Capacity of Data Collection in Arbitrary Wireless Sensor Networks

Transcription

1 This fu text pape was pee eviewed at the diection of IEEE Communications Society subject matte expets fo pubication in the IEEE INFOCOM 2010 poceedings This pape was pesented as pat of the Mini-Confeence at IEEE INFOCOM 2010 Capacity of Data Coection in Abitay Wieess Senso Netwos Siyuan Chen Shaojie Tang Minsu Huang Yu Wang Depatment of Compute Science, Univesity of Noth Caoina at Chaotte, Chaotte, Noth Caoina, USA Depatment of Compute Science, Iinois Institute of Technoogy, Chicago, Iinois, USA Abstact How to efficienty coect sensing data fom a senso nodes is citica to the pefomance of wieess senso netwos. In this pape, we aim to undestand the theoetica imitations of data coection in tems of possibe and achievabe maximum capacity. Peviousy, the study of data coection capacity [1] [6] has ony concentated on age-scae andom netwos. Howeve, in most of pactica senso appications, the senso netwo is not depoyed unifomy and the numbe of sensos may not be as huge as in theoy. Theefoe, it is necessay to study the capacity of data coection in an abitay netwo. In this pape, we deive the uppe and constuctive owe bounds fo data coection capacity in abitay netwos. The poposed data coection method can ead to ode-optima pefomance fo any abitay senso netwos. We aso examine the design of data coection unde a genea gaph mode and discuss pefomance impications. I. INTRODUCTION A wieess senso netwo consists of a set of senso devices which spead ove a geogaphica aea. The utimate goa of senso netwos is often to coect the sensing data fom a sensos to a sin node and then pefom futhe anaysis at the sin node. In this pape, we study some fundamenta capacity pobems aising fom data coection scenaio in wieess senso netwos. We conside a wieess senso netwo whee n sensos ae abitaiy depoyed in a finite geogaphica egion. Each senso measues independent fied vaues at egua time intevas and sends these vaues to a sin node. The union of a sensing vaues fom n sensos at a paticua time is caed snapshot. The tas of data coection is to deive these snapshots to a singe sin. Due to spatia sepaation, sevea sensos can successfuy tansmit at the same time if these tansmissions do not cause any destuctive wieess intefeences. As in the iteatue, the cassica potoco intefeence mode is used in ou anaysis, whie a anaysis esuts can aso be extended to physica intefeence mode by appying the technique intoduced in [7]. We aso assume that a successfu tansmission ove a in has a fixed data-ate W bit/second. The pefomance of data coection in senso netwos can be chaacteized by the ate at which sensing data can be coected and tansmitted to the sin node. In paticua, the theoetica measue that captues the imitations of coection pocessing in senso netwos is capacity fo the many-toone data coection, i.e., the maximum data ate at the sin This wo is suppoted in pat by the US Nationa Science Foundation (NSF) unde Gant No. CNS and No. CNS to continuousy eceive the snapshot data fom sensos. Data coection capacity efects how fast the sin can coect sensing data fom a sensos unde existence of intefeence. It is citica to undestand the imitation of many-to-one infomation fows and devise efficient data coection agoithms to maximize the pefomance of wieess senso netwos. Capacity imits of data coection in andom wieess senso netwos have been studied in the iteatue [1] [6]. In [1], [2], Duate-Meo et a. fist intoduced the many-to-one tanspot capacity in dense and andom senso netwos unde potoco intefeence mode. E Gama [] studied the capacity of data coection subject to a tota aveage tansmitting powe constaint whee a node can eceive data fom mutipe souce nodes at a time. Baton and Rong [4] aso investigated the capacity of data coection unde genea physica aye modes (e.g. coopeative time evesa communication mode) whee the data ate of individua in is not fixed as a constant W but vaious depending on the tansmitting powes