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Optimization of Common Air Interface in Cellular Multihop

Wireless Networks in the Presence of Traffic Variation

Beatriz Lorenzo, Student Member, IEEE Savo Glisic, Senior Member, IEEECWC, University of Oulu Telecom Laboratory, University of Oulu

Oulu, Finland Oulu, Finland

Abstract- In this paper we define the jointly optimum topology

for the duplex transmission (uplink/downlink) in multihop

cellular networks which is aware of the intercell interference and

a protocol that reconfigures the optimum topology based on the

observation of the temporal traffic in the network. In addition we

also consider the application of network coding in cellular

networks to combine the uplink and downlink transmissions and

incorporate it into the optimum bidirectional relaying with

intercell interference awareness resulting in a comprehensive

solution for 4G common air interface.

Keywords-cellular network, common air interface, intercell

interference, network coding, relay, topology control.

I. INTRODUCTIONRecently, relaying has been studied intensively for

applications in multihop cellular networks [1-5]. In [6]

relaying techniques that increase the throughput in multihop

wireless networks are analyzed by applying network coding

over bidirectional traffic flows. This technique has been

included in our approach with the focus on mitigating the

intercell interference and adapting the relaying topology to

traffic variations across the network.

Two basic coding strategies for the one-relay case were

proposed by Cover and El Gamal in [7]: decode-and-forward

(DF) and compress-and-forward(CF). Furthermore, [7, Th. 7]

provides a general lower bound on the capacity of one-relay

networks which can be achieved using a combination of DF

and CF.The relaying concept is the basis of cooperative and virtual

antenna transmission too [8-12]. The bounds of the

information theoretic capacity of a discrete memoryless

channel are given by [13] based on a timesharing approach.

The capacity analysis for the special case of degraded relay

channels by the use of superposition block Markov encoding

is presented in [14]. For other type of channels the capacity is

upper-bounded (max-flow-min-cut theorem [15]) by the

minimum of mutual information obtained by the broadcast

channel (transmission from the source to relay and

destination) and the multiple access channel (independent and

simultaneous transmission from the source and relay to the

destination). Capacity bounds and power allocation for

wireless relay channels are presented in [16] for halfduplex

relay and single-antenna terminals. Algorithms for finding the

capacity bounds for the multi-antenna terminals are given in

[17], [18].Currently considered relays are assumed to work under

half-duplex mode, by using an orthogonal duplexing (in time

or frequency) between the relay-receive phase and the relay-

transmitphase. This phase separation allows defining several

half-duplex relay protocols. The number of options leads to

the four protocol definition [19]-[20] referred to as protocol 1,

2 and 3, and forwarding. In protocol 1 the source

communicates with the relay and destination during the relay-

receive phase and in the relay-transmit phase, the relay

terminal communicates with the destination. In protocol 2

during the relay-receive phase the source only transmits to the

relay. It is assumed that the destination is not able of receiving

the message from the source in that phase. In the relay-

transmit phase source and relay transmit simultaneously to thedestination. Hence in the relay-transmit phase the channel

becomes a multiple access channel (MAC). Protocol 3 is a

combination of protocols 1 and 2. The source transmits to the

relay and the destination in the relay-receive phase and in the

relay-transmit phase, the source and the relay transmit to the

destination.Notice that the relay is transmitting during the

second phase, so that it cannot be aware of the signal

transmitted by the source in the second phase. This protocol

can achieve a betterspectral efficiency than previous ones.

Finally, the traditional forwarding protocol consists of a

transmission from the source to the relay during the relay-

receive phase and a transmission from the relay to the

destination in the relay-transmitphase.Having in mind the above results, in this paper we present

the design of a relaying protocol jointly optimizes relaying

topology, routing and scheduling in the presence of intercell

interference. By using newly developed TSL algorithm for the

search of the optimum topology we show that the optimum

choice of the relaying topology can provide significant

performance improvements. We apply network coding to

bidirectional links (uplink/downlink) and combine it with the

optimum relaying to define a new cognitive common air

interface for 4G cellular networks. We also demonstrate that a

reconfigurable relaying topology provides better network

utility and presents the framework for quantifying these

improvements for spatially and temporally varying traffic.Numerical examples show that a combination of these

components provides a flexible optimal solution for future 4G

common air interface in cellular networks.

