Performance Analysis of Wireless Dynamic Cooperative Relay Networks Using Fountain Codes

JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 4, APRIL 2010 307

Weijia Lei 1, Xianzhong Xie 2 and Guangjun Li 1 1 School of Comm. and Info. Engineering, University of Electronic Science and Technology of China, Chengdu, China 2 Institute of Personal Communications, Chongqing University of Posts and Telecommunications, Chongqing, China Email: [email protected], [email protected], [email protected]

Abstract—The use of fountain codes in wireless cooperative fallen in and how others fill theirs. With the digital relay networks can improve the system performance in fountain code, the transmitter is just like a fountain, aspects such as transmission time, energy consumption, which encodes the original information to get potentially transmission efficiency and outage probability etc. This has infinite encoded symbols and sends them out; for the been proved when the number of the relay nodes and their receivers, only if they have received enough encoded relaying capabilities are stationary. But in practical networks the relay nodes are variable and dynamic. This symbols, they can reconstruct the original information. paper proposes a simple relay scheme adopting fountain An idealized digital fountain code should have two codes, and its performance in the case when the number of properties [6]: (1) infinite encoded data can be generated the relay nodes and their capabilities to relay data are from finite source data. (2) if the length of the source data randomly changing is analyzed. The performance of a is M, the receiver can reconstruct the source data from normal relay scheme in the same network condition is also any M encoded data, and the decoding process should be analyzed for comparison. The average number of fast enough. It is difficult for any digital fountain code in transmitted data packets and transmission time required to use to satisfy both requirements, with either a limited transfer a certain number of original data packets from the ability to generate infinite encoded data or a requirement source node to the destination node are derived. We carry out numerical calculation and simulation for the two of data greater than M to decode the encoded data. A schemes. The numerical results and simulation results practical digital fountain code can work well if it can match very well, and they show it clearly that the fountain generate enough encoded data and the required data are transmission scheme can cut down the transmission time not too larger. greatly. The use of fountain codes in the wireless relay system has attracted a lot of attention in recent years. Molisch et Index Terms—cooperative relay, wireless dynamic relay al. studied the use of fountain codes in the cooperative networks, fountain codes, transmission time relay network [8][9], proposed quasi-synchronous and asynchronous transmission schemes, and analyzed their I. INTRODUCTION performances based on the mutual information. The The basic idea of cooperative communication lies in authors proposed a cooperative relay scheme based on the the cooperation among the terminals (or nodes) of the information accumulation by employing fountain codes wireless network. Cooperation can improve the system [10], analyzed its performance based on the block error performance in aspects such as power consumption, bit rate, and scaled the performance by the transmission time error rate, outage probability, cover region, etc [1]-[4]. In and energy consumption. Liu introduced several single the decentralized wireless network such as Ad hoc relay node schemes based on acknowledgment signals network, data transmission from the source node to the and derived their information theoretic achievable rates destination node is achieved via relay nodes located [11]. Castura and Mao introduced a relaying protocol in between. It is the cooperation that makes the which a relay successful in decoding the source’s communication between any two nodes possible. In such message collaborates with the source by forming a centralized network as cellular network, the technique of distributed space-time coding scheme [12]. The schemes relay transmission can also be used to extend the of Liu and Castura studied the scenario in which there is coverage of the center station. only one relay node and a direct link exists between the Fountain codes [5]-[7], also called rateless code, have source node the destination node. Nikjah and Beaulieu the capabilities of generating limitless encoding symbols, proposed two other protocols built upon previous works and the source information can be recovered from any of Liu and Castura, which select the relay nodes with the subset of the encoding symbols as long as the information best channels to the destination node to relay the conveyed by them is enough. The digital fountain code information [13], and their schemes have better energy has the properties similar to those of a fountain of water: efficiency. The results of these papers show that rateless anyone wanting water from the fountain only needs to fill codes can help to improve the performance of the his own bucket without caring what drops of water have wireless relay system. But in all of them the statuses of relay nodes are supposed to remain constant, including Manuscript received August 27, 2009; revised December 20, 2009; the number of relay nodes and their relay capabilities. In accepted January 15, 2010.

