Performance Analysis of Single Source and Single ... - Semantic Scholar

Performance Analysis of Single Source and Single Relay Cooperative ARQ protocols under Time Correlated Rayleigh Fading Channel Hong Il Choa , Chan Yong Leea , Gang Uk Hwanga,∗ a Department

of Mathematical Sciences and Telecommunication Engineering Program Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea

Abstract In this paper, we analyze the performance of a cooperative Automatic-Repeat-reQuest (ARQ) protocol under Poisson arrivals and time-correlated Rayleigh fading. The conventional non-cooperative ARQ protocol under the same channel environment is also analyzed to compare its performance with that of the cooperative ARQ protocol. A two dimensional discrete time Markov chain is constructed for the analysis. Using the steady state analysis of the Markov chain, the average frame latency and the probability generating function of the frame service time are derived. We validate our analytical results by comparing them with simulation results. From our analysis, we show that the cooperative ARQ protocol outperforms the non-cooperative ARQ protocol and the effect of the time correlation in the fading channel is not negligible. We also see that the performance metrics which we consider in this paper are almost 2 remains constant, where dS R is the distance between the source node and the relay node constant as long as dS2 R + dRD and dRD is the distance between the relay node and the destination node. Keywords: ARQ; Cooperative diversity; Performance analysis; Queueing performance; Relay network; Time correlated fading channel

1. Introduction Next generation wireless networks such as fourth generation wireless systems are required to support quality of service (QoS) requirements such as delay constraints [1]. To meet these requirements, new advanced schemes are needed and performance analysis of these schemes is also required to check their validity. Different from the channels in wired networks, wireless channels in wireless networks have two main features, fluctuation and fading in received signal strength, and the inherent broadcasting nature of the channel. Various diversity schemes have been applied to alleviate fading effect and to improve performance. In this paper, we consider a cooperative system utilizing spatial diversity. Although the broadcasting nature of the wireless channel is typically considered harmful because simultaneous transmitted signals from nodes collide, it is possible to exploit this characteristic by having an intermediate node overhear transmissions from the source node to the destination node. After a node who is willing to help the source node eavesdrops the transmission from the source node, the helping node, generally called relay node, transmits a replica or a mixture of its data frame and the eavesdropped data frame to the destination node. This method, which is known as cooperative communication, is depicted in Fig. 1. There have been a number of works on cooperative communication, e.g. [2–5, 35, 36]. Most of these have mainly focused on the physical layer and little attention has been paid to the performance of upper layers. Here, we consider the MAC (Medium Access Control) layer performance of a cooperative system and particularly concentrate on a cooperative ARQ (Automatic Repeat reQuest) protocol. Recently, a few works have studied the performance ∗ Corresponding author. Postal address: Dept. of Mathematical Sciences, KAIST, 373-1 Guseong-dong, Yuseong-gu, Daejeon, South Korea; Phone number: +82-42-350-2714; Fax number: +82-42-350-5710. An earlier version of this paper was presented at IFIP International Conference on New Technologies, Mobility and Security, Tangier, Morocco, November, 2008. Email address: [email protected] (Gang Uk Hwang )

Preprint submitted to Elsevier

December 29, 2010

Relay

Destination

Source

Figure 1: A single source node and a single relay node cooperate in order to send data to a single destination. of cooperative ARQ protocols. [7–10, 34] propose different cooperative ARQ protocols and show some gains in terms of spectral efficiency and packet error rate. In more detail, [8] proposes a cooperative ARQ protocol and shows some advantages in terms of both frame error rate and frame service time. [11] compares the performance of several cooperative ARQ protocols based on simulation. [12] suggests a new cooperative ARQ protocol that can be not only implemented in wireless LAN but is also compatible with the legacy IEEE 802.11 protocol [29]. [13] considers cooperative retransmission strategies to minimize the average frame service time which is defined as the average time duration between the time instant of a new frame transmission and the time instant of its successful reception by the destination. Regarding QoS metrics such as delay constraints and throughput, [14] and [16] analytically show that their cooperative ARQ protocol dramatically reduces the average frame service time and the jitter of the frame service time compared with a conventional non-cooperative ARQ protocol under saturated traffic conditions. [15] considers the frame latency (or the system time for a frame), which is defined as the time duration between the time instant of a frame arrival and the time instant of its successful reception by the destination, when the fading channel is time-uncorrelated. Motivated by the works in [14–16], we develop an analytical model of a cooperative ARQ protocol under general traffic conditions including both unsaturated and saturated traffic conditions and a time-correlated fading channel, which is a generalization of the model found in [11, 14, 15]. Furthermore, in this work, we analyze not only the average frame service time and the frame latency, but also the saturated throughput and the probability generating function of the frame service time by using Markov renewal theory and Little’s Theorem. The main contributions of this paper are the following. 1. Our analytical model considers general traffic conditions including both unsaturated and saturated traffic, while the model in [14, 16] assumes saturated traffic only. In many practical applications, the traffic of a network is generally unsaturated. Hence, our model can predict the actual performance in more cases. 2. The model in [15] considers a time-uncorrelated fading channel. However, our model considers a timecorrelated fading channel. Since the wireless fading channels are time-correlated in most cases [31], our model is based on a more realistic assumption on the fading channel. 3. We develop a novel two dimensional Markov chain to analyze the performance of the cooperative ARQ protocol. Based on the Markov chain, we can obtain the probability mass function of the frame service time and analyze the delay performance and the effects of the time-correlation on the fading channel. 4. For the time-correlated fading channel environment, we investigate how the position of the relay node affects the performance. It is worthwhile to mention that our results can be used to determine the locations of the relay node that guarantee given QoS requirements, e.g., [28]. The rest of this paper is organized as follows. In Section 2, the fading channel model is given and then the cooperative ARQ protocol is explained. Section 3 provides our analytical model for the cooperative ARQ protocol. The 2

G

B

Figure 2: State transition diagram for ϵi (t) for i = S D, S R, RD. performance analysis based on our analytical model is given in Section 4. Validation of our analytical model through simulation studies is given in Section 5. We give our conclusions in Section 6. 2. System Model We assume that the network consists of a source node, a relay node and a destination node as depicted in Fig. 1. We consider a network which operates using time division multiple access (TDMA). So we consider a discrete-time system where the time axis is divided into equal size slots. We assume that the time duration of a slot is equal to T seconds. The ACK or the NAK for the frame transmission is sent by the destination node in the same time slot with the frame. We assume that the ACK and the NAK are decoded perfectly. We consider a cooperative ARQ protocol which will be explained in detail later. For comparison purpose, we also consider and analyze the conventional noncooperative ARQ protocol. For simplicity, we call the cooperative ARQ protocol and the conventional non-cooperative ARQ protocol the Coop-ARQ and the Non-Coop-ARQ protocol, respectively. 2.1. Fading channel model We consider three wireless channels - S D, S R and RD1 . All three wireless channels S D, S R and RD are assumed to be flat and time-correlated Rayleigh fading channels. We assume that the source node and the relay node are sufficiently apart, so that all channels S D, S R and RD are mutually independent. In our discrete-time model, time instants at slot boundaries are indexed by t = 0, 1, 2, · · · . In the flat Rayleigh fading channel, the success-failure process of a frame transmission can be modeled as a Markov process as follows [22, 32, 33].2 Let ϵi (t) be the successfailure process of channel i in the t-th slot, for i = S D, S R, RD. Then, for a given channel reception threshold Pth and the complex envelope of the received fading gain gi (t) at the received node, we have { G, if |gi (t)|2 > Pth, ϵi (t) = (1) B, otherwise, for i = S D, S R, RD. States G and B denote the success and the failure of the transmission, respectively. Since the frame transmission starts only at slot boundaries and the channel is assumed to be invariant during a slot, ϵi (t) only depends on the channel state at the beginning of the t-th slot for each t. So ϵi (t) is a discrete time two-state Markov chain, shown in Fig. 2. Let ai and bi be the state transition probabilities of ϵi (t) such that ai = Pr {ϵi (t + 1) = G|ϵi (t) = B}, bi = Pr {ϵi (t + 1) = B|ϵi (t) = G}, 1 SD,SR

and RD are abbreviations of ’Source to Destination’, ’Source to Relay’ and ’Relay to Destination’, respectively. Markovian success-failure process of a frame transmission in this paper is adopted from [14]. One can find the original works on the Markovian success-failure process in [32, 33]. The Markovian assumption of this process is verified later in [22]. 2 The

