Performance Analysis of Random Relaying of Partitioned MDS

Codeword Block Applied to Persistent Relay CSMA

over Random Error Channels

Katsumi Sakakibara and Jumpei Taketsugu

Department of Information and Communication Engineering, Okayama Prefectural University, 719-1197, Soja, Japan

Keywords:

Cooperative Multi-hop Networks, Markov Model, MDS Codes, Performance Analysis, Persistent Relay

CSMA.

Abstract:

We propose incorporation of Random relaying of Partitioned Maximum Distance Separable codeword blocks

(RP-MDS), which has been proposed for multi-hop cooperative relay networks, to Persistent Relay Carrier

Sense Multiple Access (PRCSMA) over noisy channels. The proposed protocol elaborately employs the pow-

erful error-correcting capability of MDS codes into cooperative communication systems and introduces the

incremental redundancy concept to PRCSMA. A destination node can reinforce an error-correcting capability

when it receives a new frame. The performance of the proposed protocol is analyzed with a Markov model in

terms of the average duration of a cooperation phase and the energy efﬁciency. Numerical results indicate that

the proposed protocol can signiﬁcantly improve the performance, compared to the original PRCSMA.

1 INTRODUCTION

Cooperative communications with relay nodes have

been recognized as one of effective and promising

techniques in wireless/mobile communication sys-

tems. Relay standards are on the way to suc-

cessful implementation in Long Term Evolution

(LTE)-Advanced by the Third Generation Partner-

ship Project (3GPP) and 802.16m by IEEE (Loa

et al., 2010; Bhamri et al., 2011). Relay tech-

niques have been enthusiastically investigated from

the viewpointof the physical (PHY) and data-link lay-

ers (Bhamri et al., 2011; G´omez-Cuba et al., 2012). In

PHY layer perspective, Multiple-Input and Multiple-

Output (MIMO) and diversity techniques are attrac-

tive. In the data-link layer perspective, a number

of Cooperative Automatic Repeat reQuest (C-ARQ)

protocols have been proposed and analyzed. Partic-

ularly, the design of Medium Access Control (MAC)

protocols employed between relay nodes and the des-

tination node inﬂuences the performance, when two

or more relay nodes collaborate on an identical chan-

nel.

MAC protocols for C-ARQ systems have been

proposed recently. Dianati et al. (Dianati et al.,

2006) proposed a Node-Cooperation Stop-and-Wait

(NCSW) ARQ protocol. The performance of NCSW

with a single relay node was analyzed over two-

state Markovian channels. Morillo and Garcia-Vidal

(Morillo and Garcia-Vidal, 2011) proposed a C-ARQ

scheme with an integrated frame combiner. They

analyzed the performance with round-robin cooper-

ation among relay nodes and with Carrier Sense Mul-

tiple Access with Collision Avoidance (CSMA/CA).

Alonso-Zarate et al. (Alonso-Zarate et al., 2009;

Predojev et al., 2012) proposed Persistent Relay

CSMA (PRCSMA), which elaborately incorporates

well-known IEEE 802.11 Distributed Coordination

Function (DCF) (IEEE Standard 802.11, 1999). In

(Alonso-Zarateet al., 2009), the performance of PRC-

SMA was analyzed based on a steady-state two-

dimensional Markovian model proposed by Bianchi

(Bianchi, 2000). In the above literature (Dianati et al.,

2006; Morillo and Garcia-Vidal, 2011; Alonso-Zarate

et al., 2009; Predojev et al., 2012), it is basically as-

sumed that a node can correctly receive a transmitted

frame if no frame collisions occur. Thus, when we

consider a scenario where a channel adds errors to

a non-colliding frame, it is expected that the use of

error-correcting codes can improve the performance.

In this paper, we propose incorporation of Ran-

dom relaying of Partitioned Maximum Distance Sep-

arable codeword block (RP-MDS) (Sakakibara et al.,

2011) to PRCSMA over noisy channels. The pro-

155

Sakakibara K. and Taketsugu J..

Performance Analysis of Random Relaying of Partitioned MDS Codeword Block Applied to Persistent Relay CSMA over Random Error Channels.