a practical relay network, because of the mobility of relay transmission and the change of the relay nodes abide by nodes, their number can not remain the same. Moreover, the following rules: a relay node also has its own data to transmit while 1. Data are transmitted in packets. relaying data for other nodes, and the number of the 2. A relay node might or might not relay data at any source nodes which require it to relay information is also time. If it relays data, it must relay a full packet. The variable. So a relay node’s capability to relay information relay capabilities of the relay nodes may vary after each for a source node may change at any time. Its capability one data packet is transmitted. is also constrained by the life of its battery. So the 3. The number of the relay nodes is fixed while M statuses of the relay nodes are dynamic. original information packets are being transferred from In this paper, a transmission scheme is proposed in a the source node to the destination node. It may change dynamic relay network employing fountain codes (called after M original information packets have been fountain transmission in this paper), and its performance transferred. Here, M is the number of the original data is analyzed by using the theory of probability and packets to be grouped into a block for fountain codes to stochastic process. To compare, we also analyze the encode. performance of a normal relay network (called normal A. Fountain Transmission Scheme transmission in this paper) in the same condition. The average number of the transmitted data packets and In the first phase, the source node encodes the original transmission time required to transfer a certain number of information using fountain codes, and broadcasts the original data packets from the source node to the encoded data to the relay nodes. The relay nodes destination node are derived. We also simulate the two accumulate the encoded data until they have got enough networks on erasure channels. The results of the analysis data to recover the original information. The source node and simulation show that the employment of fountain terminates the transmission when all the relay nodes have codes can help to improve the capability of the network finished receiving. In the second phase, the relay nodes to adapt to the dynamic change of relay nodes, and re-encode the information using fountain codes. Then shorten the transmission time greatly. they transmit the encoded data to the destination node The rest of the paper is organized as follows. We when they can relay data. The number of the data packets describe the system model of the dynamic relay network transmitted by any relay node is not fixed. Due to the and the transmission scheme employing fountain codes in property of fountain codes that the original information section II, and a normal transmission scheme is also can be recovered from any unordered subset of the introduced; we analyze the performances of the two encoded symbols, the encoded data packets can be transmission schemes with fixed number of relay nodes distributed to all relay nodes to transmit, and in section III, and their performances with variable synchronization among the relay nodes is not required. number of relay nodes in section IV; in section V, we The destination node accumulates the data received from give the numerical results and the simulation results of the relay nodes, and recovers the original data. the two relay schemes on erasure channels; section VI is B. Normal Transmission Scheme the conclusion of the paper. To compare the performance of fountain transmission, a normal relay network which does not adopt fountain II. THE SYSTEM MODEL AND TRANSMISSION SCHEMES code is analyzed too. To avoid synchronization among The system model of a cooperative relay system is the relay nodes, we use the following transmission shown in Fig. 1. We assume that there is no direct link scheme. In the first phase, if at least one relay node can between the source node and the destination node. Data relay data, the source node sends data packets. Only the transferred from the source node to the destination node relay nodes which can relay data receive data. In the are relayed by L relay nodes. The data are transmitted in second phase, only the relay node which first completes two phases: firstly, the source node broadcasts the data, the data receiving process in the first phase relays data and the relay nodes receive; then, the relay nodes transmit packets. So the source node sends the same packet the data to the destination node. In a dynamic relay continuously until one relay node has correctly received it. network, the number of the relay nodes and their relay If no relay node can relay data in the first phase, the capabilities are variable. We assume that data source node suspends the transmission until there is at least one relay node that can relay data coming out. The suspension can help to decrease the energy consumption. In the second phase, the relay node which relays data sends the packet repeatedly until it has been correctly received by the destination node.

… In the next two sections we will analyze the … performances of the fountain relay network and the … normal relay network. First, we analyze the performances when the relay capabilities of the relay nodes are variable and the number of the relay nodes is fixed. Then we analyze the performances when both of them are variable. Fig. 1. The system model of a cooperative relay system