and tansmitting distances of a simutaneous tansmissions. Both [] and [4] adopted compex physica aye techniques, such as antenna shaing, channe coding and coopeative beamfoming, in thei modes. Liu et a. [5] ecenty studied the capacity of a genea some-to-some communication paadigm unde potoco intefeence mode in andom netwos whee thee ae mutipe andomy seected souces and destinations. They deived the uppe and constuctive owe bounds fo such a pobem. Chen et a. [6] studied the capacity of data coection unde potoco intefeence mode with mutipe sins. Howeve, a the above eseach shaes the common assumption whee age numbe of senso nodes ae eithe ocated on a gid stuctue o andomy and unifomy distibuted in a pane. Such assumption is usefu fo simpifying the anaysis and deiving nice theoetica imitations, but may be invaid in many pactica senso appications. To ou best nowedge, ou pape is the fist one to study data coection capacity fo abitay netwos. In this pape, we focus on deiving capacity bounds of data coection fo abitay netwos, whee senso nodes ae depoyed in any distibution and can fom any netwo topoogy. We summaize ou contibutions as foows: Fo abitay senso netwo unde potoco intefeence mode, we popose a data coection method based on Beadth Fist Seach (BFS) tee. We pove that this method can achieve coection capacity of Θ(W ) which matches the theoetica uppe bound /10/$ IEEE

2 This fu text pape was pee eviewed at the diection of IEEE Communications Society subject matte expets fo pubication in the IEEE INFOCOM 2010 poceedings This pape was pesented as pat of the Mini-Confeence at IEEE INFOCOM 2010 Since dis gaph mode is ideaistic, we aso conside a moe pactica mode: genea gaph mode. In genea gaph mode, two neaby nodes may be unabe to communicate due to vaious easons such as baie and path fading. We fist show that Θ(W ) may not be achievabe fo a genea gaph. Then we pove that BFS-based method can sti achieve capacity of Θ( W Δ ) whee Δ is a new intefeence paamete defined in Section IV. The esuts above not ony hep us to undestand the theoetica imitations of data coection in senso netwos, but aso povide pactica and efficient data coection methods (incuding how to constuct data coection stuctue and how to schedue data coection) to achieve nea-optima capacity (within constant times of the optima). Even though we ae focusing on abitay netwos, a of ou soutions can be appied to andom netwos since any andom netwo is just a specia case of abitay netwos. II. NETWORK MODELS AND COLLECTION CAPACITY A. Basic Netwo Modes In this pape, we focus on the capacity bound of data coection in abitay wieess senso netwos. Fo simpicity, we stat with a set of simpe and yet genea enough modes that ae widey used in the community. We conside an abitay wieess netwo with n senso nodes v 1,v 2,,v n and a singe sin v 0. These n sensos ae abitaiy distibuted in a fied. At egua time intevas, each senso measues the fied vaue at its position and tansmits the vaue to the sin. We adopt a fixed data-ate channe mode whee each wieess node can tansmit at W bits/second ove a common wieess channe. We aso assume that a pacets have unit size b bits. The time is divided into time sots with t = b/w seconds. Thus, ony one pacet can be tansmitted in a time sot between two neighboing nodes. TDMA scheduing is used at MAC aye. Unde the fixed data-ate channe mode, we assume that evey node has a fixed tansmission powe P. Thus, a fixed tansmission ange can be defined such that a node v j can successfuy eceive the signa sent by node v i ony if v i v j. Hee, v i v j is the Eucidean distance between v i and v j. We ca this mode dis gaph mode. We can futhe define a communication gaph G =(V,E) whee V is the set of a nodes (incuding the sin) and E is the set of a possibe communication ins. In this pape, we aways assume gaph G is connected. Due to spatia sepaation, sevea sensos can successfuy tansmit at the same time if thei tansmissions do not cause any destuctive wieess intefeences. As in