Performance Analysis of Single Source and Single Relay Cooperative ARQprotocols under Time Correlated Rayleigh Fading Channel

Hong Il Choa, Chan Yong Leea, Gang Uk Hwanga,∗

aDepartment of Mathematical Sciences and Telecommunication Engineering ProgramKorea 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 Pois-son arrivals and time-correlated Rayleigh fading. The conventional non-cooperative ARQ protocol under the samechannel environment is also analyzed to compare its performance with that of the cooperative ARQ protocol. A twodimensional discrete time Markov chain is constructed for the analysis. Using the steady state analysis of the Markovchain, the average frame latency and the probability generating function of the frame service time are derived. Wevalidate our analytical results by comparing them with simulation results. From our analysis, we show that the co-operative ARQ protocol outperforms the non-cooperative ARQ protocol and the effect of the time correlation in thefading channel is not negligible. We also see that the performance metrics which we consider in this paper are almostconstant as long as d2

S R + d2RD remains constant, where dS R is the distance between the source node and the relay node

and dRD is the distance between the relay node and the destination node.

Keywords: ARQ; Cooperative diversity; Performance analysis; Queueing performance; Relay network; Timecorrelated fading channel

1. Introduction

Next generation wireless networks such as fourth generation wireless systems are required to support qualityof service (QoS) requirements such as delay constraints [1]. To meet these requirements, new advanced schemesare needed and performance analysis of these schemes is also required to check their validity. Different from thechannels in wired networks, wireless channels in wireless networks have two main features, fluctuation and fadingin received signal strength, and the inherent broadcasting nature of the channel. Various diversity schemes havebeen applied to alleviate fading effect and to improve performance. In this paper, we consider a cooperative systemutilizing spatial diversity. Although the broadcasting nature of the wireless channel is typically considered harmfulbecause simultaneous transmitted signals from nodes collide, it is possible to exploit this characteristic by having anintermediate node overhear transmissions from the source node to the destination node. After a node who is willingto help the source node eavesdrops the transmission from the source node, the helping node, generally called relaynode, transmits a replica or a mixture of its data frame and the eavesdropped data frame to the destination node. Thismethod, 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 havemainly focused on the physical layer and little attention has been paid to the performance of upper layers. Here, weconsider the MAC (Medium Access Control) layer performance of a cooperative system and particularly concentrateon 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: guhwang@kaist.edu (Gang Uk Hwang )

Preprint submitted to Elsevier December 29, 2010

Source

Relay

Destination

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 interms of spectral efficiency and packet error rate. In more detail, [8] proposes a cooperative ARQ protocol and showssome advantages in terms of both frame error rate and frame service time. [11] compares the performance of severalcooperative ARQ protocols based on simulation. [12] suggests a new cooperative ARQ protocol that can be notonly implemented in wireless LAN but is also compatible with the legacy IEEE 802.11 protocol [29]. [13] considerscooperative retransmission strategies to minimize the average frame service time which is defined as the average timeduration between the time instant of a new frame transmission and the time instant of its successful reception bythe destination. Regarding QoS metrics such as delay constraints and throughput, [14] and [16] analytically showthat their cooperative ARQ protocol dramatically reduces the average frame service time and the jitter of the frameservice 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 timeinstant of a frame arrival and the time instant of its successful reception by the destination, when the fading channelis time-uncorrelated.

Motivated by the works in [14–16], we develop an analytical model of a cooperative ARQ protocol under generaltraffic 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 theaverage frame service time and the frame latency, but also the saturated throughput and the probability generatingfunction of the frame service time by using Markov renewal theory and Little’s Theorem. The main contributions ofthis paper are the following.

1. Our analytical model considers general traffic conditions including both unsaturated and saturated traffic, whilethe model in [14, 16] assumes saturated traffic only. In many practical applications, the traffic of a network isgenerally 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 time-correlated fading channel. Since the wireless fading channels are time-correlated in most cases [31], our modelis 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 analyzethe 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 affectsthe performance. It is worthwhile to mention that our results can be used to determine the locations of the relaynode 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 coop-erative 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 throughsimulation 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-timesystem where the time axis is divided into equal size slots. We assume that the time duration of a slot is equal to Tseconds. The ACK or the NAK for the frame transmission is sent by the destination node in the same time slot withthe frame. We assume that the ACK and the NAK are decoded perfectly. We consider a cooperative ARQ protocolwhich will be explained in detail later. For comparison purpose, we also consider and analyze the conventional non-cooperative ARQ protocol. For simplicity, we call the cooperative ARQ protocol and the conventional non-cooperativeARQ 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 assumedto be flat and time-correlated Rayleigh fading channels. We assume that the source node and the relay node aresufficiently apart, so that all channels S D, S R and RD are mutually independent. In our discrete-time model, timeinstants at slot boundaries are indexed by t = 0, 1, 2, · · · . In the flat Rayleigh fading channel, the success-failureprocess of a frame transmission can be modeled as a Markov process as follows [22, 32, 33].2 Let ϵi(t) be the success-failure process of channel i in the t-th slot, for i = S D, S R,RD. Then, for a given channel reception threshold Pth andthe complex envelope of the received fading gain gi(t) at the received node, we have