3

respectively. The state transition probability matrix P(i) of ϵi (t) is [ ] 1 − bi bi (i) P = . ai 1 − ai

(2)

To capture the effect of the locations of nodes on channel quality, the path loss related to the distance between nodes is considered. In (1), gi (t) is the complex envelope of the channel gain and E[|gi (t)|2 ] represents the path loss for i = S D, S R, RD. By [31] we have E[|gi (t)|2 ] = Kdi−β ,

(3)

where K is a constant, di is the distance between the two nodes of channel i and β is the path loss exponent. By (3), if di = Ai dS D , where Ai is a constant for i = S R, RD and AS D = 1, the following relation holds: 2 E[|gi (t)|2 ] = A−β i E[|gS D (t)| ].

(4)

To consider the effects of distance in terms of the channel reception threshold and to simplify the fading gain, we define the normalized fading gain γi (t) and the channel reception modified threshold Pthi by γi (t) =

|gi (t)|2 E[|gi (t)|2 ]

and Pthi =

Pth 1 = diβ Pth, 2 E[|gi (t)| ] K

for i = S D, S R, RD.

(5)

Then (1) is written in terms of γi (t) and Pthi as { G, if γi (t) > Pthi . ϵi (t) = B, otherwise. We note that Pthi is a function of the distance of the two nodes of channel i, i = S D, S R, RD. As the distance increases, Pthi also increases, and therefore frame errors occur more frequently. So, Pthi can be regard as a measure of the channel quality related to the distance of the two nodes of channel i. By (4) and (5), PthS D and PthS D are PthS R = AβS R PthS D , PthRD = AβRD PthS D . For example, for β = 2, which is used for free space [31], and AS R = ARD = 12 , the relay node is located in the middle between the source node and the destination node and PthS R = PthRD = 41 PthS D . Let fDi denote the maximum Doppler shift of channel i for i = S D, S R, RD. Then the exact values of elements in the transition probability matrix P(i) in (2) can be found in [23] as follows by using the bivariate distribution of the Rayleigh process: ai =

Q(θi , ρθi ) − Q(ρθi , θi ) ePthi − 1

and bi =

1 − e−Pthi , e−Pthi

where Q(·, ·) is the Marcum Q function, √ 2Pthi θi = , 1 − ρi 2 ρi = J0 (2π fDi T ), and J0 (·) is the zero-order Bessel function of the first kind. 4

Source

i

Relay Destination

ACK

i+1

Time

ACK

i

Time

ACK

Source

i

NAK

Relay

i

NAK

i

NAK

i

Destination

Time

(a) Source success

Source

i

NAK

Relay

i

NAK

Destination

NAK

ACK

i+1

Time Time

ACK

Time

(b) Relay aid success

NAK

i

i

Source

Time

Relay

Time NAK

Destination

Time

(c) Relay aid failure

i

NAK NAK NAK

i

Time Time Time

(d) No Relay aid

Figure 3: Four possible cases of the Coop-ARQ protocol. 2.2. Coop-ARQ protocol In the Coop-ARQ protocol, if the source node has a frame to transmit, it initially transmits the frame to the destination node. If the transmitted frame is successfully received by the destination node, the destination node will send an ACK to the source node announcing the success of the transmission. Otherwise, the destination node will transmit a NAK to the source node reporting the failure of the transmission. While the source node transmits the frame, the relay node can eavesdrop the transmission from the source node to the destination node. In contrast to the Non-Coop-ARQ protocol where an erroneously received frame is retransmitted by the source node, if the relay node catches correctly the frame transmitted previously by the source node, it retransmits the replica of the frame in the next slot instead of the source node. In the Coop-ARQ protocol, if the relay node successfully retransmits the frame, the destination node announces the successful reception to both nodes. But, if the frame sent by the relay node is also erroneous, the destination node sends a NAK. In this case, since the RD channel is likely to remain in the fading state in the next slot, it is better that the source node retransmits the frame in the next slot. So, in the Coop-ARQ protocol, this retransmitting procedure is initiated by the source node. The retransmission procedure mentioned above is carried out repeatedly until the successful reception of the frame by the destination node. For the sake of simplicity of protocol operation, the relay node discards the frame whenever the cooperation takes place and/or the destination node successfully receives the frame. Based on the elementary operation of the Coop-ARQ protocol mentioned above, if the source node has a frame to transmit, one of the following four cases can occur. 1. Source success: In Fig. 3a, the source node successfully transmits a frame without the aid of the relay node. If the source node has another frame to transmit in its queue, it sends the new frame at the beginning of the next slot. 2. Relay aid success: In Fig. 3b, the destination node successfully receives a frame with the help of the relay node. In this case, the initial frame transmission by the source node is erroneous. However, the relay node eavesdrops 5

the transmission correctly, it retransmits the replica to the destination node at the beginning of the next slot and the destination node receives the replica correctly. 3. Relay aid failure: In Fig. 3c, the initial transmission by the source node is erroneous but the relay node can eavesdrop the transmission. However, the retransmission by the relay node in the next slot is also erroneous. So the destination node cannot receive the frame correctly. 4. No Relay aid: In Fig. 3d, the initial transmission by the source node is erroneous and the relay node cannot eavesdrop the transmission from the source node. So the relay node cannot retransmit the replica. In this case, the source node waits for the relay node’s transmission (which does not occur) at the beginning of the next slot. 2.3. Non-Coop-ARQ protocol In the Non-Coop-ARQ protocol, there is no relay assistance. If the source node has a frame to transmit, it initially transmits the frame to the destination node. If the destination node receives correctly the frame, it replies with an ACK. Otherwise, it transmits a NAK to request a retransmission of the erroneous frame. Unlike the Coop-ARQ protocol, the retransmission procedure in the Non-Coop-ARQ protocol is always done by the source node. 3. Markov Models In this section, we develop Markov models for the Coop-ARQ and the Non-Coop-ARQ protocols given in Section 2. Before the development of the Markov models, the following common assumptions for both the Coop-ARQ and the Non-Coop-ARQ protocols are made : • There is a queue at the source node which accommodates arriving frames. • The arrivals of frames to the source node follow a Poisson process. Frames arriving at the source node during a slot are stored its queue just before the end of the slot. • The source node starts to transmit a frame only at the beginning of a slot. • The time needed for transmitting a frame and receiving an ACK or a NAK for that frame is equal to one slot. So the source node and the relay node know the success or failure of the frame transmission at the end of the slot where the frame transmission is performed. • There is no limit in the number of retransmission trials. • The time slots of all nodes in the network are synchronized. • During a slot, the channel state is assumed to be invariant. So the success of a frame transmission only depends on the channel state of the slot boundary where the transmission starts. • ACK and NAK frames from the destination node are always decoded correctly by the source and relay nodes. The following parameters are also used to develop the Markov models for the Coop-ARQ and Non-Coop-ARQ protocols. • λ: the arrival rate in frames per second. • αm : the probability of m frames arriving at the source node during one slot. • βm : the probability of m frames arriving at the source node during two slots. Since we assume that the frame arrivals follow a Poisson process with rate λ, αm and βm depend on the rate λ and slot duration T as follows: e−λT (λT )m , m! e−2λT (2λT )m βm = . m!

