the capacity of multi-channel multi-interface wireless

WIRELESS COMMUNICATIONS AND MOBILE COMPUTINGWirel. Commun. Mob. Comput. 2014; 14:803–817

Published online 17 May 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/wcm.2237

RESEARCH ARTICLE

The capacity of multi-channel multi-interface wirelessnetworks with multi-packet reception anddirectional antenna†

Jian Liu1, Fangmin Li1*, Xinhua Liu1 and Hao Wang2

1 School of Information Engineering, Wuhan University of Technology, Wuhan 430070, China2 Department of Information Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

ABSTRACT

The capacity of wireless networks can be improved by the use of multi-channel multi-interface (MCMI), multi-packetreception (MPR), and directional antenna (DA). MCMI can provide the concurrent transmission in different channels foreach node with multiple interfaces; MPR offers an increased number of concurrent transmissions on the same channel;DA can be more effective than omni-DA by reducing interference and increasing spatial reuse. This paper explores thecapacity of wireless networks that integrate MCMI, MPR, and DA technologies. Unlike some previous research, whichonly employed one or two of the aforementioned technologies to improve the capacity of networks, this research capturesthe capacity bound of the networks with all the aforementioned technologies in arbitrary and random wireless networks.The research shows that such three-technology networks can achieve at most 2�

�

pk capacity gain in arbitrary networks

and�2��

�2k capacity gain in random networks compared with MCMI wireless networks without DA and MPR. The paper

also explored and analyzed the impact on the network capacity gain with different cm , � , and k-MPR ability. Copyright

© 2012 John Wiley & Sons, Ltd.

KEYWORDS

network capacity; multi-channel multi-interface; multi-packet reception; directional antenna

*Correspondence

Fangmin Li, 122, Luoshi South Road, Hongshan district, School of Information Engineering, Wuhan University of Technology,Wuhan, Hubei Province, 430070, China.E-mail: [email protected]

1. INTRODUCTION

It is commonly accepted that the presence of multi-channelmulti-interface (MCMI)-enabled devices is highly possi-ble in the near future. A network based on multi-channelas compared with one single common channel, enablesthe full use of the network resources. The rapid growthof the IEEE 802.11 technology [1], which is widely usedin wireless communications, has led to the sharp pricedecline of the MCMI devices. This price decline hasresulted in wider acceptance and use of these devices.When using either the traditional medium access con-trol (MAC) or the new MCMI technology, simultaneous

†Please ensure that you use the most up to date class file, available from

the WCM Home Page at http://www3.interscience.wiley.com/journal/

76507157/home

transmissions may cause collisions, and the collision pack-ets are retransmitted later. Ghez et.al proposed multi-packet reception (MPR) technology [2,3], which makes itpossible for the node to receive multiple packets simul-taneously to solve this problem. This technology couldincrease the number of concurrent transmissions in the net-work and thus effectively improves network capacity andreduces the delay of received packets. In addition, recentliterature [4–7] found that using directional antennas (DAs)instead of omni-DAs in wireless networks can greatlyimprove the network capacity, because DAs can reducethe collision area. In [8], a countermeasure is proposed byusing DAs to prevent wormhole attacks. As demonstratedin a previous research, there are a series of benefits inusing DAs.

The combined using of MCMI, MPR, and DA is moreefficient than using just one or two of these three tech-niques. It is important to note that although benefits of

Copyright © 2012 John Wiley & Sons, Ltd. 803

The capacity of MCMI wireless networks with MPR and DA J. Liu et al.

such combinations do exist in the majority of cases,caution must always be exercised when using any com-bination under any condition. For example, in the regionwhere the density of mobile users is relatively low, thedistribution of users is sparse and the demand of networkresources is not very high. Thus, the number of basestations is generally small and the density is low. Underthis circumstance, omni-DA would be a better choice. Yet,the bound of network capacity which uses such combinedtechnology is required necessarily by further research forMCMI networks with MPR and DA, and it motivates us tocapture the bound of capacity in such networks.

This paper studied the capacity of a network thatcombined the three technologies: MCMI, MPR, and DA.In this network, each node is equipped with multiple inter-faces and each interface is associated with one DA that canoperate on different channels. Considering MPR nodes’decode ability, the number of decode transmissions islimited by one node at a time. The assumption is that awireless interface could decode at most k, k > 1, concur-rent transmissions within its receiving range and each nodecould use m interfaces, each of which can work on oneof the total c channels. In addition, the beamwidth of allDAs is identical � . We call such networks .m; c; �; k/wire-less networks. Similarly, MCMI networks with omni-DAsand no-MPR are called .m; c; 2�; 1/ wireless networks. Tothe best of our knowledge, there is no theoretical analysison the capacity of such .m; c; �; k/ wireless networks.This paper focused on finding the capacity bound for a.m; c; �; k/ network in both arbitrary and random networksand on exploring the benefits of such networks.

The remainder of the paper is organized as follows.We summarize the related work in Section 2. The net-work model, antenna model, and main results are describedin Section 3, and the proof for the results of arbitrarynetworks is presented in Section 4. Section 5 presentsthe proof for the results of random networks. Finally, wediscuss the results in Section 6 and conclude the paper inSection 7.

2. RELATED WORK

The capacity of the network is essential to networkresearch regardless of any layer on which those researchersfocus, because it provides the theoretical bound reflectingthe capacity of transmission. Literature [9] is known as themilestone of research on network capacity. It establisheda classic network model of wireless networks and gave aclear definition of network capacity. It demonstrated thecapacity scales as ‚.W

pn/ bit-m/s in an arbitrary net-

work and the network capacity scales as ‚�Wq

nlogn

�bits/s in a random network, and it also considered multi-channel conditions; the research proved that the capacityof a network with a single channel and one interface pernode is equal to the capacity of a network with c chan-nels and mD c interfaces per node. However, the researchdid not capture the impact on capacity by using fewer than

c interfaces per node. Kyasanur and Vaidya [10] exploredthe capacity of MCMI networks, especially when c � m.The results showed that the capacity is dependent on theratio of c

m , not on the exact value of either c or m.Furthermore, the current research trend is to improve

the network capacity by using MPR. Garcia-Luna-Aceveset.al [11] have shown that a 3D random MPR-based wire-less network has a capacity gain of ‚.logn/. M.F Guoet.al [12,13] addressed the capacity of single channelsingle interface (SCSI) and MCMI networks by usingk-MPR, respectively. For MCMI networks, the net-work capacity degrades as the ratio of c