DOI: 10.5220/0004708501550164

In Proceedings of the 3rd International Conference on Sensor Networks (SENSORNETS-2014), pages 155-164

ISBN: 978-989-758-001-7

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

S D

Source node Destination node

SER =

Relay nodes

R D

Figure 1: System model with N relay nodes.

posed protocol elaborately takes advantage of the

powerful error-correcting capability of MDS codes.

Incorporating RP-MDS into PRCSMA may intro-

duce effective performance improvement in accor-

dance with the concept of incremental redundancy

(Pursley and Sandberg, 1989). A destination node

can reinforce an error-correcting capability when it

receives a new frame, even if it includes channel er-

rors. The performance of the proposed protocol is an-

alyzed with the aid of a Markov model. The accuracy

of the model is veriﬁed by means of computer simu-

lation.

The rest of the present paper is organized as fol-

lows: Section 2 presents a system model with relay

nodes. PRCSMA is brieﬂy reviewed in Section 3.

In Section 4, after a short reminder of useful prop-

erties of MDS codes, the proposed protocol is de-

scribed. Performance of the proposed protocol is an-

alyzed in Section 5, based on the analysis in (Alonso-

Zarate et al., 2009). Numerical results are presented

in Section 6 in comparison with results obtained from

computer simulation. Finally, Section 7 concludes the

present paper.

2 SYSTEM MODEL

Consider a wireless network consisting of a pair of

source node S and destination node D with N relay

nodes; R

, R

,. .. , R

, as shown in Fig. 1. All chan-

nels are half-duplex, so that a node can not transmit

and receive simultaneously. All nodes are located

within their transmission range. Hence, each node can

overhear ongoing transmission originating from other

nodes. Let ε

, ε

, and ε

be the symbol error

probabilities on channels between source node S and

destination node D, between source node S and relay

node R

, and between relay node R

and destination

node D, respectively, for n = 1,2,...,N.

If frame

Using the symbol error rate ε, we can evaluate the bit

error rate as 1−

√

1−ε when a symbol consists of m bits.

transmission from source node S resulted in erroneous

reception at destination node D and if one or more

relay nodes succeeded in error-free reception of the

frame, then such relay nodes can collaborativelyserve

as supporters for frame retransmission. For effective

use of cooperative communications, we generally as-

sume that ε

> ε

. The duration in which relay

nodes collaborate frame retransmissions is referred to

as a cooperation phase (Alonso-Zarate et al., 2009).

Note that every frame is assumed to include an ap-

propriate header and an ideal Frame Check Sequence

(FCS) for error/collision detection,

in addition to the

payload.

3 PERSISTENT RELAY CSMA

(PRCSMA)

PRCSMA (Alonso-Zarate et al., 2009; Predojevet al.,

2012) is a MAC protocol which elaborately re-

solves frame collisions among transmission from re-

lay nodes, based on IEEE 802.11 DCF (IEEE Stan-

dard 802.11, 1999). Similarly to IEEE 802.11 DCF,

each relay node in PRCSMA inserts random back-

off delay before every frame transmission in a dis-

tributed manner according to its own contention win-

dow (CW). Let m denote a message block of k-

symbol length, which is generated at source node S.

A DATA frame consists of a header, payload m, and

FCS. Note that the terms “message block m” and

“DATA frame” are used interchangeably hereafter,

unless ambiguity arises.

The operation in PRCSMA is summarized as fol-

lows. The detailed description can be found in

(Alonso-Zarate et al., 2009). After erroneous recep-

tion of a DATA frame, destination node D broadcasts

a Call For Cooperation (CFC) frame. If one or more

relay nodes receive both the DATA frame and the CFC

frame, then the cooperation phase is invoked. Relay

nodes which join in the cooperation phase is referred

to as active relay nodes. Active relay nodes simulta-

neously start the DCF operation, after the reception of

the CFC frame followed by DIFS (Distributed Inter-

Frame Space). When destination node D correctly

receives a frame, it broadcasts an ACK frame to an-

nounce not only correct reception of the DATA frame

to source node S but also completion of the coopera-

tion phase to all the nodes.

An illustrative operational example with two ac-

tive relay nodes, R

and R

, is shown in Fig. 2.

