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Throughput analysis of Ad-Hocnetworks with directional antenna at60 GHzMohammadreza Alimadadia, Abbas Mohammadia & MohammadDehghani Soltaniaa Electrical Engineering Department, Amirkabir University ofTechnology, 424 Hafez Ave, Tehran, Islamic Republic of IranPublished online: 25 Nov 2013.

To cite this article: Mohammadreza Alimadadi, Abbas Mohammadi & Mohammad Dehghani Soltani(2014) Throughput analysis of Ad-Hoc networks with directional antenna at 60 GHz, Journal ofElectromagnetic Waves and Applications, 28:2, 228-241, DOI: 10.1080/09205071.2013.862188

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Throughput analysis of Ad-Hoc networks with directional antennaat 60 GHz

Mohammadreza Alimadadi, Abbas Mohammadi* and Mohammad Dehghani Soltani

Electrical Engineering Department, Amirkabir University of Technology, 424 Hafez Ave, Tehran,Islamic Republic of Iran

(Received 12 July 2013; accepted 30 October 2013)

In this paper, we investigate the throughput of wireless mobile Ad-Hoc networkswith directional antennas at 60 GHz unlicensed band. A new model is proposed tocharacterize the performance of directional antennas operating in millimeter-wavefrequency bands. Based on this, we introduce an appropriate interference model andderive a lower bound on the throughput capacity of millimeter-wave wirelessAd-Hoc networks. Monte-Carlo simulation is conducted to evaluate the validity ofthe analytical results.

Keywords: Ad-Hoc network; directional antenna; 60 GHz unlicensed band;millimetre-wave frequency band

1. Introduction

Wireless Ad-Hoc networks gained increased research interest over the last decade. Thisis mainly due to their wide range of applications in civilian and military operations.Without the need for centralized infrastructure support, these networks have manyimportant features such as low cost, ease of deployment and low maintenance. A criticalfactor affecting the future extension of such networks for practical implementation isnetwork capacity.

The capacity of wireless Ad-Hoc networks is constrained mainly by the interferencebetween simultaneous transmissions from neighboring nodes. A tremendous amount ofefforts has been devoted by the research community to investigate the asymptotic per-formance of wireless Ad-Hoc networks when the number of nodes increases. Guptaand Kumar have shown that when using omnidirectional antennas, the capacity of anAd-Hoc network does not scale proportional to the increasing number of nodes in thesystem.[1] They obtain the frustrating result that the throughput available to eachsource-destination pair decreases at least as 1ffiffi

np , even allowing optimal strategy and

node placement.Lots of works tried to improve this disheartening scalability of throughput capacity

by introducing various characteristics into Ad-Hoc network. Ozgur et al. [2] study thethroughput capacity of wireless Ad-Hoc networks and show that by intelligent nodecoordination and distributed MIMO communication, these networks can scale linearlywith the number of nodes. Furthermore, some other works such as [3] and [4], studythe capacity of hybrid wireless networks where fixed stations are placed to help

*Corresponding author. Email: abm125@aut.ac.ir

© 2013 Taylor & Francis

Journal of Electromagnetic Waves and Applications, 2014Vol. 28, No. 2, 228–241, http://dx.doi.org/10.1080/09205071.2013.862188

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improve the network capacity. They show that hybrid wireless networks could alsoprovide a constant per-node throughput at the expense of high infrastructure invest-ment. Fang et al. [5] evaluate the capacity of three-dimensional wireless Ad-Hoc net-works and find some lower and upper bounds for it.

Grossglauser and Tse [6] presented a two-phase packet relaying algorithm formobile Ad-Hoc networks (MANET), utilizing multiuser diversity and show that, in con-trast to the fixed node case, the throughput per S-D pair can be kept constant while thenumber of nodes is increased. A tremendous amount of other efforts (e.g. [7] and [8])are also made on the capacity of mobile Ad-Hoc networks, indicating that mobility cansignificantly improve network capacity.

As a way to breakthrough in capacity limit proven in [1] for omnidirectional case,the use of directional antennas has begun to draw a lot of attention recently. Thepotential benefits of using directional antennas include range extension, betterfrequency reuse, multipath diversity, interference reduction, and capacity increase.

