THROUGHPUT OF LARGE WIRELESS NETWORKS ON SQUARE, HEXAGONAL AND TRIANGULAR GRIDS. Kezhu Hong, Yingbo Hua

Size: px

Start display at page:

Download "THROUGHPUT OF LARGE WIRELESS NETWORKS ON SQUARE, HEXAGONAL AND TRIANGULAR GRIDS. Kezhu Hong, Yingbo Hua"

Bryan Jones
5 years ago
Views:

1 THROUGHPUT OF LARGE WIRELESS NETWORKS ON SQUARE, HEAGONAL AND TRIANGULAR GRIDS Kezhu Hong, Yingbo Hua Dept. of Electical Engineeing Univeity of Califonia Riveide, CA 9252 ABSTRACT Capacity analyi of eno netwok o wiele ad hoc netwok ha attacted a geat attention in the pat yea. But the eeach in thi aea ha pimaily focued on caling law of abitay o andom netwok intead of the exact capacity of given topologie. While the inight into how the capacity of an abitay o andom netwok cale with the numbe of node in a given aea i impotant, the exact capacity of a netwok depend on the netwok topology and can be moe deiable in pactice. In thi pape, we compae the thoughput of a lage netwok with thee poible topologie: quae (ectangula), hexagonal and tiangula. With a given topology, the netwok thoughput alo depend on the choice of outing potocol. We follow a ynchonou aay method (SAM) that i known o fa to yield the highet thoughout of a netwok on a ectangula gid.. INTRODUCTION Until the ecent wok [], capacity analyi of eno netwok o wiele ad hoc netwok ha pimaily focued on a andom o abitay netwok. The wok by Gupta and Kume [2] pioneeed a eie of eeach activitie on andom netwok a hown in [], [4], [5], [], [7]. Thee wok emphaize the caling law of the netwok capacity with epect to the numbe of node in a given aea. The exact capacity of a netwok howeve i alo influenced by the exact topology of the netwok. To bette undetand the capacity of wiele ad hoc netwok, ufficient attention mut alo be paid to netwok of known topologie. The ecent wok hown in [] follow thi pinciple. Netwok of known topologie ae diectly elevant in ome application. Wiele meh netwok that ae eceiving an inceaing inteet in induty ae an impotant example. Seno netwok ae anothe. A lage wiele meh netwok can alo be fomed by low flying aiplane in ai o by pecially equipped mobile node on gound. Each node in the meh netwok can eve a a oute (vitual bae tation) fo a neighbohood of mobile client on the gound. Thi tieed achitectue of ad hoc netwok doe not have the outing ovehead that eveely limit the capacity of a pue ad hoc mobile netwok. In thi pape, we will tudy the thoughput of a lage netwok that ae located on a ectangula gid, hexagonal gid o a tian- Thi wok wa uppoted in pat by the U. S. Amy Reeach Office unde MURI Gant No. W9NF , the U. S. Amy Reeach Laboatoy unde CTA Pogam Coopeative Ageement DAAD , and the National Science Foundation unde Gant No. TF gula gid. Thee gid ae illutated in Figue, 2, and. We will evaluate and compae the thoughput of a netwok on each of the thee gid with a fixed node denity. The thoughput of a netwok alo depend on the choice of medium acce potocol. A tudy hown in [] (and ou ecent eeach) ugget that a potocol called ynchonou aay method (SAM) i o fa known to yield the highet thoughput of a netwok on a ectangula gid. Fo thi eaon, we will apply the SAM potocol to all othe gid conideed in thi pape. The SAM potocol i imple. Duing a time inteval, all node in a ubet of the netwok tanmit towad thei neighboing node in a given diection. Duing anothe time inteval, all node in anothe ubet of the netwok do the ame. Thi poce epeat until all node in the netwok have tanmitted to thei neighboing node in one diection. The above poce alo epeat fo each of all poible diection available in the netwok. By following the SAM potocol, the ignal to intefeence and noie atio (SINR) at each eceiving node can be explicitly expeed. The dependence of SINR on the paene of each ubet of the netwok can be ued to optimize the netwok thoughout. 2. NETWORK ON SQUARE GRID A dicued in [], a netwok on quae gid i illutated in Figue whee a ubet of the netwok i hown by black and gay node. The black node ae the tanmitting node and the gay node ae the eceiving node. The paene of the ubet i detemined by pd and qd whee p and q ae intege and d i the ditance between two adjacent node. The capacity of each eceiving node i limited by the intefeence fom all tanmitting node, except the deied one, in a coeponding ubet. The eceive at the cente of the netwok ae the mot intefeed. The netwok thoughout pe node i lowe bounded by the thoughput of the node at the cente of the netwok. When lage enough, a netwok may appea to be infinite fo the node at the netwok cente. Fo convenience, we will aume that the netwok i infinite. Alo dicued in [], the SINR at a eceiving node can be expeed a: SINR = /SNR 0 δ () whee SNR 0 = P t(σ 2 d n ), P t i the tanmitted powe fom each tanmitting node, σ 2 i the noie vaiance, n i the powe decaying exponent, and δ i efeed to a the intefeence facto fo the quae gid. When P t i lage, SINR become atuated at it uppe bound δ. Auming that the noie and intefeence ae

