Capacity of Wireless Ad Hoc Networks Using Practical Directional Antennas

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1 Capaity of Wireless Ad Ho Networks Using Pratial Diretional Antennas Jue Wang Fudan University Shanghai, China Linghe Kong Shanghai Jiao Tong University Shanghai, China Min-You Wu Shanghai Jiao Tong University Shanghai, China Abstrat Capaity is one fundaental proble in wireless Ad ho networks. Deploying diretional antennas to wireless networks an redue interferene aong onurrent transissions and inrease spatial reuse, while the tehnology of ulti-hannel an separate onurrent transissions. Therefore, obining these two tehnologies into one wireless network is apable of great iproveent on the network apaity. Reent studies proposed a ulti-hannel network arhiteture that equips eah wireless node with ultiple diretional antennas, whih is alled MC-MDA network. The apaity in MC-MDA network is derived under arbitrary and rando plaeents. However, they only used a siplified diretional antenna odel. For approahing the ore aurate apaity in real senario, in this paper, we onsider a hybrid antenna odel, whih takes the effet of side lobe and bak lobe into aount. We derive the apaity upper-bounds of MC-MDA networks in arbitrary and rando network with the hybrid odel to find the effet of bak lobe and side lobe. We show that the network apaity is losely related to s the ratio of the radiuses of side lobe to ain lobe. The apaity dereases when s inreases. Moreover, we opare the network apaity of MC-MDA using the siplified antenna odel with our results. When onsidering the fator of interferene onstraint, the apaity gain using the hybrid antenna odel is +(4π )s over the one with the siplified odel. Keywords Ad ho networks, apaity, diretional antennas, ultiple hannels, ultiple interfaes I. INTRODUCTION Capaity oputing is one of the ost iportant probles in wireless Ad ho networks, whih provide the theoretial guideline to tehnial developent. Sine Gupta and Kuar [1] proposed the ethod of evaluating the apaity perforane on wireless networks, any researh studies have been arried out on this topi. They oputed the upper and lower bounds of the network apaity based on different tehnologies and tried to find out the ethod to iprove the throughput. In reent years, a great nuber of works reported on the ulti-hannel networks and the appliation of diretional antennas. A node equipped with ultiple network interfaes an proeed ultiple siultaneous transissions and reeptions in ultiple hannels. The analytial results in [] disovered that the apaity of ulti-hannel networks have different bounds depending on the ratio of the nuber of interfaes to hannels. At the sae tie, the harateristis of diretional antennas in wireless networks were investigated quite intensively. Based on the diretions of transission, different pairs of nodes loated in eah other s viinity an ouniate siultaneously. This an redue interferene aong onurrent transission and inrease spatial reuse of the wireless hannel. Nowadays, ephases are plaed on the integration of the ulti-hannel and diretional antennas, whih is ore benefiial. H.-N Dai et al. [14] identified the apaity of ulti-hannel Ad ho using diretional antennas whih they defined as MC-MDA network. However, few studies onsidered the real antennas radiation pattern when alulating the network apaity. Most of the used a setor odel to forulate the transit area of diretional antennas. On the ontrary, the effet side lobe and bak lobe is inevitable under reent tehnology. Thus, their results ay be not so lose to the theoretial value. In order to obtain a ore aurate apaity to reality, in this paper, we derive the apaity in MC-MDA network with a hybrid diretional antennas odel. In our hybrid odel, the side lobe and bak lobe is forulated as a irle, while the ain lobe is still a setor. A paraeter s is used to desribed the ratio of the radiuses of side lobe to ain lobe. After alulating the reeive-based interferene area with pratial antenna odel for eah obination of antenna odes, we derive the apaity upper-bounds of MC-MDA networks in arbitrary and rando network with the reeiver-based interferene odel. To observe ore learly, we also opare the network apaity of MC-MDA using the siplified antenna odel with the one using the hybrid odel. We show that the network apaity is losely related to s,i.e., the theoretial result exists gap between setor odel and hybrid odel. The rest of the paper is organized as follows. In Setion II, we do the literature of related work. In Setion III, we present the ontributions and ain results. The hybrid odel and interferene odel are forulated in Setion IV. The throughput apaity of arbitrary networks and rando networks are derived in Setion V and Setion VI, respetively. In Setion VII, we analyze the perforane. We onlude and disuss our future work in Setion VI. II. RELATED WORK Most of the studies related to this paper foused on the iproveent of the network apaity. Soe researhes presented theoretial and experiental studies on the apaity of ulti-hannel wireless networks /10/$ IEEE