the iteatue, we mode the intefeence using potoco intefeence mode. A nodes have a unifom intefeence ange R. When node v i tansmits to node v j, node v j can eceive the signa successfuy if no node within a distance R fom v j is tansmitting simutaneousy. Hee, fo simpicity, we assume that R is a constant α which is age than 1. Letδ(v i ) be the numbe of nodes in v i s intefeence ange (incuding v i itsef) and Δ be the maximum vaue of δ(v i ) fo a nodes v i, i =0,,n. B. Capacity of Data Coection We now fomay define deay and capacity of data coection in wieess senso netwos. Reca that each senso geneates a fied vaue with b bits at egua time intevas, and ties to tanspot it to the sin. We ca the union of a vaues fom a n sensos at paticua samping time a snapshot of the sensing data. Then the goa of data coection is to coect these snapshots fom a sensos. It is cea that the sin pefe to get each snapshot as quicy as possibe. In this pape, we assume that thee is no coeation among a sensing vaues and no netwo coding o aggegation technique is used duing the data coection. Definition 1: The deay of data coection D is the time used by the sin to successfuy eceive a snapshot, i.e., the time needed between competey eceiving one snapshot and competey eceiving the next snapshot at the sin. Definition 2: The capacity of data coection C is the atio between the size of data in one snapshot and the time to eceive such a snapshot (i.e., nb D ) at the sin. Thus, the capacity C is the maximum data ate at the sin to continuousy eceive the snapshot data fom sensos. Hee, we equie the sin to eceive the compete snapshot fom a sensos (i.e., data fom a sensos need to be deiveed). Notice that data tanspot can be pipeined in the sense that futhe snapshots may begin to tanspot befoe the sins eceiving pio snapshots. In this pape, we focus on capacity anaysis of data coection in an abitay senso netwo. III. COLLECTION CAPACITY UNDER DISK GRAPH MODEL Uppe Bound of Coection Capacity: It has been poved that the uppe bound of capacity of data coection fo andom netwos is W [1], [2]. It is obviousy that this uppe bound aso hods fo any abitay netwo. The sin v 0 cannot eceive at ate faste than W since W is the fixed tansmission ate of individua in. Theefoe, we ae inteested in design of data coection agoithm to achieve capacity in the same ode of the uppe bound, i.e. Θ(W ). We now popose a BFS-based data coection method and demonstate that it can achieve the capacity of Θ(W ) unde ou netwo mode. Ou data coection method incudes two steps: data coection tee fomation and data coection scheduing. A. Data Coection Tee - BFS Tee The data coection tee used by ou method is a cassica Beadth Fist Seach (BFS) tee ooted at the sin v 0.The time compexity to constuct such a BFS tee is O( V + E ). Let T be the BFS tee and v1,,vm be a eaves in T.Fo each eaf vi, thee is a path P i fom itsef to the oot v 0.Let δ Pi (v j ) be the numbe of nodes on path P i which ae inside the intefeence ange of v j (incuding v j itsef). Assume the maximum intefeence Δ i on each path P i is max{δ Pi (v j )} fo a v j P i. Heeafte, we ca Δ i path intefeence of path P i. Then we can pove that T has a nice popety that the path intefeence of each banch is bounded by a constant.

3 This fu text pape was pee eviewed at the diection of IEEE Communications Society subject matte expets fo pubication in the IEEE INFOCOM 2010 poceedings This pape was pesented as pat of the Mini-Confeence at IEEE INFOCOM 2010 /2 vj R Pi Fig. 1. Poof of Lemma 1: on a path P i in BFS T, the intefeence nodes fo a node v j is bounded by a constant. Lemma 1: Given a BFS tee T unde the potoco intefeence mode, the maximum intefeence Δ i on each path P i is bounded by a constant 8α 2, i.e., Δ i 8α 2. Poof: We pove by contadiction with a simpe aea agument. Assume that thee is a v j on P i whose δ Pi (v j ) > 8α 2. In othe wods, moe than 8α 2 nodes on P i ae ocated in the intefeence egion of v j. Since the aea of intefeence egion is πr 