On the exact multicast delay in mobile ad hoc networks with f-cast relay

Ad Hoc Networks xxx (2015) xxx–xxx

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Ad Hoc Networks journal homepage: www.elsevier.com/locate/adhoc

On the exact multicast delay in mobile ad hoc networks with f-cast relay Bin Yang a,b,⇑, Ying Cai c, Yin Chen d, Xiaohong Jiang a,1 a

School of Systems Information Science, Future University Hakodate, Hokkaido 041-8655, Japan School of Computer and Information Engineering, Chuzhou University, Chuzhou 239000, PR China c Beijing Key Laboratory of Internet Culture and Digital Dissemination Research, Beijing Information Science & Technology University, Beijing 100101, PR China d Graduate School of Media and Governance, Keio University, Kanagawa 252-8520, Japan b

a r t i c l e

i n f o

Article history: Received 16 December 2014 Received in revised form 16 March 2015 Accepted 18 April 2015 Available online xxxx Keywords: Mobile ad hoc networks Multicast delay Two-hop relay Packet replication

a b s t r a c t The study of multicast delay performance in mobile ad hoc networks (MANETs) is critical for supporting future multicast-intensive applications in such networks. Different from available works that mainly focus on the study of asymptotic scaling laws of the multicast delay in MANETs, this paper explores the exact multicast delay achievable in MANETs under a general multicast two-hop relay (M2HR)-ðf ; gÞ algorithm with packet replication limit f and multicast fanout g. In such an algorithm, each packet can be replicated up to f distinct relay nodes and it should be delivered to its g destination nodes through either its source node or these relay nodes. We first develop a Markov chain-based theoretical framework to model the complicated packet delivery process under the M2HR-ðf ; gÞ algorithm and then determine some basic probabilities related to packet delivery process. With the help of the theoretical framework and related basic packet delivery probabilities, the analytical models are further derived for both the mean value and variance of exact multicast delay. Finally, simulation and numerical results are provided to illustrate the accuracy of the multicast delay models as well as our theoretical findings. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Mobile ad hoc networks (MANETs) represent a class of self-organized networks where mobile devices communicate with each other via point-to-point wireless channel without any infrastructure support. Multicast in MANETs is a fundamental routing service for supporting many practical applications with one-to-many communication pattern [1–10], like the information exchanges among a group of soldiers in battlefield communication, emergency

⇑ Corresponding author at: Beijing Key Laboratory of Internet Culture and Digital Dissemination Research, Beijing Information Science & Technology University, Beijing 100101, PR China. E-mail addresses: [email protected] (B. Yang), [email protected] (Y. Cai), [email protected] (Y. Chen), [email protected] (X. Jiang). 1 Principal corresponding author.

communications among the rescuers in disaster relief, video conferencing, real-time monitoring, and VoIP. For an efficient support of these critical multicast-intensive applications in the future MANETs, multicast delay analysis in such networks has been a critical research issue, where multicast delay is defined as the time it takes for a packet to be delivered out to all its destination nodes. However, the multicast delay analysis is extremely complicated because of dynamic network topology and multiple destination nodes associated with each node. By now, the multicast delay performance still remains largely unexplored in MANETs. Recently, some research has reported the asymptotic bounds on the multicast delay in MANETs. Wang et al. showed in [11,12] that by adopting packet replication pffiffiffiffiffiffiffiffiffiffiffiffiffiffi technique in MANETs, a multicast delay of Hð n log kÞ is achievable under a two-hop relay algorithm, which is

http://dx.doi.org/10.1016/j.adhoc.2015.04.005 1570-8705/Ó 2015 Elsevier B.V. All rights reserved.

Please cite this article in press as: B. Yang et al., On the exact multicast delay in mobile ad hoc networks with f-cast relay, Ad Hoc Netw. (2015), http://dx.doi.org/10.1016/j.adhoc.2015.04.005

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algorithms: one-hop relay, two-hop relay without packet replication and two-hop relay with packet replications, respectively, where the network area is first evenly divided into n2c cells and each cell is then divided into n2x equal subcells ðc; x P 0; c þ x > 1=2Þ. The main difference between the M2HR-ðf ; gÞ algorithm in this paper and those two-hop relay algorithms in Refs. [14,15] is that the direct source-to-destination transmission is incorporated into the M2HR-(f, g) algorithm, while it was neglected in Refs. [14,15]. We note that although asymptotic results can help us understand how the multicast delay varies with network size and the number of destination nodes associated with each source node, they cannot be used to estimate the actually achievable delay performance, which provides more meaningful insights for network designers. Recently, Li et al. in [16] studied the exact multicast delay with the help of a Makov chain model and showed how the selfish behaviors of nodes affect the delay performance in DTNs, i.e., a class of very sparse MANETs where the interference is neglected. It is notable that the available research on MANET multicast delay investigated either the asymptotic multicast delay or the exact multicast delay in special MANETs where the interference and medium access contention are largely neglected, therefore these results cannot be used to estimate the actual multicast delay performance in general MANETs. In this paper, we study the exact multicast delay performance in a general MANET where both the interference and medium access control are taken into account. The main contributions of this paper are summarized as follows.  We first develop a finite-state absorbing Markov chain-based theoretical framework to model the complicated packet delivery process under the M2HR-ðf ; gÞ algorithm in the considered MANET.  We determine some basic probabilities related to packet delivery process, where the important issues of wireless interference and medium access contention in such network are carefully considered in these

probabilities. Based on the theoretical framework and these basic packet delivery probabilities, the analytical models are further derived for both the mean value and variance of exact multicast delay.  Extensive simulation and numerical results are presented to validate the accuracy of the multicast delay models and to explore how the multicast delay performance varies with system parameters. The rest of this paper is organized as follows. In Section 2, we introduce system models. Section 3 first introduces the M2HR-ðf ; gÞ algorithm, and then develops a finite-state absorbing Markov chain-based theoretical framework and also determines some basic packet delivery probabilities. Section 4 further derives the analytical models for both the mean and variance of exact multicast delay. Simulation/numerical studies are provided in Section 5. Finally, we conclude this paper in Section 6. 2. System models In this section, we first introduce network, communication, traffic and transmission scheduling models and then give the definition of multicast delay involved in this study. 2.1. Network model We consider a network consisting of n nodes that move inside a unit square region with torus boundaries, i.e., a node goes across one edge and then appears on the opposite edge of the square region. Similar to previous studies [17–23], the network is divided into m  m non-overlapping cells of equal size (see Fig. 1). The time is slotted and the nodes move independently from cell to cell in the network according to the independent and identically distributed (i.i.d.) mobility model [17,22,24]. Under this mobility model, each node selects a cell from the m2 cells with equal probability (1=m2 ) at the beginning of each time slot, and then moves into and stays at it during the time slot. Each node will repeat this mobility process in every subsequent time slot.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

(1 + Δ) ⋅ r •

better than the Hðn log kÞ delay reported without packet replication, where n represents the number of nodes in the considered networks and k is the number of destination nodes associated with each source node.2 Wang et al. also showed in [13] packet replication technique can improve the multicast delay performance in MANETs under two different mobility models, where nodes move either in a local region or in a global region. Later, Wang et al. found in [14] that under the two-hop relay algorithm with packet replications, cooperations among destination nodes can achieve pffiffiffi the multicast delay smaller than Hð nÞ in MANETs. More recently, Liu et al. studied in [15] the asymptotic multicast delay in sparse MANETs and showed that the multicast delay can achieve Xðlog k  n2ðcþxÞ Þ; Xðlog k  n2ðcþxÞ Þ and  n o  nk O max log nkf ; logf k  n2ðcþxÞ under three packet delivery

•

R0

S1

•

2

S0

α α

2

Let f ðnÞ and gðnÞ denote two non-negative functions. Then we say that: (1) f ðnÞ ¼ OðgðnÞÞ if there exist a positive integer N and a positive constant c such that for all n > N; f ðnÞ 6 cgðnÞ; (2) f ðnÞ ¼ XðgðnÞÞ if gðnÞ ¼ Oðf ðnÞÞ; (3) f ðnÞ ¼ HðgðnÞÞ if f ðnÞ ¼ OðgðnÞÞ and gðnÞ ¼ Oðf ðnÞÞ.