II. SYSTEM MODEL2.1 Network and Intercell Interference Model

A. Uplink

We consider a cellular network with a set ofI={i} base

stations. Let us assume that in a reference cell with index i=r,

there is a user m(r) connected to the access point AP(r) (base

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2010 proceedings.

978-1-4244-6398-5/10/$26.00 2010 IEEE

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station) with channel gain( )

( )r

r

mg . At the same time a cochannel

interfering userm(i)is connected to access point AP(i) in cell i,

i I-r { }i r= , with channel gain ( )( )

i

i

mg . Assuming that all the

users are expected to reach their respective access points with

the same required received power( )

( )i

i

mS , we have for the

transmitted powers( )

( )i

i

mP of the useful and interfering signal

( ) ( ) ( )( ) ( ) ( ) ;i i ii i i

m m mS g P i= I (1)

We denote by( )

( )i

r

mI the interference power level at the

position of the reference receiverAP(r) due to the interfering

cochannel signal transmitted in cell i. This can be presented as

( ) ( ) ( )

( ) ( ) ( );i i i

r r i

m m mI g P i= I-r (2)

where( )

( )i

r

mg is the gain of the channel between the interfering

userm(i) and AP(r). The signal to interference plus noise ratio

( )

( )r

r

mSINR atAP

(r) in the presence of all interfering users m(i) is

( ) ( )

( )

( ) ( )

1( ) ( )

( ) 1

( ) ( )

r i

r

i i

r r

r m m

rr imi rr m m

i r

S g

SINR N n I g

= = +

+ (3)

where ( )( ) /rr

r rmN S n SNR= = , and the channel capacity per

unit spectra can be represented as

( ) ( )

( ) ( )log(1 )r rr r

m mc SINR= + (4)

The network capacity is then given by

( )

( )r

r

mr

C c= (5)

If the radio resource management is defined as channel

assignment function A(m(r)) responsible to allocate to each

user m(r) proper channel then the optimum assignment is

defined as

( )

( )

( )( ) max ;r

r

A mA m C r= I (6)

B. Downlink

In this scenario the reference access point AP(r)is providing

power( )

( )

rm

rS to the reference user m

(r). At the same time,

interfering AP(i) providing the same signal level( )

( )

im

iS to the

userm(i), is producing interference to the useful signal of user

m(r). So, we have

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1

( ) 1 ( ) ( )

( ) ( ) ( )

,

/ ,

/

i i i

r r i i r i

r r i

m m m

i i i

m m m m m m

i i i i i i r

m m m

r r i i

i r

S g P i

I g P S g g i

SINR N g g

=

= =

= +

I

I (7)

where( )

( )

( )

im

ig is the channel gain between( )iAP and userm(i),

( )( )

( )

im

iP is the power needed at

( )iAP to provide power P for

userm(i),( )( )

( )

rm

iI is the interference power at m(r) produced by

( )iAP and( )

( )

( )

rm

rSINR is SINR at m

(r). The optimum radio

resource management is again defined by (4)-(6).

Fig. 1. Modeling interfering users positions for 2-cells.

