Throughput Analysis and Scheduling of Two-Way Relaying with Network Coding

Abstract

This paper addresses the fundamental characteristics of information exchange via network coding over two-way relaying in a wireless network. The end-to-end (Shannon) rate regions achieved by time-division (TD), MAC-layer network coding (MLNC) and PHY-layer network coding (PLNC) protocols are first characterized. It is shown that MLNC does not always achieve a better rate than TD, opportunistic MLNC is able to achieve a larger rate region, and PLNC dominates the rate regions achieved by TD and MLNC. An opportunistic scheduling algorithm for MLNC and PLNC is then respectively proposed to stabilize the two-way relaying system for Poisson arrivals whenever the rate pair is within the rate regions of MLNC and PLNC. To understand the two-way transmission limits of network coding, the sum-rate optimization with or without certain traffic pattern is analyzed, and the traffic patterns that respectively maximize the sum rate of MLNC and PLNC are found as well.

Full text

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Throughput Analysis and Scheduling of
Two-Way Relaying with Network Coding
Chun-Hung Liu, Feng Xue and Jeffrey G. Andrews
Abstract
This paper addresses the fundamental characteristics of information exchange via network coding
over two-way relaying in a wireless network. The end-to-end (Shannon) rate regions achieved by time-
division (TD), MAC-layer network coding (MLNC) and PHY-layer network coding (PLNC) protocols are
first characterized. It is shown that MLNC does not always achieve a better rate than TD, opportunistic
MLNC is able to achieve a larger rate region, and PLNC dominates the rate regions achieved by TD
and MLNC. An opportunistic scheduling algorithm for MLNC and PLNC is then respectively proposed
to stabilize the two-way relaying system for Poisson arrivals whenever the rate pair is within the rate
regions of MLNC and PLNC. To understand the two-way transmission limits of network coding, the
sum-rate optimization with or without certain traffic pattern is analyzed, and the traffic patterns that
respectively maximize the sum rate of MLNC and PLNC are found as well.
Index Terms
Network Coding, Two-way Relaying, Achievable Rate, and Opportunistic Scheduling.
I. INTRODUCTION
Users in a multihop wireless network convey information to each other with the help of
intermediate routing nodes. Generally, packets are forwarded by the relays towards their respec-
tive destinations in a decode-and-forward fashion. After the seminal paper by Ahlswede et al.
[1] on network coding for wireline networks, it is known that better throughput performance
Liu and Andrews are with the Department of Electrical and Computer Engineering, the University of Texas at Austin,
Austin TX 78712-0204, USA. Xue is with Intel Corp. at Santa Clara CA, USA. Parts of this work were presented at the IEEE
International conference on communications, Beijing China, May 2008 and at the Allerton conference on communication, control
and computing, IL USA, Sep. 2007. The contact author is J. G. Andrews (Email: jandrews@ece.utexas.edu). Manuscript date:
December 24, 2009.

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is possible if intermediate nodes are allowed to change the content of their packets [2], [3].
Wireless network coding can improve throughput and reliability due to the broadcast nature
of the wireless medium, and the resulting opportunities to gather information from all audible
transmissions [4]–[9]. Since two-way traffic is inherent to peer-to-peer communication, the two-
way relaying channel is a key building block for information exchange over multiple hops. This
paper provides fundamental characterization and limits of the two-way relaying channel with
(and without) wireless network coding.
Specifically, we consider two multihop network coding protocols as illustrated in Fig. 1(a) in
this work, i.e. MAC-Layer network coding (MLNC) and PHY-Layer network coding (PLNC).
MLNC is a coding operation that happens at or above the media access layer [4], [7], [8],
[10]–[12]. PLNC is based on the coding protocol proposed in [13]–[15]. In order to perform
MLNC and PLNC, the two source nodes respectively transmit their packets to the relay node in
the first two time slots. Then the relay node constructs a new network-coded packet from the
received packets and broadcasts it to the two source nodes in the third time slot. If MLNC is
adopted – whereby the packets are manipulated before channel coding – the transmission rate
of the relay node is limited by the weaker broadcast channel [13]. Since the network coding
procedure of PLNC is performed instead on channel coded data, PLNC can individually achieve
the broadcast channel capacities from the relay node to the source/destination nodes [8], [13]–
[16]. Therefore, PLNC always achieves better throughput than MLNC, especially when the two
channels are highly asymmetric. Nonetheless, PLNC’s throughput gain comes at the cost of
transceiver complexity.
A. Motivation and Related Work
Traditionally (pre-network coding), information exchange between two users via a relay has
been accomplished by a time-division (TD) protocol in four time slots1, as shown in Fig. 1(a).
Intuitively, the MLNC and PLNC protocols save one time slot. In [6], two-way relaying for
cellular systems was considered, while [4] and [17] proposed an MLNC algorithm effective for
wireless mesh networks in heavy traffic. The network coding protocol in Fig. 1(b) can be further
1Frequency-division could be used as well. In thispaper the comparisons focus on time-division systems only, and comparisons
can be applied to frequency-division systems similarly.

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reduced to two slots if advanced joint coding/decoding – i.e. analog network coding (ANC)2–
is allowed. In this case, both source nodes send their packets to the relay node simultaneously
during the first slot; then the relay node either amplifies and broadcasts the signals, or broadcasts
the XOR-ed packets after decoding by successive interference cancellation [9], [18]–[20]. The
throughput for analog network coding were studied in [9], [21].
Although MLNC and analog network coding have been shown to achieve throughput gains in
certain environments [6], [7], [19], it is unclear how channel realizations influence the achievable
rates and whether MLNC and/or PLNC are always better than TD in terms of end-to-end
throughput. In terms of implementation complexity, the four techniques can be ranked from low to
high as TD, MLNC, PLNC and ANC. Analog network coding requires stringent synchronization
and joint decoding and could suffer from decoding error propagation due to channel estimation
and quantization errors. Characterizing the exact achievable rate regions of TD, MLNC and
PLNC will provide guidance on the tradeoffs between them.
A frequently neglected but very important consideration for network coding is that the traffic
patterns can vary significantly, and have a significant effect on the ability to achieve the gains
promised by network coding. For example, data downloads are essentially one-way traffic, while
peer-to-peer conversations are fairly symmetric. Analyzing the gains from network coding in a
two-way relay network in view of the traffic pattern allows insight into how such systems should
be designed. In this paper we explicitly consider the traffic pattern in our results.
B. Contributions and Paper Organization
In this paper, we first characterize the Ergodic achievable rate regions of TD, MLNC and PLNC
over a single relay and show that MLNC is not always superior to TD, and an opportunistic
scheduling algorithm is then presented to achieve a larger rate region than those achieved
independently by TD and MLNC. PLNC has the largest rate region among the three. The rate
regions in the multiple relay case are also discussed. According to the achievable rate regions
of MLNC and PLNC, two opportunistic scheduling algorithms are respectively proposed for
MLNC and PLNC, and we show that they are able to stabilize the two queues at the two source
nodes when the Poisson arrival rate pair is within the rate regions of MLNC and PLNC.
2In some works, this analog network coding is also called PHY-layer network coding. It should be pointed out that PLNC in
this paper is completely different from ANC or PLNC in some other works.