III. PERFORMANCE ANALYSIS – WHEN THE NUMBER OF  RELAY NODES IS FIXED NkPNkfs() M  fs () M  kM kkL 1 L . (3) In this section we fix the number of the relay nodes to ML  L. In the analysis of the whole paper, we assume all MP 1 eB  k  P() Mi   P () Mi  channels in the networks are erasure channel, and all kM1  iM  iM   channels share the same erasure probability PeB; the The average time that the source node needs to number of the original data packets to be grouped into a transmit the data packets is block and encoded by fountain codes is M, and the TNfs() M fsM()1 N fsM (). (4) receivers need any M encoded data packets to recover the original data; the probability for any relay node to relay 2) From the Relay Nodes to the Destination Node data at any moment is Pr; the time that the source node Because data packets might be erased in the and the relay nodes need to transmit a data packet is transmission process, so the number of the data packets normalized to 1. The meanings of the subscripts in the sent by the relay nodes may be greater than that the formulas of this paper are: n – normal transmission, f – destination node needs. The condition for the relay nodes fountain transmission, (M) – the transference of M to send at least K = k data packets is: the destination original data packets, s – source node, r – relay node, d – correctly receives M data packets among the k data destination node. We use the number of the transmitted packets, and the M-th correctly received packet is the k-th data packets and the transmission time to scale the packet sent by the relay nodes. Its probability is performances of the transmission schemes. k 1 M PK k PkM 1, P k M. (5) M 1 eB eB A. Fountain Transmission  The K data packets are relayed by all L relay nodes, 1) From the Source Node to the Relay Nodes but at some time, there may be some relay nodes that can In the first phase of transmission, the source node not relay data. Denote the number of the relay nodes that groups M original data packets into a block, and encodes can relay data as L at time i, and L is a random variable them using fountain codes. The source node sends the ri ri with binomial distribution: encoded data continuously until all L relay nodes have received M packets correctly. Assume the number of the L l Ll PLri l P r 1,0,1,,  P r  l L (6) data packets sent by the source node to make sure that all l relay nodes can recover the original data is Nfs(M). The where Pr is the probability that one relay node can relay data. condition for Nfs(M) = k is: at least one relay node does not In the duration of a unit time of a data packet to be succeed in receiving M data packets until the source node transmitted (it is normalized to 1), the Lri relay nodes will has transmitted k data packets, while other relay nodes do transmit Lri data packets. Assume the required time for after the source node has transmitted M to k−1 encoded relaying the K data packets is TfrK, which is a random data packets. The probability for Nfs(M) = k is variable. The conditions for TfrK = t are: the number of the  ML data packets transmitted by the relay nodes from time 1 to PN fs() M M 1 P eB  t1 t1 kk11LL12  LL  2 t−1 is N fri , KL  Nfri min K 1, Lt 1 , and it PN k P P  P P i1 i1 fs() M12 () M k () M i   () M k  () M i  iM   iM t1 k1  L LL1 is NKfrt N fri at time t. Here, Nfri is the number of the  PPP    L1 ()Mk () Mi  () Mk i1   iM data packets transmitted by all relay nodes at time i, which  Lk1 Lj L j  equals to the number of the relay nodes which can relay data  PP ()Mk () Mi at that time, i.e. N = L . So the probability for T = t is  jiM1 j  fri ri frK   LL PT t  kk1  frK  PPkM()Mi () Mi ,1 tt11 iM iM  PK L Nfrimin K  1, L t 1 , N frt K N fri (1) ii11 tt11K  n n! PK L Lmin K  1, L t 1 , L K L , t where  is the binomial coefficient. P(M)i ri rt ri   m mnm!( )! ii11 L  in (1) is the probability that a relay node correctly (7) receives M packets among the i (i ≥ M) packets sent by Here, the x rounds the elements of x to the nearest the source node, and the M-th correctly received packet is integer greater than or equal to x. For the convenience of the i-th packet sent by the source node: u writing, we denote L as L , which is the sum of u i1 iM M  ri rau PPPiM1,. (2) i1 ()MiM 1 eB eB  random variables. We develop its probability distribution To enable all relay nodes to recover M original data from the transformation of its characteristic function. We packets, the average number of the encoded data packets have known that Lri (i = 1, 2, 3, …) are a serial of sent by the source node N fsM() is binomial distribution random variables independent of

each other. All random variables have the same During the time from 1 to T frM(), the number of the packets probability mass functions (6). The characteristic function T fr() M of Lri is relayed by the relay nodes is NLLfr() M ri raT . We L  fr() M  Pej  1 P  . (8) i1 LLrri r  r have known it is a binomial distribution random variable: Since the characteristic function of the sum of u statistically PN l PL  l independent random variables is equal to the product of  fr() M a  raT fr() M a  . (14) characteristic functions of the individual random variables  LTfr() M l LT fr() M la a PP1,0,1,,   l  LT  frM() [14], we get the characteristic function of Lrau: rr a la u u j Lu LLLrr  Pe 1 P  . (9) Its mean is rau ri r  i1 LT fr() M

From the unique relationship between the distribution NlPNlLTPfr() M   afrMa() fr() M r. (15) function and the characteristic function, we know that la 0 Lrau is also a binomial distribution random variable, and It should be noted that the number of the data packets its probability mass function is sent by the relay nodes may be greater than that the

Lu Lul destination node needs. This is brought about by the last la a PLrau l a P r 1,0,1,,  P r  l a  Lu . (10) data transmission process of these data packets from the la relay nodes to the destination node. In the last So we get transmission process, if the number of the remaining data PTfrk  t packets is smaller than that of the relay nodes which can relay data, there will be more data packets than the PK  L  L min K  1, L  t  1 , L  K  L ra t-1rt ra t -1 destination needs to be transmitted.1 minKLt 1, 1 Thus, the overall number of the transmitted data PL  l, L  K  l  ra t1 art a packets and the transmission time needed for the lKLa  destination node to correctly obtain the M original data minKLt 1, 1 packets are:  PLra t1  larta  PL  K  l . (11) lKL  NNNf ()MfsMfrM () () a . (16) minKLt 1, 1 L  TTTfM() fsM () frM () PLra(1) t  l a  PL rt  l lKLaa lKl  B. Normal Transmission minKLt 1, 1 Lt 1 lLtlaa(1) 1) From the Source Node to the Relay Nodes   PPrr(1 )  l In the normal relay system, the data are dealt with and lKLa  a transmitted in packets. According to the scheme L  L lLl K introduced afore, the source node sends a data packet PPrr(1  ) ,  t   l L continuously until at least one relay node has correctly lKla    received the packet. If there is no relay node that can The mean of the time to relay K data packets TfrK is  relay data, the source node suspends the transmission. TtPTtfrK  frK The number of the relay nodes which can relay data Lr is tKL/ a binomial distribution random variable:   minKLt 1, 1  Lt1 lLtl(1) L l Ll aa PL l P1,0,1,,  P  l L. (17) tPP rr(1  ) . (12) rrr   l l tKL/  lKLa a Assume the times that the source node needs to send a L L  PPlLl(1  )  packet to make at least one relay node which can relay  l rr lKla   data correctly receive it is Nns. The condition for Nns= k is: The number of the data packets sent by the relay nodes at least one relay node correctly receives the packet after the source node has sent it k-th times, while the other K (≥ M) is a random variable. So is T frK . The average relay nodes do not receive it correctly. Its probability is transmission time T frM() of the relay nodes is the PN ns kL| r l frK expectation of T : llll122    l  ll  1  PPkek    P k P ek     P k P ek  P k  12   l 1  TETPKkTfr() M frK  frk kM  minkLt 1, 1 k 1 kM M  1 PPeB1  eB  t  . (13) Actually, K in (5) is the minimum of the number of data packets sent kMM 1 t kL/ l kL by the relay node. Its mean is   a  L k 1 kM M M Lt1  KEK[] kPKk k PeB  1  P eB   lLtlaa(1) L lLl   kM kM M 1 1P  PPrr(1  ) PP rr (1  )  eB l  l a lkla    K is not affected by L or Pr, and is not greater than N frM(). The more