2 Gupta and Kuar [1] started the analysis of network apaity, whih guided Kyasanur and Vaidya [] to adapt the ultihannel tehnology to wireless networks where nodes ay not have a dediated interfae per hannel. [3][4][5][6] proved that ulti-hannel wireless networks an ahieve perforane iproveent over single-hannel networks, though the nodes are equipped with onidiretional antennas. Reent results showed that the apaity of wireless networks an be enhaned by using diretional antennas instead of onidiretional antennas. S. Yi et al. [7] applied diretional antennas to the single-hannel wireless Ad-ho networks, so that transission and reeption of nodes an both be diretional. Soe other works suh as [8][9][10][11][1][13] onentrated on MAC protools of Ad ho wireless networks with diretional antennas. The obination of the tehnology of ulti-hannel and diretional antennas brought us a new doain of investigation on apaity iproveent. H.-N Dai et al. [14] identified the apaity of ulti-hannel Ad ho using diretional antennas whih they defined as MC-MDA network. In this paper, we also attept to get the upper bound of MC-MDA network apaity, but under the pratial diretional antennas odel instead. All of the researhes entioned above used siplified antennas odel where the effet of side lobe and bak lobe is ignored exept [7]. Little attention has been paid on the real antenna odel before, whih does liit the upper bound of apaity in fat. S. Yi et al. [7] defined a hybrid antennas odel to ahieve a better odel of real diretional antennas for the first tie. They evaluated the apaity of single-hannel Ad ho wireless networks using diretional antennas with sender-based interferene odel. Nevertheless, our work ais to detet the upper bound of MC-MDA network apaity with the hybrid antennas odel, so that we an estiate the ipat of side lobe and bak lobe on the network perforane. III. CONTRIBUTIONS AND MAIN RESULTS A. Contributions In suary, the ontributions of this paper are as follows: s λ n W MIN O (f(n),g(n)) TABLE I NOTATIONS The ratio of the radius of the side lobe and the ain lobe in the hybrid antenna ode. Eah node sends λ bits per seond. The nuber of nodes. The nuber of available hannels. The nuber of interfaes at eah node where eah interfae is assoiated with a diretional antenna. The total data rate by using all hannels. Eah hannel an support the data rate w. The beawidth of a diretional antenna. is equal to f(n), iff(n) =O(g(n)). Fig. 1. The upper bound for the transport apaity of arbitrary networks (figure is not to sale), =60 o For our best knowledge, it is the first work that addresses the proble of the network apaity with the onsideration of radius of the side lobe and bak lobe of antennas. We alulate the reeive-based interferene area with pratial antenna odel for any obination of antenna odes. The upper bound on the apaity of ulti-hannel wireless networks under both arbitrary and rando networks with pratial antenna odel is derived. Fro our results, we verify that the apaity dereases when the radius of side lobe and bak lobe s inreases. When s is saller, the ipat on the network apaity is greater. For arbitrary networks, the effet of side lobe and bak lobe is unignorable. To get our results, we onsider a stati ulti-hannel wireless network ontaining n nodes. Eah node is equipped with the sae diretional antennas, whih have the sae bea width (generally less than π). We also take the side lobe and bak lobe of the antenna into aount. We list all the notations in Table I. B. Results We onlude the results in this paper as follows. 1) Arbitrary Networks: The transport apaity of the network is that the network transports one bit-eter per seond when one bit has been transported a distane of one eter within one seond [1]. The transport apaity of arbitrary networks is presented as follows. When is O(n), the transport apaity of the network is O(W n ( +(4π )s) ) bit-eters/se. When is Ω(n), the transport apaity of the network is O( Wn ) bit-eters/se. We illustrate the upper bound for the throughput apaity of arbitrary networks in Fig.1. ) Rando Networks: The aggregate throughput apaity of the whole network is easured in bits/se [1]. We provide the upper bound of the throughput apaity of rando networksasfollow. When is O(log n), the throughput apaity of the network is O(W n log n ) bits/se.