2 whee ɛ 0 i the powe attenuation facto along a non-lineof-ight of a tanmitte o a eceive, p/2 denote the laget intege no geate than p/2. The powe attenuation facto of a tanmiion pai become ɛ 2 when neithe the tanmitte no the eceive i in the line-of-ight with epect to each othe. Fo omnidiectional antenna, ɛ =. Howeve, when p =,wehave [(i )q ] 2 (pj) 2 n 2 δ = ɛ 2 [(i )q ] 2 (pj) 2 n 2 [(i )q ] 2 n 2 2ɛ 2 j= (pj) 2 n 2 (4) Fig.. A netwok on quae gid. Data packet ae tanmitted fom black node to thei neighboing gay node within each time lot. The pacing between two adjacent node i d mete. The vetical pacing of a tanmiion pai i pd mete and the hoizontal pacing i qd mete. The thoughput i denoted by γ in bit//hz/node o α in bit m//hz/node. all Gauian, the netwok thoughput in bit//hz pe node along any of the fou poible diection i Fo each given pai of n and ɛ, γ can be optimized ove (p, q). In Table, ample of the optimal γ and the coeponding optimal (p, q) ae given. Table. Optimized γ in bit//hz pe node fo one of fou diection of the netwok on quae gid γ,opt, (p, q) opt ɛ = ɛ =0. ɛ =0.0 n = 0.2, (2, ).794, (, 2) 2.8, (, 2) n = , (2, ) 2.780, (, 2).0442, (, 2) n =5 0.20, (2, ) , (, 2).889, (, 2) γ = G log 2 ( δ ) (2) whee G = pq i the numbe of time lot needed fo each of all node to tanmit once to it neighboing node in one of the fou poible diection on the quae gid. Thi i actually an uppe bound of the netwok thoughput, and achievable when P t i lage. Hee, each node i aumed to have a ingle antenna. The election of each ubet of the netwok influence SINR. We have choen each ubet in uch a way that any two adjacent column of tanmiion pai ae maximally offet fom each othe a hown in Figue. Ou analyi how that fo p>, δ = ɛ 2 g=0 [(2i )q ( ) g ] 2 (pj p/2 ) 2 n 2 [2(i )q ] 2 (pj) 2 n 2 [2(i )q ] 2 (pj) 2 n 2 [2(i )q ] 2 n 2 2ɛ (pj) 2 2 n 2 j= (). NETWORK ON HEAGONAL GRID A netwok on the hexagonal gid i hown in Figue 2 whee a ubet of the netwok i maked by the black (tanmitting) node and the gay (eceiving) node. The two adjacent column of tanmiion pai in each ubet ae maximally offet fom each othe. The vetical pacing of adjacent tanmiion pai i denoted by pd, and the hoizontal pacing i qd. Hee, p take all natual intege. But q can be eithe q =mo q =m.5whee m i any natual intege. The SAM potocol i applied to a ubet duing each time lot. In ode fo each of all node in the netwok to tanmit once to it neighboing node in one of the thee poible diection (of a given node), we need G h =2p (2q/) time lot if q =mo G h = 2p [2(q.5)/] time lot if q =m.5. Then, the netwok thoughput in bit//hz pe node in one of thee poible diection unde the hexagonal gid i given by γ h = G h log 2 δ h.it can be hown that if q =mand p>, then δ h = ɛ 2 g=0 [(2i )q ( ) g ] 2 ( pj p/2 ) 2 n 2 [2(i )q ] 2 ( pj) 2 n 2 [2(i )q ] 2 ( pj) 2 n 2 [2(i )q ] 2 n 2 2ɛ ( 2 pj) 2 n 2 j=