2.2 Relaying and scheduling

We will use notation ( ) ( ) ( ) ( )2 1 2 1

( , , , )r r i ir m m m m , to denote

simultaneous transmission (relaying) on reference cell rI

from user ( ) ( )1 2

r rm to m and interfering users from

( ) ( )

1 2

i im to m

( i I-r) position in all interfering cells. Under these conditions

the corresponding link capacity will be denoted as( ) ( ) ( ) ( ) ( )

2 1 2 1( , , , )r r r i ic m m m m . This capacity can be calculated by

the following set of equations

( ) ( ) ( ) ( ) ( ) ( )

2 2 2 2 2 2( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1 1

( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2

( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1 1

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

;

/

r r r i i i

r r r i i i

r i r i r i

i i i r i i

m m m m m mm m m m m m

m m m m m m

m m m m m m

S P g S P g

I P g S g g

= =

= =

( )

( )2

( )( )12

( ) ( )1 2

( )1

( ) ( )2 2

( ) ( )1 1

( )

( ) ( ) ( )

2 1 ( )

1

( ) ( )1

,

/

r

rr

r r

i

r i

i i

m

mm i i

m m

r mi r

m m

r m mi r

SSINR

n I

N g g

= =+

= +

m m

(8)

( ) ( )( ) (1) (2) ( ) (1) (2)

1 1 1 1 2 2 2 2( , ,..., ); ( , ,..., );c cN Ni im m m m m m i= = m m I-r

( )( )2

( )1

( ) ( ) ( ) ( ) ( ) ( ) ( )

2 1 2 1 2 1( , , , ) log 1 ( , ) ;

r

r

mr r r i i i i

mc m m SINR i= + m m m mI

-r

where Nc is the number of cells. We define now the multihop

(Hhops) route as a series of relaying transmissions

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2 1 1 2 1( , ,..., , , , ,..., , )r r r r r i i i i

H H H Hm m m m m m m m (9)

The capacity of the route is defined as( )

( )

( ) ( ) ( ) ( ) ( ) ( )

1 1min ( , , , ); 2,...,r

r

r r r i i

h h h h hc c m m h H

= =m m (10)

( )( )2

( ) ( )1

( ) ( ) ( ) ( ) ( )

1 1log 1 ( , , , )

r

r r

mr r r i i

h h h hmc SINR m m

= + m m

-a-

-b-

1 2 2 13

1 2 23

R

3 1 3 2

1 3

2 3

AP(r) AP(i)

1 2 2 13

1 2 23

3 1 3 2

1 3

2 3

-a-

-b-

3 1

22

11

22

3 3

m(r)= 0

uplink

downlink

0 = m(i)

m(r)= 0 0=m(i)

AP(r) AP(i)


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which is equal to the minimum link capacity on the route.

The optimum set of relaying routes is defined as

{ }( )

( )

( ) ( ) ( ) ( )

,max ; where ;

r

r

r

Hr

C C c r

= = I (11)

In order to reduce the interference produced by the

concurrent transmission of the different relaying segments, a

scheduling in different time slots is introduced in the sequel.

This scheduling will also solve the constraint that nodes in awireless network can not receive and transmit the signal

simultaneously on the same channel.

2.3. Two Dimensional Relaying Topology

The cochannel interference can be further reduced by

scheduling different transmissions in different channels (time

slots). All necessary transmissions between all users and their

respective access points should be completed in B slots

(scheduling cycle) in both directions: uplink and downlink.

As an illustration, for the two cells scenario and notation

shown in Fig.1, a possible (feasible) topology is shown in

Fig.1b. For a systematic presentation of the problem the cell

area is divided into concentric rings (three rings for each cell

in Fig.1). It is assumed that one cochannel user from each ring

has unidirectional connection with the corresponding access

point. For the uplink, the topology consists of four partial

topologies representing transmissions in four consecutive time

slots (B=4). In the first time slot (the first partial topology)

there are two simultaneous transmissions: packet originating

from ring 3 in cell #ris transmitted from ring 3 to ring 2 and at

the same time packet originating from ring 1 of cell #i is

transmitted from ring 1 to access point AP(i). In the second

time slot (the second partial topology), packet originating from

ring 3 in cell #ris transmitted from ring 2 to AP(r) and at the

same time packet originating from ring 2 of cell #i is

transmitted from ring 2 to access point AP(i)

. Similarly the

same notation is then used for transmission in time slot 3 and

4. Similar topology for the downlink is presented in the lower

part of Fig.1b. These seven partial topologies together are

referred to as apossible orfeasibletwo dimensional(time and

space) topology and will be represented in the sequel by a

given topology index t. For this concept (11) becomes

{ }(2) (2) (2, )

( 2, )

(2, ) ( ) ( ) ( )

, ,max ;where

r

r

r

B Hr

C C c

= = (12)

rIand (2 ) is the two dimensional relaying topology to be

elaborated in more detail in the next section.