2, we conside the numbe of intefeence nodes inside a sma dis with adius 2. See Figue 1 fo iustation. πr The numbe of such sma diss is at most 2 π( = 4α 2 2 )2 inside πr 2. By the Pigeonhoe pincipe, thee must be moe than 8α2 4α =2nodes inside a singe sma dis with adius 2 2. In othe wods, thee nodes v a, v b and v c on the path P i ae connected to each othe as shown in Figue 1. This is a contadiction with the constuction of BFS tee, since one of such nodes wi be visited on othe path (i.e. on a singe path a node can ony connected to two othe nodes (its paent and chid on the path)). As shown in Figue 1, if v a and v c ae connected in G, then v c shoud be visited on the othe path instead of P i. This finishes ou poof. Fig. 2. sot 1 sot 2 sot data Path P i with Δ= i Sot 1 Sot 2 Sot v a v b v c Scheduing on a path: afte Δ i sots the sin gets one data. B. Scheduing Agoithm We now iustate how to coect one snapshot fom a sensos. Given the coection tee T, ou scheduing agoithm basicay coects data fom each path P i in T one by one. Fist, we expain how to schedue coection on a singe path. Fo a given path P i, we can use Δ i sots to coect one data in the snapshot at the sin. See Figue 2 fo iustation. In this figue, we assume that R = and ony adjacent nodes intefee with each othe. Thus Δ i =. Then we coo the path using geen, ed, and bue as in Figue 2(a). Evey node (a) (b) (c) (d) on the path has unit data to tansfe. Geen, ed and bue ins ae active in the fist sot, the second sot and the thid sot, espectivey. Afte thee sots (Figue 2(d)), the eaf node has no data in this snapshot and the sin gets one data fom its chid. Theefoe, to eceive a data on the path, at most Δ i P i time sots ae needed. We ca this scheduing method Path Scheduing. Now we descibe ou scheduing agoithm on the coection tee T. Remembe T has m eaves which define m pathes fom P 1 to P m. Ou agoithm coects data fom path P 1 to P m in ode. We define that i-th banch B i is the pat of P i fom v i to the intesection node with P i+1 fo i =[1,m 1] and m-th banch B m = P m. Fo exampe, in Figue (b), thee ae fou banches in T : B 1 is fom v 1 to v a, B 2 is fom v 2 to v 0, B is fom v to v b, and B 4 is fom v 4 to v 0. Remembe that the union of a banches is the whoe tee T. Agoithm 1 shows the detaied scheduing agoithm. Agoithm 1 Data Coection Scheduing on BFS Input: BFS tee T. 1: fo each snapshot do 2: fo t =1to m do : Coect data on path P i. A nodes on P i tansmit data towads the sin v 0 using Path Scheduing. 4: The coection teminates when nodes on banch B i do not have data fo this snapshot. Notice that the tota sots used ae at most Δ i B i, whee B i is the hop ength of B i. Figue (c)-(j) give an exampe of scheduing on T.Inthe fist step (Figue (c)), a nodes on P 1 paticipate in the tansmission using the scheduing method fo a singe path (evey Δ 1 sots, sin v 0 eceives one data). Such tansmission stops unti thee is no data in this snapshot on banch B 1,as shown in Figue (d). Then in the second step data on path P 2 is tansmitted. This pocedue epeats unti a data in this snapshot each v 0. C. Capacity Anaysis We now anayze the achievabe capacity of ou data coection method by counting how many time sots the sin needs to eceive a data in one snapshot. Theoem 2: The BFS-based data coection method can achieve data coection capacity of Θ(W ) at the sin. Poof: In Agoithm 1, the sin coects data fom a m pathes in T. In each step (Lines -4), data ae tansfeed on path P i and it taes at most Δ i B i time sots. Reca that Path Scheduing needs at most Δ i time sots to coect pacets fom path P i. Theefoe, the tota numbe of time sots needed fo Agoithm 1, denoted by τ, isatmost m i=1 Δ i B i. Since the union of a banches is the whoe tee T, i.e., m i=1 B i = n. Thus, τ m i=1 Δ i B i m Δ B i=1 i Δn. Hee Δ = max{δ 1,, Δ m }. Then, the deay of data coection D = τt Δnt. The capacity C = nb D Δnt nb = W Δ. Fom Lemma 1, we now that Δ is bounded by a constant. Theefoe, the data coection capacity is Θ(W ).