ϵi(t) ={

G, if |gi(t)|2 > Pth,B, otherwise,

(1)

for i = S D, S R,RD. States G and B denote the success and the failure of the transmission, respectively. Since theframe transmission starts only at slot boundaries and the channel is assumed to be invariant during a slot, ϵi(t) onlydepends on the channel state at the beginning of the t-th slot for each t. So ϵi(t) is a discrete time two-state Markovchain, 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},

1SD,SR and RD are abbreviations of ’Source to Destination’, ’Source to Relay’ and ’Relay to Destination’, respectively.2The 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].

3

respectively. The state transition probability matrix P(i) of ϵi(t) is

P(i) =

[1 − bi bi

ai 1 − ai

]. (2)

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

E[|gi(t)|2] = Kd−βi , (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), ifdi = AidS D, where Ai is a constant for i = S R,RD and AS D = 1, the following relation holds:

E[|gi(t)|2] = A−βi E[|gS D(t)|2]. (4)

To consider the effects of distance in terms of the channel reception threshold and to simplify the fading gain, wedefine 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

E[|gi(t)|2]=

1K

dβi Pth, for i = S D, S R,RD. (5)

Then (1) is written in terms of γi(t) and Pthi as

ϵi(t) ={

G, if γi(t) > Pthi.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 distanceincreases, Pthi also increases, and therefore frame errors occur more frequently. So, Pthi can be regard as a measureof 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 RPthS D,

PthRD = AβRDPthS 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 =14 PthS D.

Let fDi denote the maximum Doppler shift of channel i for i = S D, S R,RD. Then the exact values of elementsin the transition probability matrix P(i) in (2) can be found in [23] as follows by using the bivariate distribution of theRayleigh process:

ai =Q(θi, ρθi) − Q(ρθi, θi)

ePthi − 1

and

bi =1 − e−Pthi

e−Pthi,

where Q(·, ·) is the Marcum Q function,

θi =

√2Pthi

1 − ρi2 ,

ρi = J0(2π fDi T ),

and J0(·) is the zero-order Bessel function of the first kind.

4

Time

Time

Time

Destination

Relay

Source i+1

ACKACKi

i ACKACK

ACKACK

(a) Source success

Time

Time

Time

Destination

Relay

Source

i

i

i

NAKNAK

NAKNAK

NAKNAK

i

ACKACK

ACKACK

i+1

(b) Relay aid success

Time

Time

Time

Destination

Relay

Source ii

i

NAKNAK

NAKNAK

NAKNAKi

NAKNAK

NAKNAK

(c) Relay aid failure

Time

Time

Time

Destination

Relay

Source ii

NAKNAK

NAKNAK

NAKNAK

(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 thedestination node. If the transmitted frame is successfully received by the destination node, the destination node willsend an ACK to the source node announcing the success of the transmission. Otherwise, the destination node willtransmit a NAK to the source node reporting the failure of the transmission. While the source node transmits theframe, the relay node can eavesdrop the transmission from the source node to the destination node. In contrast to theNon-Coop-ARQ protocol where an erroneously received frame is retransmitted by the source node, if the relay nodecatches correctly the frame transmitted previously by the source node, it retransmits the replica of the frame in thenext 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 isalso erroneous, the destination node sends a NAK. In this case, since the RD channel is likely to remain in the fadingstate in the next slot, it is better that the source node retransmits the frame in the next slot. So, in the Coop-ARQprotocol, this retransmitting procedure is initiated by the source node. The retransmission procedure mentioned aboveis carried out repeatedly until the successful reception of the frame by the destination node. For the sake of simplicityof protocol operation, the relay node discards the frame whenever the cooperation takes place and/or the destinationnode successfully receives the frame.

Based on the elementary operation of the Coop-ARQ protocol mentioned above, if the source node has a frame totransmit, 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. Ifthe source node has another frame to transmit in its queue, it sends the new frame at the beginning of the nextslot.