2.4 Bidirectional Relaying and Network Coding

In this section we additionally introduce network coding andcombine it with the previous results on optimum relaying to

achieve further improvements in system performance.

Let us assume that the hops are indexed in increasing order

for uplink as h(up)

and for downlink as h(down)

. By combining

the uplink and downlink traffic from the previous hop at hop h

as ( , ) ( ) ( )1 1

down up down up

h h hy y y = the number of overall time slots

needed for transmission in cycle B can be reduced. The

optimization process defined by (12) now becomes

{ }( 2) ( 2) ( 2, )

( 2, )

(2, ) ( ) ( ) ( )

, , ,max ; where

r

r

r

B Hr

C C c

= = (13)

and ( ) ( )up down = .

To elaborate this concept in more detail an example of

possible topology that includes network coding is shown in

Fig.3 for two cell scenario from Fig.1a. The traffic between

users and access point is bidirectional, so given a schedule that

alternates the transmissions between the different rings, after

certain number of time slots all intermediate users (m(i)

, i I)

have information frames buffered for transmission in both

directions. Whenever an opportunity arises, the intermediate

users combine two information frames, one for each direction,

with a simpleXOR operation and send it to its neighbors in a

single omnidirectional transmission. Both receiving nodes

already know one of the frames combined, while the other

frame is new. Thus, one transmission allows two users to

decode a new packet, effectively doubling the capacity of the

path, reducing the power consumption of the transmitter node

and reducing the number of time slots required to complete the

transmission.

If we denote by n(i)

the number of rings in cell i and

( )

1

cNi

i

N n=

= the total number of rings in the network, the

vector( )

( )

( ) ( ) ( ) ( )

1( ,..., ),i

i

i i i i n

n = R defines the amount of

generated source traffic by the users situated in the different

rings in cell i to be transmitted to the access point AP(i)

on the

uplink, and( )

( )

( ) ( ) ( ) ( )

1( ,..., ),i

i

i i i i n

n = R the traffic that the

access pointAP(i)

is transmitting to the users on the downlink.

For the same traffic vectors (i)

, (i)

the base station can

schedule the transmission through different channels (time

slots) resulting in temporal and spatial MAC protocol. The

network traffic on the uplink and downlink is defined as( )(1) (2)

1( , ,..., ) ( ,..., )cN

N = = and(1) (2)

( , ,...,=

( ) 1) ( ,..., )cN N = respectively.The base stations jointly assign an access vector

( )(1) (2)

1( , ,..., ) ( ,..., )cN

Na a= =a a a a , where each component

(0,1)na , to the different rings to give them permission to

transmit. With an=1 the users from ring n are allowed to

transmit otherwise not. In the two cell case a=(a(1),a(2)), the

first half of the coefficients represents the permissions to

transmit for the rings in cell #rand the second half for rings in

cell #i, i I-r. We consider symmetric bidirectional

transmission (duplex connection) in the sense that the access

point will only transmit to the users situated in the rings

activated by a.The transmission schedule presented in Fig. 2 defines a

possible topology for two cell scenario and access vectora=1.