Fig. 1. Network model and an example of transmission scheduling with the case of m ¼ 15 and a ¼ 5.

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2.2. Communication model A half-duplex medium is shared by all the nodes for data communication. To address the interference among simultaneous transmissions over the shared medium, we adopt the protocol model defined in [25] to decide whether a transmission between a pair of nodes is successful or not due to the interference among simultaneous transmissions. The protocol model is defined as follows: at an arbitrary time slot t, a transmission from a transmitting node i (transmitter) to its receiving node j (receiver) is successful iff the following two conditions hold for any other simultaneously transmitting node k: (1) The node j is within the transmission range of the node i. (2) dkj ðtÞ P ð1 þ DÞdij ðtÞ.

opportunity) alternately. At a time slot, if more than one nodes are residing in an active cell, only one node is randomly selected as the transmitter. To guarantee that one node in each active cell can transmit simultaneously without corrupting each other’s transmission, the parameter a should be selected appropriately. Suppose that at some time slot, a transmitting node S0 can successfully transmit a packet to a receiving node R0 . First, as shown in Fig. 1, another simultaneously transmitting node S1 is with a distance at least ða  2Þ=m away from R0 . Then, according to the protocol model [25], we have pffiffiffi ða  2Þ=m P ð1 þ DÞ  r. Since r ¼ 8=m represents the possible maximum distance between S0 and R0 , then pffiffiffi a P ð1 þ MÞ 8 þ 2. a is an integer less than or equal to m. To maximize the number of active cells ðm2 =a2 Þ in one transmission-group, we obtain

a ¼ min Here, dij ðtÞ denotes the Euclidean distance between nodes i and j at time slot t, and D > 0 models the guard zone surrounding each receiver. Similar to previous studies [26,27], the transmission range of each node covers the cell where it resides and the eight neighbor cells (two cells are said to be neighbor cells if they share a common point). Therefore, the possible maximum distance r between any two nodes can be deterpffiffiffi mined as r ¼ 8=m. The total amount of data transmitted is fixed and normalized to one packet during a successful transmission. 2.3. Traffic model We consider a multicast traffic pattern similar to that of [11,28], where all nodes in the network are divided into different multicast groups, each of which consists of g þ 1 nodes3 and in a specific multicast group, each node is a source node that transmits its packets to other g destination nodes within this multicast group, and is also a relay node that helps to forward packets from other multicast groups. We called a source node and its g destination nodes as a multicast session. Therefore, there are g þ 1 multicast sessions in a multicast group and n multicast sessions in the network.

3

nl o pffiffiffim ð1 þ DÞ 8 þ 2; m ;

ð1Þ

where de is the ceiling function. Notice that m is a design parameter. We will discuss its impact on the multicast delay performance in Section 5.3. The parameter g is determined by applications’ requirement of multicast group size. The guard factor D in protocol model is a design parameter mainly determined by the communication bit rate and physical environment [25]. 2.5. Performance metric Definition of multicast delay: For a packet at a source node, the multicast delay of the packet is defined as the time duration from the time slot when the source node starts to transmit the first copy of the packet to the time slot when all the g destination nodes have received the packet. 3. M2HR-ðf ; gÞ algorithm and Markov chain-based theoretical framework In this section, we will present a general M2HR-ðf ; gÞ algorithm and develop a Markov chain-based theoretical framework and some basic results. This framework and related results will help us to study the exact multicast delay performance.

2.4. Transmission scheduling model 3.1. M2HR-ðf ; gÞ algorithm To schedule as many simultaneous transmissions as possible, a transmission-group based scheduling scheme [17,23,26,27,29] is adopted for the medium access control (MAC). With this scheduling scheme, all m2 cells in the network are divided into a2 transmission-groups. Each group consists of those cells with a vertical or horizontal distance between any two cells is some integer multiple of a cells. Fig. 1 shows an example of m ¼ 15 and a ¼ 5, where there are 25 transmission-groups in total, and the dark gray cells all represent one transmission-group marked by 1. In every a2 time slots, each transmission-group (and also each cell inside it) will become active (i.e., get transmission 3 The number of nodes in the network is approximately equal to some integer multiple of g þ 1.

Without loss of generality, we focus on a tagged multicast session of a multicast group and denote its source node and destination nodes as S and D1 ; D2 ; . . . ; Dg , respectively. As illustrated in Fig. 2, under the M2HR-ðf ; gÞ algorithm, the source node will replicate a packet P to at most f distinct relay nodes (i.e., R1 ; R2 ; . . . ; Rf ). Each of the destination nodes may receive the packet from either the source node or one of the relay nodes that carry this packet. Notice that each node can be a potential relay for the n  ðg þ 1Þ multicast sessions of other multicast groups (except the multicast group including itself). To support the operation of the M2HR-ðf ; gÞ algorithm, we assume that each node has n  g þ 1 individual queues in its buffer: One

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Dg

P

R4

P

P

P

S

P

D5

R2

P

P P

P

R1

Rf D6

Procedure 1. Source-to-destination transmission:

D1

P

P

D2

D3

R3

D4

P

1. S randomly selects a node Di over all possible destination nodes in its transmission range; 2. S initiates a handshake with Di to obtain the RNðDi Þ; 3. if TNðSÞ ¼¼ RNðDi Þ then 4. S transmits a copy of the packet with TNðSÞ to Di from its local-queue; 5. else if TNðSÞ > RNðDi Þ then 6. S transmits a copy of the packet with RNðDi Þ to Di from its already-transmitted queue; 7. else 8. S transmits a copy of the packet with RNðDi Þ to Di from its local-queue; 9. end if

Fig. 2. Illustration of the M2HR-ðf ; gÞ algorithm.