m increases inboth arbitrary and random networks. However, there isno capacity degradation as the ratio of c

m increases

when cm D O.log.n// and k D O

�qm log.n/

c

�in

random networks.Besides, the use of DAs is a key technique to reduce

the interference area, Yi et.al [7] have shown that the

capacity gain is 2��

in arbitrary networks and 4�2

�2in ran-

dom networks for SCSI networks. Literature [14] showedthat the capacity gain using the hybrid antenna mode

is

r�2

�2Cs.4�2��2/over the antenna with the simplified

model for MCMI networks. H. Dai et.al [15] studied thethroughput capacity of multi-channel wireless networkswith a DA. The research showed that the use of DAsachieves higher capacity than omni-DAs when c

m D O.n/

in arbitrary networks and cm D O

�n�

log lognlogn

�2�in

random networks.In view of the survey of related work, this paper is the

first work to study the capacity of 2D MPR-based MCMIwireless networks by using DAs. The goal of this workis to study the impact of the ratio of c

m , beamwidth ofDA, and k-MPR ability in a network which integrates threetechnologies.

3. MODELS AND MAIN RESULTS

3.1. Network model

This paper considers the setting on a planar disk of unitarea. There are c non-overlapping channels and each nodeis equipped with m wireless DAs, 1 � m � c, and thebeamwidth of DA is identical � . The assumption is thata wireless interface could decode at most k, k > 1, con-current transmissions within its receiving range. We alsoassume that c channels can support a data rate of Wbits/s and any one of c channels can support Wc bits/s. Inaddition, there are n nodes located in the unit area and thenetwork transports �nT bits over T seconds.

Similar as [9], two network models are established forarbitrary network and random network in order to capturethe capacity of .m; c; �; k/ wireless networks. X , Y

is used to represent that node X points its antenna beamtowards node Y and vice versa.

804 Wirel. Commun. Mob. Comput. 2014; 14:803–817 © 2012 John Wiley & Sons, Ltd.DOI: 10.1002/wcm

J. Liu et al. The capacity of MCMI wireless networks with MPR and DA

3.1.1. Arbitrary network.

In the arbitrary .m; c; �; k/ network setting, each nodehas an arbitrarily chosen destination, to which it could sendtraffic at an arbitrary rate. Each node can choose arbitrarypower level for each transmission. Under these assump-tions, we give the protocol interference model for arbitrary.m; c; �; k/ wireless networks as follows.

Suppose k nodes, fXi j1 � i � kg, transmit packetsto node Xj simultaneously. Packets can be successfullyreceived by node Xj if

jXq �Xj j �max1�i�k.1C�/jXi �Xj j and Xi ,Xj

for every other node Xq , where Xq , Xj , simultane-ously transmitting to other node over the same channel.The quantity � > 0 models a guard zone that preventsinterfering nodes from transmitting on the same channel atthe same time.

3.1.2. Random network.

In a random .m; c; �; k/ wireless network, nodes arerandomly located, independently and uniformly dis-tributed. Each node has one flow to a randomly chosendestination, to which it wishes to send at �.n/ bits/s. Eachnode employs the same receiving range r.n/. Under theseassumptions, we give the protocol interference model forrandom .m; c; �; k/ wireless networks as follow.

Suppose k nodes, fXi j1 � i � kg, transmit packetsto node Xj simultaneously. Packets can be successfullyreceived by node Xj if

(1) max1�i�k jXi �Xj j � r.n/ and Xi ,Xj

(2) jXq �Xj j � .1C�/r.n/

for every other nodeXq , whereXq,Xj , simultaneouslytransmitting over the same channel.

3.2. Antenna model

Antenna systems may be broadly classified into omni-directional and DA systems. In omni-directional mode, anode is capable of transmitting signals to all directions,and signals can also interfere with other unrelated nodes. Inthe directional mode, a node can point its beam towards aspecified direction. The direction in which the main lobeshould point for a given transmission is specified to theantenna by the MAC protocol.

In the DA model of this paper, we adopt the unidirec-tional pattern [16] and extend it to a circle sector to modelthe DA pattern with radius r and angle � . In unidirec-tional pattern, the main lobe is the direction of maximumradiation or reception. In addition to the main lobe, thereare also sidelobes and backlobes. It is well-known thatgood antenna designs attempt to minimize them, becausethese lobes represent lost energy. This paper assumes thatthe DA gain is within the main lobe, and the gain outsidethe main lobe is assumed to be zero. As shown in Figure 1,in our antenna pattern, radius r represents the transmissionrange or receiving range, and angle � is the beamwidth ofantennas. In Figure 1(a), two nodes’ antenna point towardseach other, Xi , Xj . Two nodes are located in eachother’s interference area, so they can build a successfullywireless link. As shown in Figure 1(b), the beam of nodeXj is not pointed towards Xi in spite that Xi ’s beam

(a) (b)

Figure 1. Directional antenna pattern and interference condition.

Wirel. Commun. Mob. Comput. 2014; 14:803–817 © 2012 John Wiley & Sons, Ltd. 805DOI: 10.1002/wcm


points towards Xj ; therefore, Xj cannot receive packetssuccessfully from Xi and vice versa.

3.3. Main results

The results of this paper are presented as follows. G1; G2

are used to denote capacity gain of 2��

, .2��/2 over

.m; c; 2�; k/ wireless networks, respectively, and G0 isused to represent no capacity gain. This paper usesKnuth’s notation [17] to represent the following asymptoticbounds:

(1) f .n/ D O.g.n// denotes that there exists someconstant d and integer N , such that f .n/ � dg.n/for n > N .

(2) f .n/D˝.g.n// denotes g.n/DO.f .n//.(3) f .n/ D ‚.g.n// denotes f .n/ D O.g.n// and

g.n/DO.f .n//.(4) MINO .f .n/; g.n// is equal to f .n/ , if f .n/ D

O.g.n//; else, is equal to g.n/.

For the results of this paper, � is used to denote �2� ,

because 12� can be regarded as a constant.

3.3.1. Arbitrary network.

The network capacity of arbitrary networks is measuredin terms of ‘bit-m/s’ (used in [9]). The upper bound andlower bound for the capacity of arbitrary networks matchexactly. The network capacity is presented as follows:

(1) When kcm D O.n�2/, network capacity is

‚

�W�

qkmnc

�bit-m/s, whose gain is G1.