4 This fu text pape was pee eviewed at the diection of IEEE Communications Society subject matte expets fo pubication in the IEEE INFOCOM 2010 poceedings This pape was pesented as pat of the Mini-Confeence at IEEE INFOCOM 2010 V a B 2 B 4 P P P P B 1 V b P 1 P 1 P 2 V B 4 V V V V V (a) BFS Tee T (b) Banches in T (c) Step 1 (d) Afte Step 1 (e) Step 2 P 4 P 4 P 2 P P V V V V V (f) Afte Step 2 (g) Step (h) Afte Step (i) Step 4 (j) Afte Step 4 Fig.. Iustations of ou scheduing on the data coection tee T. Reca that the uppe bound of data coection capacity is W, thus ou data coection agoithm is ode-optima. Consequenty, we have the foowing theoem. Theoem : Unde potoco intefeence mode and dis gaph mode, data coection capacity fo abitay wieess senso netwos is Θ(W ). IV. COLLECTION CAPACITY UNDER GENERAL GRAPH So fa, we assume that the communication gaph is a dis gaph whee two nodes can communicate if and ony if thei distance is ess than o equa to tansmission ange. Howeve, a dis gaph mode is ideaistic since in pactice two neaby nodes may be unabe to communicate due to vaious easons such as baie and path fading. Theefoe, in this section, we conside a new genea gaph mode G =(V,E) whee V is the set of sensos and E is the set of possibe communication ins. Evey senso sti has a fixed tansmission ange such that the necessay condition fo v j to eceive coecty the signa fom v i is v i v j. Notice that v i v j is not the sufficient condition fo an edge v i v j E. Some ins do not beong to G because of physica baies o the seection of outing potocos. Thus, G is a subgaph of a dis gaph. Unde this mode, the netwo topoogy G can sti be any genea gaph (fo exampe, setting = and putting a baie between any two nodes v i and v j if v i v j / G). A. Data Coection unde Genea Gaph Mode In the new genea gaph mode, the capacity of data coection coud be W n in the wost-case. We conside a simpe staight-ine netwo topoogy with n sensos as shown in Figue 4(a). Assume that the sin v 0 is ocated at the end of the netwo and the intefeence ange is age enough to cove evey node in the netwo. Since tansmission on one in wi intefee with a othe nodes, the ony possibe scheduing is tansfeing data aong the staight-ine via a ins. The tota time sots needed ae n(n +1)/2, thus the capacity is at most Θ( W n ). Notice that in this exampe, the maximum intefeence Δ of gaph G is n. It seems the uppe bound of data coection capacity coud be W Δ. We now show an exampe whose capacity can be much age than W Δ.Again we assume a n nodes with the sin intefeing with each othe. The netwo topoogy is a sta with the sin v 0 in cente, as shown in Figue 4(b). Ceay, a scheduing which ets evey node tansfe data in ode can ead to a capacity W which is much age than W Δ = W n. V n V n Fig. 4. (a) Staight-ine Topoogy (b) Sta Topoogy The optimum of BFS-based method unde two exteme cases. Fotunatey, the BFS-based data coection agoithm sti wos we unde genea gaph mode. It is easy to see that the capacity is sti W Δ. Hee, Δ is the maximum path intefeence among a pathes. Howeve, in genea gaph mode we can not bound Δ by a constant any moe, and it coud be O(1) o O(n). Thus, thee is a gap between ou owe bound of data coection W Δ and the natua uppe bound W. Consideing both exampes shown in Figue 4, the BFS-based method matches thei tight uppe bounds W n and W. Fo the sta topoogy, even though the sin has the maxima intefeence Δ=n, each individua path has the path intefeence Δ i =1 which eads to capacity of W 1 = W. Fo the staight-ine topoogy, the path intefeence of the singe path Δ i = n, thus the capacity is W n. B. Tighte Lowe Bound Actuay W Δ is not a tight owe bound by BFS-based method. Now we ae eady to show a tighte owe bound by econsideing how to do the Path Scheduing. In Section III we caimed that the path scheduing fo a path P i can be done in Δ i P i time sots. Howeve, we can pefom path scheduing in the foowing way to save moe sots. Assume that path P i = v 0,v 1,v 2,,v Pi. Letδ Pi = max{δ Pi (v 1 ),,δ Pi (v )},