3 Fig. 4. Antenna odel Fig. 5. Interferene odel Fig.. The upper bound for the transport apaity of rando networks (figure is not to sale), =60 o, front view Fig. 3. The upper bound for the transport apaity of rando networks (figure is not to sale), =60 o, bak view When is Ω(log n) and also O(n( logn ) ), the throughput apaity of the network is O(W n ( +(4π )s) ) bits/se. When is Ω(n( logn ) ), the throughput apaity of the network is O(W n log n ) bits/se. We illustrate the upper bound for the throughput apaity of rando networks in Fig. and Fig.3. Copared to the results in [14], our results verify that the effet of side lobe and bak lobe of diretional antennas annot be ignored in apaity evaluating. IV. MODEL A. Antenna Model Previous works usually study the network apaity based on the siplified diretional antenna odel, where side lobes and bak lobes are ignored. However, we fous on the pratial diretional antenna so that we use a hybrid antenna odel[7] in this paper. Aording to [8], when the ain beawidth is ore than 40 o, the effet of side lobes and bak lobes are so onsiderable that we an take it into aount. Moreover, the upper bound of the apaity is onstrained by the in reality. Besides, the goal of this paper is to analyze the differene of the network apaity with and without side lobes and bak lobes. We siulate the real diretional antenna with a hybrid antenna odel. The beaforing patterns of the odel are a ix of onidiretional and diretional antenna odels. In the odel as shown in Fig.4, the ain lobe of beawidth is haraterized as a setor. Side lobes and bak lobes for a irle. We define paraeter s as the ratio of the radius of the irle and the setor, whih is generally less than 1. The probability that the antenna bea is pointed to a ertain diretion is π. Hene a node an reeive fro transitters within the area of both the irle and the setor. B. Interferene Model Sine the onurrent transission an ause interferene, the nodes should be separated to avoid ollision in the intersetion of the transission zones. Thus, we assue there is an interferene area that an ensure the suessful transission. We use the reeiver-based interferene odel whih depends on the protool odel proposed by Gupta and Kuar [1]. The transission fro node to X i over X j a hannel is suessful if for every other node X k siultaneously transitting over the sae hannel, the following ondition holds. Sine X k is within the bea of X j, we get d(k, j) (1 + Δ)d(i, j), Δ > 0 (1) where d(k, j) is the distane between X k and X j, and the guard zone Δ is a paraeter that ensures that onurrently transitting nodes are suffiiently far away fro the reeiver to prevent exessive interferene. C. Interferene Area Now we oe to the alulation of the area of the interferene zone with hybrid antenna odel for eah obination of antenna odes. For onidiretional transission and onidiretional reeption, the interferene zone area A OO is πr, where r is the radius of the onidiretional antenna radiation pattern. For onidiretional transission and diretional reeption shown in Fig.5, the interferene zone area A OD is the area of

4 Fig. 6. Interferene odel for diretional transission and onidiretional reeption Fig. 7. Interferene odel for diretional transission and diretional reeption the hybrid antenna radiation pattern, whih is alulated as: A OD = π(sr) + π πr π π(sr) = πr +πs s () π = r + s (π ) where r is the radius of the ain lobe of diretional antennas. For diretional transission and onidiretional reeption shown in Fig.6, if the reeiver is within the transission range of other senders, the transission ay be interfered. So we alulate the interferene area as: A DO = πr (P { T x R v sr} + P { T x R v >sr} P {T x R v }) (3) = πr (s +(1 s ) π ) where P {x} is probability of x, and T x R v is used to desribe that the ain lobe of T x pointing to R v. For diretional transission and diretional reeption shown in Fig.7, whih is used in our analysis of network apaity in this paper, we divide the reeption area into two parts to siplify the alulation: one is a sall irle with radius sr and the other is an annulus setor with radius r. Sine the beawidth of all the nodes is. The onditional interferene area is: A DD = π(sr) + π (πr π(sr) ) P {T x R v }) = πr (s + 4π s 4π ) = πr (s +(1 s ) 4π ) Copared with the sender-based interferene areas in [7], the reeiver-based results are just the inverse, i.e. the reeiverbased A DO is the sae as the sender-based A OD. V. TRANSPORT CAPACITY FOR ARBITRARY NETWORKS We derive different upper bounds of throughput apaity for the networks using diretional antennas fro two fators, i.e., the interferene and the liited transission on an interfae. The iniu bound of the is an upper bound on the network apaity. As for the hybrid antenna odel, the transport apaity for arbitrary networks is losely related to s. (4) A. Interferene Constraint Aording to the hannel odel in [1], we ake assuption that there are n nodes