3 and if q =m and p =, then δ h = δ h (n) =ɛ 2 [(i )q ] 2 ( pj) 2 n 2 [(i )q ] 2 ( pj) 2 n 2 [i ]q ] 2 n 2 2ɛ ( 2 pj) 2 n 2 j= Futhemoe, if q =m.5, then (5) [(2i )q ] 2 [ pj (/2 p/2 ) ] 2 n 2 Table 2. Optimized γ h in bit//hz pe node fo one of thee diection of the netwok on the Hexagonal gid γ h,opt, (p, q) opt ɛ = ɛ =0. ɛ =0.0 n = , (,) 2.297, (,.5) 2.797, (,.5) n = , (,) 2.58, (,.5).845, (,.5) n = , (,) , (,.5) 4.82, (,.5) fo all node in the netwok to tanmit once in one of ix poible diection i G t =2pq if q = m, og t = p[2(q 0.5) ] if q = m 0.5. The maximal aveage thoughput in bit//hz pe node in one diection i theefoe γ t = G t log 2 δ t. whee the intefeence facto δ t can be deived in a imila way a befoe. When q =, 2,... and p =2,, 4... ɛ 2 [(2i )q ] 2 [ pj (/2 p/2 ) ] 2 n 2 [2(i )q ] 2 ( pj) 2 n 2 [2(i )q ] 2 ( pj) 2 n 2 [2(i )q ] 2 n 2 2ɛ 2 j= ( pj) 2 n 2 () Fig.. A netwok on tiangula gid. The pacing along the tanmiion diection i qd t and the pacing along the diection pependicula to the tanmiion diection i pd t. The thoughput i denoted by γ t in bit//hz/node o α t in bit m//hz/node. δ t(n) = [(2i )q ( ) g ] 2 ( pj p/2 ) 2 n 2 Fig. 2. A netwok on hexagonal gid. The pacing along the tanmiion diection i qd h and the pacing along the diection pependicula to the tanmiion diection i pd h. The thoughput i denoted by γ h in bit//hz/node o α h in bit m//hz/node. Sample of the optimal γ h and the coeponding optimal (p, q) ae hown in Table NETWORK ON TRIANGULAR GRID The tiangula gid i hown in Figue whee a ubet of the netwok i maked by black and gey node. The vetical pacing of tanmiion pai i pd t, and the hoizonal pacing i qd t. Hee, p take any natual intege, but q can be eithe m o m 0.5 whee m i a natual intege. The numbe of time lot equied ɛ 2 A p =, g=0 [2(i )q ] 2 ( pj) 2 n 2 [2(i )q ] 2 ( pj) 2 n 2 [2(i )q ] 2 n 2 2ɛ ( 2 pj) 2 n 2 j= δ t(n) =ɛ 2 [(i )q ] 2 ( pj) 2 n 2 [(i )q ] 2 ( pj) 2 n 2 [i ]q ] 2 n 2 2ɛ ( 2 pj) 2 n 2 j= (8) (7)

