received the packet in the current tie slot. Thus the probability that a successful counity-transission happens is g k g k g k n g μ ϕi k + i + P Tr Tr g Tr. Since μ relay nodes is carrying the packet currently there are μ i=0 distinct counitytransissions. By suing up the probabilities of these μ 7 counity-transissions we have P RD μ μ = μ μ g P Tr Tr Tr. 3 The basic idea to deterine P Tr Tr Tr is that we first treat the event Tr Tr Tr as siultaneous but utually independent relay-to-destination transissions then deterine the probability that each transission happens c 07 IEEE. Personal use is peritted but republication/redistribution requires IEEE perission. See for ore inforation.

11 and finally ultiply the probabilities of all transissions. First consider a general transission Tr i perfored by a specific relay node e.g. R i i. This transission will happen if and only if the following five events happen in the sae tie slot: R i oves into an active cell; There are in total other nodes in the sae cell as R i and the eight neighbor cells of the cell ecept source node S its g destination nodes specific relay nodes g destination nodes of R i s local traffic and i j= k j other nodes in the sae cells as these i specific relay nodes and their neighbor cells 0 n g i k j aong the k nodes are j= in the sae cell as R i and the other k nodes are in the eight neighbor cells of the cell; 3 There are destination nodes of S in the cell and the eight neighbor cells of the cell aong the h destination nodes are in the cell and the other h destination nodes are in the eight neighbor cells g i h j ; 4 R i and one destination node j= are chosen as a transitter and a receiver respectively; 5 R i chooses to perfor relay-to-destination transission Tr i with the receiver. The probabilities that the above events happen are calculated as α i l i α li ki k k k=0 8 ki k w i wi hi h h 8 hi h h i k+h+ + = h=0 and respectively. Here l i = n g w i = g i j= i j= k j and h j. By ultiplying the probabilities of these events the probability of the transission Tr i is derived i. Siilarly we can also get the probability of the transission Tr. P Tr Tr Tr follows by ultiplying the probabilities of these transissions. Then is derived by substituting P Tr Tr Tr into 3. Proof of Lea 4: To derive P DD μ we focus on specific destination nodes each of which has received the packet. We use Tr i to denote a destination-to-destination transission and use P Td Td Td to denote the probability that these destination nodes will perfor such transissions siultaneously a counity-transission for short in the net tie slot i. In a successful counity-transission each transitting destination node can deliver a copy of the packet to other destination node. Thus the probability that a successful counity-transission happens can be deterined as μ g P Td Td Td.Sinceg μ destination nodes have received the packet in the current tie slot there are in total g μ distinct counity-transissions. By suing up the probabilities of these g μ counity-transissions we have μ g μ P DD μ = g P Td Td Td. 3 The basic idea to deterine P Td Td Td is that we first treat the event Td Td Td as siultaneous yet utually independent destination-to-destination transissions then deterine the probability that each transission happens and finally ultiply the probabilities of all transissions. First consider a general transission Td i perfored by a specific destination node e.g. D i i. This transission will happen if and only if the following five events happen in the sae tie slot: D i oves into an active cell; There are in total other nodes in the sae cell as D i and the eight neighbor cells ecept the g destination nodes of the tagged ulticast group and i k j j= other nodes in the sae cells as these i specific destination nodes and their neighbor cells0 n g i j= k j aong the k nodes are in the sae cell as D i and the other k nodes are in the eight neighbor cells of the cell; 3 There are other destination nodes in the cell and the eight neighbor cells of the cell ecept the specific destination nodes aong the h destination nodes are in the cell and the other h destination nodes are in the eight neighbor cells g = i j= h j ;4D i and one destination node are chosen as a transitter and a receiver respectively; 5 D i chooses to perfor destination-to-destination transission Td i with the receiver. The probabilities that the above events happen are calculated as α i l i α li ki k k k=0 8 ki k w i wi hi h h 8 hi h h i k+h+ + h=0 and p respectively. Here l i = n g g i j= i j= k j and w i = h j. By ultiplying the probabilities of these events the probability of the transission Td i is deterined i. Siilarly we can also get the probability of the transission Td. By ultiplying the probabilities of these transissions P Td Td Td follows. Then 7 follows by substituting P Td Td Td into 3. Other Basic Results: We use P SDRD μ μ to denote the probability that a successful source-to-destination transission and successful relay-to-destination transissions will happen siultaneously in the net tie slot in{μ μ }.Thenwehave P SDRD μ μ = μ μ + g + ρ ρ ρ i ρ c 07 IEEE. Personal use is peritted but republication/redistribution requires IEEE perission. See for ore inforation.