arbitrarily loated in a disk of unit area on the plane and eah node has interfaes. The network transports λnt bits over T seonds. If the average distane between the soure and destination of a bit is L, then a transport apaity of λnl bit-eters per seond is ahieved. We onsider any tie period of length T. In this tie interval, onsider a bit b, 1 b λnt. We assue that bit b traverses h(b) hops on the path fro its soure to its destination, where the h th hop traverses a distane of rb h. Sine the distane traversed by a bit fro its soure to its destination is at least equal to the length of the line joining the soure and the destination, we have rb h λnt L (5) Let us define H to be the total nuber of hops traversed by all bits in T, i.e. H = λnt b=1 h(b). Therefore, the nuber of bits transitted by all nodes in T (inluding bits relayed) is equal to H. There are hannels in the networks where eah node has interfaes, and eah interfae transits over a hannel with rate W. Moreover, transporting a bit aross one hop requires two interfaes, one eah at the transitting and the reeiving nodes. So, the total nuber of bits that an be transitted by all nodes over all interfaes is at ost WTn. Hene, we have H WTn It is shown in [] that eah hop onsues a disk of radius Δ ties the length of the hop around eah reeiver, i.e. rb h. When we use diretional antennas at both transitter and reeiver ends, fro the Eqn. (4), we an get the onditional interferene zone area Δ 4 π(rh b ) (s + 4π (1 s )). Therefore, Δ 4 π(rh b ) (s + 4π (1 s )) WT (7) (rb h) H 4WT Δ Hπ(s + 4π (1 s )) Sine the expression on the left hand side is onvex, we have, r h b (r ( H ) b h ) (8) H Fro Eqn. (7) and Eqn. (8), rb h 16πWTH Δ ( +(4π )s ) Substituting for H fro Eqn. (6), and using Eqn. (5) we have, (6) (9)

5 B. Interfae Constraint λnl W 8πn Δ ( +(4π )s ) (10) The axiu nuber of bits that an be transitted siultaneously over all interfaes is also one of the iportant eleents of apaity onstraints. Sine there are totally n interfaes in the network, and eah interfae an support at ost W bits/se. So if all the interfaes are working, the whole network an support Wn bits/se. Meanwhile, the axiu distane that a bit an travel in the network is O(1) eters. Thus, the bound of the network is O( Wn ) bit-eters/se. Cobining the bound of interferene onstraint with interfae onstraint, we take MIN O (W n ( +(4π )s ), Wn ) as the result, whih has been desribed in Setion III.B. VI. THROUGHOUT CAPACITY FOR RANDOM NETWORKS In rando networks, the nodes are randoly plaed and the traffi patterns are randoly hosen. Capaity for rando networks is ainly onstrained by network onnetivity, interferene, and destination bottlenek. We derive the three bounds respetively and obine the into the final result. A. Connetivity Constraint Suessful transission fro any soure to destination depends on great onnetivity of the network, whih liits the network apaity to soe extent. As a basi onstraint, the rando network should ensure to be onneted whih eans that a network is onneted whp, i.e. with probability no less than (1 1 n ). Sine the nodes are randoly plaed, eah node should keep the range of the transission ore than a ertain value so that the nuber of transission is liited. For this onstraint, Gupta and Kuar [1] found that the upper bound of a rando network using diretional antennas at both the transitter and the reeiver is O(W n log n ) bits/se. This bound is also appliable to ulti-hannel networks. B. Interferene Constraint Sine a rando ulti-hannel wireless network is a speial kind of arbitrary networks, the upper bound of apaity for arbitrary networks is appliable in rando networks. Therefore, ulti-hannel rando networks are also affeted by interferene of onurrent transission. The upper bound of the arbitrary networks under interferene onstraint is O(W n ( +(4π )s )) bit-eters/se derived in Setion V.A. The average distane between eah soure and destination in a rando network is Θ(1) eter. Thus, the throughput apaity of rando network is O(W n ( +(4π )s ) ) bits/se. C. Destination Bottlenek Constraint The axiu data flow that a node an reeiver as destination is liited, whih onstrains the network apaity as well. The apaity of a wireless network is also onstrained by the axiu nuber of bits that an be transitted siultaneously over all interfaes in the network. Consider a node X whih is the destination of the axiu nuber, D(n), of flows. Eah node has interfaes whih an transit W W bits/se, so a node an ahieve at ost bits/se. Aording to [], the axiu nuber of flows D(n) is Θ( log n ). Hene, the data rate of the flow with the iniu W rate is at ost D(n) bits/se, whih iplies that apaity for Wn the rando networks is at ost O( log n ) bits/se. Considering the bound of onnetivity onstraint, interferene onstraint and destination bottlenek onstraint, we take MIN O (W n log n,w n Wn ( +(4π )s ), log n ) as the result, whih has been desribed in Setion III.B. VII. PERFORMANCE