Sensor Network with Multiple Mobile Access Points

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1 Sensor Newor wih Mulile Mobile Access Poins Parvahinahan Veniasubramaniam, Qing Zhao and Lang Tong School of Elecrical and Comuer Engineering Cornell Universiy, Ihaca, NY 4853, USA Absrac We consider sensor newors wih mobile access oins where daa a sensor nodes are colleced by mulile mobile access oins. Using hroughu and energy efficiency as erformance measures, we address he oimal coverage areas of and he cooeraion among he mobile access oins. We show ha when he mobile access oins do no cooerae in demodulaion, disjoin coverage areas are oimal for hroughu while comleely overlaed coverage areas are oimal for energy efficiency. When he mobile access oins decode heir received aces joinly, he oimal configuraion aears o have a hase ransiion. Secifically, in order o maximize hroughu, he coverage areas of he mobile access oins should be comleely overlaed when SNR is smaller han a hreshold and disjoin oherwise. Lengh: Keywords: Regular Paer. Sensor Newor. Mobile Access Poin. Throughu. Energy Efficiency. I. INTRODUCTION A. Sensor Newors wih Mobile Access Poins Sensor newor wih mobile access oins (SENMA is an archiecure roosed for low-ower large-scale sensor newors []. As shown in Figure, SENMA consiss of wo yes of nodes: sensors and mobile access oins. Sensors are low ower and low cos nodes ha are limied in rocessing and communicaion caabiliy. In conras, mobile access oins are equied wih owerful rocessors and sohisicaed ransceivers, caable of raversing he sensor field wih carefully designed rajecory. Examles of mobile access oins include manned/unmanned aerial and ground vehicles and secially designed ligh nodes ha can ho around in he newor. They may form a small ad hoc newor and erform collaboraive daa collecion and os rocessing adding a new dimension in he sace-ime domain. In SENMA, sensors communicae direcly wih he mobile access oins. This avoids much of he overhead associaed wih medium access conrol and rouing and effecively shifs he criical newor newor oeraions away from he sensor nodes o he more owerful APs. The mobiliy of he APs adds a significan dimension o he daa collecion and rovides scoe for furher oimizaion of newor erformance by an aroriae choice of flying aern and cooeraive receion. Fig.. B. Summary of Resuls Sensor Newors wih Mobile Access The erformance of sensor newors wih a single mobile AP is well discussed in []. In his aer, we sudy he coverage areas of and he ineracion among mulile mobile access oins in a disribued mulile access scenario. As illusraed in Figure, a mobile access oin (AP acivaes a segmen of he newor in is viciniy a a given insan of ime. All sensors in he coverage area of he AP wae u uon acivaion and sar daa ransmission according o a cerain roocol. We see answers o he following quesions: should he coverage areas of he mobile access oins overla? is here an oimal fracion of overlaing for a given SNR? should he mobile access oins cooerae in demodulaion? Fig.. SENMA wih wo Mobile Access Poins The analysis is conduced using a ace framewor and answers are derived under wo erformance measures: hroughu defined as he average number of successfully received

2 aces er ime slo and efficiency defined as he average number of successfully received aces er ransmission. The former is a measure of he newor laency while he laer a measure of energy efficiency. When wo mobile access oins decode heir received aces searaely, we show ha he oimal configuraion under hroughu is for he mobile access oins o have disjoin coverage areas. Under he meric of efficiency, however, he coverage areas of he mobile access oins should comleely overla. When he aces received by he mobile access oins are decoded joinly, maximum raio combining can be used o imrove he receion of he aces ransmied by