12 ρ i = α i α α i α ki+hi l i l i+g li w i wi h ψki i = li+g w i = if i wi h ψki i 9+ z ki hi if i = We use P SRRD μ μ μ 3 to denote the probability that a successful source-to-relay transission and successful relay-to-destination transissions will happen siultaneously in the net tie slot in{μ μ }.Thenwe have P SRRD μ μ μ 3 = μ μ μ3 g θ θ θ i θ + 35 p α l i+g+ i li+g+ w i wi α = ψ if i ε i = l i+g+ w i α i α ψ ki+hi li+g+ = wi 9+ z ki hi if i = We use P RDDD μ μ to denote the probability that successful relay-to-destination transissions and successful destination-to-destination transissions will happen siultaneously in the net tie slot in{μ μ } and in{μ g }. Thenwe have θ i = α i α α i α l i l i+g li w i wi h ψki i = if i li+g ki k k=0 k i+ r r=0 k+r 8 ki+ k r k+r+ 9+ z ki if i = We use P SRDD μ μ 3 to denote the probability that a successful source-to-relay transission and successful destination-to-destination transissions will happen siultaneously in the net tie slot in{μ g }. Then we have μ g μ μ3 P SRDD μ μ 3 = g δ δ δ i δ + 37 p α i α l i+g+ ψ if i l δ i = i+g+ r α i α k+r 8 ki+ k r k+r+ k i+ 9+ z ki if i = We use P SDDD μ to denote the probability that a successful source-to-destination transission and successful destination-to-destination transissions will happen siultaneously in the net tie slot in{μ g }. Then we have μ g μ P SDDD μ = ε ε ε i ε + 39 li+g+ li+g+ g w i = k=0 ki k wi r=0 P RDDD μ μ = ε i = α i α p α i α p α i α l i ψ 9+ μ g μ μ + g + 4 ϑ ϑ ϑ i ϑ ϑ + 4 li w i l i +g l i +g = wi ψki if i li +g w i = li +g w i = + n j= wi ψki if <i + wi h ψki i if i = +. + h j j= k j 43 We use P SRRDDD μ μ μ 3 to denote the probability that a source-to-relay transission successful relayto-destination transissions and successful destination-todestination transissions will happen siultaneously in the net tie slot in{μ μ } and in{μ g }. Thenwehave μ g μ μ3 μ P SRRDDD μ μ = g + ζ ζ ζ i ζ + ζ c 07 IEEE. Personal use is peritted but republication/redistribution requires IEEE perission. See for ore inforation.

13 3 ζ i = α i α p α i α α i α l i l i +g li w i = if i li +g w i wi ψki = if <i + r wi ψki l i +g li +g ki k=0 k r=0 k+r n + h j k j j= j= k+r 8 ki+ k r 9++ if i = We use P SDRDDD μ μ to denote the probability that a source-to-destination transission successful relay-to-destination transissions and successful destination-to-destination transissions will happen siultaneously in the net tie slot in{μ μ } and in{μ g }. Thenwehave μ μ g μ P SDRDDD μ μ = α i α l i + + g + + ϱ ϱ ϱ i ϱ + ϱ li w i = wi ψki if i l p α i +g i li +g w i wi α k ϱ i = i=0 = ψ if <i + l α i +g i li +g w i wi α = ψ ki+hi 9++ z + + if i = + +. REFERENCES 47 [] S. Shakkottai X. Liu and R. Srikant The ulticast capacity of large ultihop wireless networks IEEE/ACM Transactions on Networking vol. 8 no. 5 pp October 00. [] Z. Qian X. Tian X. Chen W. Huang and X. Wang Multicast capacity in MANET witnfrastructure support IEEE Transactions On Parallel and Distributed Systes vol. 5 no. 7 pp July 04. [3] Z. Li C. Wang C. Jiang and X. Li Multicast capacity scaling for inhoogeneous obile ad hoc networks Ad Hoc Networks vol. no. pp January 03. [4] S. Zhou and L. Ying On delay constrained ulticast capacity of largescale obile ad-hoc networks in INFOCOM 00. [5] J. P. Jeong T. He and D. H. C. Du TMA: Trajectory-based ultianycast forwarding for efficient ulticast data delivery in vehicular networks Coputer Networks vol. 57 no. 3 pp Septeber 03. [6] X. Ge J. Yang H. Gharavi and Y. Sun Energy efficiency challenges of 5G sall cell networks IEEE Counications Magazine vol. 55 no. 5 pp May 07. [7] T. Han X. Ge L. Wang K. S. Kwak Y. Han and X. Liu 5G converged cell-less counications in sart cities IEEE Counications Magazine vol. 55 no. 3 pp March 07. [8] X. Ge S. Tu