Li et al. [9] study the connectivity of wireless networks using directional antennas.In addition, Yi et al. [10] analyze the capacity improvement of Ad-Hoc networks usingdirectional antennas. For arbitrary networks, by using a simple directional antennamodel ignoring the side lobe gain, they show that the capacity gain is 2pffiffiffiffi

abp when using

both directional transmission and directional reception (DTDR), where a and b are thebeamwidths of a transmitter and a receiver, respectively.

In [11] authors employed a more practical directional antenna model with the beamnumber N , the main lobe gain Gm, and the side lobe gain Gs. Based on this model theyfound some bounds on the capacity of wireless Ad-Hoc networks. In [12] authors paidattentions in maximizing the throughput capacity of wireless mesh networks usingcooperative technique and directional antenna.

The 60 GHz frequency band, which provides several GHz of unlicensed bandwidth,is a highly promising resource for future wireless short-range transmission. It isexpected to be used for high-rate wireless networks as well as for ultra-high ratepoint-to-point links, addressing applications like high-definition (HD) video streaming,wireless in-vehicle entertainment, and ultra-high speed content download.

In addition to the high-data rates that can be accomplished in this spectrum, radiopropagation in the 60 GHz band has special characteristics that make possible manyother benefits such as excellent immunity to co-channel interference, high security, andfrequency re-use. The use of high-gain directional antennas can be especially desirablein the 60 GHz band, given the above propagation characteristics. They can significantlyincrease communication range and this can be especially valuable given the highpropagation loss in typical 60 GHz environments.

By considering the advantages and facilities that wireless Ad-Hoc networks offerand also the unique characteristics of 60 GHz unlicensed band, it seems that thecombination of these two promising field of technology can bring a new wide range ofapplications. Hence, in recent years the idea of using wireless Ad-Hoc networks in 60GHz band begun to draw more attentions. In this paper, we evaluate the performanceof wireless mobile Ad-Hoc networks with directional antennas. First, we propose a newmodel for directional antenna at 60 GHz band. This model is based on the referencemodel of IEEE 802.15.3c standard which defines the physical layer (PHY) specificationof millimeter-wave (mmW) WPANs. Moreover, it matches better with the realperformance of antennas at millimeter-wave band.

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Based on this antenna model, we evaluate the throughput capacity of wirelessmobile Ad-Hoc networks with directional antennas at 60 GHz band. We find a lowerbound on the throughput capacity of mmW MANETs. Also, we propose a specialsimulation approach based on the Monte-Carlo methods to realize the validity of theobtained lower bound.

The rest of this paper is organized as follows: we introduce antenna model andrelated definitions in Section 2. Also, we introduce an appropriate interference model inthis section. In Section 3, we derive a lower bound on the throughput capacity ofmmW directional MANETs based on the proposed antenna model. In Section 4, a spe-cial simulation approach based on Monte-Carlo methods is introduced and conductedto evaluate the validity of the obtained bound. Finally, Section 5 concludes the paper.

2. Antenna model

The first step in evaluating the performance of wireless Ad-Hoc networks at 60 GHz isintroducing antenna model. To this end, we have chosen a model based on the IEEE802.15.3c reference antenna model [13]. This standard that is also called MillimeterWave WPAN or High rate WPAN works on the physical and data link layer standardi-zation for wireless personal area networks which operate on 57–64 GHz band. TheIEEE TG3c task group released the final version of this standard in September 2009. Inorder to select an antenna model for mmW WPANs, it should be considered that thedemand of low-power consumption from the one hand and high propagation loss ofmmW signals from the other hand makes the use of directional antenna an unavoidablerequirement.

The reference antenna model defined in IEEE 802.15.3c is based on the conceptshown in Figure 1.[14] In this figure, the red line shows the antenna gain of a typicaldirectional antenna. The blue line shows the antenna gain of the reference model. Thehorizontal axis plots the angular offset from the direction of antenna peak gain. In thereference model of 802.15.3c standard, the antenna gain in the main lobe is modeled asa Gaussian function and it has a side lobe of constant gain.