In this case all rings have duplex connection and the topology

consists of eight partial topologies representing transmissions

in eight consecutive time slots. In the first time slot (the first

partial topology) there are two simultaneous transmissions;

packet originating from the access point AP(r) (addressed to

user in ring 3) is transmitted to ring 2 in cell #r and at the

same time packet originating from ring 2 (addressed to access

pointAP(i)) of cell i is transmitted from ring 2 to ring 1. In the


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second time slot (the second partial topology), packet

originating from access point AP(r) (addressed to user in ring

1) is transmitted to ring 1, at the same time packet originating

from ring 3 (addressed to access point AP(r)

) is transmitted

from ring 3 to 2 and, packet originating at AP(i) (addressed to

user in ring 2) is transmitted to ring 1 in the adjacent cell.

Similarly the same notation is then used for transmission in

time slot from 3 to 8. As already discussed earlier these eight

partial topologies together are referred to as a possible twodimensional(time and space) topology and will be represented

in the sequel by a given topology index (t).

So there will be limited interference transmission for 3

users per cell in 8 channels (8 slots in Fig. 2) giving the

intercell throughput 6/8=3/4, as opposed to the 6/12=1/2 in a

conventional TDMA system where each cell uses a half of

available channels (slots). Although scheduling in Fig.1b

requires 7 time slots it also assumes transmissions over three

rings which requires higher power.

Fig. 2. Possible schedule by using network coding.

III. PROTOCOL DESCRIPTIONThe examples of topologies presented in the previous

section are based on intuition and we need a systematic

approach to the system optimization. For these purposes we

introduce the following definitions. Fori

I,

- We define topology matrix ( ) ( ) ( )2 1( , )

i i iT m m = T with

( ) ( )

2 1( , ) 1,i iT m m = if user

( )

1

im is transmitting to ( )

2

im and 0,

otherwise with indexes ( ) ( ) ( )1 2, 0,1, 2,...,

i i im m n= , ( ) ( )2 1

i im m< .

Each ( ) ( )2 1

( , )i im m pair is represented by a specific link index l

as shown in Fig. 3 for the case of two cells. With this notation

the vector of equivalent (source + relayed) rates in cell i is

2 2 1( )1

( ) ( ) ( ) ( ) ( )

2 1( , )i

i i i i i

m sm m

m

x x T m m x= + (14)

where2

( ) ( )i i

mx = x for each direction of the traffic. The

overall topology matrix will be formally defined asT=diag[T

(i)] and ( )i = x x is the vector of the overall

aggregate rate.

Fig.3. Link notation

For simplicity, in the sequel we will describe the protocol

for only one direction of the traffic and then at the end of the

section make comments on how we extend the protocol for

bidirectional case.

- The routing matrix (or relaying matrix) R=[rln] has

entries rln=1 if source n (n=1,2,..,N) is using linkl(l=1,2,..,L)

and 0 otherwise. Recall that, parameterN is the number of

overall rings in the network. Parameters( )1

iln lmr r= are

calculated as ( ) ( )1 2

( )2

( ) ( )2 1( , )i i

i

i i

lm lmm

r r T m m= and,

( )1

( ) ( )

2 1( , )i

i i

m

L T m m= .

- The scheduling set will be combined with the routing

matrix R resulting into two dimensional routing protocol

characterized by extended routing matrix (2) R . By

assuming that the scheduling cycle within the maximum

clique has B steps, the optimization process will include:

a) Utility function

( )1/ log( ) /n n nn

U B a x P = (15)

where Pn is the aggregate power needed for transmission of

information from the source n to the access point on the uplink

or viceversa on the downlink, an is the access parameter as

defined in the previous section.

b) Constraint )( )2()2()2()2( RcxR with the following

definitions ofextended system parameters

( ) ( )(2) (2)

(2) (2)

(1), (2).. ( ) ; (1), (2).. ( )

( ) , 1,2,.., ; ,

T T T T T T T T B B

diag b b B R

= =

= =

x x x x c c c c

R R R x R

(15a)

where c are the logical link capacities calculated as discussed

in Section II which capture the functional dependency of

power control and interference level in the network.