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In addition, the sum rates of the three protocols with or without traffic constraint are deter-
mined. The optimal sum rate with or without a traffic pattern constraint – defined as the ratio
between the rates of the two directions – is also characterized. It indicates that without a traffic
pattern constraint the optimal sum rate happens at one of the vertices of the achievable rate
regions. If traffic is subject to a certain pattern, MLNC achieves the maximum network coding
(throughput) gain when the pattern is close to one, i.e. the two rates are close. Whereas PLNC
achieves its maximum network coding gain when the traffic pattern is equal to the ratio of the
backward broadcast capacity to the forward broadcast capacity (see Fig. 1(b) for the definitions
of forward and backward directions).
The rest of the paper is organized as follows. Section II introduces wireless network coding for
a two-way relaying system and states the system assumptions. Section III presents the achievable
rate regions of the TD, MLNC and PLNC protocols. The opportunistic packet scheduling
algorithms for MLNC and PLNC are respectively proposed in Section IV. Section V provides
sum rate and network coding gain analysis, and finally Section VI summarizes our findings.
II. SYSTEM MODEL AND ASSUMPTIONS
Consider a multihop wireless network in which information exchange by multihop routing can
be characterized by considering the two-way relaying system illustrated in Fig. 1, where the two
source nodes A and B would like to exchange their data packets WAand WB, denoted as two
binary sequences. In the sequel, we assume there is no direct channel between the two source
nodes, otherwise, mutihop is not needed [22], [23]. All nodes in the network are assumed to be
single-antenna and half-duplex, i.e. no nodes can transmit and receive at the same time.
The core idea of MLNC is that the relay constructs a new network-coded packet WDafter
receiving WAand WB, and then it broadcasts WDto both source nodes. WDis obtained by
directly XOR-ing WAand WBbitwise, i.e. WD=WA⊕WB. As a source node receives WD,
it can decode the new packet by XOR-ing WDwith its side information, i.e. WAor WB. In
contrast to XOR-ing data contents in MLNC, PLNC does network coding on channel codes, i.e.,
after channel encoding to individual data contents [13]–[15]. Taking binary symmetric channel
for example, the relay broadcasts XA⊕XBafter receiving WAand WB, where XAand XBstand
for the channel codes of WAand WBrespectively, as shown in Fig. 2. The channel encoder of
WA(WB) which generates XA(XB) is designed according to the channel condition from relay

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node D to source node A (B). So when the XOR-ed channel code is received by a source node,
it can subtract XBor XAbefore channel decoding, as shown in Fig. 2.
Due to the different constructions, the available broadcast rates for MLNC and PLNC are
different. For MLNC, the broadcast rate is limited by the smaller broadcast capacity, due to the
fact that both ends need decode the same (XOR-ed) message. For PLNC, as shown in [13]–[15],
as long as there exists a common input distribution at the relay node, both of the broadcast
channel (e.g., binary symmetric channel and AWGN channel) capacities are achievable. In this
paper, we assume the existence of such common input distribution. In other words, each broadcast
direction can achieve its individual channel capacity if PLNC in [13] is used. Let CXY denote
the Ergodic channel capacity from nodes Xto Y. The broadcast rates of MLNC and PLNC can
be concluded from [13] [15] as follows.
Lemma 1: The achievable broadcast rate of MLNC for the relay node is Cmin for both
directions, where Cmin ,min{CDA , CDB }. If there exists a common input distribution that
maximizes the mutual information from relay node Dto nodes Aand B, then the achievable
broadcast rates of PLNC for both directions are CDA and CDB , respectively.
In addition, we call RAB the forward rate and RBA the backward rate as shown in Fig. 1(b).
µis called the traffic pattern parameter which is defined as the ratio RBA/RAB . It is a good
indicator of practical traffic such as web downloading (µ∼0), gaming (µ∼1), etc. Finally,
some important variables and parameters are defined in Table I.
III. END-TO-END ACHIEVABLE RATE REGION
In this section, we are interested in determining the end-to-end rate pair (RAB , RBA)achieved
by the aforementioned three protocols. We first characterize the end-to-end rate regions achieved
by TD, MLNC and PLNC for the single relay network in Fig. 1, and then we discuss how the
rate regions change in the multiple relay case.
A. Two-way Relaying over a Single Relay
For the two-way relaying system in Fig. 1, we assume that CDA and CD B are achieved by
the same input distribution. The achievable rate region is basically constructed by the forward
and backward rate pairs (RBA , RAB ). First consider TD that needs four time slots to exchange
packets. Since its achievable Shannon rate pairs are constrained by time allocations in the four