αm =

6

Time

Figure 4: The operation of the Coop-ARQ protocol. Circle marks on the time axis denote embedded epochs of the Markov chain. An upward vertical arrow means a frame departure at the time instant and a right horizontal arrow means an arrival at the source node. Queue denotes the queue length at the source node. 3.1. Markov model for Coop-ARQ protocol For the analysis, the stochastic process of the number of frames at the source node is considered. However it is obvious that the process is not a Markov chain because the transmission of a frame depends on the channel state between nodes. So, we consider a novel two dimensional Markov chain consisting of the number of frames in the queue at the source node and the channel states between nodes. The slot boundaries which satisfy one of the following conditions are considered as the embedded epochs of the Markov chain. • The initial time, i.e., t = 0 is the first embedded epoch of the Markov chain. • If the source node does not have any frames to transmit in a slot, then the end of the slot is an embedded epoch of the Markov chain. • If a frame transmission by the source node is successful in a slot, then the end of the slot is an embedded epoch of the Markov chain. • If a frame transmission by the source node fails in a slot, then the end of the following slot is an embedded epoch of the Markov chain. The embedded epochs are indexed by k = 0, 1, 2, · · · . An example evolution of the network showing the embedded epochs of the Markov chain appears in Fig. 4. Let qC [k] denote the number of frames in the source node just after the k-th embedded epoch and ChC [k] denote the channel state of the Coop-ARQ protocol at the k-th embedded epoch. Let tk be the actual slot index of the k-th embedded epoch. The states of ChC [k] are defined as given in Table 1. With this novel definition we can construct a two dimensional Markov chain later. Assume that qC [k] ≥ 1. Then, a frame is successfully transmitted at slot tk from the source node to the destination node directly if 1 ≤ ChC [k] ≤ 4. A frame is successfully transmitted via the relay node if ChC [k] = 5. A frame transmission fails if 6 ≤ ChC [k] ≤ 8. Using the state diagram in Table 1, four different kinds of state transition probability matrices E1, E2, E3 and F of ChC [k] are constructed. The (i, j)-th elements of E1, E2, E3 and F denote the following probabilities: E1i, j = E2i, j =

for 1 ≤ i, j ≤ 8, for 1 ≤ i, j ≤ 8,

for 1 ≤ i ≤ 5, 1 ≤ j ≤ 8, for 6 ≤ i ≤ 8, 1 ≤ j ≤ 8, = Pr {ChC [k + 1] = j|ChC [k] = i, qC [k] ≥ 2 or (qC [k] ≥ 1, D(tk ) ≥ 1)} , for 1 ≤ i, j ≤ 8,

E3i, j = Fi, j

Pr {ChC [k + 1] = j|ChC [k] = i, qC [k] = 0, D(tk ) = 0} , Pr {ChC [k + 1] = j|ChC [k] = i, qC [k] = 0, D(tk ) ≥ 1} ,    Pr {ChC [k + 1] = j|ChC [k] = i, qC [k] = 1, D(tk ) = 0} ,   0,

where D(t) denotes the number of arriving frames at the source node in the t-th slot, respectively. Then it is straightforward to see that the state transition probability matrices E1, E2, E3 and F of ChC [k] can be derived from the matrices P(S D) , P(S R) and P(RD) defined in (2) as follows: E1 = P(S D) ⊗ P(S R) ⊗ P(RD) , 7

Table 1: Definition of States of ChC [k]

ChC [k] 1 2 3 4 ChC [k] 5 6 7 8

E2 =

[

qC [k] = 0 ϵS D (tk ) ϵS R (tk ) ϵRD (tk ) G G B G G B B ϵS D (tk ) ϵS R (tk ) ϵRD (tk ) G G B B G B B

P(S D) ⊗ P(S R) ⊗ P(RD) ((1,...,8),(1,...,4))

  P(S D) ⊗ P(S R) ⊗ P(RD) ((1,...,4),(1,...,8)) )2 ( )2  ( E3 =  P(S D) ⊗ P(S R) ⊗ P(RD) ((5),(1,...,8))  0   P(S D) ⊗ P(S R) ⊗ P(RD) ((1,...,4),(1,...,4))  F =  ( ) ( )  P(S D) 2 ⊗ P(S R) 2 ⊗ P(RD)

ϵS D (tk )

G

ϵS D (tk )

B

( )2 P(S D) ⊗ P(S R) ⊗ P(RD)

qC [k] ≥ 1 ϵS R (tk ) ϵRD (tk ) G G B G B B ϵS R (tk ) ϵRD (tk + 1) G G B G B B

] ((1,...,8),(5,...,8))

,

(6)

    , 

((5,...,8),(1,...,4))

( )2 P(S D) ⊗ P(S R) ⊗ P(RD) ((1,...,4),(5,...,8)) ( )2 ( )2 ( )2 P(S D) ⊗ P(S R) ⊗ P(RD)

    ,

(7)

((5,...,8),(5,...,8))

where ⊗ and A(η,ξ) denote the Kronecker product of matrices and the submatrix notation of a matrix A for vectors η and ξ, respectively. See Chapter 13 of [37] and Chapter 1 of [38] for the definitions of the Kronecker product and the submatrix notation, respectively. The two dimensional stochastic process {qC [k], ChC [k]} forms a Markov chain. If qC [k] is equal to zero, i.e., there are no frames to transmit by the source node at the k-th embedded epoch, qC [k + 1] is equal to the number of frames arriving during one slot, because the time interval between two consecutive embedded epochs is equal to one slot time T in this case. However, the time duration between two consecutive embedded epochs can be varying when qC [k] ≥ 1. If 1 ≤ ChC [k] ≤ 4, then the time duration between two consecutive embedded epochs is equal to one slot time T and the frame transmission started at the k-th embedded epoch by the source node is successful. When ChC [k] = 5, the time duration between two consecutive embedded epochs is equal to 2T and the frame transmission started at the k-th embedded epoch is successful with the help of the relay node. If 6 ≤ ChC [k] ≤ 8 , then the time duration between two consecutive embedded epochs is equal to 2T and the frame transmission started at the k-th embedded epoch fails. By the above arguments the evolution equation of the queue at the source node is obtained as follows:  D(tk ),        q  C [k] + D(tk ) − 1, qC [k + 1] =    qC [k] + D(tk ) + D(tk + 1) − 1,      qC [k] + D(tk ) + D(tk + 1),

if qC [k] = 0, if qC [k] ≥ 1, 1 ≤ ChC [k] ≤ 4, if qC [k] ≥ 1, ChC [k] = 5, if qC [k] ≥ 1, 5 ≤ ChC [k] ≤ 8. 8

(8)