c) Each component of the set of feasible routes in should provide directional connection for each terminal to the

corresponding access point. This means that the sequence of

links generated in a clique cycle must provide connection for

all terminals to the corresponding access point. To define thisconstraint explicitly we introduce the link hopping distance hl

and the vectorh = {hl}. hl represents the number of rings thatlink l is hoping over, from its transmitter/receiver to the

corresponding receiver/transmitter. Similarly the source

hopping distance is denoted as d = {hn}. The sum of linkhopping distances on the route from source n to the access

point should be equal to the source hopping distance

( ) ( )T

b

b b =R h d (15b)

( )1

( ) ( ) ( ) ( ) ( )

2 2 2 1 1( ) ( , )i

i i i i i

n m sm m

m

x x T m m x = = + I T x x

(15c)

The formulation of the problem obtained by equations (15a)-

(15c) can be summarized as:

,

(2) (2) (2) (2)

(2 )

: maximize

subject to ( ); ( ) ( )

( ) ; ,

T

b

n

U

b b

R

=

=

T x

P

R x c R R h d

I T x x R x R

(16)

-c-

AP(r) AP(i)

m(r)=0 1 2 2 1 0=m(i)3

- -

( )2

i

( )

1

r ( )3

r

( ) ( )

3 3

r r

( )3

r

( )

2

r

( )

2

r

( )

1

r

( ) ( )

2 2

r r

slot: 12

3

4

5

6

7

8

( )

3

r

( )

2

i

( ) ( )

2 2

i i

( )

1

i

( )

3

i

( )3

i

( ) ( )

3 3

i i

( )

3

i

( )

1

i

l3 l9

1

2

3

4

(i) (i)

2 1m ,m 3 2

l6l4

l2

l31 5

1

2

3

l1

l5

1

AP(r) AP(i)

l11

l12 l10

l8

l7

(r) (r)

2 1m ,m


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In the case of bidirectional traffic an independent set of

equations (15) should be written for both directions and (16)

should be modified to include overall utility function

U=U(up)

+U(down)

with separate set of constrains for both

directions.

IV. SIMULATION RESULTSIn this section, we provide several numerical examples to

illustrate the performance of the proposed protocol. Wecalculate the link capacities ( ) ( ) ( ) ( ) ( )

2 1 2 1( , , , )r r r i ic m m m m as

specified in Section II. While the analysis is general, for

simplicity in this section, the channel gains used to calculate

( )( )2

( )1

( ) ( ) ( )

2 1,r

r

m i i

mSIR m m are

( )2

( ) ( ) ( )1 2 1

( )1 /

r

i r i

m

m m mg d and

( )2

( ) ( ) ( )1 2 1

( ) 1/i

i i i

m

m m mg d , where ( ) ( )

2 1r i

m md is the distance between the

reference receiver in ring2m and interfering transmitter in ring

( )

1

im , analogous for ( ) ( )2 1i i

m md and, is the propagation constant.

In the simulations we use =4, and SNR=10. The calculation of

the distances is straightforward from the geometry presented in

Fig.1a.

In the sequel we present the utility for different access

vectors a versus the topology index (t) for the scenario

presented in Fig. 1a. The resulting topologies, indexed by t,

represent a certain combination of the active in B slots and

will be represented formally as (2) ( ) ( ) ( )b b

b b

T L l = = .

In Fig. 4, the utility function is shown for a=[010100].

With this access vector user from ring 2 in cell # r and user

from ring 1 in cell #i have permission to transmit. The

maximum utility is obtained for topology index t=478

(u478=0.8739) by using network coding. We see a significant

improvement compared with the maximum utility with no

coding obtained for topology index t=215 (u215=0.6640).

The optimum topologies for both cases are given by

{ }478 1, 7, 4, 1, 4, 7,{ },{ },{ },{ , }down down up up down upT l l l l l l = and

{ }215 1, 4, 4, 7, 7, 1,{ },{ },{ , },{ },{ }down up down up down upT l l l l l l =

100 200 300 400 500 600-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Topology Index (t)

UtilityFunction

a=[010100]

no coding

coding

Fig. 4. Utility function for access vectora=[010100].