local-queue to store the locally generated packets waiting for their copies to be transmitted, one already-transmitted queue to store the packets whose f copies have already been transmitted to distinct relay nodes but this node has not confirmed that its g destination nodes have received the packet, and n  ðg þ 1Þ relay-queues to store the packets for the multicast sessions of other multicast groups (one queue per multicast session). In each multicast session, each node, e.g., S, labels each packet in its local-queue with a transmit number and let TNðSÞ denote the transmit number of the head-of-line packet. Similarly, each destination node, e.g., Di (1 6 i 6 g), also holds a request number RNðDi Þ equal to the transmit number of the packet it is currently requesting, so that each packet will be received in order at the destination node Di and Di has already received all packets with transmit numbers less than RNðDi Þ. When S obtains a transmission opportunity via the transmission scheduling scheme introduced in Section 2.4, it will perform the operation of the M2HR-ðf ; gÞ algorithm summarized in Algorithm 1. Algorithm 1. M2HR-ðf ; gÞ algorithm: 1. if there exists destination node(s) in the transmission range of S then 2. S conducts source-to-destination transmission (see Procedure 1); 3. else 4. S randomly selects a node Ri (1 6 i 6 n  ðg þ 1Þ) from the nodes in its transmission range; 5: S flips a fair coin (i.e., the probability of head or tail is 1/2); 6. if it is the head then 7. S conducts source-to-relay transmission (see Procedure 2); 8. else 9. S conducts relay-to-destination transmission (see Procedure 3); 10. end if 11. end if

Procedure 2. Source-to-relay transmission: 1. if Ri (as a relay) does not carry a copy of the head-of-line (HOL) packet in the local-queue of S then 2. S transmits a copy of the HOL packet to Ri ; 3. if f copies of the HOL packet have been transmitted to distinct relay nodes then 4. S removes the HOL packet from its local-queue and then inserts it into the end of its already-transmitted queue; 5. end if 6. else 7. S remains idle; 8. end if

Procedure 3. Relay-to-destination transmission: 1. S initiates a handshake with Ri to obtain the RNðRi Þ; 2. if there exists a packet with RNðRi Þ in S’s relay-queue that is randomly selected from its g relay-queues intended for Ri then 3. S transmits a copy of the packet to Ri ; 4. else 5. S remains idle; 6. end if

3.2. Markov chain-based theoretical framework For a given packet associated with the tagged multicast session, we use a 2-tuple ði; jÞ to denote a general transient state during the packet delivery process, where the i ð0 6 i 6 f Þ and j ð0 6 j 6 gÞ denote the number of relay nodes that carry a copy of the packet and the number of destination nodes that have received the packet at current time slot, respectively. According to the M2HR-ðf ; gÞ algorithm, as illustrated in Fig. 3, when the considered multicast session is in state ði; jÞ at the current time slot, one of the following transition cases will happen:

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Fig. 3. The transition diagram of a general transient state (i; j).

 SD case: source-to-destination transmission only, i.e., S successfully transmits the packet to some destination node that does not receive it, while none of relay nodes transmits the packet to any of destination nodes. Under such a transition case, the state ði; jÞ will transit to ði; j þ 1Þ.  SR case: source-to-relay transmission only, i.e., S successfully transmits a copy to some relay node that does not carry it, while none of relay nodes transmits the packet to any of destination nodes. The state ði; jÞ will transit to ði þ 1; jÞ under the SR case.

5

 ðRDÞk case: k relay-to-destination transmissions only, i.e., k relay-to-destination transmissions happen simultaneously where each transmission represents that a relay node successfully transmits the packet to some destination node that does not receive it, while other transition cases such as SD and SR do not happen. Under the ðRDÞk case, the state ði; jÞ will transit to one element of state set {ði; j þ kÞ : 1 6 k 6 i and j þ k 6 g}.  SD þ ðRDÞk case: a source-to-destination transmission and k relay-to-destination transmissions only, i.e., these k þ 1 transmissions happen simultaneously, while the SR case does not happen. Under this transition case, the state ði; jÞ will transit to one element of state set {ði; j þ k þ 1Þ : 1 6 k 6 i and j þ k þ 1 6 g}.  SR þ ðRDÞk case: a source-to-relay transmission and k relay-to-destination transmissions only, i.e., these k þ 1 transmissions happen simultaneously, while the SD case does not happen. Under such a transition case, the target state is one element of state set {ði þ 1; j þ kÞ : 1 6 k 6 i and j þ k 6 g}. If we use ði; gÞ to denote an absorbing state that each of the g destination nodes has received the packet when there are i relay nodes carrying a copy of the packet, then the transition diagram in Fig. 3 indicates that we can develop a discrete-time finite-state absorbing Markov chain illustrated in Fig. 4 to model the packet delivery process. The transitions of SD; SR; ðRDÞk ; SD þ ðRDÞk and SR þ ðRDÞk cases in Fig. 4 correspond to the M2HR-ðf ; gÞ transmissions of source-to-destination, source-torelay, k relay-to-destination, source-to-destination and k relay-to-destination, and source-to-relay and k relay-to-destination, respectively.

Fig. 4. Absorbing Markov chain for the general M2HR-ðf ; gÞ algorithm. For simplicity, the transition back to itself is not shown for each transient state.

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3.3. Basic results Based on the Markov chain framework in Fig. 4, we have the following results. Lemma 1. For a time slot and a tagged multicast session with source node S and g destination nodes, if we denote by p1 the probability that S conducts a source-to-destination transmission, and denote by p2 the probability that S conducts a source-to-relay transmission, then we have

k  gk ng1 g   X 1 X g 1 1 1 p1 ¼ 2 1   uðiÞ 2 2 m kþiþ1 a k¼1 k m i¼0 ! k  gk ng1 g   X X g 8 9 1 ; 1   u ðiÞ þ m2 m2 iþ1 k i¼0 k¼1 ð2Þ p2 ¼

 g ng1 ng1i X  n  g  1  1 i  9 1 1 1 2 2 2 a m m m i þ 1 i i¼1 ng1i ! ng1 X  n  g  1  8 i  9 1 2 ; þ m2 m i i¼1 1

2



1

ð3Þ

where uðiÞ ¼



 ng1i n  g  1  1 i  1  m12 . m2 i

the probability that S will successfully deliver a copy of the packet to a relay node (i.e., a successful relay-to-destination transmission). Then we have

PSD ðl1 Þ ¼ PSR ðl3 Þ ¼

l1 g

p1 ;

ð7Þ

l3 2ðn  g  1Þ

p2 :

ð8Þ

Lemma 4. For a tagged multicast session and a given packet, we use PRD ðx; l1 ; l2 Þ to denote the probability that x successful relay-to-destination transmissions will occur simultaneously in the next time slot, where 1 6 x 6 minfl1 ; l2 g. The probability P RD ðx; l1 ; l2 Þcanbe determined as

PRD ðx; l1 ; l2 Þ ¼



l2



x

l1

x   k1 k2    ki    kx ; g x

where 8   li  X X > li wi wi 2 2 > > m a2 ði1Þ wðki ;hi Þ; if 1 6 i 6 x  1; > 2 a m 2 > < ki h ¼1 hi ki ¼0 i ki ¼   li  X > 2 2 X >  z li wi wi > m a ði1Þ > wðki ;hi Þ 1  m9x2 ; if i ¼ x: > 2 2 : 2a m h k i i h ¼1 k ¼0 i

i

ð10Þ

Pi

Lemma 2. For a transient state ði; jÞ of the Markov chain framework in Fig. 4 ð0 6 i 6 f ; 0 6 j 6 g  1Þ and a given packet, we use l1 to denote the number of destination nodes that do not receive the packet, use l2 to denote the number of relay nodes that carry a copy of the packet, and use l3 to denote the number of relay nodes that do not carry a copy of the packet under the transient state. Then we have

l1 ¼ g  j; l2 ¼ i; l3 ¼ n  g  1  i:

ð9Þ

ð4Þ ð5Þ ð6Þ

In current time slot, suppose that the Markov chain in Fig. 4 is in the transient state ði; jÞ, then we establish the following Lemmas. Lemma 3. For the next time slot, we use PSD ðl1 Þ to denote the probability that S will successfully deliver a copy of the packet to a destination node (i.e., a successful source-to-destination transmission), use PSR ðl3 Þ to denote

Pi

Here, li ¼ n  2g  x  j¼1 kj1 ; k0 ¼ 0; wi ¼ g  j¼1 hj1 ;   Px Pki ki h0 ¼ 0; z ¼ n  x  j¼1 ðkj þ hj Þ and wðki ; hi Þ ¼ k¼0 k   Phi hi  1 kþh  8 ki þhi kh 1 hi . h¼0 kþhþ1 ki þhi m2 h m2 Lemma 5. For a tagged multicast session and a given packet, we use P SD;RD ðx; l1 ; l2 Þ to denote the probability that a successful source-to-destination transmission and x successful relay-to-destination transmissions will occur simultaneously in the next time slot, where 1 6 x 6 minfl1  1; l2 g. Then we have  

PSD;RD ðx; l1 ; l2 Þ ¼



l2 x



l1

xþ1   q1 q2    qi    qxþ1 ; g xþ1 ð11Þ

where

    8 wi li  1 Pwi > m2 a2 ði1Þ Pli 1 > wðki ; hi Þ; if 1 6 i 6 x; > 2a2 m2 hi ¼1 ki ¼0 > > hi k i > >     < Pw i Pli þg1 li þ g  1 wi qi ¼ m2 a2 2 ði1Þ wðki ; hi Þ 2 ¼0 h ¼1 k a m > i i > hi k i > > >   > zki hi 1 > :  ki þhi 1  9ðxþ1Þ ; if i ¼ x þ 1: h m2

ð12Þ

i

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Here, li ; wi , z and wðki ; hi Þ are defined after (10). Lemma 6. For a tagged multicast session and a given packet, we use P SR;RD ðx; l1 ; l2 ; l3 Þ to denote the probability that a successful source-to-relay transmission and x successful relay-to-destination transmissions will occur simultaneously in the next time slot, where 1 6 x 6 minfl1 ; l2 g. Then we have

PSR;RD ðx; l1 ; l2 ; l3 Þ ¼



  l1  l3 l2 x   h1 h2    hi    hxþ1 ; g x x ð13Þ

where

Let t i denote the time that the Markov chain in Fig. 4 takes to arrive at an absorbing state given that the chain starts from the ith transient state ð1 6 i 6 bÞ. It is notable that the 1st transient state ð0; 0Þ indicates that the source node starts to transmit the first copy of the packet, and an absorbing state corresponds to that all the g destination nodes have received the packet. Thus, the expectation E½t 1  just corresponds to the expected packet multicast delay under the M2HR-ðf ; gÞ algorithm. To derive E½t 1 , we first determine the vector t ¼ ðE½t 1 ; E½t2 ; . . . ; E½tb ÞT . We define a matrix P ¼ ðqi;j Þðbþf þ1Þðbþf þ1Þ and its submatrix Q which consists of rows 1 through b and columns 1 through b of P, where the ij-entry qi;j of P denotes the transition probability from the ith state to the jth state in Fig. 4 ð1 6 i; j 6 b þ f þ 1Þ. Based on the definition of t i , we have

    8 wi li  2 Pwi > m2 a2 ði1Þ Pli 2 > wðki ; hi Þ; if 1 6 i 6 x; > 2 2 hi ¼1 ki ¼0 > 2a m > hi k i > >       < Pki Pli þg2 li þ g  2 1  1 kþr  8 ki þ1kr ki P1 hi ¼ m2 a22 ði1Þ 2 2 r¼0 k¼0 ki ¼0 m2 2a m > > r m ki k > > >  zki 2 > > 1 :  kþrþ1 k 1þ1 1  9ðxþ1Þ ; if i ¼ x þ 1: m2

ð14Þ

i

Here, li ; wi , z and wðki ; hi Þ are defined after (10). The basic idea of the proof of Lemma 4 is summarized as follows. For x relay nodes carrying a copy of the packet of the tagged multicast session, we first show how the probability P RD ðx; l1 ; l2 Þ is related to the probability that these x relay nodes conduct x relay-to-destination transmissions simultaneously. Then the probability is derived as the product of the probabilities that one of these x relay nodes conducts a relay-to-destination transmission, which is determined based on the probabilities of its sub-events. Finally, by summarizing these results, P RD ðx; l1 ; l2 Þ can be derived. The derivations of Lemmas 5 and 6 are similar. The detailed proofs of these Lemmas can be found in Appendix A. 4. Packet multicast delay modeling In this section, we analyze both expected value and variance of packet multicast delay under the M2HR-ðf ; gÞ algorithm. 4.1. Expected packet multicast delay

E½t i  ¼

bþf þ1 X

qi;j ð1 þ E½tj Þ ¼ 1 þ

j¼1

ð15Þ

j¼1

Notice that since the jth state is an absorbing state when b þ 1 6 j 6 b þ f þ 1, we have E½t j  ¼ 0. Then, (15) can be expressed as

t¼bþQ t

ð16Þ

where b is the b  1 column vector with all entries being 1, i.e., b ¼ f1; 1; . . . ; 1gT . Thus, we have

t ¼ ðI  Q Þ1  b

ð17Þ

where I is a b  b identity matrix. We use N ¼ ðNi;j Þbb to denote the fundamental matrix of the Markov chain in Fig. 4 ð1 6 i; j 6 bÞ. According to the Markov chain theory [30], we have

N ¼ ðI  Q Þ1 :

ð18Þ

By substituting (18) into (17), we have

t ¼ N  b: For the Markov chain in Fig. 4, we use b to denote the total number of transient states, which can be determined as b ¼ gðf þ 1Þ. These b transient states are arranged into g rows and indexed as 1, 2, . . . ; b in a left-to-right and top-to-down fashion. Similarly, the f þ 1 absorbing states are indexed as b þ 1; b þ 2; . . . ; b þ f þ 1 in a left-to-right fashion.

b X qij E½t j :

ð19Þ

From (19), the expected packet multicast delay E½t1  is determined as

E½t 1  ¼

b X N1;j :

ð20Þ

j¼1

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8

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qc;cþ1 ¼ PSR ðl3 Þ;

4.2. Delay variance The variance Var½t 1  of packet multicast delay is given

ð26Þ When ði; jÞ transits to ði; j þ 1Þ, then (

by



Var½t1  ¼ E t21  ðE½t1 Þ2 :

ð21Þ

if c mod ðf þ 1Þ – 0 ði:e:; i! ¼ f Þ:

qc;cþf þ1 ¼

P SD ðl1 Þ;

if c  ðf þ 1Þ

j

c f þ1

k

¼ 1 ði:e:; i ¼ 0Þ;