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 andthe 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 caneavesdrop the transmission. However, the retransmission by the relay node in the next slot is also erroneous. Sothe 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 cannoteavesdrop 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 protocolIn 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, theretransmission 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 Section2. Before the development of the Markov models, the following common assumptions for both the Coop-ARQ andthe 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 aslot 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 theslot 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 dependson 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 proto-cols.

• 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 slotduration T as follows:

αm =e−λT (λT )m

m!,

βm =e−2λT (2λT )m

m!.

6

Time

Figure 4: The operation of the Coop-ARQ protocol. Circle marks on the time axis denote embedded epochs of theMarkov chain. An upward vertical arrow means a frame departure at the time instant and a right horizontal arrowmeans 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 itis obvious that the process is not a Markov chain because the transmission of a frame depends on the channel statebetween nodes. So, we consider a novel two dimensional Markov chain consisting of the number of frames in thequeue 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 theMarkov 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 epochof 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 epochof the Markov chain.

• If a frame transmission by the source node fails in a slot, then the end of the following slot is an embeddedepoch of the Markov chain.

The embedded epochs are indexed by k = 0, 1, 2, · · · . An example evolution of the network showing the embeddedepochs 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] denotethe channel state of the Coop-ARQ protocol at the k-th embedded epoch. Let tk be the actual slot index of the k-thembedded epoch. The states of ChC[k] are defined as given in Table 1. With this novel definition we can construct atwo dimensional Markov chain later. Assume that qC[k] ≥ 1. Then, a frame is successfully transmitted at slot tk fromthe source node to the destination node directly if 1 ≤ ChC[k] ≤ 4. A frame is successfully transmitted via the relaynode 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 Fof ChC[k] are constructed. The (i, j)-th elements of E1, E2, E3 and F denote the following probabilities:

E1i, j = Pr {ChC[k + 1] = j|ChC[k] = i, qC[k] = 0,D(tk) = 0} , for 1 ≤ i, j ≤ 8,E2i, j = Pr {ChC[k + 1] = j|ChC[k] = i, qC[k] = 0,D(tk) ≥ 1} , for 1 ≤ i, j ≤ 8,

E3i, j =

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

for 1 ≤ i ≤ 5, 1 ≤ j ≤ 8,for 6 ≤ i ≤ 8, 1 ≤ j ≤ 8,

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

where D(t) denotes the number of arriving frames at the source node in the t-th slot, respectively. Then it is straightfor-ward to see that the state transition probability matrices E1, E2, E3 and F of ChC[k] can be derived from the matricesP(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]

qC[k] = 0 qC[k] ≥ 1ChC[k] ϵS D(tk) ϵS R(tk) ϵRD(tk) ϵS D(tk) ϵS R(tk) ϵRD(tk)

1

GG

G

GG

G2 B B3

BG

BG

4 B BChC[k] ϵS D(tk) ϵS R(tk) ϵRD(tk) ϵS D(tk) ϵS R(tk) ϵRD(tk + 1)

5

BG

G

BG

G6 B B7

BG

BG

8 B B

E2 =[

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

(P(RD)

)2((1,...,8),(5,...,8))

], (6)

E3 =

P(S D) ⊗ P(S R) ⊗ P(RD)

((1,...,4),(1,...,8))(P(S D)

)2 ⊗ (P(S R))2 ⊗ P(RD)

((5),(1,...,8))0

,

F =

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

(P(RD)

)2((1,...,4),(5,...,8))(

P(S D))2 ⊗ (P(S R)

)2 ⊗ P(RD)((5,...,8),(1,...,4))

(P(S D)

)2 ⊗ (P(S R))2 ⊗ (P(RD)

)2((5,...,8),(5,...,8))

, (7)

where ⊗ and A(η,ξ) denote the Kronecker product of matrices and the submatrix notation of a matrix A for vectors ηand ξ, respectively. See Chapter 13 of [37] and Chapter 1 of [38] for the definitions of the Kronecker product and thesubmatrix notation, respectively.