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time slots, its achievable rate region is
RTD ,(RAB , RBA) : RAB ≤min{λ1CAD, λ2CDB },
RBA ≤min{λ3CBD , λ4CD A},
4
X
k=1
λk= 1,(1)
where {λk} ∈ [0,1] are the time-allocation parameters for four transmission time slots. Therefore,
referring to Table I for the Σvariables, we have the following:
Theorem 1: RTD is the triangle with vertices 0,(ΣAB ,0) and (0,ΣBA), as shown in Fig. 3.
Proof: See Part A of the Appendix.
Vertices (ΣAB ,0) and (0,ΣBA)represent the situation of forward and backward traffic, respec-
tively. Therefore, RTD is easily grasped by doing time sharing between these two vertices. For
MLNC, by Lemma 1 its achievable rates are defined in similar fashion as in (1) as follows
RMLNC ,(RAB , RBA) : RAB ≤min{λ1CAD, λ3Cmin},
RBA ≤min{λ2CBD , λ3Cmin },
3
X
k=1
λk= 1.(2)
Then we have the following theorem characterizing the achievable rate region of MLNC.
Theorem 2: If CDB ≤CDA, then RMLNC is the quadrilateral with vertices 0,(ΣAB ,0),
(0,ΣBB )and (ΣABB ,ΣAB B ), as shown in Fig. 3(a). If CDB > CDA, then RMLNC is the quadri-
lateral with vertices 0,(ΣAA,0),(ΣABB ,ΣAB B )and (0,ΣBA), as shown in Fig. 3(b).
Proof: See Part B of the Appendix.
According to Theorems 1, 2 and Fig. 3(a)(b), MLNC is not always better than TD. For example
in Fig. 3(a), MLNC is worse than TD when CDB < CD A and µis greater than certain value.
This is because the broadcast rate of MLNC is limited by the worse broadcast channel, as stated
in Lemma 1. A hybrid protocol of time sharing between MLNC and TD, called opportunistic
MLNC (OMLNC, as indicated in Fig. 3 (a)(b) by a dashed line), can achieve a larger rate region
CovH(RTD,RMLNC), the convex hull of RTD and RMLNC. Thus, in practice the relay can decide
to use TD or MLNC according to the results in Fig. 3.
Remark 1: In the achievable rate region of MLNC, Vertex (5) in Fig. 3(a) and Vertex (6)
in Fig. 3(b) may not make sense in practice since relay Djust needs to forward the received
single packet either the source node. Thus, the transmission rate should not be limited by the
weaker link capacity in this case. However, the objective of presenting the achievable rate region

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of MLNC by this way is to clarify that “always” blindly using MLNC could render us a worse
throughput than TD. So the better strategy for a relay is to opportunistically use MLNC based
on the two-way traffic pattern.
For PLNC, its achievable rate region is constructed as follows according to Lemma 1:
RPLNC ,(RAB , RBA) : RAB ≤min{λ1CAD, λ3CDB },
RBA ≤min{λ2CBD , λ3CD A},
3
X
k=1
λk= 1.(3)
Then we have the following theorem.
Theorem 3: RPLNC is the quadrilateral with vertices 0,(ΣAB ,0),(ΣABA ,ΣBAB )and (0,ΣBA).
Proof: See Part C of the Appendix.
Given the above results in Theorems 1-3 and illustrated in Fig. 3, we know the achievable
rate region of PLNC is larger than CovH(RTD,RMLNC)as long as CDA 6=CDB. Also, it should
be noticed that time sharing between TD and PLNC does not help achieve a larger rate region
as in the case of MLNC and TD.
B. Two-Way Relaying over Multiple Relays
The results in the previous subsection are for the single relay case. If there are multiple relay
nodes available for exchanging packets as shown in Fig. 4, how will the achievable rate regions
change? We first discuss the cooperative relay case. Suppose all the relay nodes in the relay
set DAB are able to cooperate such that they can share their information to attain coherently
receiving and transmitting (i.e. receive and transmit maximum ratio combining (MRC)). Hence,
the relay set DAB can be viewed as a big relay equipped with |D|AB antennas, where |D|AB
is the cardinality of DAB . In this case, the Ergodic rate regions of TD, MLNc and PLNC are
enlarged due to a SIMO/MISO channel formed by the relay nodes. However, it should be pointed
out that MLNC and PLNC could have a smaller rate region at “certain” time if all relay nodes
in DAB simultaneously broadcast. This is because bidirectional transmit diversity and array gain
are difficult to be exploited at the same time [7] [24].
For the case of relay noncooperation, an optimal relay node should be selected to route packets.
For TD, the forward traffic and backward traffic do not necessarily pass through the same optimal
relay node. In this case, the achievable rate region of TD is
R∗
TD(D∗
f, D∗
b) = CovH RTD(D∗
f),RTD(D∗
b),(4)

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where {D∗
f, D∗
b} ∈ DAB are the optimal relay nodes for forward and backward transmission,
respectively, i.e. D∗
f= arg maxD∈DAB ΣAB (D)and D∗
b= arg maxD∈DAB ΣBA(D). For MLNC
and PLNC with two-way traffic, their optimal relay nodes D∗
MLNC and D∗
PLNC are selected as
follows





D∗
MLNC = arg maxD∈DAB ΣABB (D)for MLNC
D∗
PLNC = arg maxD∈DAB ΣABA(D) + ΣB AB (D)for PLNC ,(5)
where 2ΣABB is the maximum sum rate within RMLNC and ΣABA + ΣBAB is the maximum sum
rate within RPLNC at relay node D∈DAB . Therefore, the achievable rate regions of opportunistic
MLNC and PLNC are
R∗
OMLNC(D∗
f, D∗
b, D∗
MLNC) = CovH(R∗
TD(D∗
f, D∗
b),RMLNC(D∗
MLNC)),(6)
R∗
OPLNC(D∗
f, D∗
b, D∗
PLNC) = CovH(R∗
TD(D∗
f, D∗
b),RPLNC(D∗
PLNC)).(7)
Fig. 5 shows an example of the achievable rate regions of Eqs. (4), (6) and (7).
IV. OPPORTUNISTIC NETWORK CODING AND SCHEDULING
The achievable rate regions for the three transmission protocols have been characterized
in Section III. In this section, we investigate the following question: If packets arrive at the
two source nodes according to a random process, how should the packets be scheduled for
transmission to maximize the rate region in which the queues are stable? We assume that the
buffer size for queuing packets at the two source nodes is infinite. Packets of length ℓ(bits) arrive
at the queue of source node A according to a Poisson process with rate ´
RAB , while packets of
length ℓ(bits) arrive at the queue of source node B according to another independent Poisson
process with rate ´
RBA . We also assume that ℓis large and channel coding is perfect.
We first consider the case where the relay node uses MLNC to route the packets. Consider
the time right after the previous transmission is complete. Let qAand qBdenote the number of
packets in the queues at nodes A and B, respectively. We propose the following opportunistic
packet scheduling algorithm for MLNC.
Algorithm 1: Opportunistic MLNC and Scheduling
Step 1 (qA·qB6= 0): Two packets (one from each queue) are sent over Relay D
by MLNC.
Step 2 (qA6= 0,qB= 0): One packet from node A is sent over Relay D by TD.
Step 3 (qA= 0,qB6= 0): One packet from node B is sent over Relay D by TD.