2.1 Proposed antenna model

The antenna model that is used in this paper is based on the reference model of IEEE802.15.3c. In the proposed model, the gain of antenna is as follows:

Figure 1. Basic concept of IEEE 802.15.3c antenna reference model (Source: [14]).

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Gð/; hÞ ¼ 1

aðr; gÞ e�ð/rÞ

2

e�jhjr (1)

in which / and h are the relative1 azimuth and elevation angles, respectively and wehave �p\/� p and � p

2\h� p2. We call r the divergence factor (DF) of antenna and

it is a criterion to measure how directive the antenna is. In other words, if we define Das the directivity of antenna, then we have: D ¼ 1

rg is the efficiency of antenna. aðr; gÞ is a function of r and g and as we will see

later, it is specified in such a way that it brings the ability to compare antennas withsame g0s and various r0s in a fair condition.

As it is clear from (1), the gain of antenna changes in a Gaussian manner in termsof azimuth angle and exponential in terms of elevation angle. If we set h ¼ 0, then thegain in the horizontal plane becomes:

Gð/Þ :¼ Gð/; 0Þ ¼ 1

aðr; gÞ e� /2

r2 (2)

Comparing this equation with the reference antenna model i.e. Equation (2) of [14]shows that if we set:

r ¼ /3dB

2ffiffiffiffiffiffiffiffiffiffiffi0:301

p ; aðr; gÞ ¼ e�0:1G0

then in the main lobe, the proposed antenna model has same gain with the referenceantenna model introduced in IEEE 802.15.3c standard. In the above expressions, /3dB

is the antenna’s half-power beamwidth and G0 is defined in Equation (5) of [14]. Forsimplicity, we assumed that the Gaussian model is also valid in the side lobe.

r ¼ þ1 corresponds to the omnidirectional antenna in which the antenna gain isconstant for all ð/; hÞ pairs. As r decreases from infinity to zero, the value of directiv-ity becomes more and more great until for r ¼ 0, the antenna becomes an ideal one.

As we stated before, aðr; gÞ should be determined in such a way that brings theability to compare antennas with same g0s and different r0s in a fair circumstance.Assume that S is the surface area of a sphere centered at the transmitter and with radiusR. So, in order that the performance of a directional antenna to be fairly compared tothat of an omnidirectional one, we should have,

Zp�p

Zp2

�p2

Gð/; hÞR2 sinðhÞdhd/ ¼ 4pR2g

¼) aðr; gÞ ¼ 1

pgre�

p22r

1þ r2ep22r þ r

� � ffiffiffiffiffiffiffiffipr2

p 1

2� Q

ffiffiffi2

pp

r

� �� �(3)

In the following of this paper, we will use the antenna model which is defined in (2)and (3) to evaluate the performance of directional MANETs.

2.2 Interference model

An interference model states the conditions under which a transmission from one nodeto another one will be successful. Until now, some interference models have been

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introduced for wireless Ad-hoc networks, including physical model, protocol model,and generalized physical model.

Gupta and Kumar proposed the physical interference model for the case ofomnidirectional antennas in [1] and the authors in [11] modified it to be applicable todirectional antennas. In order to provide the necessary clarity and background, here wegive a brief description of this model. Let T be the subset of nodes which simulta-neously transmit at the same time interval. Also, suppose that all of these nodes use acommon transmission power namely P. Node Ti 2 T is transmitting to node Ri. In thiscase, according to the physical interference model, the transmission from Ti to Ri issuccessfully received if

PC GtGr

jTi�Rija

N0 þP

Tj2T ;j6¼i PCGtGr

jTj�Rija� b

in which Tj and Ri also denote nodes’ locations, P is the transmission power, a is thepath loss exponent, N0 is the noise power level at the receiver, Gt and Gr are the gainof the transmitter’s and receiver’s antenna, respectively, C is a constant depends on thewavelength, antenna heights, etc. and b is the minimum acceptable signal tointerference plus noise ratio (SINR) for successful reception.