4 When q =0.5,.5, , and p =, 2,..., δ t(n) = ɛ 2 [(2i )q ] 2 [ pj (/2 p/2 ) ] 2 n 2 [(2i )q ] 2 [ pj (/2 p/2 ) ] 2 n 2 [2(i )q ] 2 ( pj) 2 n 2 [2(i )q ] 2 ( pj) 2 n 2 [2(i )q ] 2 n 2 2ɛ ( 2 pj) 2 n 2 j= The optimal γ t and (p, q) ae illutated in Table. Table. Optimized γ t in bit//hz pe node fo a given diection of the netwok on tiangula gid γ t,opt, (p, q) opt ɛ = ɛ =0. ɛ =0.0 n = 0.8, (, ).498, (,.5).85, (,.5) n =4 0.20, (, ).7209, (,.5) 2.57, (,.5) n =5 0.50, (, ).8, (,.5).2088, (,.5) An altenative choice of a ubet of the netwok on the tiangula gid i hown in Figue 4. Thi netwok of node alo fall on a paallelogam gid. The two adjacent paallel et of tanmiion pai (fom uppe-ight to lowe left) can be offet with epect to each othe in a fahion like that in Figue. The intefeence facto δ p can be imilaly found. The netwok thoughput in bit//hz pe node in any given diection i γ p = G p log 2 δ p whee G p = pq. The optimized γ p and (p, q) ae hown in Table 4. (9) Table 4. Optimized γ p in bit//hz pe node fo a given diection of the netwok on paallelogam gid γ p,opt, (p, q) opt ɛ = ɛ =0. ɛ =0.0 n = 0.79, (,).944, (,2) 2.48, (,2) n =4 0.8, (2,) 2.597, (,2).074, (,2) n = , (2,) 2.8, (,2).8459, (,2) 5. THROUGHPUT COMPARISON In the peviou ection, we have evaluated the netwok thoughput γ in bit//hz pe node. But in ode to compae the diffeent topologie, we need to ue a metic α in bit mete//hz pe node. Futhemoe, we need to aume that the node denity i the ame fo all topologie. The mallet quae aea uounded by fou node in the quae topology i denoted by A. The mallet hexagonal aea uounded by ix node in the hexagonal topology i denoted by A h. The mallet tiangula aea defined by thee node in the tiangula topology i denoted by A t. Then, a imple analyi how that A =, 2 A h =, 0.5 A t = (0) and A = d 2, A h = 2 d2 h, A t = 4 d2 t () Theefoe, d =, d 4 h =, dt = 2 (2) On the quae gid, the numbe of hop equied fo a packet to move ove a long ditance D (with D d ) in any diection θ i given by D co (/4 θ) N = 2 () 2d whee θ [0, ]. So, the aveage numbe of hop i given by 4 N = 4 Z 4 0 N dθ = 4 D d (4) Similaly, we can how that fo the hexagonal gid, N h = D co φ d h 4, φ [0, ] Fig. 4. A netwok on paallelogam gid - a vaiation of the tiangula gid. The pacing along the tanmiion diection i qd t and the pacing along the diection 20 o fom the tanmiion diection i pd t. The thoughput i denoted by γ p in bit//hz/node o α p in bit m//hz/node. Z N h = N h dφ = 4 0 D (5) d h and fo the tiangula gid, N t = D co ( ϕ) dt 2, ϕ [0, N t = Z 0 ] N t dφ = D d t ()