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It will be shown later in Section V-B that symmetric traffic through the relay node achieves the
best network coding gain for MLNC. So when both buffers have packets, Algorithm 1 schedules
symmetric traffic; when one queue is empty certainly one-way traffic should be scheduled. By
doing this, Algorithm 1 is able to opportunistically achieve CovH(RTD,RMLNC)by time sharing
between TD and MLNC. Moreover, the attractive feature of this algorithm is that it does not
require any arrival rate information of the two queues to achieve the system stability as stated
in the following theorem.
Theorem 4: Algorithm 1 stabilizes the two-way relaying system for any Poisson arrivals if the
(bit-arrival) rate pair (RAB , RBA )∈CovH(RTD,RMLNC)where RAB =´
RAB ℓand RBA =´
RBA ℓ.
Proof: See Part D of the Appendix.
Next, we consider the opportunistic packet scheduling algorithm for PLNC. We want to
show that any rate pair in Theorem 3 is stabilizable. Consider a pair (QA, QB)∈N2
+equal
to (max Qf,max Qb),where Qf, Qb∈N+are subject to Qf≤λ1ΣAB /ℓ,Qb≤λ2ΣBA /ℓ, and
Qbλ1ΣABA =Qfλ2ΣBAB 3. Let (˜
ΣABA,˜
ΣBAB )∈RPLNC be the closest point to (ΣAB A,ΣBAB )
with constraint QAλ2˜
ΣBAB ≈QBλ1˜
ΣABA . Also, choose Q∗∈N+large enough such that the
following inequalities are satisfied:
˜
ΣABA
ΣBA −˜
ΣBAB RBA
ΣBA
−1+RAB
ΣBA
+QA
Q∗<0,(8a)
˜
ΣBAB
ΣAB −˜
ΣABA RAB
ΣAB
−1+RBA
ΣAB
+QB
Q∗<0,(8b)
max{QA, QB} − Q∗<0.(8c)
Such Q∗must exist because the region confined by (8a) and (8b) in the first quadrant is enclosed
by the achievable rate region specified in Theorem 3. We propose the following algorithm that
opportunistically schedules packet transmissions over the relay node by PLNC.
Algorithm 2: Opportunistic PLNC and Scheduling
3Theoretically, we should choose a positive integer pair (Qf, Qb)large enough such that (Qfℓ/λ1, Qbℓ/λ2) =
(ΣABA ,ΣBAB ). However, such a pair may not exist. So it is acceptable to select a pair with appropriate large values of
Qfand Qbsuch that (Qfℓ/λ1, Qbℓ/λ2)≈(ΣABA ,ΣBAB ).

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Step 1 (qA≥QA,qB≥QB): QApackets from node A and QBpackets from node
B are sent over Relay D by PLNC.
Step 2 (qA< QA,qB> QB): min{qB, Q∗}packets from node B are sent over
Relay D by TD.
Step 3 (qA> QA,qB< QB): min{qA, Q∗}packets from node A are sent over
Relay D by TD.
Step 4 (qA< QA,qB< QB): qApackets from node A and qBpackets from node
B are sent over Relay D by PLNC.
It will be also discussed in Section V-B that PLNC achieves its maximum network coding
gain when the traffic pattern over a relay node is µ=CDA
CDB . So when the two buffers respectively
have at least QAand QBpackets, Algorithm 2 schedules the two-way traffic such that it has
a traffic pattern µ≈CDA
CDB . If one of the two buffers does not have enough packets to achieve
µ≈CDA
CDB , then one-way traffic is scheduled. However, if both of the buffers do not have enough
packets then PLNC is adopted because it has a better throughput than TD for any traffic pattern
as shown in Fig. 3.
Theorem 5: Algorithm 2 stabilizes the two-way relaying system for Poisson arrivals with the
(bit-arrival) rate pair (RAB , RBA )within the region constructed by (8a)-(8c)where RAB =´
RAB ℓ
and RBA =´
RBA ℓ.
Proof: See Part E of the Appendix
Remark 2: As shown in Theorem 5, Algorithm 2 is able to stabilize the queues at the source
nodes with Poisson arrivals whose rate pair is within the rate region constructed by (8a)-(8b)
in the first quadrant, which is enclosed by RPLNC in Theorem 3 for all Q∗satisfied with (8c).
Therefore, any rate pair within RPLNC in Theorem 3 is stabilizable since Q∗can be chosen
sufficiently large such that the region constructed by (8a)-(8b) approaches to RPLNC as closely
as possible. However, this might come with some queueing delay.
V. SUM-RATE OPTIMIZATION AND NETWORK CODING GAIN
In practice end-to-end sum-rate is usually the choice to characterize overall performance. In
this section, we first consider the sum-rate optimization without any constraint on traffic pattern.
Then we consider how traffic pattern influences the sum-rates of the three transmission protocols
with a traffic pattern constraint. Here we only consider the sum-rate optimization problem for a

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single relay since it is straightforward to extend to the multiple relay case.
A. End-to-End Sum Rate Optimization
We first consider the sum-rate optimization problems without any constraint on traffic pattern
for TD, MLNC and PLNC. Then we consider how traffic pattern influences the sum rates. In
addition, in order to compare the throughput performance we introduce the notion of network
coding gain, and characterize under what kind of traffic pattern the network coding gain is
maximized. Considering Theorems 1-3 and Fig. 3, then the following result is immediate for
sum rate without traffic pattern constraints.
Corollary 1: (a) The maximum sum rate achieved by TD is max{ΣAB ,ΣBA}. (b) The max-
imum sum rate achieved by MLNC is max {ΣAB ,ΣBB ,2ΣABB }if CD B < CDA , and
max{ΣBA,ΣAA,2ΣAB B }if CDB ≥CDA . (c) The maximum sum rate achieved by PLNC is
max{ΣAB,ΣBA ,ΣABA + ΣB AB }.
Proof: Since the end-to-end sum rate is RAB +RBA and all constraints in (1), (2) and (3)
are linear, the maximum sum rate must be achieved at a corner point of the achievable rate
region according to the linear programming theory.
In practice, traffic is typically of certain pattern. In a simple way, it can be described by the
ratio between forward traffic and backward traffic, i.e. µ=RBA/RAB as defined in Section II.
For example, web browsing has µ∼0while gaming has µ∼1. Here we consider the sum rate
when the ratio µis fixed. For the sum rate optimization problem with a traffic pattern, we have
the following.
Corollary 2: Consider a certain value of µ. We have
(a) The maximum sum rate achieved by TD is
R∗
TD =1 + µ
Σ−1
AB +µΣ−1
BA
.(9)
(b) The maximum sum rate achieved by MLNC is
R∗
MLNC =1 + µ
C−1
AD +µC−1
BD + max{1, µ}C−1
min
.(10)
(c) The maximum sum rate achieved by PLNC is
R∗
PLNC =1 + µ
C−1
AD +µC−1
BD + max{µC−1
DA, C −1
DB }(11)
Proof: See Part F of the Appendix.