Since, the network we are interested to analyze is using a new antenna model, weneed also to introduce a new interference model. Again suppose a situation in which anode Ti 2 T is transmitting to its intended receiver Ri. Node Tj 2 T causes interferenceto Ri reception. Figure 2 shows this situation. In the figure, we have also shown thevarious parameters that will be used in the following. Here, we define these parameters: hRi ¼ the direction that Ri selects for its antenna, hTj : the direction that Tj selects forits antenna, /ij: the angle between the x-axis and the running line from Ri to Tj. rij: thedistance between Ri and Tj and by definition /i :¼ /ii, ri :¼ rii.

According to the proposed antenna model introduced in the previous subsection, wecan define and calculate the gain of nodes in direction of each other:

Figure 2. Model parameters and their definitions.

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GRij : gain of Ri in direction of Tj, which is equal to GR

ij ¼ 1aðrR;gÞ e

�/2ij

r2R ;

GTji : gain of Tj in direction of Ri, which is equal to GT

ji ¼ 1aðrT ;gÞ e

�ðpþ/ij�hT

jÞ2

r2T ;

in which rT and rR stand for the divergence factors of transmitter and receiverantennas, respectively. By definition GR

i :¼ GRii and GT

i :¼ GTii .

Now, considering the above definitions, we introduce the following interferencemodel for millimeter wave wireless Ad-hoc networks.

2.3 mmW protocol interference model

Inspired by the above physical model, here we introduce a protocol model which isapplicative to the new antenna model. Suppose that Ti is transmitting to Ri. In order tothis transmission to be successful, two conditions should be satisfied:

(1) The distance between Ti and Ri should be less than or equal to rðnÞ i.e.ri � rðnÞ, in which rðnÞ denotes the communication range of nodes in terms ofthe nodes number.

(2) The position of every other transmitters Tj simultaneously transmitting shouldsatisfy:

GTi G

Ri

raðnÞGT

ji GRij

jTj�Rija� b ¼) b

GTjiG

Rij

GTi G

Ri

!1=a

rðnÞ� rij

As stated earlier, GTji and GR

ij are functions of /ij and hTj . So, the left side of the aboveinequality is a function of these two parameters. Therefore, we can see that for eachvalue of the ð/ij; h

Tj Þ pair, there is a corresponding range of rij values such that the

simultaneous transmitter Tj can reside, in order to have a successful transmission. Inthe remaining part of this paper, we will use the mmW protocol model to evaluate theperformance of wireless mobile Ad-Hoc networks.

3. A lower bound on the throughput capacity of mmW mobile AD-HOCnetworks

In this section, we want to derive a lower bound on the throughput capacity of wirelessMANETs. Suppose that n nodes are randomly located, i.e. independently and uniformlydistributed on a disk of unit area. Each node selects a destination to which it wishes tosend kðnÞ bits per second. The destination is chosen in a uniform manner. The nodesare moving and locations of them constitute an i.i.d stationary and ergodic process witha uniform distribution. The nodes use directional antennas to transmit and receive datawith divergence factors rT and rR, respectively, and all of them have a same communi-cation range, namely rðnÞ. It is assumed that a source node Si transmits only when thedestination Di is in its communication range. So, the transmissions are performed inone-hop delivery scheme and there is no relaying.

We assume that source and destination have the ability of beamforming, i.e. theycan change the direction of their antennas in order to point toward each other, whentheir distance is smaller than rðnÞ. So, we have:

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hDi ¼ /i ¼ hSi � p

¼) GDi ¼ 1

aðrR; gÞ; GS

i ¼1

aðrT ; gÞ(4)

in which GSi is the gain of Si in direction of Di and GD

i is the gain of Di in direction ofSi. According to the mmW protocol model, the transmission from Si to Di will be suc-cessful if every other source Sj simultaneously transmitting satisfies:

½aðrR; gÞ aðrT ; gÞ bGDij G

Sji�

1a rðnÞ� rij

in which GDij is the gain of Di in direction of Sj, GS

ji is the gain of Sj in direction of Di

and rij is the distance between Sj and Di. The values of GSij and GD

ij are as follows,

GDij ¼

1

aðrR; gÞe� /2

r2R ; GS

ji ¼1

aðrT ; gÞe�

ðpþ/�hSjÞ2

r2T

In order to derive a lower bound on the throughput capacity, we do the following steps.First of all, we compute interference area for Di. This is an area in which no othersimultaneous transmitter Sj should lie. Also, we calculate interference area for Si. Thisis an area in which no other simultaneous receiver Dj should lie. The sum of these twois the area which is dedicated to this Si � Di pair and no other simultaneously commu-nicating pairs should reside in it. Then, we should see that how many potential pairslie in this area on average.