Both active relay nodes independently set their back-

The term “ideal” implies that the probability of unde-

tected errors can be neglected.

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156

start of backoff

SIFS DIFS ACKtimeout ACKtimeout SIFS DIFS

erroneous reception frame collision error-free reception

cooperation phase

CFC ACK

erroneous reception

error-free reception

time

discard of DATA

freeze

start of backoff

Figure 2: Illustrative example of PRCSMA.

off counter to seven and a cooperation phase is in-

voked. The ﬁrst DATA frame transmission from these

relay nodes results in collision. The second transmis-

sion from relay node R

suffers from channel errors.

Finally, an ACK frame is returned by destination node

D corresponding to error-free reception of the second

transmission from R

. It completes the cooperation

phase. Notice that source node S does not participate

in a cooperation phase (Alonso-Zarate et al., 2009).

4 PRCSMA WITH RANDOM

RELAYING OF PARTITIONED

MDS CODEWORD BLOCK

In a cooperation phase in PRCSMA over noisy chan-

nels, destination node D may successively receive er-

roneous frames one by one in between backoff in-

tervals. It suggests possibility to effectively utilize

the concept of incremental redundancy (Pursley and

Sandberg, 1989), where the error-correcting capabil-

ity at a receiving node is reinforced upon frame re-

ception. In this context, we propose incorporating

RP-MDS into PRCSMA. RP-MDS has been proposed

for multi-hop cooperative relay networks over noisy

channels (Sakakibara et al., 2011). The proposed

protocol, designated as PRCSMA+RP-MDS, is de-

scribed after some properties of MDS codes are re-

viewed.

4.1 MDS Codes

Denote a linear block code of length n and dimension

k over a certain ﬁnite ﬁeld by an [n,k] code. An [n,k]

code is MDS if its minimum distance is n −k + 1. A

class of MDS codes, including Reed-Solomon codes,

is known to be fruitful in advantageous properties

(Wicker, 1995). Among them, the following two the-

orems; Theorems 8-4 and 8-6 in (Wicker, 1995), re-

spectively, are used afterward:

Theorem 1. For an [n,k] MDS code, a receiver can

recover the encoded message of length k, if it receives

at least k code symbols with no errors. 

Theorem 2. Punctured MDS codes are also MDS,

that is, the minimum distance of an [n − p,k] punc-

tured MDS code is n− p−k + 1, if n− p ≥ k. 

Suppose a systematic [Lk,k] MDS code C .

Let

G be a generator matrix of C . It is clear that G is a

k×Lk matrix. Let

G =



|{z}



|{z}



|{z}



···



L−1

{z}



(1)

be a partition of G into L blocks of identical size,

where I and G

ℓ

are an identity matrix and a square

matrix of order k for ℓ = 1,2,. .. ,L −1, respectively.

Then, for a message block m of length k to be en-

coded, a codeword of C can be also partitioned into L

codeword blocks c

ℓ

of length k;

c = mG =



|{z}



|{z}



|{z}



···



L−1

{z}



, (2)

where c

= m and c

ℓ

= mG

ℓ

for ℓ = 1,2,...,L−1.

From Theorem 1 and Theorem 2, the following corol-

lary holds at a receiver when one or more codeword

blocks c

ℓ

are received:

Corollary 1. Assume that u distinct codeword blocks,

ℓ

, c

ℓ

,. .. , c

ℓ

, are received and that a receiver

Using a systematic code, an encoded message appears

explicitly in the corresponding codeword vector. It implies

that its generator matrix includes an identity matrix, as its

submatrix. In the case that a given generator matrix is non-

systematic, we can convert it into a systematic form with

the aid of appropriate elementary row operations (Peterson

and Weldon, 1972).