From (8), the state transition probability of {qC [k], ChC [k]} is derived as follows: Pr {qC [k + 1] = j, ChC [k + 1] = m|qC [k] = i, ChC [k] = l}   α0 E1l,m , if i = 0, j = 0,      α E2 , if i = 0, j ≥ 1,  j l,m     α E3 , if i = 1, 1 ≤ l ≤ 4, j = 0,  0 l,m     if i = 1, l = 5, j = 0,  β0 E3l,m , =   α F , if i ≥ 2, 1 ≤ l ≤ 4, j ≥ i − 1, j−i+1 l,m      β F , if i ≥ 2, l = 5, j ≥ i − 1, j−i+1 l,m      β F , if i ≥ 2, 6 ≤ l ≤ 8, j ≥ i,  j−i l,m    0, otherwise, where E1l,m , E2l,m , E3l,m and Fl,m denote the (l, m)-th elements of matrices E1, E2, E3 and F, respectively. Let PC be the state transition probability matrix of {qC [k], ChC [k]}. It is straightforward to show that {qC [k], ChC [k]} is a queueing process of the M/G/1 type and its state transition matrix PC is given by  (C) (C) (C) (C)   B0 B1 B2 B3 · · ·  (C) (C) (C) (C)  A˜0 A1 A2 A3 · · · C (C) P =   , A0(C) A(C)   1 A2 · · ·   .. .. . .  . . . where (C) A˜0 = diag(α0 , α0 , α0 , α0 , β0 , 0, 0, 0)E3, (C) A0 = diag(α0 , α0 , α0 , α0 , β0 , 0, 0, 0)F, A(C) k = diag(αk , αk , αk , αk , βk , βk−1 , βk−1 , βk−1 )F, B(C) 0 = α0 E1, B(C) k = αk E2,

for k ≥ 1, for k ≥ 1.

Let πCi,j denote the steady state probability that the Markov chain {qC [k], ChC [k]} is in state (i, j) for i ≥ 0 and 1 ≤ j ≤ 8. The steady state probability vector that there are i frames in the source node is denoted as πCi , (πCi,1 , πCi,2 , · · · , πCi,8 ) for i ≥ 0. Then the steady state probability vector of the Markov chain {qC [k], ChC [k]} is given as ∑ C πC , (πC0 , πC1 , πC2 , · · · ). Since πC is the steady state probability vector, πC PC = πC and ∞ i=0 πi e1 = 1 hold, where e1 is an 8 × 1 column vector whose elements are all equal to 1. However it is not easy to solve the above equations for πC by finding an eigenvector, since PC is not a finite dimensional matrix. So the steady state probability vector πC is obtained by the matrix analytic method of [24, 25]. 3.2. Markov model for Non-Coop-ARQ protocol In the Non-Coop-ARQ protocol, there are only one source node and one destination node. The initial transmission and retransmission processes by the source node depend only on the process ϵS D (t). Let { 1, if ϵS D (tk ) = G, ChNC [k] , 2, if ϵS D (tk ) = B. The state transition probability matrix for ChNC [k] is equal to P(S D) . Let qNC [k] denote the number of frames in the source node just after the k-th slot boundary in the Non-Coop-ARQ protocol. Then {qNC [k], ChNC [k]} forms a two dimensional discrete time Markov chain. By using arguments similar to those given in Subsection 3.1, the following state transition probabilities are derived: Pr{q 1] = j, ChNC [k + 1] = m|qNC [k] = i, ChNC [k] = l}  NC [k(S + D)  α P , if i = 0,  j l,m    (S D)   if i ≥ 1, l = 1, j ≥ i − 1,  α j−i+1 Pl,m , = (S D)   α P , if i ≥ 1, l = 2, j ≥ i,  j−i l,m     0, otherwise. 9

Let PNC be the state transition probability matrix of {qNC [k], ChNC [k]}. It is straightforward to show that {qNC [k], ChNC [k]} is a queueing process of the M/G/1 type and the state transition probability matrix PNC is given by

PNC

 (NC) (NC) (NC) (NC)   B0 B1 B2 B3 · · ·  (NC) (NC) (NC) (NC)  A0 A1 A2 A3 · · ·  =  A(NC) A(NC) A(NC) · · · ,  0 1 2   .. .. . .  . . .

where A(NC) = diag(α0 , 0)P(S D) , 0 (NC) Ak = diag(αk , αk−1 )P(S D) , B(NC) = αk P(S D) , k

for k ≥ 1, for k ≥ 0.

Let πi,NC j denote the steady state probability that the Markov chain {qNC [k], ChNC [k]} is in state (i, j) for i ≥ 0 and 1 ≤ NC NC j ≤ 2. The steady state probability vector that i frames are in the source node is denoted as πiNC , (πi,1 , πi,2 ) for i ≥ 0. NC NC NC NC Then the steady state probability vector of the Markov chain {qNC [k], ChNC [k]} is given as π , (π0 , π1 , π2 , · · · ). ∑ NC Since πNC is the steady state probability vector, πNC PNC = πNC and ∞ i=0 πi e2 = 1 hold, where e2 is a 2 × 1 column vector whose elements are all equal to 1. The value of πNC can be obtained by using the matrix analytic method of [24, 25]. 4. Performance Analysis In this section we analyze the frame latency, saturated throughput and the frame service time for the Coop-ARQ and Non-Coop-ARQ protocols by using the Markov chains developed in Section 3. The definitions of these performance metrics are as follows. • TC : frame service time in the Coop-ARQ protocol in slots, i.e., the time duration between the time instant of a new frame transmission and the time instant of the successful reception of the frame by the destination node. Here new means that the transmission is not a retransmission. • T NC : frame service time in the Non-Coop-ARQ protocol in slots. • LC : frame latency in the Coop-ARQ protocol in slots, i.e., the time duration between the time instant of a frame arrival at the source node and the time instant of the successful reception of the frame by the destination node. • LNC : frame latency in the Non-Coop-ARQ protocol in slots. • T hC : saturated throughput of the network in the Coop-ARQ protocol, i.e., the average number of successfully transmitted frames per slot under the saturated traffic condition. • T hNC : saturated throughput of the network in the Non-Coop-ARQ protocol. Throughout this section, we assume that the Markov chains {qC [k], ChC [k]} and {qNC [k], ChNC [k]} are in the steady state. 4.1. Coop-ARQ protocol 4.1.1. The average frame service time E [TC ] and probability mass function of TC To get the average frame service time in the Coop-ARQ protocol, the steady state probability of the channel state at a new frame transmission instant should be derived. This probability is obtained in Lemma 1.

10

Lemma 1. Let pˆ j denote the steady state probability of the channel state (the state of the process ChC [k]) at a new frame transmission instant in the Coop-ARQ protocol for 1 ≤ j ≤ 8. Then we have   8 4 ∞ ∑ 5 ∑ ∑ ∑  1  −λT C −λT C −2λT C C (1 − e ) pˆ j = π0,l E2l, j + (1 − e ) π1,l Fl, j + (1 − e )π1,5 F5, j + πi,l Fl, j  , MC i=2 l=1 l=1 l=1 where MC = (1 − e

−λT

)