Using diretional antenna in the ulti-hannel Ad ho wireless networks an largely inrease network onnetivity and redue radio interferene, thereby iproving the network perforane greatly. We opare the perforane of our results with that of [14] in the following two tables, through whih we an analyze ore learly. For arbitrary networks, listed in Table II, when is O(n), the transport apaity for arbitrary networks has a apaity gain +(4π )s of over an MC-MDA network using siplified antennas odel in [14]. The network apaity depends on s the ratio of the radiuses of side lobe to ain lobe. Sine is not ore than π, the iniu value of is π, whih is saller than one. Therefore, there always exist soe values of that satisfy the ondition of O(n), TABLE II COMPARISON OF TRANSPORT CAPACITY FOR ARBITRARY NETWORKS Siplified antennas odel Hybrid antennas odel O(n) Θ( W n ) O(W n ( +(4π )s ) ) Ω(n) Θ( Wn ) O( Wn ) TABLE III COMPARISON OF TRANSPORT CAPACITY FOR RANDOM NETWORKS O(log(n)) Ω(log n) and O(n( log n ) ) Siplified antennas odel Θ( W n log n ) Θ( W Hybrid antennas odel O(W n log n ) n ) O(W n ( +(4π )s ) ) Ω(n( log n ) Wn Wn ) Θ( ) O( ) log n log n

6 whih eans that the network apaity ust be onerned with s in soe ondition. The apaity dereases with s inreasing. When s tends to zero, the upper bound is just the sae as the orresponding onditionin [14]. Meantie, the network apaity is at its peak of W nπ, when reahes its axiu value π. When s tends to 1, the antenna ode is equal to the onidiretional antennas. As a result, the apaity reahes iniu. So, our results are ore general opared with [14]. When is Ω(n), the transport apaity of the network is independent with s. For rando networks, listed in Table III, the apaity is onerned with s only when is Ω(log n) and also log n O(n( ) ). In this ondition, we find the sae apaity gain over MC-MDA network that we get in the arbitrary network under interferene onstraint. It is indiated that in the range within the bounds under the interferene onstraint, the ipat of side lobes and bak lobes on apaity is unignorable. On the other hand, the upper bound is not related to s in the rest of the range, beause other onstraints are tighter than the interferene. Fro the results obtained so far, it see that when the radius of side lobes and bak lobes are saller, the effet on the network apaity is greater. When s is in the different range fro zero to one, the speed of derease of apaity is also different. Fro Fig.1, Fig. and Fig.3, we believe that the apaity dereased faster when s is saller, onsidering the apaity is assoiated with s. In reality, s is usually sall so that the theoretial upper bound of apaity proposed before is hardly ahieved. VIII. CONCLUSION In this paper, we have derived the upper-bounds of MC- MDA networks in arbitrary and rando networks using hybrid antenna odel that inludes the effet of side lobe and bak lobe. We have also alulated the reeive-based interferene area with pratial antenna odel for eah obination of antenna odes. We opared our results with [14], whih foused on the apaity of MC-MDA using the siplified antennas odel, and evaluated the effet of sidelobe and bak lobe on the network perforane. The results indiate that the apaity dereases when s the ratio of the radiuses of side lobe to ain lobe inreases under the interferene onstraint. The apaity gain using the hybrid antenna odel is +(4π )s over the one with the siplified odel in [14]. The apaity dereases faster when s is larger. For arbitrary networks, the network apaity ust be onerned with s in soe ondition. In reality, s is usually so large that its ipat on the network apaity is worth studying. It should be noted that this preliinary study has not deteted the effet of side lobe and bak lobe of real antennas on the networks apaity thoroughly. The results an not be used to deterine the exat data about the bound of apaity. However, we have got the general rule of it. One of our future works ay inlude onsidering other fators that eliinate side lobe and bak lobe with the apaity alulation under hybrid antennas odel. Another future work is to develop pratial protools for wireless Ad ho networks to approahing the upper bound apaity when transitting with diretional antennas. ACKNOWLEDGMENT This researh was partially supported by NSF of China under grant No , 973 Progra of China under grant No. 006CB303000, 863 Progra of China under grant No. 006AA01Z47, the Key Projet of China NSFC Grant The authors would like to thank Mingfei Guo and anonyous reviewers who give us highly valuable oents to iprove the paper. REFERENCES [1] P. Gupta and P. R. Kuar, The Capaity of Wireless Networks, IEEE Transation on Inforation Theory, vol. 46, no., Marh 000, pp [] P. Kyasanur and N. H. Vaidya, Capaity of ultihannel wireless networks: Ipat of nuber of hannels and interfaes, ACM MOBICOM, 005. [3] A. Nasipuri, J. Zhuang, and S. 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