PerformanceAnalysisofRandomRelayingofPartitionedMDSCodewordBlockAppliedtoPersistentRelayCSMAover

RandomErrorChannels

157

systematic MDS coding

header

DATA frame (source node S) Relayed frame (relay node R )

1 l -1L

c c

message m message m

k kk

PHY MAC message m FCS

header

PHY MAC FCS

Figure 3: Frame format.

start of backoff

SIFS DIFS ACKtimeout ACKtimeout SIFS DIFS

erroneous reception frame collision successful decoding

cooperation phase

CFC ACK

decoding failure

error-free reception

time

discard of DATA

freeze

start of backoff

Figure 4: Illustrative example of proposed protocol (PRCSMA+RP-MDS).

can identify the received codeword block number,

ℓ

,ℓ

,. .. ,ℓ

, for u ≤ L and 0 ≤ ℓ

< ℓ

< ··· < ℓ

≤

L −1. Then, a k-symbol message m can be recov-

ered, if either of the following conditions is satisﬁed:

(i) at least one codeword block c

ℓ

is error-free; and

(ii) the total number of errors occurred in the u code-

word blocks is less than or equal to



(u−1)k



, (3)

where ⌊x⌋ is the maximum integer not greater than x.



Proof. Since every codeword block c

ℓ

consists of k

symbols, it is straightforward from Theorem 1 that

a receiver can recover the message m from one or

more error-free codeword blocks. This leads to the

ﬁrst condition.

Next, aggregation of the u distinct received code-

word blocks results in a codeword of a [uk,k] punc-

tured MDS code. Thus, t

or less errors can be cor-

rected according to Theorem 2, which provides the

second condition.

4.2 Proposed Protocol

(PRCSMA+RP-MDS)

In PRCSMA, as described in Section 3, what a re-

lay node transmits is a replica of the message block

m. Therefore, it is required for destination node D

to receive a frame with no errors in order to complete

the cooperation phase. By contrast, in the proposed

protocol, an active relay node randomly transmits

one out of L −1 redundant MDS codeword blocks;

,. .. ,c

L−1

, after encoding the received message

block m by C , as in (2). Furthermore, destina-

tion node D stores erroneously received frames in the

buffer rather than discard.

A frame format used in the proposed protocol is

depicted in Fig. 3. The codeword block number ℓ

should be appropriately embedded in a header part,

which can be digitized by ⌈log

L⌉ bits, where ⌈x⌉ is

the minimum integer not less than x. For small L, it

can be negligible.

We describe the proposed protocol with the aid of

an illustrative operational example with the same sce-

nario as shown in Fig. 4. Destination node D stores an

erroneous message block m into its buffer. Two ac-

tive relay nodes R

and R

independently encode m

and randomly select one codeword block. In Fig. 4,

selects c

and R

selects c

. After frame colli-

sion occurs, each relay node re-selects one codeword

block; R

does c

again and R

, c

. Upon a recep-

tion of c

from relay node R

, destination node D

aggregates the received c

and the m in the buffer,

and then, decodes [ m



] by a [2k,k] punctured

MDS code of C . According to Corollary 1, the mes-

sage block m can be retrieved if c

is received with

no errors or if the total number of symbol errors in

[ m



] is not greater than ⌊k/2⌋ . However,it fails in

Fig. 4. At this time, destination node D stores two er-

SENSORNETS2014-InternationalConferenceonSensorNetworks

158

roneous blocks, m and c

. Subsequently to reception

of c

from R

, the message block m is successfully

recovered by decoding [ m



] with a [3k,k]

MDS code, which can correct up to k errors. Finally,

an ACK frame is returned from destination node D. It

completes the cooperation phase.

Notice that source node S does not take part in

a cooperation phase similarly to PRCSMA (Alonso-

Zarate et al., 2009). Furthermore, for L = 1 the pro-

posed protocol is reduced to the original PRCSMA,

since no error-correctingcapability is available at des-

tination node D.

5 PERFORMANCE ANALYSIS

5.1 Assumptions and Markov Model

In this section, we analyze the performance in the

cooperation phase, based on the Markov model in

(Alonso-Zarate et al., 2009). We impose identical as-

sumptions with (Alonso-Zarate et al., 2009). Since

we focus on the cooperation phase, it is presumed

that destination node D has stored an erroneous mes-

sage block m. We assume that a cooperation phase

start with N active relay nodes. We ignore erroneous

reception of control frames; ACK frames, and of a

header part in each frame. The CW value at each re-

lay node remains constant W all the time, that is, no

doubling procedure is carried out even if frame trans-

mission failure occurs, as opposed to the legacy DCF

(IEEE Standard 802.11, 1999). All frames involved

in collision are to be retransmitted, until the coopera-

tion phase is completed. We assume symmetric chan-

nels between relay node R

and destination node D,

that is, the symbol error rates between each relay node

and destination node D are identical and independent;

= ε

= ··· = ε

= ε

Then, a Markov model with respect to the value

of backoff counter at a relay node is quoted in Fig. 5

from (Alonso-Zarate et al., 2009). In Fig. 5, P

repre-

sents the probability that the cooperation phase ends

in a slot. Note that a slot duration varies depending

on frame transmissions in the slot.