A potential pair is a Sj � Dj pair in which the distance between Dj and Sj is lessthan rðnÞ. As we stated previously, only these pairs have the ability to exchange data.Other pairs that don’t satisfy this condition in a certain time instant, have no effect onthe network performance and we can suppose that they are not in the network in thatinstant.

Then, we should find the density of the potential pairs, we call it np. In order to dothis, we need the cumulative distribution function (CDF) of random variableRi :¼ jSi�Dij.

Now, if we show the interference area of Si � Di pair with �Sm, then the averagenumber of the potential pairs in the �Sm area is m ¼ np � �Sm. These pairs should besilent in the time instant that Si transmits to Di. So, there is a scheduling policy underwhich from every mþ 1 time slots, we can allocate one slot to each node such that alltransmissions become successful. Since, all the above conclusions have been derivedunder the basic assumption that the destination Di resides in the communication range

of Si, we can conclude that the throughput kðnÞ ¼ min cpW1þm ;

W2

n ois feasible, in which

p :¼ PrfjSi � Dij � rðnÞg and W is the channel capacity. Now, we want to implementthe above method to derive a lower bound on the throughput capacity of directionalmmW MANETs in a step by step manner.

Step 1: average interference area of the Si � Di pair

First of all, we want to calculate pðr;/Þ, the probability of the event that a transmitterSj which is located at rij ¼ r and /ij ¼ /, that causes interference to Di reception. Inall the followings, we assume hDi ¼ 0. Because of symmetry, this condition places nolimits on the final results. Now, it is clear that ðr;/Þ denotes the location of the inter-fering node Sj in a cylindrical coordinate system, in which Di is at the center.

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In order to calculate pðr;/Þ, it is necessary to determine the range of hSj values thatbased on the mmW protocol model leads to interference. Since, hSj is chosen uniformlyfrom ð�p; p� interval, we can calculate pðr;/Þ by dividing the length of the desiredrange by 2π. According to Equation (1), we can write the interference condition at thepoint ðr;/Þ as:

½aðrR; gÞ aðrT ; gÞ bGRijG

Tij �

1a rðnÞ[ r (5)

in which GRij ¼ 1

aðrR;gÞ e� /2

r2R and GT

ji ¼ 1aðrT ;gÞ e

�ðpþ/�hT

jÞ2

r2T .

Substituting in (2) and then by some simple manipulations the interference probabil-ity is computed as follows:

pðr;/Þ ¼

1; r\rminð/Þ

1p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiar2T ln

b1ae

� /2

ar2R rðnÞr

24

35

vuuut rminð/Þ� r� rmaxð/Þ

0; rmaxð/Þ\r

8>>>>><>>>>>:

(6)

in which by definition we have :

rminð/Þ ¼ b1ae

� p2

ar2T e

� /2

ar2RrðnÞ;

rmaxð/Þ ¼ b1ae

� /2

ar2RrðnÞ (7)

Figure 3 shows pðr;/Þ for various points of the x–y plane.

Figure 3. pðr;/Þ for various points of the x–y plane.

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Averaged interference area of Di is calculated by integration of pðr;/Þ over the x-yplane.

�SMD ¼Zp�p

Zrmaxð/Þ0

pðr;/Þrdrd/ (8)

It can be shown that after computing the above integral, �SMD has an upper bound as:

�SMD ¼ O b2aGðrT ÞFðrRÞr2ðnÞ

� �(9)

in which we defined,

GðrT Þ ¼ e� 2p2

ar2T

FðrRÞ ¼ffiffiffiffiffiffiffiffiffiffipar2R2

r1

2� Q

2pffiffiffiffiffiffiffiffiar2R

p !" #

(10)

In a similar way, we can determine an upper bound on the interference area of Si byexchanging rT and rR in (9).