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B. Network Coding Gain
Given the results of the sum-rate optimization, the throughput of the three transmission
protocols can be computed. We introduce the network coding (throughput) gain ρAB (in dB
scale) as
ρAB ,10 log10
R∗
A
R∗
B(dB),(12)
where R∗
Aand R∗
Bare the maximum sum rates for protocol A and protocol B, respectively.
Consider TD as the baseline. The network coding gain of MLNC is easily found by (9) and
(10) as
ρMT = 10 log10 1 + C−1
DB +µC−1
DA −max{1, µ} · C−1
min
C−1
AD +µC−1
BD + max{1, µ} · C−1
min ,(13)
while the network coding gain of PLNC can be found by (9) and (11) as
ρPT = 10 log10 1 + C−1
DB +µC−1
DA −max{µC−1
DA, C −1
DB }
C−1
AD +µC−1
BD + max{µC−1
DA, C −1
DB }.(14)
Note that ρPT >0and it is easy to see ρPM ,ρPT −ρMT >0, which shows PLNC is always
superior to TD and MLNC. Now we would like to know what values of µcan maximize ρMT
and ρPT because they can give us some insight about how to schedule the two-way traffic such
that the maximum throughput of MLNC and PLNC can be achieved if compared to TD. The
optimal traffic patterns of maximizing ρMT and ρPT, as shown in the following theorem, inspire
the ideas of the opportunistic scheduling in Section IV.
Theorem 6: The gain ρMT achieves its maximum when the two-way traffic is perfectly sym-
metric, i.e. µ= 1. The gain ρPT achieves its maximum when the two-way traffic has the pattern
such that µ=CDA
CDB .
Proof: We only prove the case of MLNC here since the proof of PLNC is similar. For
µ≤1, we can take the derivative of ρMT(µ)with respect to µand obtain
dρMT
dµ =1/CADCDA + 1/CDA Cmin + 1/CBD (1/Cmin −1/CDB )
(1/CAD +µ/CBD + 1/Cmin)2>0.
That means µhas to increase in order to increase ρMT. Similarly, we can show that dρMT
dµ <0
when µ > 1. Thus, the maximum of ρMT must happen at µ= 1.
Remark 3: The results in Theorem 6 indicate a potential weakness of PLNC. That is, the
coding gain of PLNC is hightly dependent on the broadcast channel conditions. When the
broadcast channels are pretty much symmetric, PLNC can only provide marginal throughput
benefit compared to MLNC in this context. Thus, PLNC may not be very attractive for real

13
applications because it cannot guarantee a significant throughput gain against MLNC and the
extra circuitry cost of implementing PLNC could be very high.
C. Simulation Results of Network Coding Gain
In this subsection, we use simulations to illustrate how the network coding gain is affected
by the traffic pattern and the relay position in a two-way relaying system. Suppose now node A
represents the base station with transmit power 20 dBm, the relay station with transmit power 15
dBm is node D, and node B denotes the user station with transmit power 10 dBm. All channels
between different nodes are flat Rayleigh fading, reciprocal and have a path loss exponent 3.
The distance between nodes A and B is d, and the distances from relay D to nodes A and B are
respectively dDA =xd and dDB = (1 −x)d, where x∈(0,1). We first look at how the network
coding gains change with the traffic patterns. Suppose relay D is positioned at 1000 m away
from node A, and the distance from relay D to node A is 300 m (i.e. x= 0.3). The noise power
is assumed as -100 dBm. This setup will result in CDA ≈1.4CDB on average. So we would like
to expect to see the simulation results coincide with the achievable rate region indicated in Fig.
3 (a).
The simulation results of the network coding gains for this case are presented in Fig. 6. Indeed,
in the figure we can observe that the maximum coding gain happens at µ≈1for MLNC and
OMLNC protocols, and it is about 1.1dB ≈1.29 (i.e. MLNC increases about 29% throughput
compared to TD). The network coding gains of MLNC and OMLNC are seriously impacted
when the traffic is very asymmetric. In addition, the maximum of ρPT happens at µ≈1.5,
which is close to our expectation 1.4. It is about 1.35 dB, which means PLNC can have 36%
throughput increment over TD when µ≈1.5. As expected, PLNC has not only the best network
coding gain among all the protocols but also the robustness against asymmetric traffic pattern.
The network coding gain of OMLNC, ρOMT is also much better than what achieved by MLNC,
especially when µ > 1. In Fig 6, we also see MLNC, OMLNC and PLNC all achieve a similar
rate when µ≤1and their rates are slightly higher than TD’s rate ( this is due to small CBD.).
Hence, the simulation results perfectly match the rate region in Fig. 3 (a).
Next, we would like to understand how the relay position between nodes A and B affects the
network coding gain. The simulation results with µ= 2 are presented in Fig. 7. Since the initial
relay position is very close to node A, CDA is initially greater than CDB and then it becomes

14
smaller than CDB after the relay moves sufficiently far away from node A. So we can see that
when x < 0.5PLNC and OMLNC both have a larger coding gain than that of MLNC since
µ= 2 and CDA > CDB (This is exactly the case in Fig. 3(a)) whereas when x≥0.5all network
coding gains are almost the same since µ= 2 and CD A ≤CDB (This corresponds to the case in
Fig. 3(b)). Fig. 8 illustrates the simulation results with µ= 1/2, and we can observe Figs. 7 and
8 are pretty much symmetric. For MLNC and OMLNC, both figures indicate their maximum
network coding gain happens at x= 0.5. This is because their gain is very much dependent
on the broadcast capacity Cmin and it increases along the increase in Cmin (see (13)). So the
maximum gain of MLNC and OMLNC must happen at the location where the maximum of Cmin
happens. The relay location which has the maximum broadcast capacity is exactly the middle
point between nodes A and B, i.e. x= 0.5. Similarly, according to (14) the maximum network
coding gain of PLNC must happen at the location where we have CDA
CDB =µ. For example, in
Fig. 8 the maximum gain of PLNC happens at the location x≈0.7where CDA
CDB ≈0.5 = µ.
VI. CONCLUDING REMARKS
In this work, we investigate the throughput limits of information exchange over a two-way
relaying channel with or without wireless network coding. The end-to-end Ergodic rate regions
for the TD, MLNC and PLNC protocols are all characterized. We found that always using
MLNC for information exchange could achieve a smaller rate region than TD. Opportunistically
using MLNC is able to achieve a larger rate region. PLNC achieves the largest rate region
since PLNC can achieve the individual broadcast channel capacity. Two opportunistic packet
scheduling algorithms for MLNC and PLNC are proposed that can stabilize the two-way relaying
system for the Poisson arrival rate pairs within the respective rate regions of MLNC and PLNC.
The maximum sum rates of the three protocols with or without a traffic pattern constraint are
found. MLNC achieves its maximum sum rate when two-way traffic is very symmetric, while
PLNC achieves its maximum sum rate when two-way traffic pattern parameter µis CDA /CDB .
Thus, the throughput performance of PLNC highly depends on the broadcast channel conditions.
Finally, we verify that for MLNC and OMLNC the optimal relay location that achieves the
maximum throughout gains is the middle point of the two source nodes, whereas for PLNC with
a specific traffic pattern parameter µ, the relay should be positioned at the location such that
CDA
CDB ≈µ.