The two dimensional stochastic process {qC[k],ChC[k]} forms a Markov chain. If qC[k] is equal to zero, i.e., thereare no frames to transmit by the source node at the k-th embedded epoch, qC[k + 1] is equal to the number of framesarriving during one slot, because the time interval between two consecutive embedded epochs is equal to one slot timeT 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 andthe frame transmission started at the k-th embedded epoch by the source node is successful. When ChC[k] = 5, thetime duration between two consecutive embedded epochs is equal to 2T and the frame transmission started at the k-thembedded epoch is successful with the help of the relay node. If 6 ≤ ChC[k] ≤ 8 , then the time duration between twoconsecutive embedded epochs is equal to 2T and the frame transmission started at the k-th embedded epoch fails. Bythe above arguments the evolution equation of the queue at the source node is obtained as follows:

qC[k + 1] =

D(tk), if qC[k] = 0,qC[k] + D(tk) − 1, if qC[k] ≥ 1, 1 ≤ ChC[k] ≤ 4,qC[k] + D(tk) + D(tk + 1) − 1, if qC[k] ≥ 1,ChC[k] = 5,qC[k] + D(tk) + D(tk + 1), 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}

=

α0E1l,m, if i = 0, j = 0,α jE2l,m, if i = 0, j ≥ 1,α0E3l,m, if i = 1, 1 ≤ l ≤ 4, j = 0,β0E3l,m, if i = 1, l = 5, j = 0,α j−i+1Fl,m, if i ≥ 2, 1 ≤ l ≤ 4, j ≥ i − 1,β j−i+1Fl,m, if i ≥ 2, l = 5, j ≥ i − 1,β j−iFl,m, if i ≥ 2, 6 ≤ l ≤ 8, j ≥ i,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

PC =

B(C)

0 B(C)1 B(C)

2 B(C)3 · · ·

A0(C) A(C)

1 A(C)2 A(C)

3 · · ·A(C)

0 A(C)1 A(C)

2 · · ·....... . .

,where

A0(C)= diag(α0, α0, α0, α0, β0, 0, 0, 0)E3,

A(C)0 = 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, for k ≥ 1,

B(C)0 = α0E1,

B(C)k = αkE2, 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, π

Ci,2, · · · , πC

i,8) for i ≥ 0. Then the steady state probability vector of the Markov chain {qC[k],ChC[k]} is given asπC , (πC

0 , πC1 , π

C2 , · · · ). Since πC is the steady state probability vector, πC PC = πC and

∑∞i=0 π

Ci 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 forπC by finding an eigenvector, since PC is not a finite dimensional matrix. So the steady state probability vector πC isobtained by the matrix analytic method of [24, 25].

3.2. Markov model for Non-Coop-ARQ protocolIn 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

ChNC[k] ,{

1, if ϵS D(tk) = G,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 thesource node just after the k-th slot boundary in the Non-Coop-ARQ protocol. Then {qNC[k],ChNC[k]} forms a twodimensional discrete time Markov chain. By using arguments similar to those given in Subsection 3.1, the followingstate transition probabilities are derived:

Pr{qNC[k + 1] = j,ChNC[k + 1] = m|qNC[k] = i,ChNC[k] = l}

=

α jP

(S D)l,m , if i = 0,

α j−i+1P(S D)l,m , if i ≥ 1, l = 1, j ≥ i − 1,

α j−iP(S D)l,m , if i ≥ 1, l = 2, j ≥ i,

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 =

B(NC)

0 B(NC)1 B(NC)

2 B(NC)3 · · ·

A(NC)0 A(NC)

1 A(NC)2 A(NC)

3 · · ·A(NC)

0 A(NC)1 A(NC)

2 · · ·...

.... . .

,where

A(NC)0 = diag(α0, 0)P(S D),

A(NC)k = diag(αk, αk−1)P(S D), for k ≥ 1,

B(NC)k = αkP(S D), for k ≥ 0.

Let πNCi, j denote the steady state probability that the Markov chain {qNC[k],ChNC[k]} is in state (i, j) for i ≥ 0 and 1 ≤

j ≤ 2. The steady state probability vector that i frames are in the source node is denoted as πNCi , (πNC

i,1 , πNCi,2 ) for i ≥ 0.

Then the steady state probability vector of the Markov chain {qNC[k],ChNC[k]} is given as πNC , (πNC0 , π

NC1 , π

NC2 , · · · ).

Since πNC is the steady state probability vector, πNC PNC = πNC and∑∞

i=0 πNCi 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 andNon-Coop-ARQ protocols by using the Markov chains developed in Section 3. The definitions of these performancemetrics are as follows.

• TC: frame service time in the Coop-ARQ protocol in slots, i.e., the time duration between the time instant of anew 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.

• TNC: 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 framearrival 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.

• ThC: saturated throughput of the network in the Coop-ARQ protocol, i.e., the average number of successfullytransmitted frames per slot under the saturated traffic condition.

• ThNC: 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 steadystate.

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 stateat a new frame transmission instant should be derived. This probability is obtained in Lemma 1.

