Performance Analysis of Rayleigh Fading Ad Hoc Networks with Regular Topology

Transcription

1 Pefomance Analysis of Rayleigh Fading Ad Hoc Netwoks with Regula Topology Xiaowen Liu and Matin Haenggi Depatment of Electical Engineeing Univesity of Note Dame Note Dame, IN 6556, USA {xliu, Abstact Fo wieless ad hoc netwoks with stationay and deteministically placed nodes, finding the optimal placement of the nodes is an inteesting and challenging poblem, especially unde enegy and QoS constaints. We study and compae the pefomance of seveal netwoks with egula topologies utilizing a Rayleigh fading link model. Fo neaest neighbo and shotest path outing, analytical expessions of the path efficiency, delay, and enegy consumption fo a given end-to-end eception pobability ae deived. Fo the intefeence analysis, the maximum thoughput and optimum tansmit pobability ae detemined, and a simple MAC scheme is compaed with an optimum schedule, yielding lowe and uppe pefomance bounds. I. INTRODUCTION In cetain wieless ad hoc netwoks, in paticula in wieless senso netwok [], many nodes will be stationay fo most of the time afte deployment. Fo example, in many applications of envionmental monitoing, chemical/biological detection, secuity in a shopping mall o paking lot, the sensos ae fixed. Moeove, to guaantee high exposue of the events of inteest [], unifom coveage is beneficial, suggesting the use of egula node placement schemes. Finding the optimal placement of nodes fo a good tade-off between enegy consumption, thoughput, and delay is an impotant and challenging poblem. In this pape, we investigate netwoks with egula topologies (squae, tiangle, hexagon) in which each node has the same numbe of neaest neighbos and the distance between all pais of neaest neighbos is the same. We call them squae, tiangle, and hexagon netwoks. Ou analysis is based on a Rayleigh fading channel model, which includes both lage-scale path loss and stochastic smallscale vaiations in the channel. As a esult, all communicationelated popeties of the netwok become andom vaiables, in paticula the signal-to-noise-and-intefeence atio (SINR) that detemines the success of a tansmission in contast to the disk model o potocol model, whee a deteministic tansmission adius is assumed []. Note that even with static nodes as assumed in this pape, the channel quality vaies because of movement in the envionment, which is easily confimed expeimentally []. The suppot of the U.S. National Science Foundation (gants ECS and CAREER CNS -7869) is gatefully acknowledged. Fo the pefomance analysis of multi-hop wieless ad hoc netwoks, the key issues ae enegy consumption, end-to-end eliability, delay and thoughput. In the disk model, the end-to-end eliability p EE is not nomally consideed since links ae assumed eithe % o % eliable. The Rayleigh fading model pemit the chaacteization of packet eception pobabilities and, consequently, the evaluation of the end-toend eliability of a path. While Rayleigh fading was consideed in ealie wok [5], its impact on p EE and the associated enegy issues have not been studied. In this pape we ae compaing all these chaacteistics fo the thee netwoks. In Section II, the Rayleigh fading link model is intoduced, and it is shown that the noise analysis and intefeence analysis can be caied out sepaately. Section III pesents the noise analysis of zeo-intefeence netwoks. The enegy consumption, path efficiency and delay of a connection fo a given end-toend eception pobability is investigated. Section IV povides the intefeence analysis fo diffeent MAC schemes. The simulation esults of a simple MAC scheme and an optimum MAC scheme detemine the lowe and uppe bound of the achievable thoughput. Section V concludes the pape. II. THE RAYLEIGH FADING LINK MODEL We assume a naowband Rayleigh block fading channel. A tansmission fom node i to node j is successful if the signal-to-noise-and-intefeence atio (SINR) γ ij is above a cetain theshold Θ that is detemined by the communication hadwae and the modulation and coding scheme. The SINR γ Q N +I is given by γ =, whee Q is the eceived powe, which is exponentially distibuted with mean Q. Ove a tansmission of distance with an attenuation d α, we have Q = P d α, whee P denotes the tansmit powe, N the noise powe, and I is the intefeence powe affecting the tansmission, i.e., the sum of the eceived powe fom all the undesied tansmittes. The analysis is simplified by the following Theoem [6]: Theoem : In a Rayleigh fading netwok, whee nodes tansmit at powe level P i ( i =,...,k), the eception pobability P[Q Θ(I + N )] of a tansmission ove a link distance d with tansmit powe P and k othe nodes at distance d i can be factoized into the eception pobability of a zeo-noise netwok and the eception pobability of a