5 This fu text pape was pee eviewed at the diection of IEEE Communications Society subject matte expets fo pubication in the IEEE INFOCOM 2010 poceedings This pape was pesented as pat of the Mini-Confeence at IEEE INFOCOM 2010 Vn og n Fig. 5. Vn ogn+1 V n ogn n og n Iustation of the advantage of a new path scheduing. i.e., δ Pi is the maximum intefeence among fist nodes v 1 to v in path P i. In the fist step, using δ Pi P i sots, evey node on the path tansfes its data to its paent. Afte the fist step, the eaf v Pi aeady finishes its tas in this ound and has no data fom cuent snapshot. In the second step, using δ Pi P sots, i 1 the cuent snapshot data wi move one moe eve up aong the path in the BFS tee. Repeat these steps unti a data aong this path each the sin. It is easy to show that the tota numbe of time sots used by the above pocedue is P i =1 δpi. Since δ Pi Δ i, P i =1 δpi Δ i P i. Figue 5 shows an exampe whee P i =1 δpi is much smae than Δ i P i.againwehave n sensos and the sin distibuted on a ine P as shown in the figue. Assume that R =. On the eft side, thee ae og n nodes cose to each othe, thus thei δ(v i ) = og n except fo δ(v n og n+1 ) = og n +1. On the ight side, evey node has δ(v i )=. Thus, Δ = og n +1 and Δ P =Θ(nog n). In addition, δ P = og n +1 fo = n og n +1,,n and δ P =fo =,,n og n, δp 2 =2, and δ1 P =1. Theefoe, P =1 δp = (og n + 1) og n +(n og n) = Θ(n). It is obvious that P =1 δp =Θ(n) is smae than Δ P =Θ(nog n) in ode. Based on the new path scheduing anaysis, we now deive a tighte owe bound fo ou BFS-based method. Reca that ou method tansfes data based on banches in BFS tee T. In T, thee ae m pathes P i and m banches B i as shown in Figue (a) and (b). Then the tota numbe of time sots used by Agoithm 1 with new path scheduing is at most m P i i=1 = P i B i +1 δ Pi. It is cea that this numbe is much smae than m i=1 Δ i B i fom pevious anaysis. Notice that fo path P i ou agoithm (Line -4 in Agoithm 1) wi teminate the tansmission unti the banch B i does not have data fo cuent snapshot and switch to next path P i+1. Thus, the index is ony fom P i to P i B i +1. Theefoe, the capacity achieved by ou agoithm is at east W m Pi i=1 = P i B i +1 δp i n m Pi i=1 = P i B i +1 δp i Let Δ = n which can be deived given the BFS tee. Hee Δ is a ind of weighted-aveage of the maximum intefeence among pathes P i and banches B i in the BFS tee. We then have the foowing eationship: n Δ Δ Δ 1,. among the maximum intefeence Δ in the whoe gaph, the maximum intefeence Δ in the pathes/baches of the BFS tee, and the aveage maximum intefeence Δ in the pathes/banches of the BFS tee. These thee intefeence numbes can be diffeent fom each othe in ode. Even though W Δ is a tighte owe bound fo data coection, thee is sti a gap between it and the uppe bound W.Thus,we eave finding a tighte bound to cose the gap as one of ou futue wo. Theoem 4: Unde potoco intefeence mode and genea gaph mode, data coection capacity fo abitay senso W netwos is at east Δ and at most W. V. CONCLUSION In this pape, we study the theoetica imitations of data coection in tems of capacity fo abitay wieess senso netwos. We fist popose an efficient data coection method to achieve capacity of Θ(W ), which is ode-optima unde potoco intefeence mode. Howeve, when the undeying netwo mode is a genea gaph, we show that Θ(W ) may not be achievabe. We pove that BFS-based method can sti achieve capacity of Θ( W Δ ) fo genea gaphs. A of ou methods can aso achieve these esuts fo andom netwos. Thee ae sti sevea open pobems eft as ou futue wo. (1) We woud ie to cose the gap of uppe and owe bounds of data coection capacity fo genea gaphs. (2) Even though the capacity of data aggegation fo abitay netwos has been studied in [8], they ony conside the wost case capacity. It is inteesting to study aggegation capacity fo any abitay netwo. () Hee we focus on achieving ode-optima capacity (i.e., constant appoximation fo minimizing deay and maximizing capacity), but how to achieve optima (o nea-optima) capacity (i.e., educe the appoximation atio) is a moe chaenging tas. We eave it as one of ou futue wo. Reca that some of the pobems (e.g. minimum deay data aggegation [9]) ae NP-had. REFERENCES [1] E.J. Duate-Meo and M. Liu, Data-gatheing wieess senso netwos: Oganization and capacity, Compute Netwos, 4, , 200. [2] D. Maco, E.J. Duate-Meo, M. Liu, and D.L. Neuhoff, On the manyto-one tanspot capacity of a dense wieess senso netwo and the compessibiity of its data, in Poc. Int Woshop on Infomation Pocessing in Senso Netwos, 200. [] H.E. Gama, On the scaing aws of dense wieess senso netwos: the data gatheing channe, IEEE Tans. on I.T., 51(): , [4] R. Zheng and R.J. Baton, Towad optima data aggegation in andom wieess senso netwos, in Poc. of IEEE Infocom, [5] B. Liu, D. Towsey, and A. 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