10

Lemma 1. Let p j denote the steady state probability of the channel state (the state of the process ChC[k]) at a newframe transmission instant in the Coop-ARQ protocol for 1 ≤ j ≤ 8. Then we have

p j =1

MC

(1 − e−λT )8∑

l=1

πC0,lE2l, j + (1 − e−λT )

4∑l=1

πC1,lFl, j + (1 − e−2λT )πC

1,5F5, j +

∞∑i=2

5∑l=1

πCi,lFl, j

,where

MC = (1 − e−λT )8∑

j=1

πC0, j +

(1 − e−λT )4∑

j=1

πC1, j + (1 − e−2λT )πC

1,5

+ ∞∑i=2

5∑j=1

πCi, j,

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 ofthe Markov chain {qC[k],ChC[k]}. A tagged embedded epoch, say the k-th embedded epoch, is the slot boundary atwhich 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 probabilityvector πC of {qC[k],ChC[k]} as follows:

MC = (1 − e−λT )8∑

l=1

πC0,l +

(1 − e−λT )4∑

l=1

πC1,l + (1 − e−2λT )πC

1,5

+ ∞∑i=2

5∑l=1

πCi,l. (9)

Then p j are derived by multiplying the state transition probabilities of the channel process ChC[k] to the respectiveprobabilities of the conditions that are mentioned above and by dividing them by MC .

�Let k 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∑j=1

p jE[zTC |ChC[k] = j, qC[k] ≥ 1

](10)

=

4∑j=1

p jz + p5z2 +

8∑j=6

p jE[zTC |ChC[k] = j, qC[k] ≥ 1

].

The above equation (10) follows from the observation that 1 ≤ ChC[k] ≤ 4 implies the successful transmission ofthe new frame by the source node and ChC[k] = 5 implies the successful transmission of the new frame via the relaynode. For 6 ≤ j ≤ 8 we have

E[zTC |ChC[k] = j, qC[k] ≥ 1

]=

8∑k=1

F j,kE[zTC |ChC[k + 1] = k,ChC[k] = j, qC[k + 1] ≥ 1, qC[k] ≥ 1

]=

8∑k=1

F j,k

(E[zTC+2|ChC[k + 1] = k, qC[k + 1] ≥ 1

])=

4∑k=1

F j,kz3 + F j,5z4 +

8∑k=6

F j,kE[zTC |ChC[k + 1] = k, qC[k + 1] ≥ 1

] z2.

(11)

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:

E[zTC∣∣∣ChC[k + 1], qC[k + 1] ≥ 1

]= E[zTC∣∣∣ChC[k], qC[k] ≥ 1

]. (12)

Using (11) and (12), it is easy to see that the value of E[zTC |ChC(k) = j, qC(k) ≥ 1

]is obtained for 1 ≤ j ≤ 8 as

follows:E[zTC |ChC[k] = 6, qC[k] ≥ 1

]E[zTC |ChC[k] = 7, qC[k] ≥ 1

]E[zTC |ChC[k] = 8, qC[k] ≥ 1

] =(I3 − z2F[(6,7,8),(6,7,8)]

)−1

z3∑4

j=1 F6, j + z4F6,5

z3∑4j=1 F7, j + z4F7,5

z3∑4j=1 F8, j + z4F8,5

, (13)

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 isderived as follows:

GTC (z) =4∑

j=1

p jz + p5z2

+

8∑j=6

p j

4∑k=1

F j,kz3 + F j,5z4 + F[( j),(6,7,8)]

(I3 − z2F[(6,7,8),(6,7,8)]

)−1

∑4

s=1 F6,s + zF6,5∑4s=1 F7,s + zF7,5∑4s=1 F8,s + zF8,5

z3

.The average frame service time E [TC] is then obtained as follows:

E [TC] =ddz

GTC (z)∣∣∣∣∣z=1.

The probability mass function of TC is also obtained by using the probability generating function GTC (z) as follows:

Pr {TC = k} = dk

dzk

GTC (z)k!

∣∣∣∣∣∣z=0.

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 actualsojourn time of the Markov chain in state (i, j) in slots, i.e., the time duration in slots between the embedded epochwith state (i, j) and the next embedded epoch. Then we have the following equations:

µi, j =

1, if i = 0,1, if i ≥ 1, 1 ≤ j ≤ 4,2, if i ≥ 1, 5 ≤ j ≤ 8.

Let{qC(t), ˜ChC(t)

}t≥0

be the corresponding semi-Markov process [26], i.e., qC(t) = qC[k] and ˜ChC(t) = ChC[k] if

tk ≤ t < tk+1. An example of the process{qC(t), ˜ChC(t)

}is shown in Fig 5.