P SD ðl1 Þ þ PRD ð1; l1 ; l2 Þ; otherwise; ð27Þ

Since the E½t1  can be determined by (20), we only need to

derived the E t 21 here. Based on the definition of t i , we have



E t2i ¼

bþf þ1 X

qi;j E½ð1 þ t j Þ2 

j¼1

¼

b b h i X X qi;j E t2j þ 2 qi;j E½t j  þ 1 j¼1

qc;cþkðf þ1Þ ¼ PSD;RD ðk  1; l1 ; l2 Þ þ PRD ðk; l1 ; l2 Þ;  c – 1: ð28Þ if 2 6 k 6 minfl1 ; l2 g and c  ðf þ 1Þ f þ1 When ði; jÞ transits to ði þ 1; j þ kÞ, then

ð22Þ

j¼1

Since the jth state is an absorbing state when h i b þ 1 6 j 6 b þ f þ 1, we have E½t j  ¼ 0 and E t2j ¼ 0.

qc;c

where bhc is the largest integer not greater than h. When ði; jÞ transits to ði; j þ kÞ,

qc;cþ1þkðf þ1Þ ¼ PSR;RD ðk; l1 ; l2 ; l3 Þ; if 1 6 k 6 minfl1 ; l2 g;  c – 1 and c mod ðf þ 1Þ – 0: ð29Þ c  ðf þ 1Þ f þ1 When ði; jÞ transits to itself,

8 j k c > if c  ðf þ 1Þ f þ1 ¼ 1; > > 1  PSD ðl1 Þ  PSR ðl3 Þ; > > > P > minf l ; l g 1 2 > 1  PSD ðl Þ  PRD ðk; l1 ; l2 Þ > 1 > k¼1 > > < Pminfl1 1;l2 g  k¼1 P SD;RD ðk; l1 ; l2 Þ; if c mod ðf þ 1Þ ¼ 0; ¼ Pminfl1 ;l2 g > > PRD ðk; l1 ; l2 Þ > 1  PSD ðl1 Þ  PSR ðl3 Þ  k¼1 > > > > > Pminfl1 1;l2 g P SD;RD ðk; l ; l Þ > 1 2 > > Pk¼1 > : minfl ;l g otherwise:  k¼1 1 2 PSR;RD ðk; l1 ; l2 ; l3 Þ;

 T Let t ¼ t21 ; t22 ; . . . ; t2b , then we can rewritten (22) as

t ¼ Q  t þ 2Q  t þ b:

ð23Þ

Substituting (19) into (23), we have 

t ¼ Nð2Q  N þ IÞb: Then

E½t21 

ð24Þ

can be determined as



E t21 ¼ e  t :

ð25Þ

where e ¼ ð1; 0; . . . ; 0Þ. To calculate the values of both E½t 1  and Var½t 1 , we only need to derive the matrix Q . 4.3. Derivation of matrix Q Recall that the entry qc;d of Q represents the transition probability from the cth transient state to the dth transient state in the Markov chain of Fig. 4 ð1 6 c; d 6 bÞ. Suppose that the cth transient state is the transient state ði; jÞ, where 0 6 i 6 f and 0 6 j 6 g. Based on the Markov chain structure and some related basic results derived in Section 3.3, we can calculate non-zero qc;d as follows. When ði; jÞ transits to ði þ 1; jÞ,

ð30Þ

5. Numerical results In this section, we first provide simulation results to validate the accuracy of the analytical multicast delay models, and then apply the theoretical results to explore how the system parameters would affect the packet multicast delay performance in the considered MANETs. 5.1. Simulator setting A simulator was developed to simulate the packet delivery process under the M2HR-ðf ; gÞ algorithm and the system models considered in this paper. In addition to the i.i.d mobility model, we also implemented the random walk and random waypoint models. Random walk model [21]: At the beginning of every time slot, each node either moves to one of its eight neighboring cells or remains in its current cell. Each of these nine cells is independently selected with equal probability (1/9) by each node. Random waypoint model [6]: At the beginning of every time slot, each node first independently selects two values equal to dh and dv uniformly from ½1=m; 3=m, and then moves a horizontal distance dh and a vertical distance of dv .

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9

(a) Expected multicast delay E[t1 ] vs. number of destination nodes g

(b) Relative standard deviation δ vs. number of destination nodes g Fig. 5. Model validation through comparison between theoretical and simulation results.

 as the expected multicast delay obtained We denote y from simulation, where is calculated as the average value of 107 random and independent simulation results. We denote s as the related sample standard deviation, which is calculated as

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X107 1  Þ2 ; s¼ ðyi  y 7 i¼1 10  1

of g.4 The corresponding simulation results and the theoretical ones are summarized in Fig. 5, where each simulation result of relative standard deviation d is obtained from (32) and each theoretical one of d is calculated as

d¼ ð31Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½t1  : E½t 1 

ð33Þ

5.2. Model validation

Fig. 5 indicates clearly that the simulation results under the i.i.d mobility model agree very well with the theoretical ones, indicating that our theoretical results can accurately capture the multicast delay performance under the M2HR-ðf ; gÞ algorithm. We can also see from Fig. 5(a) that as the number of destination nodes g increases, the multicast delay E½t1  will increase. This is because for the network scenario there, the time that all g destination nodes take to receive an identical packet, will increase with g, so the multicast delay of the packet will increase. On the

To validate the accuracy of multicast delay analysis, we conduct simulations for a network scenario with n ¼ 100; m ¼ 16; f ¼ 4; D ¼ 1 and different values

4 Since D is set as 1, the transmission-group parameter a is determined as a ¼ minf8; mg paccording nl o to the following formula defined in (1): ffiffiffim a ¼ min ð1 þ DÞ 8 þ 2; m .

where yi is the multicast delay in the ith simulation. The simulated relative standard deviation d is then obtained according to the following formula:

s d¼ :  y

ð32Þ

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(a) E[t1 ] vs. f

(b) δ vs. f Fig. 6. Multicast delay vs. packet replication limit f.

other hand, we can see that the increase of g leads to the decrease of the corresponding relative standard deviation d in Fig. 5(b). It is notable that in Figs. 5(a) and (b), the simulation results under the random walk model show very similar multicast delay behaviors with the theoretical results under the i.i.d. mobility model. While those under the random waypoint model are slightly different from those under the i.i.d. mobility model, but they well approximate the general trends of E½t1  and d. 5.3. Performance analysis Based on our theoretical results, we first explore the impact of f on the performance ðE½t 1 ; dÞ. For the network scenarios of n ¼ f50; 300; 600g and setting of m ¼ 16; g ¼ 4 and D ¼ 1, we summarize in Fig. 6 how

E½t 1  and d vary with f. It can be observed from Fig. 6 that as f increases, both E½t1  and d monotonously decrease. This is mainly due to that the number of relay nodes that carry a copy of an identical packet increases with f, which leads to more opportunities that the destination nodes receive the packet from the relay nodes, and thus a lower multicast delay. The result in Fig. 6 indicates that the packet replication technique could efficiently support these important applications with stringent multicast delay/variance requirements in future MANETs, such as military communication, emergency disaster relief, real-time monitoring and video streaming. To understand the impact of n on the performance ðE½t1 ; dÞ, we summarize in Fig. 7 how E½t 1  and d vary with network setting of m ¼ f16; 24; 32g; f ¼ 5; g ¼ 4 and D ¼ 1. We can see from Fig. 7 that as n increases, both E½t 1  and d first decrease and then increase, and there exists