5 Then, the aveage netwok thoughput in bit mete//hz pe node fo the quae, hexagonal and tiangula gid ae given by α = γ D = N 4 γ D α h = γ h N = h 4 4 γ h α t = γ t D 2 = γ t N t (7) Howeve, unde the paallelogam gid, we can allow each node to tanmit in ix poible diection a hown in Figue 4 o in only fou poible diection a hown in Figue 5. Given ix diection, Np = N t. But with fou diection, we need to ecalculate the aveage numbe N p of hop equied fo a long ditance D a follow: N p = ( D co ( / ϑ) dt 2, ϑ [, 0] D co (/ ϑ) d t 2, ϑ [0, Z ] (8) and N p = 2 N p dϑ = 8 D (9) d t A compaion of α p and α p i α p = γp D N p = γp dt / >α p = γp D = N p We will ignoe α p but only conide α p: α p = 2 γ p γp dt 8/ (20) Table 5 illutate α, α h, α t and α p that ae all nomalized by Fig. 5. A netwok on paallelogam gid with fou poible diection of tanmiion/eception at each node. The pacing along the tanmiion diection i qd t and the pacing along the diection 20 o fom the tanmiion diection i pd t. The thoughput i denoted by γ p = γ p in bit//hz/node o α p in bit m//hz/node. q the common facto. We can ee that α h i the laget when omnidiectional antenna ae ued, and α p i the laget when diectional antenna ae ued. Table 5. Compaion of the (optimized) netwok thoughput in bit m//hz/node unde unit denity of node α,α h α t,α p ɛ = ɛ =0. ɛ =0.0 n = n =4 n = CONCLUSION We have evaluated the thoughput in bit m//hz/node of a lage wiele netwok on thee diffeent topological gid - quae, hexagonal and tiangula. The outing potocol ued i called ynchonou aay method (SAM). Ou eult ugget that the hexagonal gid i the mot efficient when omnidiectional antenna ae ued, and that the (-diection) paallelogam vaiation of the tiangula gid i the mot efficient when diectional antenna ae in ue. Howeve, the cell patition fo each node on a tiangula gid i hexagon, and the cell patition fo each node on a hexagonal gid i tiangle. If the netwok i ued a a vitual netwok of bae tation, the tiangula gid may have an advantage. Anothe obevation fom thi tudy i that if the node denity of a lage netwok i kept contant, then the netwok thoughput doe not eem to change much a the topology change. A new challenge now i to find the optimal outing potocol fo a given topology. So fa, we have not been able to wok out a potocol to beat the SAM cheme unde non-fading channel condition. Futhe eeach fo fading channel i undeway. 7. REFERENCES [] Y. Hua, Y. Huang, and J. Gacia-Luna-Aceve, On maximum thoughput of lage ad hoc communication netwok, ubmitted to IEEE Signal Poceing magazine Special Iue on Signal Poceing fo Wiele Ad hoc Communication Netwok, 200. [2] P. Gupta and P. Kuma, The capacity of wiele netwok, IEEE Tanaction on Infomation Theoy, vol. IT-4, no. 2, pp , Mach [] A. Agawal and P. R. Kuma, Capacity bound fo ad hoc and hybid wiele netwok, ACM SIGCOMM Compute Communication Review, pp. 7 82, [4] U. C. Kozat and L. Taiula, Thoughput capacity of andom ad hoc netwok with infatuctue uppot, Mobicom 0, pp. 55 5, 200. [5] B. Liu, Z. Liu, and D. Towley, On the capacity of hybid wiele netwok, Infocom 0, pp , 200. [] S. Toumpi, Capacity bound fo thee clae of wiele netwok: aymmetic, clute, and hybid, Mobihoc 04, pp. 44, [7] S. Yi, Y. Pei, and S. Kalyanaaman, On the capacity impovement of ad hoc wiele netwok uing diectional antenna, Mobihoc 0, pp. 08, 200.

Chapter 19 Webassign Help Problems

Chapter 19 Webassign Help Problems Chapte 9 Webaign Help Poblem 4 5 6 7 8 9 0 Poblem 4: The pictue fo thi poblem i a bit mileading. They eally jut give you the pictue fo Pat b. So let fix that. Hee i the pictue fo Pat (a): Pat (a) imply