(2) When kcm D �.n�2/, network capacity is

‚�Wmnc

�bit-m/s.

The capacity of arbitrary networks is illustrated inFigure 2. The capacity of arbitrary networks with differentvalues of k is also illustrated in Figure 3 to make theseresults more clearly.

Figure 3. The capacity of arbitrary .m; c; �; k/ wireless net-works with different values of k (figure is not to scale).

3.3.2. Random network.

The network capacity of random networks is measuredin terms of ‘bits/s’. The upper bound and lower bound forthe network capacity are presented as follows:

(1) Upper bound

�1 . When k DO�

lognlog logn

�

i. When cm D O

��2 lognk

�, �nL D O

�Wk�2

qn

logn

�bits/s, whose gain is G2.

ii. When cm D �

��2 lognk

�and c

m D

O

��2nk

�log logn

logn

�2�, �nL D O

�W�

qkmnc


iii. When cm D �

��2nk

�log logn

logn

�2�, �nL D

O�Wmnk log logn

c logn

�bits/s.

�2 . When k D��

lognlog logn

�and k DO.n/

i. When cmDO

��2 log logn

�, �nLDO

�W�2

qnk

log logn


Figure 2. The capacity of arbitrary .m; c; �; k/ wireless networks (figure is not to scale).



ii. When cm D �

��2 log logn

�, cm D O

�n�2

k

�and

k D O�

nlog logn

�, �nL D O

�W�

qkmnc

�bits/s,

whose gain is G1.

iii. When cm D�

�n�2

k

�, �nLDO

�Wmnc

�.

�3 . When k D�.n/

i. When cmDO.�

2p

log logn/, �nLDO�

Wn�2p

log logn


ii. When cm D �.�2

plog logn/, �nL D O

�Wmnc

�bits/s.

The capacity is illustrated in three figures with differentranges of k (Figures 4, 6, and 8) to clearly compre-hend the upper bound for capacity of random networks.The upper bound for capacity of random networks withdifferent values of k is also illustrated in three figures(Figures 5, 7 and 9) to depict the results with differentranges of k, respectively.

(2) Lower bound

�1 . When kcm DO.�

2 logn/

i. When cm D O

��2q.logn/3n

�and k D O

�qn

logn

�,

�nLD��Wk�2

qn

logn

�bits/s.

ii. When cm DO

��2q.logn/3n

�and kD�

�qn

logn

�,

�nLD��

Wn�2 logn

�bits/s.

iii. When cmD�

��2q.logn/3n

�and kDO

��2m logn

c

�,

�nLD��Wk�2

qn

logn

�bits/s.

�2 . When kcm D �.�2logn/ and also

O

��2n

�log logn

logn

�2�, �nLD�

�W�

qmnc

�bits/s.

Figure 5. The upper bound for the capacity of random.m; c; �; k/ wireless networks with different values of k (k D

O�

log nlog log n

�, and figure is not to scale).

�3 . When kcm D �

��2n

�log logn

logn

�2�, �nL D

��Wmn log logn

c logn

�bits/s.

4. CAPACITY OF ARBITRARYNETWORK

4.1. Upper bound

In an arbitrary .m; c; �; k/ wireless network, we supposethat the whole network transmits �nT over T seconds. Theaverage distance between the source and destination of abit is L. Hence, a capacity of �nL is achieved.

Lemma 1. Under the protocol model, the capacity ofarbitrary .m; c; �; k/ wireless networks is bounded asfollows:

(1) When k D O.n/, arbitrary .m; c; �; k/ network

capacity is O

�W�

qkmnc

�bit-m/s.

Figure 4. The upper bound for the capacity of random .m; c; �; k/ wireless networks when k DO�

log nlog log n

�(figure is not to scale).



Figure 6. The upper bound for the capacity of random .m; c; �; k/ wireless networks when k D ��

log nlog log n

�and also k D O.n/

(figure is not to scale).


��

log nlog log n

�, k DO.n/ and figure is not to scale).

(2) When k D �.n/, arbitrary .m; c; �; k/ network

capacity is O�Wn�

qmc

�bit-m/s.

Proof . When k D O.n/, considering bit b, 1 � b � �nT .Supposing that it moves from its source to its destination

in a sequence of h.b/ hops, where the h-th hop traverses adistance of rh

b. Then we have

�nTL�

�nTXbD1

h.b/XhD1

rhb : (1)

We should group the nodes into�j

nkC1

kC 1

�groups

to utilize the k-MPR ability. Note that we are trying toupper bound the network capacity; hence, such nodegrouping is only an ideal case and need not to be feasible.Each of these groups has k transmitters and one receiver.Because each node is equipped with m interfaces, the

firstj

nkC1

kgroups totally contain at most

jnkC1

kmk

transmission pairs and each of these groups containsat mostmk transmission pairs. The last one group contains

m�n�

jnkC1

k.kC 1/� 1

�transmission pairs. Conse-

quently, there are totally at most m�n�

jnkC1

k� 1

�transmissions in any time slot. Hence, we have

Figure 8. The upper bound for the capacity of random .m; c; �; k/ wireless networks when k D�.n/ (figure is not to scale).




�.n/ and figure is not to scale).

H WD

�nTXbD1

h.b/�W Tm

c

�n�

�n

kC 1

� 1

��W Tmn

c:

(2)

It is shown in [9] that each hop consumes a disk ofradius �2 times the length of the hop around each receiver.What is more, when a node points its beam towards areceiver and the receiver is only affected by the nodeswithin its antenna beam. On the average, �

2� proportionof the nodes inside the reception beam could interferewith the receiver. Thus, the conditional interference zone

area is ��2 rhb

�2 ��2�

�2D �2�2

16�

�rhb

�2. Hence, the

number of simultaneous transmissions T .n/ multiplied by�2�2

16�

�rhb

�2must be less than the area of network (equal

to 1).