the relay nodes are and the higher is the probability Pr, the more redundant data packets are sent by the relay nodes.

l l jlj l l The mean of Lr0 is PPk ek  P k  P ek  P ek ,1  k  . (18)   L k L j 1 j   LkPLkkPr0  111  P rrr0    Here, Pk is the probability that one relay node correctly kk00 receives the packet when the source node sends it k-th L . (26) 1Pr times, and P is the probability that one relay node still  L ek 11P has not received the packet correctly after the source node r has sent it k times. They are The average time for accomplishing the first phase is k1 P PP1 TNLnsns r001 NL ns r . (27) keB eB. (19)   PP k ek eB 2) From the Relay Nodes to the Destination Node Substitute (19) into (18), we can simplify (18) to l There is only one relay node that transmits data to the PN kL|1 l Pkkkl1  P P  P destination node. Assume the times that the relay node ns r eB eB eB eB . (20) (1)kl kl  (1) kl  l transmits a data packet to make the destination node PPPPkeB  eB  eB1,1  eB   correctly receive it is Nnr. The probability that Nnr = k is According to the total probability theorem we get k1 PNkPnr  eB1,1 P eB k. (28) L The mean of N is PNns k PN  ns  kL|,1 r  l  PL r  l  k . (21) nr l0  k1 1 We notice that the source node suspends transmission NkPNkkPPnr nr  eB 1  eB . (29) kk11 1PeB if Lr = 0, so Lr = 0 should be excluded from (21), and (17) needs to be mended. There are The average time for the relay node to transmit the data packets is LLllLl  Ll PPrr11  PP rr TNnr nr1 N nr . (30) ll  PL l ,1,,  l L, (22) Therefore, the overall number of the transmitted data r 10PL L r 11Pr packets and the transmission time needed for the and destination to correctly obtain one original data packet are: L  NNnnsnr N PNns k PN ns  kL| r  l  PL r  l . (31) l1 TTnnsnr T L . (23) 1 L Ll  PPPPk(1)kl 11,1  l l    L  eB eBl r r 11Pr l1  IV. PERFORMANCE ANALYSIS – WHEN THE NUMBER OF The average number of the data packets sent by the RELAY NODES IS VARIABLE source node to ensure at least one relay node correctly The composition of the relay nodes may change after receives a data packet is the expectation of Nns: an information block has been transferred from the source  node to the destination node. Assume that the number of NkPNkns    ns the relay nodes L is limited in the range of [L , L ], k1 min max  L the change process of the number of the relay nodes is a 1 (1)kl lL l Ll kPeB11 P eB PP r  r EL,1, L L l Markov chain, and its state space is  min min 1(1Pr ) kl11 L  1 L Ll LLmin2, , max . Its transition probability matrix is 11PPPkPll(1) kl . (24) L eB r r eB lk11l  pp p p 11Pr  LLmin,, min LL min min 1 LL min , min 2 LL min , max   L Ll ppLL,, LL p LL , p LL , PPl 1  min 1 min min 1 min 1 min 1 min 2 min 1 max  1 L l rr Ppp p p . (32)  LLmin 2,, min LL min 2 min 1 LL min 2 , min 2 LL min 2 , max Ll    1P  11Pr l1 eB    pp p p Now we calculate the time required for the first phase.  LLmax,, min LL max min 1 LL max , min 2 LL max , max  Since the source node suspends the transmission if Lr = 0, Here pij is the probability that the number of the relay the total time of first phase is the sum of the time to nodes changes from i to j, and it satisfies transmit the packets and the suspension time. The former Lmax simply equals to the average number of the transmitted pij0,  pijLLL ij  1,  , min , min  1, , max . (33) packets. Now we analyze the suspension time. Denote the jL min time of Lr = 0 as Lr0, and Lr0 = k means that the stretch of It is easy to know that this Markov chain is ergodic, so time during which there is no relay node can relay data is we can get its stationary distribution: k. So the suspension time equals to the value of Lr0. The pPLqqLLq  ,,1,,min min  L max . (34) condition for Lr0 = k is: Lr = 0 occurs consecutively k Our ultimate purpose is to get the mean values of the times before Lr ≠ 0. The probability of Lr0 = k is number of the transmitted data packets and the required k L L time, so it is not necessary to derive their distribution PLrr0  k111  P  P r  (25) functions. We can derive the values from the results of kL(1) k L 11,0 PPkrr    