15
APPENDIX
PROOFS OF THEOREMS
A. Proof of Theorem 1
Vertex (0,ΣBA)corresponds to the case of one-way backward traffic, it is achieved by setting
λ1=λ2= 0,λ3=ΣBA
CBD and λ4=ΣBA
CDA . Similarly, vertex (ΣAB,0) corresponds to the case of
one-way forward traffic, achieved by setting λ3=λ4= 0,λ1=ΣAB
CAD and λ2=ΣAB
CDB . Since RTD
is described by linear constraints, it is convex and thus achievable. Now we show that RTD is
also an outer bound for TD. Consider the four linear constraints of transmission rates in (1).
Dividing each of them by their corresponding channel capacity and adding them up, we obtain
RAB
CAD
+RAB
CDB
+RBA
CBD
+RBA
CDA
=RAB
ΣAB
+RBA
ΣBA
≤
4
X
k=1
λk= 1.
This is exactly the region below the line connecting vertices (0,ΣBA)and (ΣAB ,0).
B. Proof of Theorem 2
First consider the case CDB ≤CDA. The achievable rate region RMLNC in (2) becomes
RMLNC =(RAB , RBA) : RAB ≤min{λ1CAD, λ3CDB },
RBA ≤min{λ2CBD , λ3CD B },
3
X
k=1
λk= 1.(15)
Vertex (ΣAB ,0) corresponds to the case of one-way forward traffic, and is achieved by setting
λ1=ΣAB
CAD ,λ2= 0 and λ3=ΣAB
CDB . Vertex (0,ΣBB )corresponds to the case of one-way backward
traffic, and is achieved by setting λ1= 0,λ2=ΣBB
CBD and λ3=ΣBB
CDB . Finally vertex (ΣABB ,ΣABB )
is achieved by setting λ1=ΣABB
CAD ,λ2=ΣABB
CBD and λ3=ΣABB
CDB . Because (15) is a convex
quadrilateral defined by the three vertices and (0,0), the region (15) is achievable by time-
sharing among the four vertices. Next, by time-sharing we show the quadrilateral RMLNC in (15)
is also an outer bound for MLNC. Consider the four linear constraints in (15) and divide each
of them by their corresponding channel capacity. We have
RAB
CAD
+RAB
CDB
+RBA
CBD
=RAB
ΣAB
+RBA
CBD
≤
3
X
k=1
λk= 1,(16a)
RBA
CBD
+RBA
CDB
+RAB
CAD
=RBA
ΣBB
+RAB
CAD
≤
3
X
k=1
λk= 1.(16b)

16
Equation (16a) corresponds to the region below the line connecting (ΣAB ,0) and (ΣAB B ,ΣABB ),
and (16b) corresponds to the region below the line connecting (ΣAB B ,ΣABB )and (0,ΣBB ). This
completes the proof for the case CDB ≤CDA . The proof for the case CDB > CDA is similar.
C. Proof of Theorem 3
Vertex (ΣAB ,0) corresponds to the case of one-way forward traffic since by Lemma 1 PLNC
can achieve the individual capacity of hDB in the third time slot, (ΣAB ,0) can be achieved
by setting λ1=ΣAB
CAD ,λ2= 0 and λ3=ΣAB
CDB . Vertex (0,ΣBA)can be achieved in a similar
way. Finally, the rate pair (ΣABA,ΣB AB )is achieved by setting λ1= (1 + CAD CDA
CBD CDB +CAD
CDB )−1,
λ2=CADCD A
CBD CDB λ1and λ3=CAD
CDB λ1. Since RPLNC is described by linear constraints, it is convex
and thus the quadrilateral is within in. Now we show that the quadrilateral RPLNC is also an
outer bound for PLNC. Consider the four rate inequalities in (3). Dividing each of them by their
corresponding channel capacity and summing them
RAB
CAD
+RAB
CDB
+RBA
CBD
=RAB
ΣAB
+RBA
CBD
≤
3
X
k=1
λk= 1,(17a)
RBA
CBD
+RBA
CDA
+RAB
CAD
=RBA
ΣBA
+RAB
CAD
≤
3
X
k=1
λk= 1,(17b)
Equation (17a) corresponds to the region below the line segment connecting (ΣAB ,0) and
(ΣABA ,ΣBAB ), and (17b) corresponds to the region below the line segment connecting (ΣABA ,ΣBAB )
and (0,ΣBA).
D. Proof of Theorem 4
Without loss of generality, assume each packet has a unit length so that RAB =´
RAB and
RBA =´
RBA . Now consider the time right after the n-th transmission. Note that each transmission
is one of the three steps in Algorithm 1. Denote the queue length vector at time nas Q(n),
[qA(n)qB(n)]. It is easy to verify that Q(n)forms a non-reducible Markov chain. Note that the
number of packets arriving at node A during a transmission time slot ∆tis a Poisson process
with parameter RAB ∆t, while the number of packets arriving at node B is also a Poisson process
with parameter RBA ∆t.
In order to show the stability of the Markov process Q(n), we define the Lyapunov function
as follows
V(n),Σmin
ΣAB −Σmin
q2
A(n) + Σmin
ΣBA −Σmin
q2
B(n) + 2qA(n)qB(n),(18)