�SMS ¼ O b2aGðrRÞFðrT Þr2ðnÞ

� �(11)

Now, we can compute the average interference area of the Si � Di pair:

�SM ¼ �SMD þ �SMS ¼ O b2aHðrT ; rRÞr2ðnÞ

� �(12)

in which we defined :

HðrT ; rRÞ ¼ GðrT ÞFðrRÞ þ GðrRÞFðrT Þ

Step 2: the average number of potential pairs in the interference area

As stated before, in order to calculate np; i.e. the average density of potential pairs, itis necessary to compute the CDF of random variable Ri ¼ jSi�Dij. Toward thisobjective, we can first condition it on the location of Si. Then, we calculate theprobability density function (PDF) of the Si location. Finally, we shall obtain thedesired probability by integrating the conditional probability over the PDF of Si. If weshow the CDF of Ri by FRðxÞ we have:

FRðxÞ ¼ PrðRi � xÞ ¼Z1ffiffi

pp

0

PrðRi � xjSi ¼ uÞfsðuÞdu (13)

in which fsðuÞ is the PDF of Si location. Since, the nodes are distributed uniformly onthe disc surface, we can say that PrðRi � xjSi ¼ uÞ is the overlapping area of theintersection between the above disc and a circle centered at Si ¼ u and of radius x.Based on the relative values of x and u, various cases may occur. Figure 4 shows twoof them.

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As is clear from the figure, computing PrðRi � xjSi ¼ uÞ and then integration of iton the fsðuÞ is complicated due to edge effects. But, here we don’t need to do so. Thisis because we are only interested in those rðnÞ values satisfying the followingcondition:

limn!þ1

rðnÞ ¼ 0 (14)

For example, rðnÞ ¼ffiffiffiffiffiffiffilnðnÞpn

qwhich is the critical range needed to guarantee the

connectivity of wireless networks, satisfies (14). Since, our aim is to evaluate theperformance of the network when n goes to infinity, if (14) is satisfied then rðnÞ willbe small enough such that the total area of the circle resides in the disc as shown inFigure (4b). So, it is clear that we only need the values of FRðxÞ for small values of x.Therefore,

PrðRi � xjSi ¼ uÞ ¼ px2 (15)

Now, for calculation of FRðxÞ, we only need to determine fsðuÞ. We know that bydefinition,

fsðuÞ ¼ limdu!0

Prðu� Si\uþ duÞdu

(16)

and since node distribution on the disc is uniform, the probability radius u and outerradius uþ du are equal to the total area, i.e. one. Hence, we can write,

Prðu� Si\uþ duÞ ¼ pðuþ duÞ2 � pu2 ¼ 2puduþ pðduÞ2

So, according to (16) we have,

fsðuÞ ¼ limdu!0

Prðu� Si\uþ duÞdu

¼ 2pu (17)

Now, substituting (15) and (17) into (13), we can determine FRðxÞ in terms of x.

FRðxÞ ¼ PrðRi � xÞ ¼Z1ffiffi

pp

0

ðpx2 � 2puÞdu ¼ px2 (18)

Figure 4. Two typical cases of overlapping area.

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According to Ri definition, we can say that Di will reside in the communication rangeof Si when we have, Ri ¼ jSi�Dij � rðnÞ and as stated before, if rðnÞ satisfies (14) thenwe can compute FRðrðnÞÞ for large values of n from (18). So, the probability of theevent that Di resides in the communication range of Si is equal to: FRðrðnÞÞ ¼ pr2ðnÞ.

Now, the density of potential pairs can be calculated as follows:

np ¼ n� FRðrðnÞÞ ¼ npr2ðnÞ (19)

Using this equation we can calculate the average number of potential pairs in theinterference area.

m ¼ np � �SM ¼ npr2ðnÞ � Oðb2aHðrT ; rRÞr2ðnÞÞ (20)

¼) m ¼ Oðnr4ðnÞb2aHðrT ; rRÞÞ (21)

Now, we are ready for the last step.