5.2 Equations in Equilibrium

In equilibrium, an in-ﬂow and an out-ﬂow are bal-

anced for every state in Fig. 5. Letting π

the steady-state probability of state w for w =

0,1,2, .. .,W −1, we obtain











w+1

W−1

∑

i=0

for w = 0,1,2,...,W −2,

W−1

∑

i=0

for w = W −1.

(4)

Solving the recursive expression and the boundary

condition in (4) under the normalizing condition π

+ ···+ π

W−1

= 1, we have

{1−(1−P

)

W−w

}

−(1−P

){1−(1−P

)

W−w

}

(5)

for w = 0,1,...,W −1. Since frame transmission oc-

curs only when the backoff counter reaches to zero,

the probability of i-frame collision can be given by

= Pr[i-frame collision] =





(1−π

)

N−i

(6)

for i = 0,1,...,N. Then, a slot is idle with probability

, one frame is transmitted in a slot with probability

, and frame collision takes place with probability

1−q

−q

Next, we evaluate the probability P

of complet-

ing the cooperation phase. Destination node D stores

an erroneous DATA frame c

= m, when the coop-

eration phase starts. The initial probability that the

stored message includes e symbol errors is

α(e) =

1−(1−ε

)





(1−ε

)

k−e

(7)

for e = 1,2,...,k. Then, when destination node D re-

ceives a non-collided frame; say c

ℓ

, ℓ > 0 if L > 1,

aggregating two blocks results in [c



ℓ

]. The co-

operation phase ends, if either of two conditions in

Corollary 1 is satisﬁed. The probability of error-free

reception of a block of length k is (1−ε

)

. Taking

into account the fact that up to ⌊k/2⌋ errors in [c



ℓ

]

can be corrected, we have the probability of success-

ful decoding at destination node D as

succ











(1−ε

)

for L = 1,

(1−ε

)

⌊k/2⌋−1

∑

j=1

⌊k/2⌋−j

∑

e=1





(1−ε

)

k−j

α(e)

for L ≥ 2.

(8)

In the case of L > 2, further gain on P

succ

can be avail-

able when other code word blocks are received. How-

ever, we omit it in (8). Finally, we obtain

= q

succ

= nπ

(1−π

)

n−1

succ

. (9)

PerformanceAnalysisofRandomRelayingofPartitionedMDSCodewordBlockAppliedtoPersistentRelayCSMAover

RandomErrorChannels

159

210

W-1W-2

1-P

W1/ W1/ W1/ W1/ W1/

+ + +

Figure 5: Markov model (Alonso-Zarate et al., 2009).

5.3 Average Duration of Cooperation

Phase

Once P

is provided, it implies that a cooperation

phase consists of 1/P

slots in average, in which the

last slot is the only successful one. Hence, the average

numbers of idle slots, of slots with 1-frame transmis-

sion, and of slots with frame collision can be evalu-

ated by

#[idle] =



−1



1−q

succ

, (10)

#[1-frame transmission]

= 1 +



−1



(1−P

succ

)

1−q

succ

, (11)

#[frame collision] =



−1



1−q

−q

1−q

succ

, (12)

respectively. Then, the average duration of a cooper-

ation phase, given that N active relay nodes collabo-

rate, is given by

E[duration



= T

succ

+ T

slot

#[idle]

+ T

fail

(#[1 frame transmission] −1

+ #[frame collision])

= T

succ



−1



slot

+ T

fail

(1−q

−q

succ

)

1−q

succ

, (13)

where T

slot

, T

succ

, and T

fail

are the idle slot dura-

tion, the duration of successful message transmission

consisting of the DATA and the ACK frames, SIFS

and DIFT, and the duration of erroneous reception

or frame collision consisting of the DATA frame and

ACKtimiout, respectively. They are given as

succ

= T

DATA

+ T

SIFS

+ T

ACK

+ T

DIFS

, (14)

fail

= T

DATA

+ T

ACKtimeout

, (15)

where T

DATA

and T

ACK

are DATA frame duration and

ACK frame duration, respectively, and other T

’s are

the duration of element x.