T .n/�16�

�2�2�rhb

�2 (3)

Because the number of concurrent transmissions withinthe disk of radius �2 r

hb

centered at a receiver is at most k,in arbitrary .m; c; �; k/ wireless networks, we have

�nTXbD1

h.b/XhD1

1 .The h� th hop of bit b over sub-channel m in slot s/�2�2

16�k

�rhb

�2�W �

c: (4)

By summing over all the channels (which can in totalpotentially transport W bits), we have the constraint,

�nTXbD1

h.b/XhD1

�2�2

16�k

�rhb

�2�W T : (5)

The equation (5) can be rewritten as

�nTXbD1

h.b/XhD1

1

H

�rhb

�2�16�kW T

�2�2H: (6)

Because the quadratic function is convex, we have

0@�nTXbD1

h.b/XhD1

1

Hrhb

1A2

�

�nTXbD1

h.b/XhD1

1

H

�rhb

�2: (7)

Combining (6) and (7) yields

�nTXbD1

h.b/XhD1

1

Hrhb �

r16�kW T

�2�2H: (8)

Then, substituting (2) in (7) gives

�nTXbD1

h.b/XhD1

rhb �W T

��

r16�kmn

c: (9)

Finally, we substitute (1) in (9) and obtain

�nL�W

��

r16�kmn

c: (10)

When k D �.n/, because there are not enough trans-mitters to utilize the k-MPR ability, the network capacitycannot be further improved compared with the casewhen k D ‚.n/. Therefore, the network capacity is

O�Wn�

qmc

�bit-m/s when k D�.n/. �

Lemma 2. The capacity of arbitrary .m; c; �; k/ wireless

networks is O�Wmnc

�bit-m/s.

Proof . Consider the interface constraint of an arbitrary.m; c; �; k/ wireless networks. Because every node has minterfaces, there are interfaces in the whole network. Eachinterface can support at most W

c bits/s, and the maxi-mum distance that a bit can travel is ‚.1/ in the network.

Thus, the interface bound of the network is O�Wmnc

�bit-m/s. �

According to Lemma 1, 2, we can obtain the network

capacity that isO

�MINO

�W�

qkmnc ; Wn

�

qmc ;

Wmnc

��bit-m/s. The minimum bound of them is an upper bound



on the network capacity. Then, we have the following the-orem on the capacity of an arbitrary .m; c; �; k/ wirelessnetwork.

Theorem 1. The upper bound on the capacity of an arbi-trary .m; c; �; k/ wireless network is shown as follows:


O

�W�

qkmnc

�bit-m/s.

(2) When kcm D �.n�2/ network capacity is

O�Wmnc

�bit-m/s.

4.2. Constructive lower bound

Lemma 3. Supposing m; c; c are positive integers suchthat c D c

m . Then, the capacity of .m; c; �; k/ wirelessnetworks is at least the capacity of .1; c; �; k/ wirelessnetworks.

Proof . Consider a .m; c; �; k/wireless network. We groupthe c channels into c groups (numbered from 1 to c),with m channels per group. Specifically, channel group i ,1� i � c, contains all channels j such that .i�1/mC1�j � im. Besides, nodes’ antenna beams are aimed toproper directions to guarantee correct transmissions in allgroups. The same transmission groups can be constructedas in .1; c; �; k/ wireless networks to simulate this behav-ior in .m; c; �; k/ wireless networks. For the nodes of agroup, we assign the m interfaces of each node to them channels in the channel group i . The aggregated data

rate of each channel group is Wmc D W

c. Therefore, a

channel group in a .m; c; �; k/ network can support thesame data rate as a channel in the .1; c; �; k/ network. Thismeans that the .m; c; �; k/ network can mimic the behav-ior .1; c; �; k/ network. Hence, the capacity of .m; c; �; k/wireless networks is at least the capacity of .1; c; �; k/wireless networks. �

Lemma 4. Supposing that m and c are positive integersand the capacity of a .m; c; �; k/ wireless network is atleast half of the capacity supported by a

�1; b cmc; �; k

�network.

Proof . When b cm c Dcm , the result directly follows from

Lemma 3. Otherwise,m< c and we use ci Dmbcmc of the

channels in the .m; c; �; k/ network and ignore the rest ofthe channels. This can be viewed as a .m; ci ; �; k/ networkwith a total data rate of Wi D

cic W bits/s. By Lemma 3,

a .m; ci ; �; k/ network with total data rate of Wi cansupport at least the capacity of a

�1;cim ; �; k

�network that

is equal to the capacity of a�1; b cm c; �; k

�network. How-

ever, when Wi < W , the .m; ci ; �; k/ network with totaldata rateWi can achieve only a fraction Wi

Wof the capacity

of a�1; b cm c; �; k

�network with total data rate W .

We have

Wi

WDm

cbc

mc Db cm ccm

�b cmc

b cm c C 1; since

c

m� b

c

mc C 1

�1

2; since b

c

mc � 1:

Hence, a .m; ci ; �; k/ network can support at least halfof the capacity supported by a

�1; b cmc; �; k

�network.

Recall that ci � c, the capacity of .m; c; �; k/ wirelessnetworks is at least the capacity of .m; ci ; �; k/ wirelessnetworks. Therefore, we can obtain that a .m; c; �; k/

network can support at least half of the capacity supportedby a

�1; b cmc; �; k

�network. This implies that asymptoti-

cally, a .m; c; �; k/ network has the same order of capacityas a

�1; b cmc; �; k

�network. �

Theorem 2. The achievable capacity of an arbitrary.m; c; �; k/ wireless networks is bounded as follows:


�

�W�

qkmnc

�bit-m/s.

(2) When kcm D �.n�2/ network capacity is

��Wmnc

�bit-m/s.

Proof . The unit-area plane is divided into a number ofequal-sized rhombuses, whose length of sides is l . In eachrhombus, there are eight positions for placing nodes asshown in Figure 10. Nodes at from T1 to T4 act as transmit-ters, and nodes from R1 to R4 act as receivers. T3 and R1are at the distance of �r from the center of the rhombus.In each rhombus, there are four position pairs to repre-sent the data flow, from T1 � R1 to T4 � R4, and eachpair can form a subrhombus whose side is l0. Subrhombusrepresents DAs interference zone, and the beamwidth of alltransmitters and receivers is � .

Consider k-MPR ability, each receiver has k transmit-ters simultaneously. Hence, we place kg nodes at each ofthe transmitter positions and g nodes at receiver positions.

Let g D min�2�c0�; n�8.kC1/�

�, where c0 D b

cmc. Par-

tition the square area into n�8�.kC1/g

equal-sized rhombus

cells, and place 8�.kC1/g�

nodes in each cell. Because the

total area is 1, each cell has an area of 8�.kC1/gn�

, and the

length of side l Dq8�.kC1/gn� sin � .

In each subrhombus, the distance from Ti to Ri is r ,and each node is equipped with DAs with beamwidth � .