q section III. In the formulas of this section, the subscript of LLmax max 1Pr “v” means that the number of the relay nodes is variable. LPLqLprv00() r . (40) q q qLmin qL min 11P A. Fountain Transmission r The average time required in the first phase is 1) From the Source Node to the Relay Nodes We have got the average number of the data packets TNLnsv nsv r00 v 1 NL nsv r v . (41) sent by the source node in (3) when the number of the relay nodes is fixed to L. L is a random variable when the 2) From the Relay Nodes to the Destination Node The change of the number of the relay nodes does not number of the relay nodes is variable. Thus, N is the fs() M affect the number of the data packets sent by the relay function of L. Using the fundamental theorem of nodes and the time required, and they are identical to expectation [14], we can get the average number of the these in (31): data packets sent by the source node when the number of 1 the relay nodes is variable: Nnrv  1PeB . (42) Lmax  NPLqNfsv() M () fs () M TNnrv  nrv qL min qq Lmax kk1 Mq  V. NUMERICAL AND SIMULATION RESULTS pMqeB1 P k P() MiMi  P () qLmin  kM1  iM iM To clearly show the performances of the networks, we . (35) carry out numerical calculations based on the results of The average time to transmit the data packets is sections III and IV. In the mean time, to verify the theoretic results, we also simulate the performances of the TNfsv() M fsv() M1 N fsv () M . (36) networks. In the calculation and simulation, the channels 2) From the Relay Nodes to the Destination Node are erasure channel, and all the channels share the same Similarly, when the number of the relay nodes is variable, erasure probability. the average number of the data packets transmitted by the Because the numerical and simulation results of the relay nodes and the average transmission time can be fountain relay network are based on the transference of M calculated on the basis of (15) and (13): original data packets, and those in the normal relay

LLmax max network are based on the transference of 1 original data NPLqNpqTPfrv() M()fr() M  qfr() M r packet, so to compare more directly the performances of qLmin qL min the two schemes, the results of the fountain relay network minkqt 1, 1 Lmax  are divided by M when they are illustrated or listed in the k 1 kM M  pqPq r P eB1  P eB  t , figures and table in this section. The parameter M is 100 qL kMM 1 t  kq/ l kq min   a in the numerical calculations and simulations. qt1 q q  lqtlaa(1)  l ql A. When the Number of the Relay Nodes Is Fixed PPrr(1  )   P r (1 Pr )  l lkl l a a  Figs. 2 and 3 are the average number of the data (37) packets sent by the source node and the transmission time

Lmax required to make the relay nodes correctly receive one TPLqTfrv() M () fr () M original data packet. Numerical results are indicated by qL min solid lines and the simulation results by dashed lines. In

Lmax minkqt 1, 1  k1 M  the fountain network, the transmission from the source kM .(38) pPPtqeBeB 1    node to the relay nodes is a broadcast transmission. It is qL kMM 1 t  kq/ l  kq min   a required that all the relay nodes correctly receive the data, qt1 q  so the number of the transmitted data packets and lqtlaa(1) q lql  PPrr(1  )  PP rr (1  )  l  l transmission time are increasing along with the increase a lkla   2  of the number of the relay nodes. The probability Pr that the relay nodes can transmit data does not affect the B. Normal Transmission results. In the normal network, only one relay node which 1) From the Source Node to the Relay Nodes can relay data at that time is required to correctly receive Similarly, we obtain the average number of the data the data packet. The number of the transmitted packets packets transmitted and the suspension time in the first decreases with the increase of the number of the relay phase based on (24) and (26): nodes and Pr. The transmission time is the time that the Lmax source node needs to send data plus the suspension time. NPLqNnsv () ns qL min q l ql , (39) 2 Lmax Because of the use of fountain codes, this increase is very slight. For q PPrr1  1 l example, when the erasure probability PeB is 0.1, and there is 1 relay  p  node, the average number of the data packets sent by the source node is qL q ql min 1P 11Pr l1 eB 1.111, the use of fountain codes does not affect the number in this case. But if the number of the relay nodes increases to 8, it is only 1.164 if fountain codes are employed – increased by 4.73% and it is 1.656 if fountain codes are not used – increased by 49% [10].