2 .5 A Nomalized enegy consumption.5.5 O Fig.. Topology of a squae netwok. zeo-intefeence netwok as follows: ( p = exp ΘN ) k P d α ( + Θ Pi d ) α }{{} i= P d }{{ i } p N p I B. () p N is the pobability that the SNR γ N := Q /N is above the theshold Θ, i.e., the eception pobability in a zeointefeence netwok as it depends only on the noise. The second facto p I is the eception pobability in a zeo-noise netwok. This allows an independent analysis of noise and intefeence issues. III. NOISE ANALYSIS Fist, we study the pefomance of zeo-intefeence netwoks, whee only one node is tansmitting at tansmit powe P in evey timeslot. Fo each connection, the souce and destination ae unifomly andomly chosen. It is assumed that the netwok is lage and dense, which implies that the distibutions of the Euclidean distance between the souce and destination ae identical fo all thee netwoks and that the diection is unifomly distibuted in [, π). We oute the packet via neaest neighbos along the shotest path towad its destination. In [7], this is called minimum-enegy outing; we call it neaest neighbo and shotest path outing. It is assumed that all the thee netwoks have the same node density λ =. A. Squae netwoks We fist analyze the squae lattice netwok with N = N x N y nodes and distance d between all pais of neaest nodes. The next-hop eceive of each packet is one of the fou neaest neighbos (top, bottom, left and ight). Fo squae netwoks with unit density, the distance between neaest nodes d =. The optimality of a path can be measued by the atio between the Euclidean distance and the tavelled distance d T. So we define the path efficiency as η = Euclidean distance tavelled distance = d T, < η. () In Fig., O is the souce, A is the destination, and is the angle between OA and the hoizontal axis. By using neaest.5.5 (a) (b) Enegy consumption Fig.. (a) and (b) nomalized enegy consumption as a function of fo squae netwoks. neighbo and shotest path outing, the Euclidean distance is OA and the tavelled distance d T is OB + BA. We have η() = = d T cos + sin = cos + sin. () If we move the destination along the line OA, the path efficiency will not change, so η is only a function of, and it is peiodic with peiod π/. Thus in the following analysis, we estict between and π/. We can see that when = π/, η min = / ; when = o π/, η max =. is unifomly distibuted based on the lage and dense netwok assumptions, so the expected value of η is π actan( ).795. Fig. (a) displays the path efficiency as a function of between and π/. We assume that evey packet has a given end-to-end eception pobability p EE, dictated by the application (o the tanspot) laye. Fom Section II, we know that fo a zeointefeence netwok, the link eception pobability ove a link of distance d is given by p N = e ΘN P d α. Solving fo P, we find the necessay tansmit enegy to achieve a link eliability p N to be E L = dα ΘN. If thee ae h hops with equal distance ln p N d, the link eception pobability p N is p /h EE. Then the tansmit enegy at each hop is E L = h d α ΘN lnp EE. () Using neaest neighbo and shotest path outing, the tavelled distance is d T = cos + sin, whee is the Euclidean distance. The numbe of hops d T /d is h = d (cos + sin). (5) The total enegy consumption of this oute is Etot() s = he L = d (cos + sin d ) α ΘN. (6) ln p EE Let E := ΘN ln p EE. Consideing the unifom distibution of, we can find the expected total enegy consumption in units of E as E s π ( tot = (cos + sin ) d = + ) E π/ π d α d α.666 d α. (7)