sensors in he overla of he coverage areas (region A in Figure. In his case, we observe ha he oimal configuraion under hroughu undergoes a hase ransiion as SNR increases; here exiss an SNR hreshold beyond which he oimal coverage areas of he mobile access oins change from being comleely overlaed o being disjoin. These findings rovide imoran insighs ino newor design. An informaion-heoreic analysis of similar newor configuraions of SENMA has been resened in [3], [4]. Our conclusions wih regard o flying aerns are similar in naure o he inferences drawn in [3]. II. SYSTEM MODEL We consider a newor where sensors are randomly disribued over a field. I is assumed ha every sensor has daa o ransmi. The newor is assumed o oerae in ime division dulex (TDD mode. Time is sloed ino inervals of equal lengh ha is equal o he ransmission ime of a ace. A he beginning of every slo, he AP ransmis a beacon, which is used by sensors o gain synchronizaion. We assume ha beacons ransmied by differen APs are synchronous and indisinguishable. Each acivaed sensor, uon receiving a beacon, ransmis is informaion wih a consan robabiliy. This robabiliy is a funcion of he channel sae disribuion and size of he newor and can be oimized o maximize he hroughu or efficiency as desired. A. Channel Model In his secion, we describe he basic channel model used in he analysis of he newor. The mobile APs are assumed o fly a a heigh d above he newor. We assume ha he radial disances of he sensors are negligible comared o d. The roagaion channel gain beween he ih sensor and he AP during slo is given by γ = R i d ( where R i is a comlex Gaussian random variable. The channel sae γ is indeenden and idenically disribued across sensors,across slos and from one AP o anoher. All sensor are assumed o ransmi using he same ransmi ower P T. Since d is a consan for all sensors, we shall henceforh incororae i ino he ransmi ower of he sensor. B. Daa Receion Model In his secion, we shall describe he receion model of a single mobile AP. Collaboraive receion of mulile APs will be discussed in secion IV-A. We assume he hysical layer is based on direc sequence sread secrum communicaion wih sreading gain N. There is a ool of N orhogonal codes, and each ransmiing sensor randomly ics a code for ransmission. The receiver erforms mached filering on each code o demodulae he received daa. The receion of a ace is successful if he received signal o inerference and noise raio (SINR is greaer han a hreshold β. In slo, if K j sensors ransmi using he j h sreading code and heir channel saes are given by (γ ( j,, γ ( j Kj, hen he crierion for successful receion of sensor i is well-aroximaed [5] by P T γ ( j i σ + K j =, i P > β, T γ ( j where σ is he variance of he bacground noise. I is assumed ha β >. C. Performance Merics In order o comare he erformance of SENMA for differen configuraions, wo merics are used - hroughu and efficiency. The hroughu of he newor is given by he average number of aces successfully received er slo. The efficiency of he newor is given by he average number of aces successfully received er ransmission. For he comarison of differen configuraions, he robabiliy of ransmission is oimized o maximize he required meric. Le denoe he robabiliy of ransmission for a sensor. Le C denoe he hroughu er slo when sensors ransmi. The maximum hroughu and efficiency are given by λ = max η = max = n = ( n C ( ( n C n where n is he number of acivaed sensors. III. SENMA WITH NON-COOPERATIVE MOBILE APS In his secion we shall analyze he erformance of SENMA wih wo non-cooeraing mobile APs. Consider he newor as shown in Figure. The mobile AP M can received aces from ransmiing sensors in A and A, whereas M can receive aces from ransmiing sensors in A and A. I is assumed ha he beacons ransmied by he MAPs are synchronized and indisinguishable. The sensors are herefore unaware of heir locaion, and hence he robabiliy of ransmission is a consan, oimized o maximize he erformance meric. The sensors are uniformly disribued across he newor and he number of sensors in he area of acivaion of an AP is assumed o be a consan n. Hence, if he fracion of overla is given by A A = A A = ρ, (3