8 ∑

πC0, j

j=1

  ∞ 5 4 ∑   ∑ ∑ −λT C −2λT C  + (1 − e ) π1, j + (1 − e )π1,5  + πCi,j , j=1

i=2 j=1

E2i, j and Fi, j are the (i, j)-th elements of the channel transition matrices given in (6), (7), respectively. Proof. The set of slot boundaries at which new frames are transmitted is a subset of the set of embedded epochs of the Markov chain {qC [k], ChC [k]}. A tagged embedded epoch, say the k-th embedded epoch, is the slot boundary at which a new frame is transmitted if and only if one of the following conditions are satisfied: • {qC [k − 1], ChC [k − 1]} = (0, l) for 1 ≤ l ≤ 8 and there is at least one arrival just before the k-th embedded epoch. • {qC [k − 1], ChC [k − 1]} = (1, l) for 1 ≤ l ≤ 5 and there is at least one arrival just before the k-th embedded epoch. • {qC [k − 1], ChC [k − 1]} = (i, l) for i ≥ 2 and 1 ≤ l ≤ 5. Let MC be the probability of the above conditions occurring. MC is then obtained by using the steady state probability vector πC of {qC [k], ChC [k]} as follows:  ∞ 5  4 8 ∑ ∑  ∑ ∑  −λT C −2λT C −λT C πCi,l . (9) π1,l + (1 − e )π1,5  + MC = (1 − e ) π0,l + (1 − e ) i=2 l=1

l=1

l=1

Then pˆ j are derived by multiplying the state transition probabilities of the channel process ChC [k] to the respective probabilities of the conditions that are mentioned above and by dividing them by MC .  Let kˆ be an arbitrary embedded epoch where a new frame transmission begins. The probability generating function GTC (z) of TC can be obtained as follows: [ ] GTC (z) , E zTC =

8 ∑

[ ] ˆ = j, qC [k] ˆ ≥1 pˆ j E zTC |ChC [k]

(10)

j=1

=

4 ∑

pˆ j z + pˆ5 z2 +

j=1

8 ∑

[ ] ˆ = j, qC [k] ˆ ≥1 . pˆ j E zTC |ChC [k]

j=6

ˆ ≤ 4 implies the successful transmission of The above equation (10) follows from the observation that 1 ≤ ChC [k] ˆ the new frame by the source node and ChC [k] = 5 implies the successful transmission of the new frame via the relay node. For 6 ≤ j ≤ 8 we have [ ] ˆ = j, qC [k] ˆ ≥1 E zTC |ChC [k] =

8 ∑

] [ ˆ = j, qC [kˆ + 1] ≥ 1, qC [k] ˆ ≥1 F j,k E zTC |ChC [kˆ + 1] = k, ChC [k]

k=1

= =

8 ∑

( [ ]) F j,k E zTC +2 |ChC [kˆ + 1] = k, qC [kˆ + 1] ≥ 1

k=1

  8 ∑ [ ] T F j,k E z C |ChC [kˆ + 1] = k, qC [kˆ + 1] ≥ 1  z2 . F j,k z + F j,5 z + 

4 ∑ k=1

3

(11)

4

k=6

11

The second equality of (11) follows from the fact that {qC [k], ChC [k]} is a Markov chain. Since the Markov chain {qC [k], ChC [k]} is in the steady state, the following equation holds: [ ] [ ] ˆ qC [k] ˆ ≥1 . E zTC ChC [kˆ + 1], qC [kˆ + 1] ≥ 1 = E zTC ChC [k], (12) [ ] ˆ = j, qC (k) ˆ ≥ 1 is obtained for 1 ≤ j ≤ 8 as Using (11) and (12), it is easy to see that the value of E zTC |ChC (k) follows: ]  [T   3 ∑4 4 ˆ = 6, qC [k] ˆ ≥ 1  E z C |ChC [k] ] ( )−1 z ∑4j=1 F6, j + z F6,5   [ ˆ = 7, qC [k] ˆ ≥ 1  = I3 − z2 F[(6,7,8),(6,7,8)] z3 j=1 F7, j + z4 F7,5  , E zTC |ChC [k] (13)   3 ∑4 ]  [ TC ˆ = 8, qC [k] ˆ ≥1 z j=1 F8, j + z4 F8,5 E z |ChC [k] where I3 denotes the 3 × 3 identity matrix. Combining (10) and (13), we have the following theorem. Theorem 2. For the Coop-ARQ protocol, the probability generating function GTC (z) of the frame service time TC is derived as follows: GTC (z) =

4 ∑ j=1

+

8 ∑ j=6

pˆ j z + pˆ5 z2   ∑4  ∑ 4 ( )−1 ∑ s=1 F6,s + zF6,5    F j,k z3 + F j,5 z4 + F[( j),(6,7,8)] I3 − z2 F[(6,7,8),(6,7,8)]  4s=1 F7,s + zF7,5  z3  . pˆ j    ∑4  k=1 s=1 F 8,s + zF 8,5

The average frame service time E [TC ] is then obtained as follows: d TC G (z) . E [TC ] = dz z=1 The probability mass function of TC is also obtained by using the probability generating function GTC (z) as follows: dk GTC (z) Pr {TC = k} = k . k! z=0 dz 4.1.2. The average frame latency E [LC ] To get the average frame latency E [LC ], the probability of having n frames in the source node at an arbitrary slot is analyzed. Then the average frame latency can be obtained by applying Little’s Theorem. Let µi, j be the actual sojourn time of the Markov chain in state (i, j) in slots, i.e., the time duration in slots between the embedded epoch with state (i, j) and the next embedded epoch. Then we have the following equations:   1, if i = 0,    1, if i ≥ 1, 1 ≤ j ≤ 4, µi, j =     2, if i ≥ 1, 5 ≤ j ≤ 8. { } Let q˜C (t), Ch˜ C (t)

be the corresponding semi-Markov process [26], i.e., q˜C (t) = qC [k] and Ch˜ C (t) = ChC [k] if { } tk ≤ t < tk+1 . An example of the process q˜C (t), Ch˜ C (t) is shown in Fig 5. { } Let π˜ C(i, j) be the steady state probability that q˜C (t), Ch˜ C (t) is in state (i, j) at an arbitrary slot time. By the t≥0 C Markov renewal theory [26], we have the following equation for π˜ (i, j) : π˜ C(i, j) , lim =

t≥0

The amount of slots which are in state (i, j) during [0, t] t

t→∞ πC(i, j) µi, j . ∑ C k,l π(k,l) µk,l

(14)

12

Time

{ } Figure 5: An example of the process q˜C (t), Ch˜ C (t) . Circle marks on the time axis denote embedded epochs of the Markov chain {qC [k], ChC [k]}. An upward vertical arrow means a frame departure at the time instant and a right horizontal arrow means an arrival at the source node. E-index[k] means the index of the embedded epoch of the Markov chain {qC [k], ChC [k]} and S-index(t) means the index of the slot. Queue(t) denotes the number of frames in the queue just after the beginning of the t-th slot. Note that π˜ C(i, j) is not the probability that i frames are in the queue at the source node at an arbitrary slot boundary. There are some chances that the arbitrary slot boundary is not an embedded epoch of the Markov chain {qC [k], ChC [k]}. In this case, the actual number of frames in the queue at the source node can be different from the number of frames denoted by the state, since there may be some new frame arrivals between the last embedded epoch and the arbitrary slot boundary. An example of the case is highlighted with a box in Fig. 5. In the (tk1 + 1)-th slot in Fig. 5, one can find that the number of frames in the queue just after the beginning of the slot is equal to 4 due to two new frames arriving during the tk1 -th slot. However we have q˜C (tk1 + 1) = 2 in Fig. 5. Using (14), the probability that n frames are in the queue at the source node at an arbitrary slot boundary is given by the following theorem. Theorem 3. Let qˆ n be the probability that n frames are in the source node at an arbitrary slot boundary. Then we have  8  ∑     π˜ (0, j) , if n = 0,      j=1  qˆ n =     4 n 8  ∑ ∑    1 ∑     π˜ (n, j) + αn−i π˜ (i, j)  , if n ≥ 1. π˜ (n, j) +    j=1 2 j=5 i=1 Proof. We first tag an arbitrary slot boundary and call it the tagged slot boundary. Note that qˆ n , Pr {n frames in the queue at the source node at the tagged slot boundary} ∑ = π˜ C(i, j) fn(i, j)