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11

(a) E[t1 ] vs. n

(b) δ vs. n Fig. 7. Multicast delay vs. number of nodes n.

an optimal value of n to achieve the minimum E½t 1  or d. This is because the effects of n on the performance are two folds. On one hand, when the network is sparse, a bigger n will result in a higher packet delivery speed at which a packet is distributed, and thus a lower multicast delay. On the other hand, when the network becomes relatively crowded, a bigger n will result in a lower packet delivery speed due to the negative effects of wireless interference and medium access contention issues, and thus a higher multicast delay. Another observation from Fig. 7 is for each fixed setting of n, a larger value of m leads to a higher E½t1 . This observation can be explained as follows. Recall that in our study, the considered network area is evenly divided into m  m cells of equal size. Under the same setting of n, a larger value of m leads to a lower node density (i.e., n=m2 ) and thus a more sparse network. Since the packet delivery speed becomes lower

in a more sparse network, the multicast delay becomes higher for a larger m. The results in Fig. 8 summarize how E½t1  and d vary with D. We can see from Fig. 8(a) that for each setting of g there, E½t 1  is a piecewise function of D, and as D increases, E½t 1  monotonically increases and there exists a threshold value of D, beyond which E½t1  will converge to a constant value. This can be explained as follows. When D is relatively small, we can see from the formula (1) that increasing D will increase the number of transmission-groups (i.e., a2 ) and will also decrease the number of cells (i.e., m2 =a2 ) in each transmission-group. This will lead to a decrease in the transmission opportunity of each node and thus a higher multicast delay. A careful observation from the formula (1) that both the numbers of transmission-groups and cells in each transmission-group remain unchanged for a small range

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B. Yang et al. / Ad Hoc Networks xxx (2015) xxx–xxx

(a) E[t1 ] vs. Δ

(b) δ vs. Δ Fig. 8. Multicast delay vs. guard factor D.

of D, which will lead to a constant value of E½t 1  in the small range of D. Thus, E½t1  is a piecewise function of D. When D further increases such that a2 ¼ m2 , the number of cells in each transmission-group achieves a minimal value 1 and remains unchanged, thus E½t1  converges to a maximal constant value. Interestingly, Fig. 8(b) illustrates that for each setting of g; d remains unchanged as D increases. 6. Conclusion In this paper, we derived the analytical models for both the mean and variance of the exact multicast delay in a general MANET where the interference and medium access control are taken into account. Extensive simulations show that our theoretical framework can efficiently capture the packet delivery process and thus accurately predicts the

packet multicast delay/variance performance. Our results indicate that packet replication technique can remarkably decrease packet multicast delay and variance, which provides an efficient support for these critical applications with stringent multicast delay/variance requirements in future MANETs. It is expected that our study will help network designers to determine a suitable network size, so as to minimize the packet multicast delay and variance while simultaneously meet a given multicast delay/variance performance requirement. Acknowledgments This work was supported by the NSF of China Grant 61472057, the NSF of Anhui Education Department Grants KJ2014A179, KJ2013B188 and KJ2011ZD06, Chuzhou University Grants 2014qd013, 2013RC005 and

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2012kj002Z, and the Talented Team of Computer System Architecture. Appendix A. Proofs of the Lemmas 1–6

1 is selected as a transmitter is iþ1 . For the latter one, the probability is 1. By summing over the joint probability of these events under the former one and that under the latter one, p2 then follows.

A.1. Proof of Lemma 1

A.2. Proof of Lemma 2

We first derive p1 . For the tagged multicast session with source node S and g destination nodes, at a time slot, S can conduct a source-to-destination transmission iff the following four events occur: (1) S is in an active cell. (2) There are k ðk P 1Þ destination nodes in the same active cell as S or in its eight neighbor cells. (3) There are i ð0 6 i 6 n  g  1Þ other nodes in the active cell (except S and its g destination nodes). (4) S is selected as a transmitter. We use E1 ; E2 ; E3 and E4 to denote these four events, respectively. Note that E2 consists of two mutually exclusive sub-events denoted by E20 and E200 , where E20 represents that these k destination nodes are in the same active cell as S and E200 represents that they are in the eight neighbor cells of the active cell. For the former sub-event, 1 the probability that S is selected as a transmitter is kþiþ1 , then the joint probability PðE1 ; E20 ; E3 ; E4 Þ can be determined as

Under the transient state ði; jÞ in the Markov chain of Fig. 4, we can see that the number of destination nodes that have received the packet is j, and a multicast session has g destination nodes, thus (4) follows. Since all the i relay nodes carry a copy of the packet under the transient state, (5) follows. For each multicast session, all the n  g  1 relay nodes help to forward copies of the packet to destination nodes. Since l2 þ l3 ¼ n  g  1; (6) then follows.

PðE1 ; E20 ; E3 ; E4 Þ ¼ PðE1 ÞPðE20 jE1 ÞPðE3 jE1 ; E20 ÞPðE4 jE1 ; E20 ; E3 Þ k  gk g   m2 X g 1 1 2 ¼ a2 1 2 2 m m m k¼1 k 

ng1 X i¼0

1 uðiÞ kþiþ1 ðA:1Þ



 ng1i n  g  1  1 i  where uðiÞ ¼ 1  m12 . For the latm2 i ter sub-event, the probability that S is selected as a trans1 , then the joint probability PðE1 ; E200 ; E3 ; E4 Þ can mitter is iþ1 be determined as

PðE1 ; E200 ; E3 ; E4 Þ ¼ PðE1 ÞPðE200 jE1 ÞPðE3 jE1 ; E200 ÞPðE4 jE1 ; E200 ; E3 Þ k  gk g   m2 X g 8 9 2 ¼ a2 1  m2 m2 m k¼1 k 

ng1 X i¼0

uðiÞ

1 iþ1 ðA:2Þ

where uðiÞ is the same as that of (A.1). Then p1 follows by summing over these two probabilities of (A.1) and (A.2). We proceed to derive p2 . S can conduct a source-to-relay transmission at a time slot iff the following four events occur: (1) S is in an active cell. (2) None of the g destination nodes is in the same active cell as S or its eight neighbor cell. (3) There are i (1 6 i 6 n  g  1) other nodes in the active cell or in its eight neighbor cell (except S). (4) S is selected as a transmitter. We note that the 3rd event consists of two mutually exclusive sub-events: these i other nodes are either in the active cell or in its eight neighbor cells. For the former one, the probability that S

A.3. Proof of Lemma 3 Given l1 destination nodes that have not received the packet in current time slot, the source node S may deliver a copy of the packet to one of the l1 destination nodes in the next time slot. Note that these l1 events are mutually exclusive. The probability that S will deliver a copy to a single destination node is pg1 . By summing over the probabilities of these

PSD ðl1 Þ ¼

l1 g

l1 events, we have p1 :

ðA:3Þ

We now derive P SR ðl3 Þ. Similarly, given l3 relay nodes that do not carry a copy of the packet in current time slot, S may deliver a copy of the packet to one of l3 relay nodes in the next time slot. Note that these l3 events are mutually exclusive. The probability that S will deliver a copy to a p2 . By summing over the probabilsingle relay node is 2ðng1Þ ities of these