Hence, we have the side of subrhombus l0 Drp2.1�cos �/2 sin � .



Figure 10. The placement of nodes within a cell. There are g nodes at each of the transmitter positions, and kg nodes atreceiver positions.

Because of triangles and patchwork, we have l D

2.1 C 2�/l0 D.1C2�/r

p2.1�cos �/

sin � . Recall that l Dq2�.kC1/gn� sin � , the transmission range can be obtained as

r D1

1C 2�

s4�.kC 1/g sin �

n�.1� cos �/: (11)

Consider one node placed at any position from R1 toR4, namely n1

R, n2R

, n3R

, and n4R

, respectively. Besides, knodes are placed at any position from T1 to T4, and wegroup those nodes into four groups by position, namelygroup1

T, group2

T, group3

T, and group4

T, respectively.

There are k nodes in each group for transmitting. Underthe protocol model of interference, it is easy to find that thedistance from n1

Tto group2

T, group3

T, and group4

T

all are greater than .1 C �/r ; therefore, n1R

can receivethe packets from k nodes in the group1

Tsuccessfully.

Similarly, niR

can receive the packets from k nodes in

the groupiT

, 1 � i � n�32�.kC1/g

, in the wholeunit-area plane.

From the aforementioned construction, there are at mostnkkC1

concurrent transmissions within the domain. Restrict-ing attention to only these transmissions, there are totallynkkC1

simultaneous transmissions, each of which trav-els r meters and the interface of each node transmitsat W

c0bits/s. By equation (11), this achieves the capac-

ity of 11C2�

q4�.kC1/g sin �n�.1�cos �/

W nkc0.kC1/

bit-m/s. Recall that

g D min�2�c0�; n�8.kC1/�

�. Hence, the capacity of the

.1; c0; �; k/ network is

2�

1C 2�

s2c0.kC 1/ sin �

n.1� cos �/

W nk

c0.kC 1/�, if g D

2�c0

�I

or1

1C 2�

ssin �

2.1� cos �/

W nk

c0.kC 1/, if g D

n�

8.kC 1/�:

Hence, the capacity of an arbitrary .1; c0; �; k/ net-

work is bounded by��MINO

�W�

qknc0; Wnc0

��bit-m/s.

According to Lemma 4, we have the capacity of an arbi-

trary .m; c; �; k/ to be �

�MINO

�W�

qkmnc ; Wmnc

��bit-m/s, which leads to the aforementioned theorem. �

5. CAPACITY OF RANDOMNETWORKS

5.1. Upper bound

Different from arbitrary networks, the capacity of randomnetworks is affected by three major factors [10]: networkconnectivity, interference, and destination bottleneck.

A. Connectivity constraintLemma 5. The number of simultaneous transmissions onany particular subchannel is no more than 16�

c1�2�2r2.n/in

the protocol model.

Proof . Supposing that node Xi in Figure 11 transmitssuccessfully to node Xj on the same subchannel. Then,no other node Xk within a distance �r.n/ of Xi cansimultaneously receive a separate transmission on the samesubchannel because of the protocol interference model andthe triangle inequality.

Therefore, disks of radius �r.n/2 centered at each

receiver on the same subchannel are disjoint. Because the

area of each disk is ��2r2.n/4 and each receiver can receive

from k transmitters simultaneously, it follows that the net-work can support no more than 4k

c1��2r2.n/simultaneous



Figure 11. Xk cannot receive at the same time as Xj on thesame subchannel.

transmissions on the same subchannel. Noting that antennatransmission and reception are both directional, the area of

conditional interference zone is��2�

�2�r2.n/. Thus, the

number of simultaneous transmissions on any particular

subchannel should not exceed 4kc1��2r2.n/

�2��

�2. �

According to Lemma 5, each transmission over eachsubchannel is W

c bits/s. By adding all the transmissionstaking place at the time over all channels, we see thattotally transmission rate cannot be more than 16�k

�2�2r2.n/W

bits/s in the protocol interference model.Recall that L denotes the mean length of a line connect-

ing two independently and uniformly distributed point onthe unit-area disk. Then, the mean number of hops taken

by a packet is at least Lr.n/

. Because each source generates�.n/ bits/s, there are n sources, and each bit needs to be

relayed by on the average Lr.n/

nodes, it follows that thetotal number of bits/s served by the entire network needs to

be at least Ln�.n/r.n/

.

The condition Ln�.n/r.n/

� 16�k�2�2r2.n/

W will be needed to

ensure that all the required traffic is carried.Thus, �.n/� 16�W k

Ln�2�2r.n/.

In order to know the range of �.n/, we should knowthe range of r.n/ to guarantee the connectivity. Accord-ing to the result in [18], we can take the transmission range

r.n/ �

qlognC2k log logn

�n to satisfy this requirement withhigh probability. Then, we can get the following upperbounds for this constraint.

(1) When k D O�

lognlog logn

�, n�.n/ is O

�Wk�2

qn

logn

�bits/s.

(2) When k D ��

lognlog logn

�and k D O.n/, n�.n/ is

O.W�2

qnk

log logn / bits/s.

(3) When k D�.n/, n�.n/ is O�

Wn�2p

log logn

�bits/s.

B. General constraintA random .m; c; �; k/ wireless network is a special case

of arbitrary .m; c; �; k/ wireless networks, so the upperbound for the capacity of arbitrary .m; c; �; k/ wirelessnetworks is applicable to random networks. In a randomnetwork, the distance between any source-destination pairis ‚.1/ meter. Hence, according to the upper bound forarbitrary networks, the capacity of random networks isbounded as follows:

(1) When kcm D O.n�2/, n�.n/ is O

�W�

qkmnc

�bits/s.

(2) When kcm D�.n�

2/, n�.n/ is O�Wmnc

�bits/s.

C. Destination bottleneck constraintConsider a node X in a random .m; c; �; k/ wireless

network. X has at mostD.n/ flows whose destinations areX , and the maximum number of D.n/ is found in [10].Under the network model we described earlier, each chan-nel supports a data rate at Wc bits/s. Because of k-MPRability, the total data rate at which X can receive over minterfaces is Wmk

c . Moreover, because X should accom-modate at least D.n/ flows, the data rate of the minimumrate flow is at most Wmk

cD.n/bits/s. Therefore, by definition

of �.n/, �.n/ � WmkcD.n/

, the network capacity n�.n/ D

O�WmnkcD.n/

�. According to Lemma 7, the network capacity

is O�Wmnk log logn

c logn

�bits/s.