17
Now consider the case in Step 1 of Algorithm 1. In this case we have qA(n+ 1) = qA(n)−
1 + δA(n)and qB(n+ 1) = qB(n)−1 + δB(n), where δA(n)and δB(n)are Poisson random
variables with parameters RAB /Σmin and RBA/Σmin, respectively. It follows that
E[V(n+ 1)|Q(n)] = V(n) + 2 ΣAB
Σmin Σmin
ΣAB −Σmin RAB
ΣAB
−1+RBA
ΣAB qA(n)
+2 ΣBA
Σmin Σmin
ΣBA −Σmin RBA
ΣBA
−1+RAB
ΣBA qB(n) + ∆a,(19)
where
∆a=Σmin
ΣAB −Σmin "RAB
Σmin
−12
+RAB
Σmin #+Σmin
ΣBA −Σmin "RBA
Σmin
−12
+RBA
Σmin #
+1−RAB
Σmin 1−RBA
Σmin .
So it is a constant depending on RAB , RB A,Σmin,ΣAB and ΣBA . Next consider the case in Step
2of Algorithm 1. We have qA(n+ 1) = qA(n)−1 + ˆ
δA(n)and qB(n+ 1) = ˆ
δB(n), where
ˆ
δA(n)and ˆ
δB(n)are Poisson random variables with parameters RAB
ΣAB and RBA
ΣAB , respectively. So
we have
E[V(n+ 1)|Q(n)] = V(n) + 2 RB A
ΣAB
+Σmin
ΣAB −Σmin RAB
ΣAB
−1qA(n) + ∆b,(20)
where ∆bis also a constant depending on RAB , RBA ,Σmin ,ΣAB and ΣBA ,and it is
∆b=Σmin
ΣAB −Σmin "RAB
ΣAB
−12
+RAB
ΣAB #+Σmin
ΣBA −Σmin "RBA
ΣAB 2
+RBA
ΣAB #
+2RBA
ΣAB RAB
ΣAB
−1.
Finally consider the case in Step 3 of Algorithm 1. We have qA(n+1) = ˇ
δA(n)and qB(n+1) =
qB(n)−1 + ˇ
δB(n), where ˇ
δA(n)is a Poisson random variable with parameter RAB
ΣBA and ˇ
δB(n)
is also Poisson with parameter RBA
ΣBA . So it follows that
E[V(n+ 1)|Q(n)] = V(n) + 2 RAB
ΣBA
+Σmin
ΣBA −Σmin RBA
ΣBA
−1qB(n) + ∆c,(21)
where ∆cis also a constant depending on RAB , RBA ,Σmin ,ΣAB and ΣBA , and it is
∆c=Σmin
ΣBA −Σmin "RBA
ΣBA
−12
+RBA
ΣBA #+Σmin
ΣAB −Σmin "RAB
ΣBA 2
+RAB
ΣBA #
+2RAB
ΣBA RBA
ΣBA
−1.

18
Since (RAB , RBA)is within the quadrilateral formed by the origin, (ΣAB ,0),(Σmin,Σmin )and
(0,ΣBA), the following inequalities are obvious:
Σmin
ΣAB −Σmin RAB
ΣAB
−1+RBA
ΣAB
<0and Σmin
ΣBA −Σmin RBA
ΣBA
−1+RAB
ΣBA
<0.
Therefore, we have E[V(n+ 1)|Q(n)] ≤V(n)−1when qA(n)or qB(n)is sufficiently large.
According to the Foster-Lyapunov criterion [25], Q(n)is stable.
E. Proof of Theorem 5
Using the same assumptions in the proof of Theorem 4, the stability of the Markov process
Q(n)can be shown by defining the following Lyapunov function:
V(n),˜
ΣBAB
ΣAB −˜
ΣABA
q2
A(n) + ˜
ΣABA
ΣBA −˜
ΣBAB
q2
B(n) + 2qA(n)qB(n).(22)
First consider the case in Step 1 of Algorithm 2. Thus, we know qA(n+1) = qA(n)−QA+δA(n)
and qB(n+ 1) = qB(n)−QB+δB(n)where δA(n)and δB(n)are Poisson random variables
with parameters RAB QA/˜
ΣABA and RBA QB/˜
ΣBAB , respectively. It follows that
E[V(n+ 1)|Q(n)] = V(n) + 2QA
ΣAB
˜
ΣABA "˜
ΣBAB
ΣAB −˜
ΣABA RAB
ΣAB
−1+RBA
ΣAB #qA(n)
+2QB
ΣBA
˜
ΣBAB "˜
ΣABA
ΣBA −˜
ΣBAB RBA
ΣBA
−1+RAB
ΣBA #qB(n) + ˆ
∆a,(23)
where ˆ
∆ais a constant depending on RAB , RBA,˜
ΣABA,˜
ΣBAB , QA, QB,ΣAB and ΣBA , and it is
ˆ
∆a=˜
ΣBAB Q2
A
ΣAB −˜
ΣABA "RAB
˜
ΣABA
−12
+RAB
˜
ΣABA QA#+
˜
ΣABA Q2
B
ΣBA −˜
ΣBAB "RBA
˜
ΣBAB
−12
+RBA
˜
ΣBAB QB#+ 2QAQBRAB
˜
ΣABA
−1 RBA
˜
ΣBAB
−1.
Next consider the case in Step 2 of Algorithm 2. So in this case we have qA(n)< QA,
qA(n+ 1) = qA(n) + ˆ
δA(n)and qB(n+ 1) = [qB(n)−Q∗]++ˆ
δB(n).ˆ
δA(n)and ˆ
δB(n)are
Poisson random variables with parameters RAB Q∗
ΣBA and RBA Q∗
ΣBA , respectively. If qB(n)< Q∗then
the Lyapunov function is always bounded. Thus only the case qB(n)> Q∗is needed to be
discussed. So if qB(n)> Q∗then we have
E[V(n+1)|Q(n)] < V (n)+2Q∗qB(n)"˜
ΣABA
ΣBA −˜
ΣBAB RBA
ΣBA
−1+RAB
ΣBA
+QA
Q∗#+ˆ
∆b,(24)