5.4 Energy Efﬁciency in Cooperation

Phase

Similarly to (13), the average of total energy con-

sumed in a cooperation phase starting with N active

relay nodes can be evaluated;

E[energy consumption



= E

succ

+ E

idle

#[idle]

+ E

fail

(1)(#[1-frame transmission] −1)

∑

i=2

fail

(i)#[i-frame collision]

= E

succ



−1



1−q

succ

(

idle

+ E

fail

(1)q

(1−P

succ

)

∑

i=2

fail

(i)q

)

, (16)

where E

succ

is the total energy consumed by N active

relay nodes, source node S and destination node D in a

successful slot, E

idle

is that in an idle slot, and E

fail

(i)

is that in an unsuccessful slot, given that i-frame col-

lision occurs for i = 1,2,... ,N, respectively. Let P

, and P

be consumed power at a node when trans-

mitting, receiving, and sensing the channel, respec-

tively. Then, three states in (16) of the energy con-

sumption in a slot are given by

succ

= P

DATA

+ P

SIFS

+ P

ACK

+ P

DIFS

+ P

DATA

+ P

SIFS

+ P

ACK

+ P

DIFS

+ N(P

DATA

+ P

SIFS

+ P

ACK

+ P

DIFS

), (17)

idle

= (N + 2)P

slot

, (18)

fail

(i) = i(P

DATA

+ P

ACKtimeout

)

+ (N + 2−i)(P

DATA

+ P

ACKtimeout

)

(19)

for i = 1,2, .. .,N, respectively. Finally, we deﬁne the

energy efﬁciency η as

η =

E[message length in bits]

E[energy consumption



(20)

for a cooperation phase starting with N active relay

nodes.

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160

Table 1: Parameters for numerical results.

(a) Frame format

PHY preamble 96 [µsec]

MAC header (incl. FCS)

34 [byte]

message length 512 [byte]

ACK length

14 [byte]

CFC length

14 [byte]

Block length: k 64 [symbol]

(b) DCF parameters

slot duration: T

slot

10 [µsec]

DIFS: T

DIFS

50 [µsec]

SIFS: T

SIFS

10 [µsec]

ACKtimeout: T

ACKtimeout

50 [µsec]

CW: W 16

Transmission P

1900 [mW]

Reception P

1340 [mW]

Channel sensing P

1340 [mW]

(d) Channel

channel rate (DATA) 54 [Mbps]

channel rate (control)

6 [Mbps]

symbol error rate:

(10

−1

,10

−2

)

(ε

,ε

)

(10

−2

,10

−3

)

6 NUMERICAL RESULTS

We examine the derived expressions with exhaustive

computer simulation and compare the performance of

the proposed protocol to that of PRCSMA. The val-

ues of parameters employed are shown in Table 1.

The frame format and the DCF parameters are basi-

cally extracted from (Alonso-Zarate et al., 2009; Pre-

dojev et al., 2012) and IEEE 802.11 standard (IEEE

Standard 802.11, 1999). The power consumption is

identical with (Predojev et al., 2012). Two pairs of

the symbol error rates are considered; (ε

,ε

) =

(10

−1

,10

−2

) and (10

−2

,10

−3

). A block length in

frame is k = 64 symbols and two types of MDS codes

C are considered; a half-rate [128,64] MDS code for

L = 2, a quarter-rate [256,64] MDS code for L = 4.

Note that for L = 2, a relay node always transmits c

since a codeword consists of two blocks; c = [c



]. The theoretical results for L = 4 are omitted

in order to avoid the complexity to derive the prob-

ability of successful decoding at destination node D,

(8). The simulation program is written in C language

and the results are obtained by averaging 10

trials of

cooperation phases. Recall that a cooperation phase

starts with destination node D which has already held

m including e errors with probability α(e), (7), for

e = 1,2,...,k.