(15)

i, j

where fn(i, j) , Pr{n frames in the queue at the source node | the semi Markov process is in state (i, j)}. For i = 0 or i ≥ 1 , 1 ≤ j ≤ 4, the tagged slot boundary is an embedded epoch of the Markov chain {qC [k], ChC [k]}. So the number of frames in the queue at the source node at the tagged slot boundary is equal to the number of frames denoted by the state. That is, { 1, if n = i, fn(i, j) = (16) 0, otherwise , 13

for i = 0 or i ≥ 1 , 1 ≤ j ≤ 4. For i ≥ 1, 5 ≤ j ≤ 8, the tagged slot boundary is an embedded epoch with probability (i, j) 1 is determined in the same way as in (16). 2 . If the tagged slot boundary is an embedded epoch, then the value of fn On the other hand, when the tagged slot boundary is not an embedded epoch, the number of frames at the tagged slot boundary is increased by the number of newly arriving frames just before the tagged slot boundary. So we have the following equation: fn(i, j) =

1 1 I{n = i} + I{n ≥ i}αn−i , 2 2

(17)

where I{·} denotes the indicator function for i ≥ 1, 5 ≤ j ≤ 8. Combining (15), (16) and (17), we obtain qˆ n .  Let E[NC ] be the average number of frames in the source node. The value of E[NC ] is obtained as follows: E[NC ] =

∞ ∑

nqˆ n .

(18)

n=0

The average frame latency E[LC ] is derived from (18) by using Little’s Theorem as follows: E [LC ] =

E [NC ] . λT

4.1.3. The saturated throughput of the network T hC The saturated throughput T hC can be defined as follows: The number of frames successfully transmitted during [0, t] t The average number of frames successfully transmitted between two embedded epochs in {qC [k], ChC [k]} = . The average sojourn time between two embedded epochs in {qC [k], ChC [k]} (19)

T hC = lim

t→∞

The denominator and numerator in (19) can be obtained by using the steady state probability of the channel state under the saturated condition. Under the saturated condition, the channel ( ) process ChC [k] is a Markov chain with the C transition probability matrix F which is defined in (7). Let ch = chCj be the steady state probability vector of 1≤ j≤8

the Markov chain ChC [k] under the saturated condition. Then chC is obtained by solving the following equations: chC = chC F, 8 ∑

chCj = 1.

j=1

∑ Since the transmission of a frame is successful only for 1 ≤ ChC [k] ≤ 5, the numerator in (19) is equal to 5j=1 chCj . ∑ ∑ The denominator in (19) is equal to 4j=1 chCj + 2 8j=5 chCj . By the above arguments, the saturated throughput of the network T hC can be obtained as follows: ∑5 C j=1 ch j T hC = ∑4 ∑ 8 C C j=1 ch j + 2 j=5 ch j ∑5 C j=1 ch j = . ∑8 1 + j=5 chCj

14

4.2. Non-Coop-ARQ protocol 4.2.1. The average frame service time E [T NC ] and probability density function of T NC As in the Coop-ARQ protocol case, the steady state probabilities of the channel state at new frame transmission instants are calculated first, in the following lemma. Lemma 4. Let pˆ NC be the steady state probability of the channel state at a new frame transmission instant in the j Non-Coop-ARQ protocol. Then we have   2 ∞ ∑ ∑  1  (S D) (S D) (S D)  −λT NC NC −λT NC NC (1 − e ) pˆ j = π0,l Pl, j + (1 − e )π1,1 P1, j + πi,1 P1, j  , MNC i=2 l=1 for 1 ≤ j ≤ 2. Proof. Assume that a new frame is transmitted at slot k. Then one of the following conditions is satisfied: • {qNC [k − 1], ChNC [k − 1]} = (0, j) for 1 ≤ j ≤ 2 and there is at least one arrival just before the k-th embedded epoch. • {qNC [k − 1], ChNC [k − 1]} = (1, j) for j = 1 and there is at least one arrival just before the k-th embedded epoch. • {qNC [k − 1], ChNC [k − 1]} = (i, j) for i ≥ 2 and j = 1. Let MNC be the probability of the above conditions occurring. MNC is obtained by using the steady state probability vector πNC of {qNC [k], ChNC [k]} as follows: MNC = (1 − e−λT )

2 ∑

NC −λT NC π0, )π1,1 + j + (1 − e

j=1

∞ ∑

NC πi,1 .

(20)

i=2

Then pˆ NC j are derived by multiplying the state transition probabilities of the channel process ChNC [k] to the respective probabilities of the conditions that are mentioned above and by dividing them by MNC  Let kˆ be an arbitrary slot boundary where a new frame transmission begins. The probability generating function GT NC (z) of T NC can be obtained as follows: [ ] GT NC (z) , E zT NC =

2 ∑

[ ] T NC ˆ = j |ChNC [k] pˆ NC j E z

(21)

j=1

[ ] ˆ =2 . = pˆ 1NC z + pˆ 2NC E zT NC |ChNC [k] [ ] ˆ = 2 is obtained by solving the following equation. The value of E zT NC |ChNC [k] 2 [ ] ∑ [ ] D) T NC ˆ =2 = ˆ =2 E zT NC |ChNC [k] P(S |ChNC [kˆ + 1] = k, ChNC [k] 2,k E z k=1

=

2 ∑

[ ] D) T NC +1 ˆ =k P(S E z |Ch [ k] NC 2,k

(22)

k=1

[ ] D) 2 (S D) T NC ˆ =2 . = P(S |ChNC [k] 2,1 z + P2,2 zE z Combining (21) and (22), we have the following theorem:

15

Theorem 5. For the Non-Coop-ARQ protocol, the probability generating function GT NC (z) of the frame service time T NC is derived as follows: G

T NC

(z) =

pˆ 1NC z

+

pˆ 2NC

D) z2 P(S 2,1 D) 1 − zP(S 2,2

.

The average frame service time E [T NC ] can be obtained as follows: d T NC G (z) . E [T NC ] = dz z=1 The probability mass function of T NC is also derived by using the probability generating function GTC (z) as follows: dk GT NC (z) Pr {T NC = k} = k . k! z=0 dz 4.2.2. The average frame latency E [LNC ] Unlike the Coop-ARQ protocol, all slot boundaries are embedded epochs of {qNC [k], ChNC [k]} in the Non-CoopARQ protocol. Hence, the steady state probability that there are i frames in the queue at the source node at an arbitrary slot boundary is equal to πiNC for i ≥ 0. Thus the average number of frames in the source node in the steady state E [NNC ] is given as follows: E [NNC ] =

∞ ∑

iπiNC .

(23)

i=0

The average frame latency in the Non-Coop-ARQ protocol E[LNC ] is derived from (23) by using Little’s Theorem as follows: E [LNC ] =

E [NNC ] . λT

4.2.3. The saturated throughput of the network T hNC The saturated throughput T hNC for the Non-Coop-ARQ can be defined as follows: The number of successfully transmitted frames during [0, t] t = The average number of frames transmitted during one slot time.

T hNC = lim

t→∞

The channel state process ChNC [k] itself forms a Markov chain with state transition probability matrix P(S D) . Let NC chNC = (chNC is obtained by j )1≤ j≤2 be the steady state probability vector of the Markov chain {ChNC [k]}. Then ch solving the following equations: chNC = chNC P(S D) , 2 ∑

chNC j = 1.

j=1

Since the transmission of a frame is successful only when ChNC [k] = 1, the saturation throughput T hNC is derived as follows: T hNC = ch1NC .