3 B O Fig.. (a) Tiangle netwok A x B O (b) Hexagon netwok Topologies of a tiangle netwok and a hexagon netwok. Fig. (b) displays the total enegy consumption in units of E. Aveaging the numbe of hops of (5) ove, the expected numbe of hops h is.7. Fo othe outing schemes that pemit longe hops, the tansmit enegy consumption is highe in most cases, in paticula fo high path loss exponents. We omit the details due to the limited space. B. Tiangle netwoks and hexagon netwoks Othe egula topologies of inteest ae the tiangle topology and its dual, the hexagon topology. Fo each tiangle, thee ae thee vetices and six neaest neighbos fo each vetex, while fo the hexagon, thee ae six vetices fo each hexagon and thee neaest neighbos fo each vetex. The distance between all pais of neaest nodes is d. We use the same assumption as fo the squae netwoks, i.e., the netwoks ae lage and dense such that the distibution of and fo all the topologies ae identical. In the tiangle netwok, each node is located in a hexagon with aea d, so d T = fo unit density. Similaly, fo hexagon netwoks, d H = fo unit density. Note the supescipt T, H denote the tiangle netwok and hexagon netwok, espectively. Fig. (a) shows a tiangle netwok. O is the souce and A is the destination. We want to find the numbe of hops h using neaest neighbo and shotest path outing. We can split the netwok into goups such that the membes of one goup ae equidistant (in hops) to the souce. These goups ae the nodes in hexagons centeed aound the souce as shown in Fig. (a). Thus, the fist goup will be 6 nodes that ae one hop away in the hexagon with peimete 6d, the second goup will be nodes that ae two hops away in the hexagon with peimete d, and so on. In Fig. (a), because the angle between OA and the hoizontal axis Ox is between and π/, we daw the vetical AB to the line with π/ to Ox. The hop numbe h is AB divided by d sin(π/). The tavelled distance is hd. We will estict within and π/ because h is a peiodic function of with peiod π/. Thus we have h T sin(π/ ) = = ( d sin(π/) d T cos + ) sin, η T () = h T d T = sin(π/ ) =, cos + sin fo π/. (8) The expected value of η T is ln π.985. A x Tiangle Hexagon.5.5 (a) Nomalized enegy consumption Tiangle Hexagon.5.5 (b) Enegy consumption Fig.. and nomalized enegy consumption as a function of fo tiangle and hexagon netwoks. TABLE I COMPARISON OF SQUARE, TRIANGLE, HEXAGON NETWORKS FOR α = Enegy (α=) AND α = (NOISE ANALYSIS). Enegy (α=) Hop numbes h/ η Squae Tiangle Hexagon Fo the hexagon topology, we use a simila method to find the goup of nodes that ae equidistant (in hops) fom the souce as shown in Fig. (b). We have h H η H () sin(π/ ) /d H = ( cos + sin d H ), / sin(π/ ) = cos + sin, (9) whee π/. The expected value of η H is 9 ln π Fig. (a) shows the elationship between the path efficiency and fo tiangle and hexagon netwoks fo within and π/. Simila to the squae netwok, we detemine the total tansmit enegy consumption fo a given end-to-end eception pobability povided the Euclidean distance fom the souce to the destination is fixed fo all the topologies. The total enegy consumption is E tot () = he L = h dα ΘN ln p EE = h d α E. Inseting the expessions of h fo the tiangle and hexagon topologies yields the enegy consumption in both topologies: Etot() T = ( cos + sin) E d α, Etot() H = ( cos + sin ) E d α, () 9 whee Etot T and Etot H denotes the enegy consumption of tiangle and hexagon netwoks fo π/. The expected total enegy consumption in units of E is ( π + )dt α fo the tiangle topology and ( π )dh α fo the hexagon topology. The expected total enegy consumption fo tiangle and hexagon topology fo α = and α = is listed in Table I. It is shown fo α =, tiangle topology consumes less enegy, while fo α =, hexagon topology gives the least enegy consumption. The diffeence comes fom the facto of d α. Fig. (b) displays the nomalized enegy consumption fo