3 he number of sensors in A, A, and A are given by ( ρn, ( ρn and ρn resecively. We oin ou ha he resuls obained below also aly o cases of randomly disribued n. This deerminisic seu is for he ease of resenaion. A. Performance Comarison We shall now comare he erformance of he newor under hree configuraions - no overla (ρ = 0, comlee overla (ρ = and arial overla (0 < ρ <. No Overla: In he case of no overla, he wo mobile APs acivae indeenden regions of he newor. Since, i is assumed ha channel sae is i.i.d across sensors and ime slos, he hroughu of he sysem is wice he hroughu of a newor wih one AP. Le P (P T, ρ = 0 [6] denoe he average robabiliy of successful receion of a ace when sensors ransmi wih ransmi ower P T. The hroughu and efficiency are given by ( n λ(, ρ = 0 = ( n P (P T, 0 (4 = n = ( n P (P T, 0 η(, ρ = 0 = (5 n P (P T, 0 = e βσ /P T ( + β (6 where is he robabiliy of ransmission. Comlee Overla: The mobile APs acivae he same region of he newor, and each ransmiing sensor can be received by boh he APs. If we le P (P T, ρ = denoe he robabiliy ha given sensors ransmi, a sensor is successfully received a aleas one AP, he hroughu and efficiency can be similarly given by ( n λ(, = ( n P (P T, (7 = n = ( n P (P T, η(, = (8 n P (P T, = P (P T, 0 (P (P T, 0 (9 where is he robabiliy of ransmission, which can be oimized deending on he meric of erformance required. The following heorem characerizes he oimal fracion of overla for he differen erformance merics discussed. Theorem : For any given P T, i max λ(, ρ max λ(, 0 ii max η(, ρ = P (P T, 0 ρ(p (P T, 0 /( ρ iii max η(, ρ max η(,. Proof: i Consider he newor as shown in figure. As saed before, le he number of sensors in A, A, A be equal o n = ( ρn, n = ρn, n 3 = n resecively. Since i is assumed ha beacons are synchronous and indisinguishable, Fig. 3. Poins Fig. 4. Poins Throughu (aces/slo Efficiency (aces/slo/sensor Non Collaboraive Receion No Overla Comlee Overla SNR (db Throughu of SENMA wih wo non-cooeraive Mobile Access Non Collaboraive Receion No Overla Comlee Overla SNR (db Efficiency of SENMA wih wo non-cooeraive Mobile Access he robabiliy of ransmission is a consan. The hroughu of he newor wih fracion of overla ρ is given by λ(, ρ = n n n 3 =0 =0 3=0 ( ( ( n n n ( n+n3+n 3 [ P P (P + (P T, 0 + P + 3 P + P + 3 ] (0 In he above equaion, we have used P o denoe P (P T, 0 for ease of resenaion. Equaion (0 can be obained from he fac ha, sensors ransmiing in each region are indeenden. Sensors in A and A ( and 3 resecively are received by only one AP, while sensors in A ( are received by boh APs. The above exression can be rewrien as λ(, ρ = n n =0 =0 + ( n+n ( + P + (P T, 0 α, α 0

4 +n = ( n+n (P (P T, 0 α = =0 ( n P (P T, 0 α =0 λ(, 0 From he above equaions, i is clear haλ(, 0 is greaer han λ(, ρ for any given. Therefore, i is rue. iisince he ransmissions in each orhogonal code are indeenden, he hroughu scales linearly wih he sreading gain, and i is sufficien o consider he efficiency for N =. Assuming ha β >, a mos one ace can be successfully received er orhogonal code [5]. Therefore he efficiency is maximized when robabiliy of inerference in minimized i.e. when he robabiliy of ransmission is close o zero. In oher words, max η(, ρ = lim 0 λ(, ρ (n + n = lim 0 n P (P T, 0 + n P (P T, (n + n = ( ρp (P T, 0 + ρp (P T, 0 ρ(p (P T, 0 ( ρ = P (P T, 0 ρ(p (P T, 0 ρ iii I is easy o see ha he exression for max η(, ρ is non-decreasing. Hence he maximum occurs a ρ =. From he above heorem, i is clear ha he oimal configuraion o maximize he