The value of the former is identical to the number of the 5.5 transmitted packets. The latter will have a very large Fountain value when the number of the relay nodes and Pr are low. 5 Numerical Normal,Pr=0.2 Figs. 4 and 5 are performances of the second phase (in 4.5 Simulation Normal,Pr=0.5 Fig. 5, because the results of numerical calculation and 4 Normal,Pr=0.8 simulation are very close, so it is difficult to distinguish 3.5 the two categories of curves.). In the normal network, 3 only the relay node that first accomplishes data receiving 2.5 in the first phase transmits data, so the change of the 2 number of the relay nodes and the probability Pr do not time transmisssion Average affect the performance of the second phase. In the 1.5 1 fountain network, the data packets are transmitted by all the relay nodes. For any one of the relay nodes that can 1 2 3 4 5 6 7 8 9 10 relay data, if there are data that need to be transmitted, it Number of relay nodes relays them. So the more the relay nodes are and the (a) PeB= 0.1 higher is P , the shorter is the transmission time. This is 5 r Numerical Fountain shown in Fig. 5. Fig. 4 shows that the number of the data 4.5 Simulation Normal,Pr=0.2 packets sent by the relay nodes in the fountain network Normal,Pr=0.5 4 increases slightly along with the increase of the number Normal,Pr=0.8 3.5 of the relay nodes and Pr. Figs. 6 and 7 are the overall number of the data packets 3 sent by the source node and the relay nodes and the time 2.5 required to make one original data packet correctly 2

transferred from the source node to the destination node. time transmisssion Average 1.5 In the fountain network, the transmission time is 1 shortened greatly with only the slight increase of the number of the transmitted data packets. For example, 1 2 3 4 5 6 7 8 9 10 when the number of the relay nodes is 10, the probability Number of relay nodes P is 0.5, P is 0.1, for a data packet to be correctly (b) PeB= 0.01 r eB Fig. 3. The average transmission time from the source node to the transferred from the source node to the destination node, relay nodes. The number of the relay nodes is fixed.

1.175 1.15 Fountain, Pr=0.2 1.15 ets 1.145 Numerical k Fountain Fountain, Pr=0.5 Simulation

Numerical pac 1.14 1.125 Normal,Pr=0.2 d Fountain, Pr=0.8

Simulation tte 1.135 Normal,Pr=0.5 i Normal 1.1 Normal,Pr=0.8 1.13 transm 1.075 f 1.125 er o b 1.12 1.05 1.115 1.025 1.11 Average number of transmitted packets 1 Average num 1.105 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Number of relay nodes Number of relay nodes (a) PeB= 0.1 (a) PeB= 0.1 1.03 1.05 1.045 Fountain, Pr=0.2 Numerical 1.025 Fountain, Pr=0.5 Simulation 1.04 Fountain, Pr=0.8 1.02 Fountain 1.035 Normal Numerical Normal,Pr=0.2 1.03 Simulation 1.015 Normal,Pr=0.5 Normal,Pr=0.8 1.025 1.01 1.02 1.015 1.005 1.01 Average number of transmitted packets Average number of transmitted packets 1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Number of relay nodes Number of relay nodes (b) PeB= 0.01 (b) PeB= 0.01 Fig. 2. The average number of transmitted packets from the source Fig. 4. The average number of the transmitted packets from the relay node to the relay nodes. The number of the relay nodes is fixed. nodes to the destination node. The number of the relay nodes is fixed.

the average number of the transmitted data packets and 2.35 the transmission time are 2.113 and 2.114 respectively in Pr=0.2 Numerical the normal network, and they are 2.302 and 1.395 in the Pr=0.5 Simulation 2.3 fountain network. The number of the transmitted data Pr=0.8 packets increases by 8.94%, but the transmission time is 2.25 cut down by 34.01%. Fountain

B. When the Number of the Relay Nodes Is Variable 2.2 In the numerical calculation and simulation for both Normal networks with a variable number of the relay nodes, we 2.15 assume that the number of the relay nodes L ranges from

1 to 10. The transition probability matrix of the number packets transmitted of number Average 2.1 of the relay nodes is 1 2 3 4 5 6 7 8 9 10 231300 000 Number of relay nodes 13 13 13 0 0 0 0 (a) PeB= 0.1 2.02 0 131313 0 0 0 Pr=0.2 Numerical P. (43) ets k 2.08  Pr=0.5 Simulation pac

0 0 0 0 13 13 13 d 2.07 Pr=0.8 tte 0000 01323 i 2.06 It is easy to get its stationary distribution, which is: transm f 2.05 1 er o

pPLq,1,2,,10 q  . (44) b q  2.04 Fountain 10 Normal Figs. 8 to 13 are the results of numerical calculation 2.03 and simulation of the two networks. In the figures, the 2.02 Average num numerical results are shown by the solid bars, and the simulation results are shown by the hollow bars. The 1 2 3 4 5 6 7 8 9 10 performances are calculated and simulated when the Number of relay nodes erasure probabilities of the channels are 0.1 and 0.01. (b) PeB= 0.01 Fig. 6. The average overall number of transmitted packets. The Figs. 12 and 13 are the average number of the overall number of the relay nodes is fixed.