G. Mao C. Wang and T. Han 5G ultra-dense cellular networks IEEE Wireless Counications vol. 3 no. pp February 06. [9] Z. Su Q. Xu Y. Hui M. Wen and S. Guo A gae theoretic approach to parked vehicle assisted content delivery in vehicular ad hoc networks IEEE DOI: 0.09/TVT [0] B. Yang Y. Cai Y. Chen and X. Jiang On the eact ulticast delay in obile ad hoc networks with f-cast relay Ad Hoc Networks vol.33 pp October 05. [] X. Wang W. Huang S. Wang J. Zhang and C. Hu Delay and capacity tradeoff analysis for otioncast IEEE/ACM Transactions on Networking vol. 9 no. 5 pp October 0. [] C. Hu X. Wang and F. Wu Motioncast: On the capacity and delay tradeoffs in MobiHoc 009. [3] X. Wang Y. Bei Q. Peng and L. Fu Speed iproves delay-capacity trade-off in otioncast IEEE Transactions on Parallel and Distributed Systes vol. no. 5 pp May 0. [4] X. Wang Q. Peng and Y. Li Cooperation achieves optial ulticast capacity-delay scaling in MANET IEEE Transactions on Counications vol. 60 no. 0 pp October 0. [5] J. Liu X. Jiang H. Nishiyaa and N. Kato Multicast capacity delay and delay jitter in interittently connected obile networks in INFOCOM 0. [6] J. Zhang Y. Li Z. Liu F. Wu F. Yang and X. Wang On ulticast capacity and delay in cognitive radio obile ad hoc networks IEEE Transactions Wireless Counications vol. 4 no. 0 pp March 05. [7] Y. Chen J. Liu X. Jiang and O. Takahashi Throughput analysis in obile ad hoc networks with directional antennas Ad Hoc Networks vol. no. 3 pp. 35 May 03. [8] C. Zhang X. Zhu and Y. Fang On the iproveent of scaling laws for large-scale MANETs with network coding IEEE Transactions on Selected Areas in Counications vol. 7 no. 5 pp June 009. [9] P. Li Y. Fang J. Li and X. Huang Sooth trade-offs between throughput and delay in obile ad hoc networks IEEE Transactions on Mobile Coputing vol. no. 3 pp March 0. [0] P. Li and Y. Fang On the throughput capacity of heterogeneous wireless networks IEEE Transactions on Mobile Coputing vol. no. pp Deceber 0. [] J. Liu X. Jiang H. Nishiyaa and N. Kato Delay and capacity in ad hoc obile networks with f-cast relay algoriths IEEE Transactions on Wireless Counications vol. 0 no. 8 pp August 0. [] Z. Kong E. M. Yeh and E. Soljanin Coding iproves the throughputdelay tradeoff in obile wireless networks IEEE Transactions on Inforation Theory vol. 58 no. pp Noveber 0. [3] M. J. Neely and E. Modiano Capacity and delay tradeoffs for ad-hoc obile networks IEEE Transactions on Inforation Theory vol. 5 no. 6 pp June 005. [4] P. Gupta and P. Kuar The capacity of wireless networks IEEE Transactions on Inforation Theory vol. 46 no. pp March 000. [5] S. R. Kulkarni and P. Viswanath A deterinistic approach to throughput scaling in wireless networks IEEE Transactions on Inforation Theory vol. 50 no. 6 p June 004. [6] M. Franceschetti O. Dousse D. N. C. Tse and P. Thiran Closing the gap in the capacity of wireless networks via percolation theory IEEE Transactions on Inforation Theory vol. 53 no. 3 pp March 007. [7] J. Zhang X. Wang X. Tian Y. Wang X. Chu and Y. Cheng Optial ulticast capacity and delay tradeoffs in MANETs IEEE Transactions on Mobile Coputing vol. 3 no. 5 pp May 04. [8] P. Li Y. Fang and J. Li Throughput delay and obility in wireless ad hoc networks in INFOCOM 00. [9] M. Grossglauser and D. N. Tse Mobility increases the capacity of ad hoc wireless networks in INFOCOM 00. [30] A. E. Gaal J. Maen B. Prabhakar and D. Shah Optial throughput-delay scaling in wireless networks-part i: The fuid odel IEEE Transactions on Inforation Theory vol. 5 no. 6 pp June 006. [3] S. Zhou and L. Ying On delay constrained ulticast capacity of large scale obile ad-hoc networks in INFOCOM c 07 IEEE. Personal use is peritted but republication/redistribution requires IEEE perission. See for ore inforation.