hroughu requires he wo mobile APs o acivae indeenden regions, whereas he configuraion needed o maximize he efficiency requires he APs o acivae he same region. The los in Figure 3 and 4 verify he above facs. These resuls also aly o a general disribuion of sensors in a field, and no only a fixed number of sensors n in he acivaed region. This is roved in he following corollary. Corollary : Le q(n, A be he robabiliy ha an area A in he newor has n sensors. Le λ(, ρ reresen he average hroughu of a newor wih fracion of overla ρ and number of sensors disribued according o q. Then, for any P T, λ(, ρ λ(, 0. Proof: Le λ n (, ρ denoe he hroughu of a sysem wih ransmission robabiliy, fracion of overla ρ and n sensors in he acivaed region. The average hroughu of a newor as shown in figure is given by λ(, ρ = q(n, A q(n, A q(n 3, A λ(, ρ n n 3 n where λ(, ρ is as given in equaion 0. We can rewrie 0 as λ(, ρ = n n + ( n+n ( + P + (P T, 0+ =0 =0 n 3 n 3=0 =0 3+ ( n3+n ( 3 + P 3+ (P T, 0 α (λ n +n (0, ρ + λ n3+n (0, ρ Therefore, λ(, ρ = q(n, A q(n, A q(n 3, A λ(, ρ n n n 3 q(n, A q(n, A q(n 3, A n n n 3 (λ n +n (0, ρ + λ n3+n (0, ρ a = q(n, A q(n, A λ n+n (0, ρ n n = q(n, A + A λ n (0, ρ n = λ(, 0 The equaliy a is because A and A are equal in area and A + A is equal o he acivaion region of one AP. The corollary herefore jusifies he claims for a random disribuion of sensors across he newor. IV. SENMA WITH CO-OPERATIVE MOBILE APS In his secion, we shall numerically analyze he erformance of SENMA wih mobile APs ha co-oerae wih each oher in receion. In reference o figure, he mobile APs can co-oerae in he receion of aces from sensors in A. A. Receion Model The collaboraive receion imlemened in his aer is in he form of maximum raio combining (MRC. Each ransmiing sensor ransmis ilos in he ace, which are used by he APs o esimae he channel gain beween he sensor and he AP. I is assumed ha here are a large number of ilos, and each sensor randomly chooses a ilo for ransmission. The robabiliy ha wo sensors simulaneously ransmi he same ilos is negligible. If R i and R i reresen he comlex gains for sensor i a he wo APs, hen he SINR a he ouu of he MRC is given by [7] γ i = ( Ri n + j i R j R i Ri + R iri + Ri The ace is successfully received if γ i > β. ( n + j i R j (

5 B. Performance Comarison Since, collaboraion is no ossible when here is no overla in he acivaion regions, he hroughu and efficiency of a newor wih no overla is he same as given by equaions (4 and (5. Comlee Overla: As discussed in he revious secion, he hroughu and efficiency of a newor wih comlee overla can be given by ( n λ c (, = ( n P c (P T, ( η c (, = = λ(, n (3 where P c (, is he average robabiliy of success of a sensor hrough coheren deecion, given sensors ransmi. For he model discussed in he revious secion, he exression for P c (, can be given by [8]: P(P c T, = ( β β + Γ (4 Γ = σ P T + ( Collaboraive Receion No Overla Comlee Overla Throughu (aces/slo Collaboraive receion SNR = 6 db SNR = db SNR = 0 db Fracion of Overla (ρ Fig. 6. Throughu of SENMA wih wo cooeraive Mobile Access Poins and arial overla assumed o have symmeric robabiliies of ransmission, hen for any robabiliy of ransmission λ c (, ρ ρ max λ c (, + max ( ρλc (, 0 (6 Proof: Consider he newor as shown in Figure. The symmery condiion requires sensors in A and A o have symmeric robabiliies of ransmission. In oher words, if sensors in A decide o ransmi, an equal number of sensors in A would also ransmi. Therefore, he newor can now be divided ino wo secions, A (ora 3 and A. The hroughu of such a configuraion can be shown o be Throughu (aces/slo λ c (, ρ = ρn ρ ( ( ρn ρn + ( n = [ = P+ (P T, 0 + P+ c (P T, ], Fig SNR (db Throughu of SENMA wih wo cooeraive Mobile Access Poins The comarison of hroughu and efficiency for he wo cases is loed wih resec o SNR in figure 5 and 7. I is eviden from