In Fig. 5, the utility function versus the topology index for

a=[010010] is shown. With this access vector user in ring 2, in

both cells have permission to transmit. As the number of

topologies obtained for this access vector is very high, we plot

the segment of topologies close to the optimum topology. The

maximum utility is u=0.6991 by using network coding, and

with no coding the maximum utility is u=0.5826. Both are

smaller than in the previous case due to higher interference

level. As several topologies provide the maximum utility, in

Fig. 6 we show the transmission pattern for one of the

optimum topologies (t=7) in the case with no coding for theprevious access vector defined by the set of links

{ }7 1, 10, 4, 7, 4, 1, 7, 10,{ },{ },{ , },{ },{ },{ },{ }down up down up up up down downT l l l l l l l l = .We can see that isolated short range transmissions are

favored which can simultaneously reduce the intercell

interference and power consumption.

In Fig. 7 we plot the transmission pattern for topology

{ }5621 1, 4, 7, 1, 4, 10, 7, 10,{ },{ },{ },{ , },{ }down up down up down up up downT l l l l l l l l = that corresponds to one of the topologies with coding that

provides the maximum utility fora=[010010]. We can see an

improvement in the number of slots needed with coding (5

slots) compared to 7 slots in the case with no coding. So for

the same type of isolated and short range transmissions the

utility function is improved by reducing the number of slots.

0 1000 2000 3000 4000 5000 6000-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Topology Index (t)

UtilityFunction

a=[010010]

no codingcoding

Fig. 5. Utility function for access vectora=[010010].

Fig. 6. Representation of the transmission pattern definedby the topology index t=7

1 221

1 2 4(r) (r)

2 1m ,m

(i) (i)

2 1m ,m32

l1

123AP(r)

AP(i)

Time slot 1

1 21

l10

23AP(r)

AP(i)

Time slot 2

1 2 332

l4

123AP(r)

Time slot 3

l7

Time slot 4

3

3 453Time slot 7

AP(r)

AP(i)

2 1

1

l10

AP(i)

1 2 332

l4

123AP(r) 1 AP

(i)

1 2 332

l1

123AP(r)

AP(i)

1Time slot 5

1 2 3 123

AP(r)

Time slot 6l7

AP(i)

1


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6/6

Fig. 7. Representation of the transmission pattern definedby the topology index t=5621

In Fig. 8 we represent the overall capacity for the previous

access vector a. For the optimum topologies with coding we

can see that the overall capacity of the system improves by a

factor 4 compared with the case without coding.

1000 2000 3000 4000 5000 60004

6

8

10

12

14

16

Topology Index (t)

OverallCapacity

a=[010010]

no coding

coding

Fig. 8. Overall capacity for a=[010010].

V. CONCLUSIONIn this paper we present some solutions on intercell

interference aware optimum relaying topology that includes

bidirectional links and network coding. The utility function

used in the optimization process drives the solution towards

the topology favoring simultaneously isolated and short range

transmissions. As expected, within these solutions further

improvements are obtained by using network coding to reduce

the number of slots needed for transmission. For example, for

access vector a=[010100], the maximum utility obtained is

u=0.8739 by using network coding, which is a significant

improvement compared with the maximum utility with no

coding u=0.6640. For a=[010010], the maximum utility is

u=0.6991 by using network coding, and with no coding

u=0.5826. Both values are smaller than in the previous case

due to higher interference level.

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Time slot 1

Time slot 2

Time slot 3

Time slot 5

1 2 3 4(r) (r)2 1m ,m

(i) (i)2 1m ,m

32l1

123AP

(r)AP

(i)

1 2 332l4

123AP

(i)

1 2 3 4

l7

123AP

(r)

34531 2

AP(r)

AP(i)2 1

1

l7l10

3 4

l153

Time slot 41 2

AP(i)2 1

l4 l10

AP(r)

AP(i)1

1

AP(r)


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