PSR ðl3 Þ ¼

l3 events, we have

l3 2ðn  g  1Þ

p2 :

ðA:4Þ

A.4. Proof of Lemma 4 To derive P RD ðx; l1 ; l2 Þ, we first consider x relay nodes carrying a copy of the packet of the tagged multicast session, and use PðF 1 ; F 2 ; . . . ; F x Þ to denote the probability that x relay-to-destination transmissions will be performed simultaneously from these x relay nodes to any x destination nodes of the tagged multicast session in the next time slot, where Fi ð1 6 i 6 xÞ denotes the ith relay-to-destination transmission. Since the number of x-combinations of the l2 relay nodes carrying a copy of   l2 , these relay nodes in each x-combinathe packet is x tion may conduct x successful relay-to-destination transmissions simultaneously. Under successful relay-to-destination transmissions, these relay nodes in each x-combination successfully deliver copies of the packet to distinct destination nodes (one copy per destina  l2 events are mutually tion node). Notice that these x

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exclusive. Given that there are l1 destination nodes that do not receive the packet, the probability of such an event is  

l1

 x  PðF 1 ; F 2 ; . . . ; F x Þ. By summing over the probabilities g x   l2 events, we then have of these x

P RD ðx; l1 ; l2 Þ ¼



l2 x





l1



x   PðF 1 ; F 2 ; . . . ; F x Þ: g

ðA:5Þ

x To derive PðF 1 ; F 2 ; . . . ; F x Þ, we apply the multiplication rule to obtain that

        PðF 1 ; F 2 ; . . . ; F x Þ ¼ P F 1 P F 2    P F i    P F x

ðA:6Þ

where F 1 denotes F 1 and F i denotes F i jF 1 F 2    F i1 ð2 6 i 6 xÞ. Now we need to determine those probabilities in (A.6). First, we derive F i when 1 6 i 6 x  1. For the event F i that represents a transmission from a given relay node (e.g., Ri ) to any destination node, it will occur in the next time slot iff the following five sub-events occur: (1) Ri is in an active P cell. (2) There are ki (0 6 ki 6 n  2g  x  ij¼1 kj1 ) other nodes in the same cell as Ri and its eight neighbor cells (except the g destination nodes of the source node S, the considered x relay nodes, the g destination nodes of Ri serving as a source node for another multicast session) and Pi those j¼1 kj1 other nodes residing in the same cells as the considered x relay nodes and their neighbor cells), and among these ki other nodes, k nodes are in the same cell as Ri and the other ki  k nodes are in the eight neigh  P bor cells. (3) There are hi 1 6 hi 6 g  ij¼1 hj1 destination nodes in the same cell as Ri and its eight neighbor cells, and among them, h nodes are in the same cell and the other hi  h nodes are in the eight neighbor cells. (4) Ri and one destination node are selected as a transmitter and a receiver. (5) Ri selects to conduct relay-to-destination transmission. The probabilities of these sub-events can be determined     Pli li Pki ki  1 k  8 ki k m2 a2 ði1Þ , as ki ¼0 k¼0 a2 m2 , m2 m2 ki k         Pwi wi Phi hi hi 1 h 8 hi h 1 , kþhþ1 , and 12, respechi ¼1 h¼0 ki þhi hi h m2 m2 P and tively. Here, li ¼ n  2g  x  ij¼1 kj1 P wi ¼ g  ij¼1 hj1 . Multiplying the probabilities of these sub-events, we can get the probabilities of the event F i ð1 6 i 6 x  1Þ.   We proceed to derive P F x . For the event F x , it can be divided into six sub-events consisting of above five sub-events and a new sub-event. The new sub-event is that all remaining nodes are in the other m2  9x cells except those cells where the considered x relay nodes reside and their neighbor cells. The probability of the new sub-event  z Px 2 is m m9x , where z ¼ n  x  j¼1 ðkj þ hj Þ. Multiplying 2

the probabilities of these six sub-events, we then get the probabilities of the event F x . By multiplying the probabilities of these events

   F 1 ; F 2 ; . . . ; F x ; ðA:6Þ then follows. (9) follows by substituting (A.6) into (A.5). A.5. Proof of Lemma 5 To derive P SD;RD ðx; l1 ; l2 Þ, we first consider the source node S and x relay nodes carrying a copy of the packet of the tagged multicast session, and use PðA; F 1 ; F 2 ; . . . ; F x Þ to denote the probability that a source-to-destination transmission from S to any destination node and x relay-to-destination transmissions from the considered x relay nodes to any x destination nodes will be performed simultaneously in the next time slot, where A and F i ð1 6 i 6 xÞ denote the source-to-destination transmission and the ith relay-to-destination transmission, respectively. Since the number of x-combinations of the l2 relay nodes   l2 , these relay nodes in carrying a copy of the packet is x each x-combination may conduct x successful relay-to-destination transmissions simultaneously. Under a successful source-to-destination transmission and x successful relay-to-destination transmissions, S successfully delivers the packet to a destination node and these relay nodes in each x-combination successfully deliver copies of the packet to distinct destination nodes (one copy per   l2 events are destination node). Notice that these x mutually exclusive. Given that there are l1 destination nodes that do not receive the packet, the probability of  

l1

xþ1 PðA; F 1 ; F 2 ; . . . ; F x Þ. By summing such an event is  g xþ1   l2 events, we then have over the probabilities of these x

PSD;RD ðx; l1 ; l2 Þ ¼



l2 x



 

l1



xþ1  PðA; F 1 ; F 2 ; . . . ; F x Þ: g xþ1 ðA:7Þ

To derive PðA; F 1 ; F 2 ; . . . ; F x Þ, we use the multiplication rule to obtain that

PðA; F 1 ; F 2 ; . . . ; F x Þ ¼ PðF 1 ÞPðF 2 jF 1 Þ    PðF x jF 1 F 2    F x1 Þ  PðAjF 1 F 2    F x Þ:

ðA:8Þ

Now we need to determine those probabilities in (A.8). Similar to the derivation process of PðF 1 ; F 2 ; . . . ; F x Þ in (A.6), we can get the probabilities of these events fF 1 ; F 2 ; . . . ; F x }. We proceed to derive PðAjF 1 F 2    F x Þ. For the event A given F 1 F 2    F x , it can be divided into five sub-events: (1) S is in an active cell. (2) There are kxþ1 (0 6 kxþ1 6 n  g P x  xþ1 j¼1 kj1  1) other nodes in the same cell as S and its eight neighbor cells (except S, the g destination nodes of P S, the considered x relay nodes and those xþ1 j¼1 kj1 other nodes residing in the same cells as the considered x relay