he figure ha a hase ransiion occurs a a aricular SNR, below which he comleely overlaed newor has a higher maximum hroughu, and above which he zero overla configuraion yields a higher hroughu. I however remains o be seen, if i is ossible o obain a higher hroughu hrough arial overla (0 < ρ <. The following heorem rovides an insigh on he erformance of such oologies Theorem : Le λ c (, ρ be he hroughu of a sysem wih fracion of overla ρ and robabiliy of ransmission. If sensors in he wo non-overlaing regions (A and A are where P (P T, 0 is as given in equaion 6 and P c (P T, is as given in equaion 4. I can be shown ha ρn ρ = = Therefore, = ( ρn λ c (, ρ = ( ρ ( ρn = ( n ρ where = +. ( n [( ρp (P T, 0 + ρp c (P T, ] ( ρ max ( n P (P T, 0 + ρ λ c (, 0 + ρ max λ c (, ( n P c (P T, The above heorem shows ha, under he given condiions, he hroughu of a sysem wih arial overla is less han he hroughu of eiher one of he exreme configuraions. In oher words, he hroughu is a convex U funcion of he fracion

6 of overla. Alhough he condiion imosed in Theorem is sringen, he inuiion is ha he resul exends o he general case as well. This is clearly illusraed in he figure 6. Fig. 7. Efficiency (aces/slo/user Collaboraive Receion No Overla Comlee Overla SNR (db Efficiency of SENMA wih wo cooeraive Mobile Access Poins Figure 7 shows ha even in he co-oeraive MAP case, he efficiency of he comleely overlaed newor is uniformly beer han oher configuraions. This is eviden from he following heorem Theorem 3: For any given P T i η c (ρ = max η c (, ρ = ( ρp(pt,0+ρp c ρ (PT, V. CONCLUSION In his aer, we analyzed he erformance of SENMA wih wo mobile access oins in differen configuraions. An imoran conclusion from he aer is ha arial or imerfec overla configuraion canno rovide he maximum hroughu. The analysis in he aer does no exlicily use he disribuion of he fading arameer, and hence hese resuls may aly o a general fading model. REFERENCES [] L. Tong, Q. Zhao, and S. Adireddy, Sensor Newors wih Mobile Agens, in Proc. 003 Miliary Communicaions Inl Sym., Boson, MA, Oc [] P. Veniasubramaniam, S. Adireddy, and L.Tong, Sensor Newors wih Mobile Agens: Oimal Random Access and Coding, o aear in IEEE Journal on Sel. Areas in Comm.: Secial Issue on Sensor Newors, h://acs.ece.cornell.edu/ubj.hml. [3] G.Mergen, Q. Zhao, and L.Tong, Sensor newors wih mobile access: Energy and caaciy consideraions, submied o IEEE Trans. on Communicaions, Mar [4] G. Mergen and L. Tong, Caaciy Consideraions for Sensor Newors wih Mobile Agens, in Proceedings of he 4s Alleron Conf. on Communicaions, Conrol, and Comuing, Monicello, IL, Oc [5] B.Haje, A.Krishna, and R.O.LaMaire, On he Caure Probabiliy for a Large Number of Saions, IEEE Trans. Communicaions, vol. 45, no., , February 997. [6] M.Zorzi and R.Rao, Caure and reransmission conrol in mobile radio, IEEE J. Selec. Areas Commun., vol., no. 8, , Ocober 994. [7] J. Winers, Oimal Combining in Digial Mobile Radio wih Cochannel Inerference, IEEE J. Selec Areas in Comm, vol., no. 4, , July 984. [8] W. Jaes, Microwave Mobile Communicaions. New Yor: Wiley, 974. ii max η c (, ρ max η c (, Proof: i By following he same argumens used in he roof of ii in heorem, i can be shown ha he efficiency is given by: η c (ρ = lim 0 λ c (, ρ (n + n = n P (P T, 0 + n P c (P T, n + n = ( ρp (P T, 0 + ρp c (P T, ρ (7 ii I remains o be shown ha η c (, ρ is a monoonically increasing funcion for 0 < ρ <. The derivaive of η c (ρ wih resec o ρ is given by d dρ η(ρ = P c (P T P (P T ( ρ (8 Since P c ( is he robabiliy of coheren receion by wo MAPs in a single user ransmission, i is sricly greaer han P (, which is he robabiliy of success a a single AP. Therefore, he derivaive in he above equaion is sricly osiive and hence η( is an increasing funcion in he given range, and ii is rue.

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