16

5. Model validation and performance results To validate our analytical model for the Coop-ARQ protocol and compare its performance metrics with those of the Non-Coop-ARQ protocol, we simulate the simple network of Fig. 1. The simulation is performed using MATLAB. In the simulation, the slot duration is set to 5 ms which is currently used in various wireless networks [27]. For the simulation of the fading channel, time-correlated Rayleigh fading channels are generated by a built-in m-file script in MATLAB which passes white Gaussian noise through a filter whose power spectrum corresponds to the Jakes Doppler spectrum [30]. Throughout this section, the maximum Doppler shift fD i for i = S D, S R, RD is set to fD . 5.1. The effect of the correlation in the channel for the Coop-ARQ protocol We first examine the effect of the time-correlation in the channel on system performance. When β = 2, λ = 30, PthS D = −3 dB and PthS R = PthRD = 41 PthS D are set, Fig. 6 shows the changes in the average frame service time and the average frame latency for both simulation and analytical results as fD increases in the Coop-ARQ protocol. In the time-correlated channel, the average frame latency of the Coop-ARQ protocol increases as fD decreases. This implies that the performance becomes worse as the fading channel is more time-correlated. Next, to investigate the effect of the time correlation in detail, we plot the frame service time distribution in Fig. 7. The figure shows that the frame service time tends to be longer when fD = 2 than when fD = 18. Note that a long frame service time increases the latencies of the other frames in the queue. So, we see that a smaller value of fD results in higher average frame latency which is in accordance with our previous result. From Fig. 7, we see that the probabilities of occurring service times with even numbers of slots are greater than those of occurring service times with odd numbers of slots. Noting that two time slots are needed for a successful transmission when the initial transmission fails, the above result implies that successful retransmissions of frames frequently occur with the aid of the relay node, i.e., the ‘Relay aid success’ case of the Coop-ARQ protocol frequently occurs, due to the bad condition of the SD channel. We note that the performance under the time-correlated channel converges to that under the independent (timeuncorrelated) channel as fD increases. This is because, as fD increases, the channel becomes less time-correlated and eventually independent. So, our model with time-correlated channel generalizes the model presented in [15] where the channel is time-uncorrelated. 5.2. Performance comparisons of the Coop-ARQ protocol and the Non-Coop-ARQ protocol To compare the performance metrics of the Coop-ARQ protocol and the Non-Coop-ARQ protocol, for β = 2 and PthS R = PthRD = 14 PthS D , we obtain simulation and analytic results. First, as PthS D varies, we plot the average frame service times and the average frame latencies of the Coop-ARQ protocol and the Non-Coop-ARQ protocol in Fig. 8 for fD = 6.6667 and λ = 30. We also plot the saturated throughput for fD = 6.6667 in Fig. 9. When the carrier frequency is 2.4 Hz and the relative speed of the mobile nodes is 5 Km/h, the maximum Doppler shift fD is equal to 6.6667. All performance results of the Coop-ARQ protocol are better than those of the Non-Coop-ARQ protocol. In particular, as PthS D increases, the average frame service time and the average frame latency of the Coop-ARQ protocol slowly increase but those of the Non-Coop-ARQ protocol dramatically increase. Since PthS D is closely related with the distance between the source node and the destination node, our results show that the relay node is needed to be positioned in a right place to satisfy delay QoS requirement of delay-sensitive multimedia traffic in a wireless network. When λ = 30 and PthS D = −3 dB are set, Fig. 10 shows the average frame service times and the average frame latencies of the Coop-ARQ protocol and the Non-Coop-ARQ protocol for different values of fD . Under the saturated condition at the source node, we also compare the saturated throughputs of both protocols for PthS D = −3 dB in Fig. 11. As seen in the figures, for all fD values, the performance metrics of the Coop-ARQ protocol are better than those of the Non-Coop-ARQ protocol. This means that the Coop-ARQ protocol still has benefits under the saturated traffic condition. Furthermore, the Coop-ARQ protocol has better performance for lower fD . It implies that the relay node is more useful for pedestrian communications which have low fD and occur more frequently in metropolitan areas. To investigate the effect of the arrival rates for the Coop-ARQ protocol and the Non-Coop-ARQ protocol, we plot in Fig. 12 the average frame latency for fD = 6.6667 and PthS D = −3 dB as λ is changed. The increase with λ in the average frame latency for the Non-Coop-ARQ protocol is significant although that for the Coop-ARQ protocol is 17

1.8 Coop−ARQ (correlated channel, theoretical) Coop−ARQ (correlated channel, simulation) Coop−ARQ (independent channel, theoretical) Coop−ARQ (independent channel, simulation)

Average Service Time (slot)

1.75

1.7

1.65

1.6

1.55

1.5

2

4

6

8

10 fD

12

14

16

18

(a) Average frame service time

3.5

Average Frame Latency (slot)

Coop−ARQ (correlated channel, theoretical) Coop−ARQ (correlated channel, simulation) Coop−ARQ (independent channel, theoretical) Coop−ARQ (independent channel, simulation) 3

2.5

2

1.5

2

4

6

8

10 f

12

14

16

18

D

(b) Average frame latency

Figure 6: Average frame service time and average frame latency of the Coop-ARQ protocol under time correlated channel when λ = 30 and PthS D = −3 dB.

18

0

10

Coop−ARQ theoretical, λ = 30, fD = 2 Coop−ARQ simulation, λ = 30, fD = 2 Coop−ARQ theoretical, λ = 30, f = 18

−1

Probabiliy of Service Time

10

D

Coop−ARQ simulation, λ = 30, fD = 18 −2

10

−3

10

−4

10

−5

10

0

5

10 Service Time (slot)

15

20

Figure 7: Frame service time distribution of the Coop-ARQ protocol when λ = 30. and PthS D = −3 dB relatively small. So, as the frame arrival rate increases, the Non-Coop-ARQ protocol becomes unstable faster than the Coop-ARQ protocol. 5.3. Relay positioning for the Coop-ARQ protocol It is helpful to study the relay node positioning problem with the help of our analytic model. Note that the relay node positioning problem is solved based on the the channel qualities of S R and RD that are respectively represented by PthS R and PthRD in this paper. Refer to (5). So we plot the average frame service time, the average frame latency and saturated throughput as functions of PthS R and PthRD in figs. 13 and 14. In the figures, we set β = 2, PthS D = −3 2 dB, λ = 50 and fD = 6.6667. From the figures, we see that if PthS R + PthRD is constant, i.e., dS2 R + dRD is constant, then all performance metrics are almost the same. 6. Conclusion In this paper, we develop an analytical model for a cooperative ARQ protocol operating under time-correlated Rayleigh fading channels. Our work is an extension of previous studies [14–16]. Based on our analytical model, we obtain the probability generating function of the frame service time, the average frame latency and the saturated throughput. From the analysis, the performance metrics of the Coop-ARQ protocol are shown to be superior to those of the Non-Coop-ARQ protocol. We also find that the effects of time-correlation in the fading channel are not negligible and accordingly should be considered in the system design. In addition, we show that if PthS D + PthRD is constant, all performance metrics are almost the same. Our analytical framework and results can be used to determine the deployment of relay nodes in a wireless communication networks. Acknowledgement The authors would like to express their thanks to the editor and reviewers for their comments and suggestions which significantly improved the presentation of this paper. This work was supported by the Korea Research Foundation (KRF) grant funded by the Korea government (MEST) (No. 2009-0072665). References [1] J. Govil and J Govil, “4G mobile communication systems : Turns, trends and transition,” in Proc. IEEE ICCIT, pp. 13-18, Nov. 2007.