Lemma 6. The maximum number of flows for which a

node in the network is a destination,D.n/, is‚�

lognlog logn

�,

with high probability. [10]

By satisfying the three constraints previously men-tioned, the upper bound for the network capacity presentedin Section 3.3 is obtained.

5.2. Constructive lower bound

A routing scheme and a transmission scheduling schemeare constructed for any random .1; ca; �; k/ wirelessnetwork, where ca D b cmc to establish a tighter bound.According to Lemma 4, the achieved capacity is a lowerbound for the capacity of random .m; c; �; k/ wirelessnetworks.

5.2.1. Cell construction.

The torus of unit area is divided into square cells, andthe area of each cell is a.n/, similar to the approach used in[19]. The size of each square denoted by a.n/ must satisfythe three constraints mentioned in Section 5.1.

It is found in [10] that when the size of each square isgreater than a certain value, each square must contain a



certain number of nodes. Therefore, it can guarantee suc-cessful transmissions from source nodes to destinationnodes. Their lemma is stated here.

Lemma 7. If a.n/ is greater than 50 lognn , each cell has

‚.na.n// nodes, with high probability. [10]

We ensure that a.n/� 100 lognn for a large n to simplify

the calculation.Consider the k-MPR ability, each node can receive k

packets simultaneously on each of the ca channels. In otherwords, we should guarantee that each node has at leastkca neighbors. In addition, it is found in [7] that in a ran-dom network, the use of DAs at both the transmitter and

the receiver can reduce the interfering area by��2�

�2, so

the cell size should be large enough,��2�

�2a.n/ � kca

n .

Thus, for a .1; ca; �; k/ wireless network, a.n/ is equal to

max

�100 logn

n ; kcan

�2��

�2�.

We take . 1D.n/

/2 as another possible value for a.n/,

where D.n/ D ‚�

lognlog logn

�(By lemma 7) to ensure the

flow bottleneck constraint.

Then, we set a.n/Dmin

�max

�100 logn

n ; kcan

�2��

�2�,�

1D.n/

�2�.

It is found in [15] that the number of cells thatinterfere with any given cell is bounded by a constant

81.2 C �/2 �2

4�2. In this paper, we would consider the

impact of � , so the following lemma can be gotten.

Lemma 8. The number of cells that interfere with anygiven cell is bounded by . �2� /

2kinter (where kinter D81.2C�/2), which is independent of a.n/ and n.

5.2.2. Routing scheme.

We construct a simple routing scheme that chooses aroute with the shortest distance to forward packets. Astraight line denoted by S-D line is passing through thecells where source node S and destination node D arelocated. Packets are delivered along the cells lying on thesource-destination line. Then, we choose a node withineach cell lying on the straight line to carry that flow. Thenode assignment is based on load balancing. In particular,the flow assignment procedure is divided into two steps.

Step 1: Source and destination nodes are assigned. For anyflow that originates from a cell, source node S isassigned to the flow. Similarly, for any flow thatterminates in a cell, destination node D is assignedto the flow. After this step, only those flows pass-ing through a cell (not originating or terminating)are left.

Step 2: We assign the remaining flows. To balance theload, we assign each remaining flow to a node that

has the least number of flows assigned to it. Thus,each node has nearly the same number of flows.

We present the following lemma to bound the numberof source-destination lines that pass through any cell. Thislemma has already been proved in [19].

Lemma 9. The number of source-destination lines pass-ing through any cell (including lines originating and termi-nating in the cell) is O.na.n// with high probability.[19]

Because a.n/ is greater than 100 lognn , each cell has

‚na.n/ nodes (by Lemma 8). We can conclude that

each intermediate node serves O�

1pa.n/

�flows. Each

node is the originator of one flow, and each node isthe destination of at most D.n/ flows, where D.n/

is ‚�

lognlog logn

�. Therefore, the total flows assigned to

any node is at most O�1CD.n/C 1p

a.n/

�. Recall that

a.n/ D min

�max

�100 logn

n ; kcan

�2��

�2�;�

1D.n/

�2�,

the total flows assigned to any node is equal to

O�

1pa.n/

�flows.

5.2.3. Scheduling transmissions.

A two-layer schedule is built as the same as [10]; how-ever, we modified the second layer to satisfy the k-MPRability and DAs. In the schedule model of this paper,any transmission in this network must satisfy the fol-lowing two additional constraints simultaneously: (i) eachinterface only allows one transmission/reception at thesame time; (ii) any two transmission groups (we groupk transmissions received by a receiver into one transmis-sion group) on any channel should not interfere with eachother.

We describe the two-layer time slots as follows to satisfythe aforementioned two constraints, respectively.

First layer: In this layer, a second is divided into anumber of edge-color slots, and at most, one transmis-sion/reception is scheduled at every node during eachedge-color slot. Hence, the first constraint is satisfied. Weconstruct a routing graph in which vertices are the nodesin the network and an edge denotes transmission/receptionof a node. In [10], it shows that this routing graph can

be edge-colored with at most O�

1pa.n/

�colors. There-

fore, we divide one second into O�

1pa.n/

�edge-color

slots and each slot has a length of ��p

a.n/�

seconds.

Each slot is stained with a unique edge-color. Because alledges connecting to a vertex use different colors, each nodehas at most one transmission/reception scheduled in anyedge-color time slot.

Second layer: In this layer, each edge-color slot can befurther split into smaller mini-slots and satisfies both thetwo aforementioned constraints. We construct an interfer-ence graph in which vertices are the transmission groups



in the network and edges denote interference betweentwo groups. According to Lemma 9, every cell has atmost a constant number of interfering cells with a factor��2�

�2, and each cell has ‚.na.n// nodes (by Lemma 8).

Thus, each node has at most O

��2�

�2na.n/k

�trans-

mission groups’ edges in the interference graph. It iswell-known that a graph with maximum degree v canbe vertex-colored with at most v C 1 colors [20].Hence, the interference graph can be vertex-colored with

at most O

��2�

�2na.n/

�colors. Therefore, we use

k1

��2�

�2na.n/k

to denote the number of vertex-colors

(where k1 is a constant). Two transmission groups assignedthe same vertex-color that do not interfere with each other,whereas two groups stained with different colors mayinterfere with each other. So, we need to schedule theinterfering groups either on different channels or at dif-ferent mini-slots on the same channel. Thus, we divide

each edge-color slot into

��2�

�2k1na.n/kca

�mini-slots on

every channel. Recall that each edge-color slot has a lengthof �.

pa.n// seconds, so each mini-slot has a length of

�

0@ p

a.n/��2�

�2 k1na.n/kca

�1A seconds.