19
where ˆ
∆bis also a constant depending on RAB , RBA,˜
ΣABA ,˜
ΣBAB , QA, QB,ΣAB and ΣBA .
Finally consider Step 3 of Algorithm 2 is applied. So we know qB(n)< QB,qB(n+ 1) =
qB(n) + ˆ
δB(n)and qA(n+ 1) = [qA(n)−Q∗]++ˆ
δA(n).ˆ
δA(n)and ˆ
δB(n)are Poisson random
variables with parameters RAB Q∗
ΣAB and RBA Q∗
ΣAB , respectively. Similarly, we only need to consider
the case qA(n)> Q∗. So if qA(n)> Q∗then we have
E[V(n+1)|Q(n)] < V (n)+2Q∗qA(n)"˜
ΣBAB
ΣAB −˜
ΣABA RAB
ΣAB
−1+RBA
ΣAB
+QB
Q∗#+ˆ
∆c,(25)
where ˆ
∆cis a constant depending on RAB , RBA ,˜
ΣABA ,˜
ΣBAB , QA, QB,ΣAB and ΣBA .
Since (RAB , RBA)and Q∗respectively satisfy (8a), (8b) and (8c), we have E[V(n+1)|Q(n)] ≤
V(n)−1when qA(n)or qB(n)is sufficiently large. By the Foster-Lyapunov criterion, Q(n)is
stable.
F. Proof of Corollary 2
(a) For TD, its achievable rate region is enclosed in the first quadrant by line RAB
ΣAB +RBA
ΣBA = 1
according to Theorem 1. So replacing RBA with µRAB in the line equation and solving for RAB ,
(9) is obtained by (1 + µ)RAB . (b) For MLNC, we first consider the case when CDB < CDA .
According to Theorem 2, the boundary line of the achievable rate region is RAB
ΣAB +RBA
CBD = 1
when µ < 1. Replacing RBA with µRAB and solving for RAB , and then we can get the desired
result for µ < 1. The boundary line is RAB
CAD +RBA
ΣBB = 1 when µ≥1, and hence the maximum
sum rate can be found by replacing RBA with µRAB . The case CD B ≥CDA can be solved by
the same fashion. (c) For PLNC, we first consider the case µ > CDA
CDB . According to Theorem 3,
the boundary line of the achievable rate region is RAB
CAD +RBA
ΣBA = 1. Replacing RBA with µRAB
and solving for RAB , and then the desired sum rate RAB (1 + µ)for µ > CDA
CDB can be obtained.
The case µ≤CDA
CDB can be proved by the same way.
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21
TABLE I
DEFINITIONS OF IMPORTANT VARIABLES
Symbol Definition
CXY channel capacity from node X to Y
RAB (RBA ) forward (backward) rate from node A (B) to B (A)
µtraffic pattern parameter, RBA/RAB
ΣAB (1/CAD + 1/CDB )−1
ΣBA (1/CBD + 1/CDA )−1
ΣBB (1/CBD + 1/CDB )−1
ΣAA (1/CAD + 1/CDA)−1
ΣABA (1/ΣAB +CDA /CBD /CDB )−1
ΣBAB (1/ΣBA +CDB /CAD /CDA )−1
ΣABB (1/ΣAB + 1/Cmin)−1
Σmin (1/CAD + 1/CBD + 1/Cmin)−1
Cmin min{CDA, CD B }
AB
D
W
A
W
B
R
AB
R
BA
(2)
(3)
(1)
(4)
(End-to-End Forward Rate)
(End-to-End Backward Rate)
TD
(2)
(1)
(3) Broadcast
MLNC/PLNC
(a) (b)
BDA
C
AD
C
DB
C
BD
C
DA
Fig. 1. (a) Two-way relaying over a single relay: Time-division multihop (TD) protocol needs 4 time slots, but wireless network
coding only needs 3 time slots. (b) End-to-end rate: RAB is called forward rate and RBA is called backward rate.

22
Channel
Encoder A
Channel
Encoder B
W
A
W
B
Bit-wise
XOR
Relay Node D
Bit-wise
XOR Channel
Decoder B
W
A
Channel
Encoder A
W
B
Source Node A
Binary
Symmetric
Channel
XA
XB
XB
XA
Fig. 2. PHY-Layer network coding (revised from [8], [13])
(a) (b)
00
RAB
R
BA
= 1
C
DB
> C
DA
= 1
RBA
R
AB
=C
DA
C
DB
(1) (2)
(3)
(4)
(5)
(4)
(3)
(1)
C
DB
< C
DA
(2)
(6)
=CDA
CDB
Fig. 3. Achievable Rate Regions for TD, MLNC and PLNC. Verteices (1)-(6) are (0,ΣBA), (ΣABA ,ΣBAB ), (ΣAB B ,ΣABB ),
(ΣAB ,0), (0,ΣBB ) and (ΣAA,0), respectively. TD is the triangular region with vertices (1), (4), 0. MLNC is the quadrilateral
region with vertices (1), (3), (4), 0in (a) and with vertices (1), (3), (6), 0in (b). PLNC is the quadrilateral region with vertices
(1), (2), (4), 0. Note that MLNC, PLNC and opportunistic MLNC achieve the same rate region with vertices (1), (3), (4), 0if
CDA =CDB .
D
AB
Fig. 4. Two-way Relaying over Multiple Relays

23
R
BA
0
R
AB
R
BA
0
R
AB
R
BA
0
R
AB
( =1)
=C
DA
C
DB
(a) (b) (c)
R
TD
(D
f
)
R
TD
(D
b
)
R
TD
R
OMLNC
R
MLNC
(D
MLNC
)
R
TD
(D
f
)
R
TD
(D
b
)
R
TD
(D
f
)
R
TD
(D
b
)
R
PLNC
(D
PLNC
)
R
OPLNC
Fig. 5. Achievable rate regions over an optimal relay: (a) is for TD, (b) is for MLNC, and (c) is for PLNC when CDB < CDA.
0 1 2 3 4 5 6 7 8 9 10
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Network Coding Gain ρ (dB)
Traffic Pattern Parameter µ
ρPT
ρOMT
ρMT
Fig. 6. Network coding gain ρvs. traffic pattern parameter µfor different transmission protocols over a single relay node.
ρMT,ρOMT and ρPT are the network coding gains for MLNC, OMLNC and PLNC.

24
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1
−0.5
0
0.5
1
1.5
Network Coding Gain ρ (dB)
x (Normalized Distance dDA/d)
ρPT
ρMT
ρOMT
µ = 2
Fig. 7. Network Coding Gain vs. Relay Position for the case of µ= 2.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1.5
−1
−0.5
0
0.5
1
1.5
Network Coding Gain ρ (dB)
x (Normalized Distance dDA/d)
µ = 0.5
ρOMT
ρPT
ρMT
Fig. 8. Network Coding Gain vs. Relay Position for the case of µ= 1/2.