The average duration of a cooperation phase

and the energy efﬁciency in a cooperation phase

are presented in Fig. 6 and in Fig. 7, respectively,

as a function of the number of active relay nodes

N. The agreement between the theoretical and

simulation results validates the accuracy of the de-

rived expressions. Evidently, the proposed protocol,

PRCSMA+RP-MDS, outperforms the original PRC-

SMA. In addition, it is revealed from computer sim-

ulation that the performance of PRCSMA+RP-MDS

for L = 4 coincides with that for L = 2, so that a half-

rate MDS code sufﬁces for PRCSMA+RP-MDS.

From Fig. 6(a) the proposed protocol can achieve

approximately 40% reduction in the average duration

of a cooperation phase for (ε

,ε

) = (10

−1

,10

−2

The Energy efﬁciency is also improved by the pro-

posed protocol, as shown in Fig. 7(a). However, it

is clear from Fig. 6(b) and Fig. 7(b) that the degree

of performance improvement by the proposed pro-

tocol decreases, as the channel quality is enhanced,

since the opportunity to take advantage of the error-

correcting capability of the MDS code decreases at

destination node D. For the values of parameters

given in Table 1, the probability of error-free recep-

tion of a frame is

(1−ε

)

≈

(

0.526 for ε

= 10

−2

0.938 for ε

= 10

−3

(21)

It implies that destination node D requires to re-

ceive a frame approximately 1/0.526 ≈ 1.90 times

and 1/0.938 ≈ 1.07 times in average before the mes-

sage m be successfully recovered for ε

= 10

−2

and

= 10

−3

, respectively. On the other hand, since

destination node D can receive a frame other than m

in the cooperation phase in the proposed protocol, the

error-correcting decoding for a half-rate [2k,k] MDS

code can be carried out. In this case, at most ⌊k/2⌋

symbol errors can be corrected. Then, the probability

of decoding failure is given as

∑

i=⌊k/2⌋+1





(1−ε

)

2k−i

≈

(

1.70×10

−36

for ε

= 10

−2

3.92×10

−96

for ε

= 10

−3

(22)

PerformanceAnalysisofRandomRelayingofPartitionedMDSCodewordBlockAppliedtoPersistentRelayCSMAover

RandomErrorChannels

161

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 5 10 15 20

Averege Duration of Cooperative Phase [msec]

Number of Active Relay Nodes:

Sim.Theory

PRCSMA

L =2)

=4)L

Proposed (

Proposed ( (omitted)

PRCSMA

Proposed

(PRCSMA + RP-MDS)

(a) For (ε

,ε

) = (10

−1

,10

−2

)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 5 10 15 20

Averege Duration of Cooperative Phase [msec]

Number of Active Relay Nodes: N

Sim.Theory

PRCSMA

L =2)

=4)L

Proposed (

Proposed ( (omitted)

PRCSMA

Proposed

(PRCSMA + RP-MDS)

(b) For (ε

,ε

) = (10

−2

,10

−3

)

Figure 6: Average duration of cooperation phase.

which is negligibly small, so that one frame reception

other than m sufﬁces for destination node D to re-

cover the message block m in most cases. Therefore,

the performance of the proposed protocol is indepen-

dent of the value of L ≥ 2.

Another observation from Fig. 6 is that the aver-

age duration slightly decreases for N ≤ 3 and then it

turns to increase. For N ≤ 3, frame collisions are rare

events. In addition, the more active relay nodes exist,

the sooner the ﬁrst transmission at a relay node takes

place in a cooperation phase. These observations de-

crease the average duration with or without the use

of RP-MDS. However, for N ≥ 4, the probability of

frame collisions can not be negligible and frame col-

lisions add another backoffinterval and frame retrans-

mission. Hence, the average duration of a cooperation

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 5 10 15 20

Energy Efficiency in Cooperative Phase [Mbps/J]

Number of Active Relay Nodes: N

Sim.Theory

PRCSMA

L =2)

=4)L

Proposed (

Proposed ( (omitted)

PRCSMA

Proposed

(PRCSMA + RP-MDS)

(a) For (ε

,ε

) = (10

−1

,10

−2

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 5 10 15 20

Energy Efficiency in Cooperative Phase [Mbps/J]

Number of Active Relay Nodes: N

Sim.Theory

PRCSMA

L =2)

=4)L

Proposed (

Proposed ( (omitted)

Proposed

(PRCSMA + RP-MDS)

PRCSMA

(b) For (ε

,ε

) = (10

−2

,10

−3

)

Figure 7: Energy efﬁciency in cooperation phase.

phase increases.