4 Received packets pe node pe timeslot α= α=5 N = x nodes Tansmit Pobability Fig. 5. Received packet pe node pe timeslot fo Θ = and α =,,, 5 fo a squae netwok with N = nodes. α = fo the two topologies. Aveaging the numbe of hops of (8) and (9) ove, the expected numbe of hops h fo fixed in units of ae π /dt.6 and π /dh.5 fo tiangle and hexagon netwoks. So fo α =,, in a zeointefeence netwok, the tiangle topology is the best one due to its lowest enegy consumption, least delay and highest path efficiency. Howeve, fo α =,5, the hexagon topology has the least enegy consumption. IV. INTERFERENCE ANALYSIS In this section, we conside a netwok of N nodes, whee evey node always has a packet to tansmit (heavy taffic assumption). The eception is only coupted by intefeence, not by noise. A. A simple MAC scheme Fist, we study a vey simple MAC scheme, with the aim of finding a lowe pefomance bound fo moe elaboate schemes. Fo the netwok, it is assumed that nodes ae tansmitting packets independently in evey timeslot with tansmit pobability p at equal tansmit powe level and the next-hop eceive of evey packet is one of its neighbos. The packets ae of equal length and fit into one timeslot. The pefomance measue is the thoughput which is the expected numbe of successful packet tansmissions in one timeslot. Hee we povide simulation esults of the thee netwoks. A detailed analysis of the thoughput fo netwoks with diffeent topologies can be found in [9]. Fig. 5 displays the simulation esult of the elationship between the pe-node thoughput and tansmit pobability fo Θ = and vaious α fo a squae netwok with nodes, whee fo α =, the maximum pe-node thoughput g max /N =.77 is achieved at the optimum tansmit pobability p max =.78. The thoughput cuve fo othe two topologies has a simila shape, but g max and p max ae diffeent. Without intefeence, we would have g max /N = %p max. So we define the tansmit efficiency as T eff = g max/n p max. Fo all the thee netwoks, the tansmission efficiency The same MAC scheme was consideed in [8]. We use MATLAB to simulate the MAC scheme and the Rayleigh fading channel. TABLE II COMPARISON OF SQUARE, TRIANGLE AND HEXAGON NETWORKS FOR α = (INTERFERENCE ANALYSIS), WHERE p max, g max/n, T eff, l h, Z DENOTE OPTIMUM TRANSMIT PROBABILITY, MAXIMUM THROUGHPUT PER NODE, TRANSMIT EFFICIENCY, EFFECTIVE HOP LENGTH AND EFFECTIVE TRANSPORT CAPACITY. p max g max/n T eff l h Z Squae Tiangle Hexagon is about.7, which is simila to that of slotted ALOHA, namely e. We define the effective hop length as l h = E[/h] fo a fixed fom the expessions in (5), (8) and (9). The effective tanspot capacity is the distance-weighted thoughput, defined as Z := gmax N l h. The compaison of squae, tiangle and hexagon netwoks fo α = is shown in Table II. The tansmit efficiency is about.7 fo thee topologies. We see that the hexagon netwok has the highest tansmit pobability, thoughput, and effective tanspot capacity. B. Compaison with optimum schedule Fo lage netwoks, nodes in diffeent locations can use the channel simultaneously (spatial euse) if they ae sufficiently sepaated so that mutual intefeence will not pevent simultaneous successful tansmissions. Exploiting spatial euse, we can devise a scheduling scheme that maximizes the thoughput. Hee we will deal with the scheduling poblem in a squae netwok. Assume that in evey squae aea with q nodes, only one node is tansmitting. Fig. 6(a) shows the optimum scheduling scheme fo q = fo the fist phases (the numbe indicates the phase numbe). Shifting the fou links connecting fou nodes in the squaes to thei ight and bottom squaes, we can get anothe phases. Since it is fo bidiectional taffic, 6 phases ae needed. The total numbe of phases is q(q )++(q ) = q. In Fig. 6(b), the thoughput as a function of q is plotted. Optimum scheduling is achieved at q = 6, 5,, fo α =,,, 5. The thoughput atio between the simple MAC scheme and the optimum one is.7,.8,. fo α =,, 5, which shows that the elative pefomance of the simple MAC scheme is bette fo lowe α than highe α. Inteestingly, the cuve of α = is quite diffeent fom the simulation esults of the simple MAC scheme shown in Fig. 5. The eason is that fo α =, the eceived intefeence powe will be infinite fo a eceive located in an infinite plane with a unifom and finite density of tansmittes, as pointed out in [7]. The esults of the simple MAC scheme is fo a netwok, while the esult of the optimum schedule is deived fo a vey lage netwok. So, in the latte case, the SIR is much smalle, confiming that the pe-node thoughput fo α = to convege to zeo with inceasing netwok size. In fact, the simple MAC scheme is vey simila to slotted ALOHA. In geneal, slotted ALOHA assumes Poisson taffic, wheeas ou simple MAC scheme assumes evey node always has a packet to tansmit in each timeslot (heavy taffic assumption).