Step 3: a lower bound on the throughput capacity of directional mmW MANETs

As stated at the beginning of this section, after the calculation of m, we can consider ascheduling policy such that in every mþ 1 time intervals, each potential pair gets oneinterval in which to transmit and all transmissions are successfully received. Weremember that all the above conclusions are based on the assumption that the distancebetween Di and Si is less than rðnÞ. The probability of this event is equal to pr2ðnÞ.According to the above discussions, we can conclude the following theorem:

Theorem 1. In random MANETs employing IEEE 802.15.3c directional antennamodel, under the (14) condition there is a constant 0 \ c \þ1 not depending onW ; a; b and n such that

kðnÞ ¼ mincr2ðnÞW

1þ pnr4ðnÞb2aHðrT ; rRÞ

;W

2

( )bits=second (22)

is feasible with high probability. If we set rðnÞ ¼ffiffiffiffiffiffiffilnðnÞpn

q, then the above bound becomes

as:

kðnÞ ¼ minclnðnÞW

nþ ln2ðnÞHðrT ; rRÞ;W

2

(23)

that for large n changes to:

kðnÞ ¼ minc lnðnÞW

n;W

2

(24)

4. Numerical results

In this section, we evaluate the validity of the results obtained in the previous sectionsusing a special kind of Monte-Carlo simulation method, we call it Layered Implementa-tion. The aim of this method is to consider the effect of all parameters in evaluation ofthe system performance. This layered simulation is implemented in such a way that ineach layer a special random parameter is realized for a certain number of times.

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Layer 1: Each node chooses randomly a destination from n� 1 other nodes. This isrepeated for N1 times.Layer 2: For each realization of layer 1, we consider N2 transmission intervals. In thebeginning of each interval, the nodes are distributed randomly and uniformly on thedisc.Layer 3: In each realization of layer 2, nS nodes are chosen as transmitters. Thesenodes transmit to their corresponding receivers if their distance be less than rðnÞ.Selection of nS transmitters among n nodes is performed N3 times.

For each final realization, we calculate the number of successful transmissionsaccording to the mmW protocol interference model and the average throughput pernode is computed. This averaging is performed backward from the lowest layers toupper ones. So, we can determine the total average throughput capacity. Figure 5 showsthe result of this procedure. The simulation parameters are as follows:b ¼ 5; a ¼ 4; rT ¼ rR ¼ p

6 ;N1 ¼ N2 ¼ N3 ¼ 1000. The blue line represents the asymp-totic lower bound obtained in (24), i.e. the values of clnðnÞ

n .2 As was expected, the lowerbound is not valid for small values of n; a fact we were ready to face, because thisbound describes the network performance only for large values of n:

Theorem 1 states that for sufficiently large values of n, the throughput will alwaysremain above the blue line. To investigate further, let us call ksðnÞ and kT ðnÞ respec-tively, the throughput obtained by simulation and the lower bound of theorem 1. In thiscase, we expect to have a n0 such that for all values of n� n0 the following relationholds:

50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Per Node Throughput Capacity of Mobile Directional Networks

Number of Nodes ( n )

Thr

ough

put

per

Nod

e (/

W)

bits

/se

c

Asymptotic Lower Bound

Simulation Result

Figure 5. Per node throughput capacity of the mmW MANET and the asymptotic lower bound.

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dksðnÞdn

� dkT ðnÞdn

) ksðniþ1ÞksðniÞ

� kT ðniþ1ÞkT ðniÞ

¼clnðniþ1Þniþ1

clnðniÞni

¼ lnðniþ1ÞlnðniÞ

:niniþ1

kðniþ1ÞkðniÞ denotes the growth rate of kðnÞ. In Table 1, we have compared the growth rateof ksðnÞ and kT ðnÞ. As it is clear from the table, the lower bound holds up for allvalues of n� 350, which confirms the validity of the obtained results.