6 7 Numerical Pr=0.2 Numerical Fountain, P r=0.2 5 Simulation 6 Simulation Pr=0.5 Fountain, P r=0.5 Pr=0.8 Fountain, P r=0.8 4 5 Normal

3 4

Normal Fountain 2 3 Average transmission time Average transmisssion time transmisssion Average 1 2

0 1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Number of relay nodes Number of relay nodes (a) PeB= 0.1 (a) PeB= 0.1 6 7 Numerical Pr=0.2 Numerical Fountain, P r=0.2 5 6 Simulation Pr=0.5 Simulation Fountain, P r=0.5 Pr=0.8 4 Fountain, P r=0.8 5 Normal 3 4

2 3 Normal Fountain Average transmission time

Average transmisssion time transmisssion Average 1 2

0 1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Number of relay nodes Number of relay nodes (b) PeB= 0.01 (b) PeB= 0.01 Fig. 5. The average transmission time from the relay nodes to the Fig. 7. The average overall transmission time. The number of destination node. The number of the relay nodes is fixed. the relay nodes is fixed.

transmitted packets and the overall transmission time to 1.5 s t Numerical Simulation Normal e

transfer a data packet from the source node to the k Normal Pr=0.2 Pr=0.8 1.25 Fountain destination node. Because the performances on the whole pac Pr=0.2 Pr=0.8

d Pr=0.5 from the source node to the destination node are more e Fountain Pr=0.5 itt 1 significant, we also list the numerical results of Figs. 12 ransm t and 13 in Table I. From Table I, we can find that the 0.75 f average number of the transmitted data packets in the er o fountain network is slightly higher than that in the normal b 0.5 network, but the transmission time is much shorter. The system performance improvement brought by the use of 0.25 verage num verage fountain code is very attractive. A 0 0.01 0.1 P VI. CONCLUSION eB Fig. 8. The average number of the transmitted packets from the source We have analyzed the performance of a wireless relay node to the relay nodes. The number of the relay nodes is variable. network employing fountain codes. For comparison, we 2.5 have also analyzed the performance of the normal relay Numerical Simulation 2.25 network. The analyses are done in the condition that the Pr=0.2 Pr=0.2 relay nodes are dynamic, including their number and their 2 Normal capabilities to relay data. Due to the mobility, power 1.75 Normal constraint, and the payload change of the relay nodes, the 1.5 Pr=0.5 Pr=0.5 statuses of the relay nodes are constantly changing in a 1.25 Fountain Fountain Pr=0.8 Pr=0.8 practical network. That the numerical results and the 1 simulation results match very well proves that our 0.75

analyses are correct. These results show that the use of Average tranmission time 0.5 fountain codes can help to cut down the transmission time 0.25 dramatically with only the tiny increase of the number of 0 the transmitted data packets. Another merit of the 0.01 0.1 P fountain relay scheme is that in the second phase of the eB transmission no synchronization among the relay nodes is Fig. 9. The average transmission time from the source node to the relay nodes. The number of the relay nodes is variable. required. This is valuable because synchronization is always a difficult problem in a practical system. In the 1.5 Numerical Simulation Fountain analyses, we have used the theory of probability and Fountain Pr=0.5 Normal 1.25 Normal stochastic process as a chief mathematical tool. The Pr=0.5 Pr=0.2 Pr=0.8 dynamic property of the relay nodes is modeled by Pr=0.2 Pr=0.8 1 binominal distribution and Markov chain, which simplifies our analyses. To more accurately analyze the 0.75 performance of the dynamic wireless relay networks employing fountain codes, a more accurate statistical 0.5 model is required to describe the dynamic characteristic of the relay nodes. To find the models will be valuable 0.25

work in the future. Average number of transmitted packets 0 0.01 0.1 P TABLE I. eB THE AVERAGE NUMBER OF THE OVERALL TRANSMITTED PACKETS AND Fig. 10. The average number of the transmitted packets from the relay THE OVERALL TRANSMISSION TIME. THE NUMBER OF THE RELAY NODES nodes to the destination node. The number of the relay nodes is variable. IS VARIABLE. 2 P = 0.1 eB Numerical 1.75 Pr = 0.2 Pr = 0.5 Pr = 0.8 Simulation Pr=0.2 Pr=0.2 Packet Packet 1.5 Packets Time Time Time me s s i Fountain on t on 1.25 Fountain 2.266 2.781 2.273 1.805 2.279 1.5604 i Fountain Normal