5 Received packets pe node pe timeslot α=5 (a). α= q 9 5 (b) Fig. 6. (a) The optimum tansmit schedule fo q =. (b) Received packets pe node and timeslot fo Θ = and α =,,, 5 fo a lage squae netwok whee evey q -th node is tansmitting in evey timeslot. Fo squae netwoks with m m nodes, the simulation esults in Fig. 7(a) show that the pe-node thoughput fo α = deceases with m (solid line). We appoximate the elationship between the pe-node thoughput and m by a/ ln(bm) with a =.65 and b =.66. The appoximation is plotted in Fig. 7(a) by dashed line. The pefect match confims the esult in [7], whee it was shown that fo α =, the total intefeence powe of a netwok of adius is given by integating c/ (fo some constant c) fom some R to. Hence, the SIR depends logaithmically on. In Fig. 7(b), the thoughput distibution ove the location of the nodes is ecoded fo a netwok with 5 5 nodes at the optimal tansmit pobability p max =. with the pe-node thoughput g max /N =.. We notice that nodes at the bounday of the netwok, especially at the cone, contibute the most to the thoughput, since they ae subject to less intefeence. This shows that even fo faily lage netwoks, the bounday effects cannot be neglected. V. CONCLUSIONS The Rayleigh fading link model does not only chaacteize the wieless link moe accuately than the disk model, but it also pemits a sepaate analysis of noise and intefeence. The esulting packet eception pobabilities simply have to be multiplied, yielding the total eception pobability P[SINR > Θ]. The noise analysis can be caied out independently of the MAC scheme, and the intefeence analysis is invaiant to a common facto in the tansmit powe, as the SIR does not change when all the nodes scale thei powe by the same facto. Received packets pe node pe timeslot x Simulation Appoximation m (a) 5 x 7 Fig. 7. Fo α =, (a) The simulation esult and appoximation of the elationship between the pe-node thoughput and m. (b) Distibution of thoughput ove 5 5 nodes. In the case of light taffic, the noise analysis alone povides accuate esults, wheeas in the case of heavy taffic, the netwok will be intefeence-limited, and both pats ae elevant. The chaacteization of the link eception pobabilities pemits the evaluation of end-to-end packet eception pobabilities p EE ; in ou analysis, p EE is assumed to be dictated by the application as pat of the QoS specification. In the noise analysis, fo α =,, the tiangle netwok gives the best pefomance due to its lowest enegy consumption, delay and highest path efficiency. Howeve, fo α =,5, the hexagon netwok has the least enegy consumption. In the intefeence analysis, the hexagon netwok exhibits the highest tansmit pobability, thoughput and effective tanspot capacity. By compaing the topologies and esults, we find that connecting with less neaest neighbos can impove the thoughput. Fo squae netwoks, fo α >, the thoughput atio between the simple MAC scheme and the optimum one is between [.,.75] the simple MAC is close to the optimum fo lowe α. The pefomance of any pactical MAC laye will lie between the bounds povided by these two MAC schemes. Fo α = (fee space popagation), spatial euse is not possible fo the inteio nodes in lage and dense netwoks, since the intefeence powe diveges to infinity as the netwok size is scaled. Received packets pe timeslot REFERENCES [] I. F. Akyildiz, W. Su, Y. Sankaasubamaniam, and E. Cayici, Wieless senso netwoks: a suvey, Compute Netwoks, vol. 8, pp. 9, Ma.. [] S. Megeian, F. Koushanfa, G. Qu, G. Velti, and M. Potkonjak, Exposue in Wieless Senso Netwoks: Theoy and Pactical Solutions, Wieless Netwoks, pp. 5, Aug.. [] P. Gupta and P. R. Kuma, The Capacity of Wieless Netwoks, IEEE Tansactions on Infomation Theoy, vol. 6, pp. 88, Ma.. [] N. H. Shepad et al., Coveage Pediction fo Mobile Radio Systems Opeating in the 8/9 MHz Fequency Range, IEEE Tansactions on Vehicula Technology, pp. 7, Feb [5] M. Zozi and S. Pupolin, Optimum Tansmission Ranges in Multihop Packet Radio Netwoks in the Pesence of Fading, IEEE Tansactions on Communications, vol., pp. 5, July 995. [6] M. Haenggi, Analysis and Design of Divesity Schemes fo Ad Hoc Wieless Netwoks, IEEE Jounal on Selected Aeas in Communications, vol., pp. 9 7, Jan. 5. [7] T. J. Shepad, A channel access scheme fo lage dense packet adio netwoks, in ACM SIGCOMM, pp. 9, Aug [8] L. Hu, Topology Contol fo Multihop Packet Netwoks, IEEE Tansactions on Communications, vol., no., pp. 7 8, 99. [9] X. Liu and M. Haenggi, Thoughput Analysis of Fading Senso Netwoks with Regula and Random Topologies, EURASIP Jounal on Wieless Communications and Netwoking, no., N y (b) N x 5