As shown in Figure 12, channels are numbered from

1 to c. First, one second is divided into O�

1pa.n/

�edge-color slots on every channel to satisfy constraint(1). Second, each edge-color slot is further divided intol. �2� /

2 k1na.n/kca

mmini-slots. In every mini-slot, we can

guarantee that every transmission satisfies the two con-straints simultaneously.

5.2.4. Utilization of k-MPR ability.

There are at most k concurrent transmissions within itsreceiving range for each receiver. However, the number of

concurrent transmissions for a receiver is less than k insome cases. If k transmissions are received by a receiversimultaneously, at least k nodes, being transmitters, willinterfere with the receiver. Besides, in the random network,the number of flows served by each node should be greaterthan k. In other words, each node has the ability of servingk flows to the least extent. We should satisfy the followingtwo fully utilization requirements (FUR) simultaneously tofully utilize k-MPR ability.

FUR(1) The number of concurrent receptions cannotexceed the number of nodes which are located at theinterfere cells of the receiver

�k DO

��2kinterna.n/

ca

��.

FUR(2) The number of concurrent receptions can-not exceed the number of flows each node serves�k DO

�1pa.n/

��.

5.2.5. The achieved capacity lower bound.

We first analyze the capacity of the .1; ca; �; k/

wireless networks and then extend the results to a.m; c; �; k/ wireless network. In the .1; ca; �; k/ wire-less network, recall that each mini-slot has a length of

�

0@ p

a.n/��2�

�2 k1na.n/kca

�1A seconds, and each channel can

transmit at the rate of Wca

bits/s. Therefore, �.n/ D

�

0@ W

pa.n/

ca

��2�

�2 k1na.n/kca

�1A bits can be transported in each

mini-slot on every channel.

Because

��2�

�2k1na.n/kca

��2�

�2k1na.n/kca

C 1, we

have �.n/ D �

0@ W

pa.n/

ca

��2�

�2 k1na.n/kca

Cca

1A bits/s. Hence,

�.n/D�

0@MINO

0@ Wk�

�2�

�2npa.n/

;Wpa.n/ca

1A1A bits/s.

Figure 12. Two-layer transmission schedule.



Table I. The capacity gain over .m; c; �; k/ networks for random networks.

Recall that each flow is scheduled in one mini-slot oneach hop during one-second interval and every source-destination flow can support a per-node throughput of�.n/ bits/s. Therefore, when k-MPR is fully utilized,the total network capacity is n�.n/ which is equal

to �

0@MINO

0@ Wk�

�2�

�2pa.n/

;Wnpa.n/

ca

1A1A bits/s. In a

.m; c; �; k/ wireless network, by Lemma 4, when k-MPRis fully utilized, the total network capacity is equal to

�

0@MINO

0@ Wk�

�2�

�2pa.n/

;Wmn

pa.n/

c

1A1A bits/s.

Lemma 10. Assume k1 D O.k2/. C1 is the capacity of.m; c; �; k1/ wireless networks, and C2 is the capacity of.m; c; �; k2/ wireless networks, then C1 DO.C2/.

Proof . The .m; c; �; k2/ wireless networks can imitate.m; c; �; k1/ wireless networks by restricting k2 to k1.In addition, the capacity of .m; c; �; k/ wireless networkswill not decrease with the increase of k. Therefore, thecapacity of .m; c; �; k2/ wireless networks is at least thecapacity of .m; c; �; k1/ wireless networks. Hence, we getthe lemma. �

According to FUR (1) and (2), we can knowwhether k-MPR can be fully utilized. Recall that

a.n/ D min

�max

�100 logn

n ; kcan

�2��

�2�;�

1D.n/

�2�,

where D.n/ D ‚�

lognlog logn

�. When fully utilizing k-MPR

ability, we can get the lower bound through substitutingfor the three values. Moreover, when k-MPR ability is notfully utilized, we can apply Lemma 10 to get the lowerbound. The lower bound for the random network capacitypresented in Section 3.3 is obtained.

6. DISCUSSION

The arbitrary network bounds may be viewed as the bestcase bounds on network capacity, because the location ofnodes and traffic patterns can be controlled in the arbitrarynetwork and it is more realistic on network conditions.

From the results of this work, when kcm D O

��2�

2�n�

,

a .m; c; �; k/ wireless network can achieve a capacity gain2��

pk over a .m; c; 2�; 1/ wireless network. However,

when kcm is too large, the MCMI network with MPR and

DAs has no capacity gain over a .m; c; 2�; 1/ wireless net-work. The value of k and m are based on the ability ofhardware and cost; the number of non-overlapping chan-

nels c would be smaller than��2�

�2mnk

to get a greatly

capacity gain.Recall the results for random networks, there are three

ranges of k; we abstractly use k-smal l , k-moderate,and k-large to denote the range of k. When ratio c

m issufficiently small in each range, a .m; c; �; k/ wireless net-

work can achieve a capacity gain at most��2�

�2k over

a .m; c; 2�; 1/ wireless network. Moreover, we use c=m-smal l , c=m-moderate, and c=m-large to denote therange of c

m . We can conclude that a .m; c; �; k/ wirelessnetwork has a capacity gain over a .m; c; 2�; 1/ wire-less network as shown in Table I. The region-A has the

greatest capacity gain�2��

�2k and the next is the region-

B�2��

�2pk, the region-C

�2��

�pk, and the region-D�

2��

�2. When c

m is c=m-large, there is no capacity gain

over a .m; c; 2�; 1/ wireless network (region-E).

7. CONCLUSION

This paper considered a wireless network that inte-grates MCMI, MPR, and DA. It studied the capacity of.m; c; �; k/ wireless networks under both arbitrary andrandom networks. The results show that using DAs andMPR in MCMI networks cannot only enhance networkconnectivity but also mitigates interferences. However,when kc

m is large enough in the arbitrary networks, thereis no capacity gain over using omni-DAs and no-MPR. Inthe random networks, when c

m is sufficiently large, thereis no gain. It also shows that combining MCMI with k-MPR and DAs can achieve at most 2�

�

pk capacity gain

in arbitrary networks compared with .m; c; 2�; 1/ wirelessnetworks and .2�

�/2k capacity gain in random networks.