5. Conclusion

The throughput capacity of millimeter wave wireless mobile Ad-Hoc networks wasinvestigated. A new directional antenna model was proposed based on the referencemodel of the IEEE 802.15.3c WPAN standard. As far as we know, this is the mostaccurate and realistic model that has ever been used in the capacity study of Ad-HocNetworks. Based on this, an appropriate interference model was introduced and a lowerbound on the throughput capacity of mmW MANETs was derived. Using a layeredimplementation of Monte-Carlo method, a simulation was conducted. The results showthe validity of the obtained bound at large values of n: In the future, we will find anupper bound and study the scalability conditions.

Notes1. That is with respect to the direction that the node select for its antenna.2. Here, the exact value of c is not the matter of our discussion. We set c ¼ 5 only for

illustration purposes.

References[1] Gupta P, Kumar P. The Capacity of Wireless Networks. IEEE Trans. Inf. Theory.

2000;46:388–404.[2] Ozgur A, Leveque O, Tse D. Hierarchical cooperation achieves optimal capacity scaling in

Ad Hoc networks. IEEE Trans. Inf. Theory. 2007;53:3549–3572.[3] Li P, Fang Y. Impacts of topology and traffic pattern on capacity of hybrid wireless

networks. IEEE Trans. Mobile Comput. 2009;8:1585–1595.[4] Li P, Zhang C, Fang Y. Capacity and delay of hybrid wireless broadband access networks.

IEEE J. Sel. Areas Commun., Special Issue on Broadband Access. 2009;27:117–125.[5] Li P, Pan M, Fang Y. Capacity bounds of three-dimensional wireless Ad Hoc networks.

IEEE/ACM Trans. Networking. 2012;20:1304–1315.[6] Grossglauser M, Tse D. Mobility increases the capacity of Ad Hoc wireless networks. IEEE/

ACM Trans. Networking. 2002;10:477–486.

Table 1. The analytical and simulated values of kðnÞ growth rates

niksðniþ1ÞksðniÞ

lnðniþ1ÞlnðniÞ

niniþ1

niksðniþ1ÞksðniÞ

lnðniþ1ÞlnðniÞ

niniþ1

10 0.828 0.341 350 0.961 0.89550 0.960 0.589 400 0.934 0.906100 0.905 0.725 450 0.953 0.916150 0.859 0.793 500 0.683 0.556200 0.843 0.834 1000 0.612 0.550250 0.942 0.861 2000 0.742 0.702300 0.857 0.880 3000 0.802 0.777

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[7] Li P, Fang Y, Li J. Throughput, delay, and mobility in wireless ad hoc networks. In Proc.San Diego, CA: IEEE INFOCOM; 2010 Mar:1–9.

[8] Wang X, Bei Y, Peng Q, Fu L. Speed improves delay-capacity trade-off in motioncast. IEEETrans. Parallel Distrib. Syst. 2011;22:729–742.

[9] Li P, Zhang C, Fang Y. Asymptotic connectivity in wireless Ad Hoc networks usingdirectional antenna. IEEE/ACM Trans. Networking. 2009;17:1106–1117.

[10] Yi S, Pei Y, Kalyanaraman S, Azimi-Sadjadi B. How is the capacity of Ad Hoc networksimproved with directional antennas? Wirel. Networks. 2007;13:635–648.

[11] Li P, Zhang C, Fang Y. The capacity of wireless Ad Hoc networks using directionalantennas. IEEE Trans. on Mobile Comput. 2011;10:1374–1387.

[12] Pan M, Yue H, Liand P, Fang Y. Throughput maximization of cooperative wireless meshnetworks using directional antennas. In: Proc. IEEE Int’l Conf. on Comm. in China (ICCC‘12); 2012 Sep.

[13] IEEE 802.15.3c. IEEE standard for information technology – telecommunications andinformation exchange between systems – local and metropolitan area networks – specificrequirements. Part 15.3: Wireless Medium Access Control (MAC) and Physical Layer(PHY) specifications for high rate Wireless Personal Area Networks (WPANs) amendment2: Millimeter-wave-based alternative physical layer extension. IEEE Std 802.15.3c-2009(Amendment to IEEE Std 802.15.3-2003), pp. c1-187; 2009.

[14] Toyoda I, Seki T. Antenna model and its application to system design in the millimeter-wave wireless personal area networks standard. NTT Tech. Rev. 2011;9:1–5.

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