Let πC(i, j) be the steady state probability that

{qC(t), ˜ChC(t)

}t≥0

is in state (i, j) at an arbitrary slot time. By theMarkov renewal theory [26], we have the following equation for πC

(i, j):

πC(i, j) , lim

t→∞

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

=πC

(i, j)µi, j∑k,l π

C(k,l)µk,l

.

(14)

12

Time

Figure 5: An example of the process{qC(t), ˜ChC(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 righthorizontal arrow means an arrival at the source node. E-index[k] means the index of the embedded epoch of theMarkov chain {qC[k],ChC[k]} and S-index(t) means the index of the slot. Queue(t) denotes the number of frames inthe queue just after the beginning of the t-th slot.

Note that πC(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 framesdenoted by the state, since there may be some new frame arrivals between the last embedded epoch and the arbitraryslot 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 findthat the number of frames in the queue just after the beginning of the slot is equal to 4 due to two new frames arrivingduring the tk1 -th slot. However we have qC(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 givenby the following theorem.

Theorem 3. Let qn be the probability that n frames are in the source node at an arbitrary slot boundary. Then wehave

qn =

8∑j=1

π(0, j), if n = 0,

4∑j=1

π(n, j) +12

8∑j=5

π(n, j) +

n∑i=1

αn−iπ(i, j)

, if n ≥ 1.

Proof. We first tag an arbitrary slot boundary and call it the tagged slot boundary. Note that

qn , Pr {n frames in the queue at the source node at the tagged slot boundary}=∑i, j

πC(i, j) f (i, j)

n(15)

where

f (i, j)n , 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 framesdenoted by the state. That is,

f (i, j)n =

{1, if n = i,0, otherwise , (16)

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 probability12 . If the tagged slot boundary is an embedded epoch, then the value of f (i, j)

n is determined in the same way as in (16).On the other hand, when the tagged slot boundary is not an embedded epoch, the number of frames at the tagged slotboundary is increased by the number of newly arriving frames just before the tagged slot boundary. So we have thefollowing equation:

f (i, j)n =

12

I{n = i} + 12

I{n ≥ i}αn−i, (17)

where I{·} denotes the indicator function for i ≥ 1, 5 ≤ j ≤ 8. Combining (15), (16) and (17), we obtain qn.

�

Let E[NC] be the average number of frames in the source node. The value of E[NC] is obtained as follows:

E[NC] =∞∑

n=0

nqn. (18)

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 ThC

The saturated throughput ThC can be defined as follows:

ThC = limt→∞

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)

The denominator and numerator in (19) can be obtained by using the steady state probability of the channel stateunder the saturated condition. Under the saturated condition, the channel process ChC[k] is a Markov chain with thetransition probability matrix F which is defined in (7). Let chC =

(chC

j

)1≤ j≤8

be the steady state probability vector of

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

chC = chC F,8∑

j=1

chCj = 1.

Since the transmission of a frame is successful only for 1 ≤ ChC[k] ≤ 5, the numerator in (19) is equal to∑5

j=1 chCj .

The denominator in (19) is equal to∑4

j=1 chCj + 2

∑8j=5 chC

j . By the above arguments, the saturated throughput of thenetwork ThC can be obtained as follows:

ThC =

∑5j=1 chC

j∑4j=1 chC

j + 2∑8

j=5 chCj

=

∑5j=1 chC

j

1 +∑8

j=5 chCj

.

14

4.2. Non-Coop-ARQ protocol

4.2.1. The average frame service time E [TNC] and probability density function of TNC

As in the Coop-ARQ protocol case, the steady state probabilities of the channel state at new frame transmissioninstants are calculated first, in the following lemma.

Lemma 4. Let pNCj be the steady state probability of the channel state at a new frame transmission instant in the

Non-Coop-ARQ protocol. Then we have

pNCj =

1MNC

(1 − e−λT )2∑

l=1

πNC0,l P(S D)

l, j + (1 − e−λT )πNC1,1 P(S D)

1, j +

∞∑i=2

πNCi,1 P(S D)

1, j

,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 embeddedepoch.

• {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 probabilityvector πNC of {qNC[k],ChNC[k]} as follows:

MNC = (1 − e−λT )2∑

j=1

πNC0, j + (1 − e−λT )πNC

1,1 +

∞∑i=2

πNCi,1 . (20)

Then pNCj 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 k be an arbitrary slot boundary where a new frame transmission begins. The probability generating functionGTNC (z) of TNC can be obtained as follows:

GTNC (z) , E[zTNC]

=

2∑j=1

pNCj E[zTNC |ChNC[k] = j

]= pNC

1 z + pNC2 E[zTNC |ChNC[k] = 2

].