ACKNOWLEDGEMENTS

This work is funded by the National Natural Science Foun-dation of China under Grant Nos.60970019 and 61170090.It is also supported by the Key Program of Hubei Provin-cial Natural Science Foundation (China) under GrantNo.2009CDA132. We are extremely grateful to the edi-tors and reviewers for their comments and suggestions whohelped us to improve the quality of this paper.

REFERENCES

1. IEEE Standard for Wireless LAN Medium AccessControl (MAC) and Physical Layer (PHY) Specifica-tions, Nov. 1997. P802.11.

2. Ghez S, Verdu S, Schwartz SC. Stability propertiesof slotted aloha with multipacket reception capabil-ity. IEEE Transactions on Automatic Control 1988; 33:640–649.

3. Ghez S, Verdu S, Schwartz SC. Optimal decentral-ized control in the random access multipacket channel.IEEE Transactions on Automatic Control 1989; 34:1153–1163.

4. Korakis T, Jakllari G, Tassiulas L. A mac protocol forfull exploitation of directional antennas in ad-hoc wire-less networks, In Proceedings of MobiHoc, Annapolis,MD, United States, 2003; 98–107.

5. Zhang Z. Pure directional transmission and receptionalgorithms in wireless ad hoc networks with direc-tional antennas, In Proceedings of IEEE InternationalConference on Communication, vol. 5, Seoul, Korea,2005; 3386–3390.

6. Ramanathan R, Redi J, Santivanez C, Wiggins D,Poli S. Ad hoc networking with directional antennas:a complete system solution. IEEE Journal on SelectedAreas in Communications 2005; 23(3): 496–506.

7. Yi S, Pei Y, Kalyanaraman S. On the capacityimprovement of ad hoc wireless networks using direc-tional antennas, In Proceedings of MobiHoc, Annapo-lis, MD, United States, 2003; 108–116.

8. Hu L, Evans D. Using directional antennas to preventwormhole attacks, In Proceedings of the Network andDistributed System Security Symposium, San Diego,California, United States, 2004; 1–11.

9. Gupta P, Kumar PR. The capacity of wireless net-works. IEEE Transactions on Information Theory2000; 46: 388–404.

10. Kyasanur P, Vaidya NH. Capacity of multi-channelwireless networks: impact of number of channelsand interfaces, In Proceedings of ACM MobiCom,Cologne, Germany, 2005; 43–57.

11. Garcia-Luna-Aceves J, Sadjadpour H, Wang Z.Challenges: towards truly scalable ad hoc networks, In

Proceeding of ACM MobiCom, Montreal, QC, Canada,2007; 207–214.

12. Guo M, Wang X, Wu M. On the capacity of k-mprwireless networks. IEEE Transactions on WirelessCommunications 2009; 8(7): 3876–3886.

13. Guo M, Wang X, Wu M. On the capacity of multi-packet reception enabled multi-channel multi-interfacewireless networks. Computer Networks 2010; 54:1426–1439.

14. Wang J, Kong L, Wu M. Capacity of wireless ad hocnetworks using practical directional antennas, In Pro-ceeding of WCNC, Sydney, NSW, Australia, 2010;1–6.

15. Dai H, Ng K, Wong R, Wu M. On the capacityof multi-channel wireless networks using directionalantennas, In Proceeding of IEEE InfoCom, Phoenix,AZ, United States, 2008; 1301–1309.

16. Carr J. Directional or omnidirectional antenna, 2011.http://www.dxing.com/tnotes/tnote01.pdf.

17. Knuth DE. The Art of Computer Programming.Addison-Wesley: MA, 1997.

18. Wan PJ, Yi CW. Asymptotic critical transmissionradius and critical neighbor number for k-connectivityin wireless ad hoc networks, In Proceeding of ACMMobiHoc, Tokyo, Japan, 2006; 1–8.

19. Gamal AE, Mamen J, Prabhakar B, Shah D.Throughput-delay trade-off in wireless networks, InProceeding of IEEE InfoCom, Hongkong, China,2004; 464–475.

20. West DB. Introduction to Graph Theory, (2nd edn).Upper Saddle River: New York, 2001.

AUTHORS’ BIOGRAPHIES

Jian Liu received his BS degree inCommunications Engineering fromthe Wuhan University of Technology,Wuhan, China, in 2011, and iscurrently pursuing his MS degreein Communication and InformationSystem at Wuhan University of Tech-nology. His current research interestsinclude information theory, wireless

ad hoc and sensor networks, and network routing.

Fangmin Li is now the vice deanof the School of Information Engi-neering, Wuhan University of Tech-nology, Wuhan, Hubei, China. Hereceived his BS degree from theHuazhong University of Science andTechnology (China) and his MSdegree from the National Universityof Defense Technology (China) in



1990 and 1997, respectively, both in Computer Science. In2001, he received his PhD degree at Zhejiang University(China) in the field of Computer Science and Engineer-ing. In 2002, he joined the School of Information Engi-neering at the Wuhan University of Technology, where heis currently a full professor. His main research interestsinclude ad hoc networks, new generation network architec-ture, and embedded system design. His past research hasbeen published in over 70 scientific journals and confer-ence proceedings and 16 China patents. His research hasbeen funded by the National Natural Science Foundationof China and other sources. He is a senior member of theChina Computer Federation and executive member of theSensor Network Technical Committee (China).

Xinhua Liu is currently an associateprofessor in the School of Informa-tion Engineering at Wuhan Univer-sity of Technology, Wuhan, China.He received his BS degree from theUniversity of South China (China)and his MS degree in Communica-tion and Information System from

the Wuhan University of Technology in 1997 and 2005,respectively. He obtained his PhD degree from theSchool of Information Engineering, Wuhan Universityof Technology in 2010. His research interests includecomputer networks, multi-hop wireless network, andembedded system.

Hao Wang received his BEngdegree in Electrical Engineeringand Automation from East ChinaUniversity of Science and Technol-ogy, Shanghai China, in 2007, andhis MSc degree in Control Theoryand Engineering from Shanghai JiaoTong University, Shanghai, China, in2010. Currently,he is pursuing his

PhD degree in the Department of Information Engineer-ing, The Chinese University of Hong Kong. His generalinterests include wireless networking and smart grid.


the capacity of multi-channel multi-interface wireless

Documents