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Figure 8: Average frame service time and average frame latency of the Coop-ARQ protocol and the Non-Coop-ARQ protocol versus PthS D when λ = 30 and fD = 6.6667.

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Figure 9: Saturated throughput of the Coop-ARQ protocol and the Non-Coop-ARQ protocol versus PthS D when fD = 6.6667. [2] T. M. Cover and A. E. Gamal, “Capacity theorems for the relay channel,” in IEEE Trans. Inform. Theory, vol. 25, no. 5, pp. 572-584, Sept. 1979. [3] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity - Part I : System Description,” in IEEE Trans. Commu., vol. 51, no. 11, pp. 1927-1938, Nov. 2003. [4] J. N. Laneman and G. W. Wornell, “Distributed space-time coded protocols for exploiting cooperative diversity in wireless networks,” in IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2415-2425, Oct. 2003. [5] T. E. Hunter and A. Nosratinia, “Cooperative diversity through coding,” in Proc. IEEE ISIT, p. 220, 2002. [6] A. Stefanov and E. Erkip, “Cooperative space-time coding for wireless networks,” in Proc. IEEE ITW, pp. 50-53, Apr. 2003. [7] L. Yi and J. Hong, “A new cooperative communication MAC strategy for wireless Ad hoc networks,” in Proc. IEEE ICIS, pp. 569-574, July 2007. [8] J. D. Morillo-Pozo and J. Garcia-Vidal, “A low coordination overhead C-ARQ protocol with frame combining,” in Proc. IEEE PIMRC, pp. 1-5, Sept. 2007. [9] T. Tabet, S. Dusad and R. Knoop, “Diversity-Multiplexing-Delay trade off in half-duplex ARQ relay channels,” in Proc. IEEE Trans. Inf. Theory., pp. 3793-3805, Oct. 2007. [10] I. Ahmed, M. Peng, and W. Wang “Performance analysis of an ARQ initialized cooperative communication protocol in shadowed Nakagamim wireless channel,” in Proc. IEEE ICC, pp. 321-325, May 2008. [11] V. Mahinthan, H. Rutagemwa, J. W. Mark, and X. Shen, “Performance of adaptive relaying schemes in cooperative diversity systems with ARQ,” in Proc. IEEE Globecom., pp. 4402-4406, Nov. 2007. [12] P. Liu, Z. Tao, S. Narayanan, T. Korakis, and S. Panwar, “CoopMAC : A cooperative MAC for wireless LANs,” in IEEE J. Sel. Areas Commun.., vol. 25, no. 2, pp. 340-354, Feb. 2007. [13] L. Xiong, L. Libman, and G. Mao, “Optimal strategies for cooperative MAC-layer retransmission in wireless networks,” in Proc. IEEE WCNC, pp. 1495-1500, Sept. Apr. 2008. [14] M. Dianati, X. Ling, K. Naik, and X. Shen, “A node-cooperative ARQ for wireless Ad Hoc networks,” in IEEE Trans. Vehic. Technol., vol. 55, no. 3, pp. 1032-1044, May 2006. [15] I. Cerutti, A. Fumagalli, and P. Gupta, “Delay models of single-source single-relay cooperative ARQ protocols in slotted radio networks with Poisson frame arrivals,” in IEEE/ACM Trans. Network., vol. 16, no. 2, pp. 371-382, Apr. 2008. [16] V. Mahinthan, H. Rutagemwa, J.W. Mark, and X. Shen, “Cross-Layer Performance Study of Cooperative Diversity System With ARQ,” in IEEE Trans. Vehic. Technol., vol. 58, no. 2, pp. 705-719, Feb. 2009. [17] E. Telatar, “Capacity of multi-antenna Gaussian channels,” in Euro. Trans. Telecommu., vol. 10, no. 5, pp. 585-595, Nov. 1999. [18] L. Zheng and D. N. Tse, “Diversity and multiplexing : A fundamental trade-off in multiple antenna channels,” in IEEE Trans. Inform. Theory, vol 49., no. 5, pp. 1073-1096, May 2003. [19] V. Tarokh, H. Jafarkhani, and A. Calderbank, “Space-time block codes from orthogonal designs,” in IEEE Trans. Inform. Theory, vol. 5, no. 5, pp. 1456-1467, July 1999. [20] L. Kleinrock and F. Tobagi, “Packet switching in radio channels: Part I-Carrier sense multiple-access modes and their throughput-delay characteristics,” in IEEE Trans. Commu., vol. 23, no. 12, pp. 1400-1416, Dec. 1975. [21] D. Bertsekas and R. Gallager, Data Networks, 2nd ed. Prentice Hall, 1992. [22] H. S. Wang and P. C. Chang, “On verifying the first-order Markovian assumption for a Rayleigh fading channel model,” in IEEE Trans. Veh. Technol., vol. 45, no. 2, pp. 353-357, May. 1996. [23] M. Zorzi, R. R. Rao, and L. B. Milstein, ”ARQ error control for fading mobile radio channels,” in IEEE Trans. Veh. Technol., vol. 46, no. 2,

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[24] [25] [26] [27]

[28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

pp. 445-455, May 1997. M. F. Neuts, Structured Stochastic Matrices of M/G/1 type and their Application. New York: Dekker, 1989. L. Breuer and D. Baum, An Introduction to Queueing Theory and Matrix-Analytic Methods, Springer, 2010. V. G. Kulkarni, Modeling and Analysis of Stochastic Systems, London: Chapman & Hall, 1995. IEEE Standard for Local and Metropolitan Area Networks, Part 16: Air Interface for Fixed and Mobile Broadband Wireless Access Systems. Amendment 2: Physical and Medium Access Control Layers for Combined Fixed and Mobile Operation in Licensed Bands, IEEE Std 802.16e 2005, Feb. 2006. IEEE Draft Standard P802.16j/D5, Part 16: Air Interface for Fixed and Mobile Broadband Wireless Access Systems-Multihop Relay Specification, May, 2008. IEEE Standard for Local and Metropolitan Area Networks, Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, ANSI/IEEE Std 802.11, 1999 Edition. M. C. Jeruchim, P. Balaban and K. S. Shanmugan, Simulation of Communication Systems, Cambridge University Press, 2005. A. Goldsmith, Wireless Communications, New York: Kluwer Academic/Plenum, 2000. E. N. Gilbert, “Capacity of a burst-noise channel.”, in Bell Syst. Tech. J., vol. 39, pp. 1253-1265, Sep. 1960. E. O. Elliot, “Estimates of error rates for codes on burst-noise channels,” in Bell Syst. Tech. J., vol. 42, pp 1977-1997, Sep. 1963. W. Xu, J. Lin and L. Wu, “Performance Analysis of Cooperative ARQ with Code Combining over Nakagami-m Fading Channel”, in Wireless Personal Communications, June 2009. E. Zimmermann, P. Herhold and G. Fettweis, “The Impact of Cooperation on Diversity-Exploiting Protocols”, in Proc. of 59th IEEE Vehicular Technology Conference (VTC 2004). A. Scaglione, D. L. Goeckel and J. N. Laneman, “Cooperative Communications in Mobile Ad Hoc Networks”, in IEEE Signal Processing Magazine, pp. 18-29, Sep. 2006. A. J. Laub, Matrix Analysis for Scientists and Engineers, Philadelphia: SIAM, 2005. G. H. Golub and C. F. Van Loan, Matrix Computations, Baltimore: The Johns Hopkins University Press, 1996.

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