(21)

The value of E[zTNC |ChNC[k] = 2

]is obtained by solving the following equation.

E[zTNC |ChNC[k] = 2

]=

2∑k=1

P(S D)2,k E

[zTNC |ChNC[k + 1] = k,ChNC[k] = 2

]=

2∑k=1

P(S D)2,k E

[zTNC+1|ChNC[k] = k

]= P(S D)

2,1 z2 + P(S D)2,2 zE

[zTNC |ChNC[k] = 2

].

(22)

Combining (21) and (22), we have the following theorem:

15

Theorem 5. For the Non-Coop-ARQ protocol, the probability generating function GTNC (z) of the frame service timeTNC is derived as follows:

GTNC (z) = pNC1 z + pNC

2

z2P(S D)2,1

1 − zP(S D)2,2

.

The average frame service time E [TNC] can be obtained as follows:

E [TNC] =ddz

GTNC (z)∣∣∣∣∣z=1.

The probability mass function of TNC is also derived by using the probability generating function GTC (z) as follows:

Pr {TNC = k} = dk

dzk

GTNC (z)k!

∣∣∣∣∣∣z=0.

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-Coop-

ARQ protocol. Hence, the steady state probability that there are i frames in the queue at the source node at an arbitraryslot boundary is equal to πNC

i for i ≥ 0. Thus the average number of frames in the source node in the steady stateE [NNC] is given as follows:

E [NNC] =∞∑

i=0

iπNCi . (23)

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

E [LNC] =E [NNC]λT

.

4.2.3. The saturated throughput of the network ThNC

The saturated throughput ThNC for the Non-Coop-ARQ can be defined as follows:

ThNC = limt→∞

The number of successfully transmitted frames during [0, t]t

= The average number of frames transmitted during one slot time.

The channel state process ChNC[k] itself forms a Markov chain with state transition probability matrix P(S D). LetchNC = (chNC

j )1≤ j≤2

be the steady state probability vector of the Markov chain {ChNC[k]}. Then chNC is obtained bysolving the following equations:

chNC = chNC P(S D),

2∑j=1

chNCj = 1.

Since the transmission of a frame is successful only when ChNC[k] = 1, the saturation throughput ThNC is derived asfollows:

ThNC = chNC1 .

16

5. Model validation and performance results

To validate our analytical model for the Coop-ARQ protocol and compare its performance metrics with those ofthe 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 thesimulation of the fading channel, time-correlated Rayleigh fading channels are generated by a built-in m-file script inMATLAB which passes white Gaussian noise through a filter whose power spectrum corresponds to the Jakes Dopplerspectrum [30]. Throughout this section, the maximum Doppler shift fDi 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 protocolWe 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 =14 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 thetime-correlated channel, the average frame latency of the Coop-ARQ protocol increases as fD decreases. This impliesthat 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 longframe 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 successfultransmission when the initial transmission fails, the above result implies that successful retransmissions of framesfrequently occur with the aid of the relay node, i.e., the ‘Relay aid success’ case of the Coop-ARQ protocol frequentlyoccurs, 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 (time-uncorrelated) channel as fD increases. This is because, as fD increases, the channel becomes less time-correlated andeventually independent. So, our model with time-correlated channel generalizes the model presented in [15] wherethe channel is time-uncorrelated.

5.2. Performance comparisons of the Coop-ARQ protocol and the Non-Coop-ARQ protocolTo 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-ARQprotocol and the Non-Coop-ARQ protocol in Fig. 8 for fD = 6.6667 and λ = 30. We also plot the saturated throughputfor fD = 6.6667 in Fig. 9. When the carrier frequency is 2.4 Hz and the relative speed of the mobile nodes is 5Km/h, the maximum Doppler shift fD is equal to 6.6667. All performance results of the Coop-ARQ protocol arebetter than those of the Non-Coop-ARQ protocol. In particular, as PthS D increases, the average frame service timeand the average frame latency of the Coop-ARQ protocol slowly increase but those of the Non-Coop-ARQ protocoldramatically increase. Since PthS D is closely related with the distance between the source node and the destinationnode, our results show that the relay node is needed to be positioned in a right place to satisfy delay QoS requirementof 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 framelatencies of the Coop-ARQ protocol and the Non-Coop-ARQ protocol for different values of fD. Under the saturatedcondition at the source node, we also compare the saturated throughputs of both protocols for PthS D = −3 dB inFig. 11. As seen in the figures, for all fD values, the performance metrics of the Coop-ARQ protocol are better thanthose of the Non-Coop-ARQ protocol. This means that the Coop-ARQ protocol still has benefits under the saturatedtraffic condition. Furthermore, the Coop-ARQ protocol has better performance for lower fD. It implies that the relaynode is more useful for pedestrian communications which have low fD and occur more frequently in metropolitanareas.