ss Normal Normal 2.181 3.091 2.144 2.304 2.129 2.1590 i 1 0.085 -0.311 0.129 -0.499 0.150 -0.599 Difference 0.75 Pr=0.5 3.89% -10.05% 6.00% -21.69% 7.07% -27.73% Pr=0.5 Pr=0.8 PeB = 0.01 0.5 Pr=0.8 P = 0.2 P = 0.5 P = 0.8 Average tranm r r r 0.25 Packets Time Packets Time Packets Time Fountain 2.036 2.505 2.043 1.617 2.050 1.3951 0 0.01 0.1 Normal 2.016 2.927 2.013 2.174 2.0116 2.0418 P 0.020 -0.422 0.030 -0.557 0.038 -0.647 eB Difference Fig. 11. The average transmission time from the relay nodes to the 1.00% -14.43% 1.50% -25.60% 1.90% -31.67% destination node. The number of the relay nodes is variable.

[8] A. F. Molisch, N. B. Mehta, J. S. Yedidia, and J. Zhang, Numerical F - Fountain 2.5 Pr=0.2 Pr=0.5 Pr=0.8 “Cooperative Relay Networks Using Fountain Codes”. In Proc. Simulation N - Normal F F F 2.25 Pr=0.2 Pr=0.5 Pr=0.8 N N N IEEE GLOBECOM 2006, Nov. 27 - Dec. 1, 2006, pp. 1-6. FNF N F N 2 [9] A. F. Molisch, N. B. Mehta, J. S. Yedidia, and J. Zhang, 1.75 “Performance of Fountain Codes in Collaborative Relay Networks”. IEEE Transactions on Wireless Communi- 1.5 cations, vol. 6, no. 11, pp. 4108-4119, Nov. 2007. 1.25 [10] W. Lei, X. Li, X. Xie, and G. Li, “Wireless Cooperative 1 Relay System Using Digital Fountain Codes,” In Proc. 0.75 IEEE ICCCAS 2009, Jul. 23-25, 2009, pp. 123-127. 0.5 [11] Y. Liu, “A Low Complexity Protocol for Relay Channels 0.25 Employing Rateless Codes and Acknowledgement,” In Average number of transmitted packets 0 Proc. IEEE ISIT 2006, Jul. 9-14, 2006, pp. 1244-1248. 0.01 0.1 P [12] J. Castura and Y. Mao, “Rateless Coding for Wireless eB Relay Channels”, IEEE Transactions on Wireless Fig. 12. The average overall number of transmitted packets. Communications, vol. 6, no. 5, pp. 11638-1642, May 2007. The number of the relay nodes is variable. [13] R. Nikjah and N. C. Beaulieu, “Novel Rateless Coded Selection Cooperation in Dual-Hop Relaying Systems”, In 3.5 Numerical Pr=0.2 F - Fountain Proc. IEEE GLOBECOM 2008, Nov. 30-Dec. 4, 2008, pp. 1-6. Pr=0.2 Simulation F N N - Normal [14] H. P. Hsu, Probability, Random Variables, and Random F N 3 Processes. New York: McGraw-Hill Companies, Inc., 1997. me i Pr=0.5 2.5 Pr=0.5 Pr=0.8 on t Pr=0.8 F N i F N F N

ss F

i N 2 Weijia Lei received his M.S. degree in communications and information system 1.5 from Beijing University of Posts and 1 Telecommunications, Beijing, China in Average tranm 1999 and B.S. degree in communication 0.5 engineering from Chongqing University of Posts and Telecommunications 0 0.01 0.1 (CQUPT), Chongqing, China in 1992. P eB He is an associate professor in CQUPT, Fig. 13. The average overall transmission time. The number and currently is studying for a Ph.D. of the relay nodes is variable. degree in University of Electronic Science and Technology of China (UESTC). His main research field is in wireless communications technology. ACKNOWLEDGMENT This work was supported by NNSF of China Xianzhong Xie received the Ph.D. (60872037), CQ CSTC (2008BB2411) and Open degree in communication and Research Fund of KLMC, CQUPT. information system from Xidian University, Xi’an, China in 2000. He is REFERENCES currently with School of Computer Science and Technology at CQUPT, [1] A. Sendonaris, E. Erkip, and B. Aazhang, “User where he is a professor and director of Cooperation Diversity Part I: System Description,” IEEE Institute of Personal Communications. Transaction on Communications, vol. 51, no. 11, pp. 1927- His research interestists include 1938, Nov. 2003. cooperative diversity, cognitive radio [2] A. Sendonaris, E. Erkip, and B. Aazhang, “User networks, MIMO and precoding, and signal processing for Cooperation Diversity Part II: Implementation Aspects and wireless communications. He is the principal author of four Performance Analysis,” IEEE Transaction on Communi- books on 3G, MIMO, Cognitive Radio, and TDD Technology. cations, vol. 51, no. 11, pp. 1939-1948, Nov. 2003. He has published more than 80 papers in journals and 30 papers [3] A. Nosratinia and T. E. Hunter, A. Hedayat, “Cooperative in international conferences. Communication in Wireless Networks,” IEEE Communi-

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