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nodes and their neighbor cells), and among these kxþ1 other nodes, k nodes are in the same cell as S and the other kxþ1  k nodes are in the eight neighbor cells. (3) There are hxþ1 ð1 6 hxþ1 6 wx  hx Þ destination nodes in the same cell as S and its eight neighbor cells, and among them, h nodes are in the same cell. (4) S is selected as a transmitter. (5) All remaining nodes are in the other m2  9ðx þ 1Þ cells except those cells where the considered x relay nodes and S reside and their neighbor cells. The probabilities of these sub-events can be determined     Plx þgkx 1 lx þ g  kx  1 Pkxþ1 kxþ1  1 k 2 2 , as m aa2 mði1Þ 2 kxþ1 ¼0 k¼0 m2 kxþ1 k      8 kxþ1 k Pwx hx wx  hx Phxþ1 hxþ1  1 h  8 hxþ1 h ,  m2 , hxþ1 ¼1 h¼0 m2 m2 hxþ1 h  2 zkxþ1 hxþ1 1 1 and m 9ðxþ1Þ , respectively. Multiplying kþhþ1 m2 the probabilities of these sub-events, we can get the probabilities of the event A given F 1 F 2    F x . By multiplying the probabilities of these events fF 1 ; F 2 jF 1 ; . . . ; AjF 1 F 2    F x g, (A.8) then follows. (11) follows by substituting (A.8) into (A.7). A.6. Proof of Lemma 6 To derive P SR;RD ðx; l1 ; l2 ; l3 Þ, we first consider the source node S; x relay nodes carrying a copy of the packet of the tagged multicast session and a relay node (e.g., R) that does not carry its copy, and use PðB; F 1 ; F 2 ; . . . ; F x Þ to denote the probability that a source-to-relay transmission from S to R and relay-to-destination transmissions from the considered x relay nodes to any x destination nodes will be performed simultaneously in the next time slot, where B and F i ð1 6 i 6 xÞ denote the source-to-relay transmission and the ith relay-to-destination transmission, respectively. Since the number of x-combinations of the u2   l2 , these relay nodes carrying a copy of the packet is x relay nodes in each x-combination may conduct x successful relay-to-destination transmissions simultaneously. Similarly, S may conduct a successful source-to-relay transmission from it to one of u3 relay nodes that do not carry a copy of the packet. Thus, P SR;RD ðx; l1 ; l2 ; l3 Þ can be determined as

PSR;RD ðx; l1 ; l2 ; l3 Þ ¼



  l1  u3 l2 x   PðB; F 1 ; F 2 ; . . . ; F x Þ: g x x ðA:9Þ

To derive PðB; F 1 ; F 2 ; . . . ; F x Þ, we use the multiplication rule to obtain that

PðB; F 1 ; F 2 ; . . . ; F x Þ ¼ PðF 1 ÞPðF 2 jF 1 Þ    PðF x jF 1 F 2    F x1 Þ  PðBjF 1 F 2    F x Þ:

ðA:10Þ

Now we need to determine those probabilities in (A.10). Similar to the derivation process of PðF 1 ; F 2 ; . . . ; F x Þ in (A.6), we can get the probabilities of these events fF 1 ; F 2 ; . . . ; F x }. We proceed to derive PðBjF 1 F 2    F x Þ. For the event B given F 1 F 2    F x , it can be divided into five sub-events: (1) S is in

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an active cell. (2) There are kxþ1 ð0 6 kxþ1 6 n  g  x Pxþ1 j¼1 kj1  2Þ other nodes in the same cell as S and its eight neighbor cells (except S, the g destination nodes of S; R, the P considered x relay nodes and those xþ1 j¼1 kj1 other nodes residing in the same cells as the considered x relay nodes and their neighbor cells), and among these kxþ1 other nodes, k nodes are in the same cell as S and the other kxþ1  k nodes are in the eight neighbor cells. (3) R is either in the same cell as S or in the eight neighbor cells. (4) S and R are selected as a transmitter and a receiver. (5) All remaining nodes are in the other m2  9ðx þ 1Þ cells except those cells where the considered x relay nodes and S reside and their neighbor cells. The probabilities of these sub-events can be determined     Plx þgkx 2 lx þ g  kx  2 Pkxþ1 kxþ1  1 k 2 2 as ma2ma2 x, kxþ1 ¼0 k¼0 m2 kxþ1 k    8 kxþ1 k     P1 1 1 r 8 1r 1 , ,  kxþ11 þ1, and r¼0 kþrþ1 m2 r m2 m2  2 zkxþ1 2 m 9ðxþ1Þ , respectively. Multiplying the probabilim2 ties of these sub-events, we can get the probabilities of the event B given F 1 F 2    F x . By multiplying the probabilities of these events fF 1 ; F 2 jF 1 ; . . . ; BjF 1 F 2    F x g; ðA:10Þ then follows. (13) follows by substituting (A.10) into (A.9).

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Bin Yang received his B.S. and M.S. Degrees both in Computer Science from Shihezi University, China, in 2004 and from China University of Petroleum, Beijing Campus, in 2007, respectively. He is currently a Ph.D. candidate at the School of Systems Information Science, Future University Hakodate, Japan and is also faculty member at the School of Computer and Information Engineering, Chuzhou University, China. His research interests include performance modeling and evaluation, stochastic optimization and control in wireless networks.

Ying Cai received her B.S., M.S. and Ph.D. Degrees in Applied Mathematics and Information Security from Xidian University, Beijing Science and Technology University and Beijing Jiaotong University, China, in 1989, 1992 and 2010, respectively. Dr. Cai is currently a full professor of Beijing Information Science and Technology University, China. She is a visiting research scholar in the Department of Electrical and Computer Engineering at University of Florida from September 2012 to September 2013. Her current research interests include wireless networks and cryptography algorithm, computer security, etc. Dr. Cai has authored and coauthored more than 30 publications in refereed professional journals and conferences.

Yin Chen received his B.S. and M.S. Degrees both in Computer Science from Xidian University, China in 2008 and 2011 and Ph.D. Degree in Systems Information Science from Future University Hakodate, Japan in 2014, respectively. From April to October 2014, he worked as a postdoctoral researcher in Future University Hakodate. He is currently an assistant professor at the Graduate School of Media and Governance, Keio University, Japan. His research interests are in wireless communication and networks. In particular, his current research is focused on the modeling and performance evaluation of wireless networks based on queuing theory and stochastic geometry.

Xiaohong Jiang received his B.S., M.S. and Ph.D Degrees all from Xidian University, China. He is currently a full professor of Future University Hakodate, Japan. Dr. Jiang was an Associate professor of Tohoku University, Japan, from February 2005 to March 2010, an assistant professor in Japan Advanced Institute of Science and Technology (JAIST), from October 2001 to January 2005. Dr. Jiang was a JSPS research fellow at JAIST from October1999 to October 2001. He was a research associate in the University of Edinburgh from March 1999 to October 1999. Dr. Jiang’s research interests include computer communications networks, mainly wireless networks, optical networks, etc. He has published over 250 technical papers at premium international journals and conferences, which include over 40 papers published in top IEEE journals and conferences like IEEE/ACM Transactions on Networking, IEEE Journal of Selected Areas on Communications, and INFOCOM. Dr. Jiang was the winner of the Best Paper Award and Outstanding Paper Award of IEEE HPCC 2014, IEEE WCNC 2012, IEEE WCNC 2008, IEEE ICC 2005-Optical Networking Symposium, and IEEE/IEICE HPSR 2002. He is a Senior Member of IEEE.

Please cite this article in press as: B. Yang et al., On the exact multicast delay in mobile ad hoc networks with f-cast relay, Ad Hoc Netw. (2015), http://dx.doi.org/10.1016/j.adhoc.2015.04.005