To investigate the effect of the arrival rates for the Coop-ARQ protocol and the Non-Coop-ARQ protocol, we plotin Fig. 12 the average frame latency for fD = 6.6667 and PthS D = −3 dB as λ is changed. The increase with λ inthe average frame latency for the Non-Coop-ARQ protocol is significant although that for the Coop-ARQ protocol is

17

2 4 6 8 10 12 14 16 181.5

1.55

1.6

1.65

1.7

1.75

1.8

fD

Ave

rage

Ser

vice

Tim

e (s

lot)

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

(a) Average frame service time

2 4 6 8 10 12 14 16 181.5

2

2.5

3

3.5

fD

Ave

rage

Fra

me

Late

ncy

(slo

t)

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

(b) Average frame latency

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

18

0 5 10 15 2010

−5

10−4

10−3

10−2

10−1

100

Service Time (slot)

Pro

babi

liy o

f Ser

vice

Tim

e

Coop−ARQ theoretical, λ = 30, f

D = 2

Coop−ARQ simulation, λ = 30, fD

= 2

Coop−ARQ theoretical, λ = 30, fD

= 18

Coop−ARQ simulation, λ = 30, fD

= 18

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 theCoop-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 relaynode positioning problem is solved based on the the channel qualities of S R and RD that are respectively representedby PthS R and PthRD in this paper. Refer to (5). So we plot the average frame service time, the average frame latencyand saturated throughput as functions of PthS R and PthRD in figs. 13 and 14. In the figures, we set β = 2, PthS D = −3dB, λ = 50 and fD = 6.6667. From the figures, we see that if PthS R + PthRD is constant, i.e., d2

S R + d2RD 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-correlatedRayleigh 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 saturatedthroughput. From the analysis, the performance metrics of the Coop-ARQ protocol are shown to be superior to those ofthe Non-Coop-ARQ protocol. We also find that the effects of time-correlation in the fading channel are not negligibleand 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 thedeployment 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 suggestionswhich significantly improved the presentation of this paper. This work was supported by the Korea Research Founda-tion (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.

19

−5 −4 −3 −2 −1 0 11

1.5

2

2.5

3

3.5

4

4.5

5

5.5

PthSD

(dB)

Ave

rage

Ser

vice

Tim

e (s

lot)

Coop−ARQ theoreticalCoop−ARQ simulationNon−Coop−ARQ theoreticalNon−Coop−ARQ simulation

(a) Average frame service time

−5 −4 −3 −2 −1 0 10

5

10

15

20

25

30

35

40

45

PthSD

(dB)

Ave

rage

Fra

me

Late

ncy

(slo

t)

Coop−ARQ theoreticalCoop−ARQ simulationNon−Coop−ARQ theoreticalNon−Coop−ARQ simulation

(b) Average frame latency

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

20

−5 −4 −3 −2 −1 0 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PthSD

(dB)

Sat

urat

ed T

hrou

ghpu

t

Coop−ARQ theoreticalCoop−ARQ simulationNon−Coop−ARQ theoreticalNon−Coop−ARQ simulation

Figure 9: Saturated throughput of the Coop-ARQ protocol and the Non-Coop-ARQ protocol versus PthS D whenfD = 6.6667.

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2 4 6 8 10 12 14 16 181.4

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Figure 10: Average frame service time and average frame latency of the Coop-ARQ protocol and the Non-Coop-ARQprotocol versus fD when PthS D = −3 dB and λ = 30.

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Figure 11: Saturated throughput of the Coop-ARQ protocol and the Non-Coop-ARQ protocol versus fD when PthS D =

−3 dB.

10 20 30 40 50 60 70 80 90 100 1100

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Figure 12: Average frame latency of the Coop-ARQ protocol and the Non-Coop-ARQ protocol versus λwhen PthS D =

−3 dB and fD = 6.6667.

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Figure 13: Average frame service time and average frame latency of the Coop-ARQ protocol over PthS R and PthRD

when fD = 6.6667, PthS D = −3 dB and λ = 50.

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0.60.62

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Figure 14: Saturated throughput of the Coop-ARQ protocol over PthS R and PthRD when PthS D = −3 dB and fD =

6.6667.

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