Next, as shown in Fig. 2 and Fig. 4, a cooperation

phase consists of consecutive and synchronized slots.

These slots are classiﬁed into three categories; idle

slots of duration T

slot

, slots with 1-frame transmission

of duration of T

succ

or T

fail

, and slots with frame colli-

sions of duration of T

fail

. Clearly, one slot in slots with

1-frame transmission is a successful slot of duration

of T

succ

which is the last slot in a cooperation phase.

Fig. 8 shows the average number of slots in a cooper-

ation phase, classiﬁed into the three categories. The

average number of these slots are theoretically eval-

uated as (10)–(12). Predictably, the average number

of slots with frame collision monotonously increases

in proportion to increment of the number of active re-

lay nodes. The average number of idle sots decreases

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162

0 5 10 15 20

Average Number of Slots

Number of Active Relay Nodes: N

Sim.Theory

Idle

Collision

1-Frame

(a) For L = 1 and (ε

,ε

) = (10

−1

,10

−2

)

0 5 10 15 20

Averege Number of Slots

Number of Active Relay Nodes: N

Sim.Theory

Idle

Collision

1-Frame

(b) For L = 2 and (ε

,ε

) = (10

−1

,10

−2

)

0 5 10 15 20

Averege Number of Slots

Number of Active Relay Nodes: N

Sim.Theory

Idle

Collision

1-Frame

,ε

) = (10

−2

,10

−3

)

0 5 10 15 20

Averege Number of Slots

Number of Active Relay Nodes: N

Sim.Theory

Idle

Collision

1-Frame

(d) For L = 2 and (ε

,ε

) = (10

−2

,10

−3

)

Figure 8: Slot distribution in cooperation phase.

on the contrary. The incorporation of RP-MDS suc-

cessfully facilitates the completion of a cooperation

phase. Therefore, the average number of slots with

1-frame transmission can be reduced by the use of the

proposed protocol. Particularly, the use of RP-MDS

can approximately halve the average number of slots

for (ε

,ε

) = (10

−1

,10

−2

), comparing Fig. 8(a) to

Fig. 8(b).

7 CONCLUSIONS

We have proposed incorporation of RP-MDS, which

has been proposed for multi-hop cooperative relay

networks (Sakakibara et al., 2011), to PRCSMA over

noisy channels. The proposed protocol elaborately

takes advantage of the powerfulerror-correctingcapa-

bility of MDS codes into cooperative communication

systems and introduces the incremental redundancy

concept to PRCSMA. A destination node can rein-

force the error-correcting capability when it receives

a new frame. Assuming symmetric relay channels,

we have analyzed the performance of the proposed

protocol in terms of the average duration of a cooper-

ation phase and the energy efﬁciency in a cooperation

phase. The accuracy of theoretical results has been

validated by means of computer simulation. Numer-

ical results have indicated that the proposed protocol

can improve the performance, compared to the orig-

inal PRCSMA, particularly over severe noisy chan-

nels. It is also revealed that the use of a half-rate MDS

PerformanceAnalysisofRandomRelayingofPartitionedMDSCodewordBlockAppliedtoPersistentRelayCSMAover

RandomErrorChannels

163

code sufﬁces in the proposed protocol.

Further study includes, for example, the consider-

ation of header errors and feedback errors, and the ex-

tension to bidirectional communication systems and

to the use of network coding.

ACKNOWLEDGEMENTS

This work was partly supported by Japan Society for

the Promotion of Science under Grant-in-Aid for Sci-

entiﬁc Research (C) (KAKENHI no. 25420379).

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