Measuring performance of ad hoc networks using timescales for information flow

Transcription

1 Measuring perfrmance f ad hc netwrks using timescales fr infrmatin flw Raissa M. D Suza, Sharad Ramanathan, and Duncan Temple Lang Bell Labratries, Lucent Technlgies Murray Hill, NJ raissa@micrsft.cm, {sharadr,dtemplelang}@lucent.cm Abstract We define metrics t characterize the perfrmance f ad hc netwrks based n timescales fr infrmatin flw, pwer cnsumptin and interference. The statistical distributin f timescales has nt been previusly cnsidered. Yet, it is imprtant fr understanding the feasibility f cmmunicating ver such netwrks, fr cmparing different algrithms fr building up netwrk tplgy and fr distinguishing regimes f ruting. We quantify the lngest timescale fr infrmatin flw and estimate its distributin. We als intrduce a decentralized adaptive pwer algrithm, that uses nly infrmatin lcal t each device, fr building ad hc netwrks. This algrithm is shwn t perfrm significantly better by all ur metrics when cmpared with a standard, cnstant pwer, algrithm. I. INTRODUCTION Understanding hw a cllectin f wireless devices that knw nly f their lcal envirnment, can rganize int a cmmunicatins netwrk with n central cntrl is an imprtant pen prblem. Such ad hc netwrks are bth dynamic and temprary since the netwrk tplgy changes as devices mve in space, as new devices jin the regin and thers leave, and as devices turn n and ff. By sending ut queries and listening fr replies, devices can learn the identity f ther devices in their transmissin range and hence the lcal netwrk cnnectivity. Knwledge f the tplgy beynd the immediate transmissin range is cnveyed alng a sequence f intermediary devices. In this manner devices build up knwledge f their cnnectivity, ptentially string the infrmatin in address bks and ruting tables. Cmmunicatin with distant regins relies fundamentally n devices cperating in relaying ne anther s data. Thus a message may hp frm device t device when fllwing a path frm surce t destinatin. Such netwrks perate entirely thrugh peer-tpeer interactins and culd be f use in a variety f situatins frm mbile military units t a cllectin f mving cars transmitting infrmatin abut upcming rad hazards. Fr an verview f sme utstanding research issues and ptential applicatins, see fr instance [1]. What kinds f algrithms wuld individual mbile cmmunicatin devices use t build up such a netwrk amngst themselves? Hw d we cmpare the relative perfrmance f such algrithms? Perfrmance can be measured primarily by the maximum achievable thrughput given a cnstrained set f resurces. We measure this indirectly in terms f three perfrmance metrics: the pwer cnsumptin, the interference frm ther users and the time t transmit messages. All three quantities are statistical in nature and their respective distributin functins rely upn the prperties f the underlying netwrk, characterized by the user density, usage patterns, ruting strategies, netwrk tplgy, etc. Thus t measure perfrmance, ne must first carefully quantify each f the perfrmance metrics and determine techniques t estimate their statistical distributins. Then the statistical advantages f different netwrk building algrithms can be cmpared. While the pwer usage and interference are easier t quantify[2], mre care is needed in determining the statistics f a characteristic time t transmit messages, defined herein by the estimated time needed t diffuse infrmatin thrughut the netwrk. Aside frm being a perfrmance metric, the statistical distributin f this characteristic time tells us the feasibility f building such netwrks and furthermre gives insight int what size t build ruting tables and hw ften t refresh them. Cnventinal netwrks are fr the mst part static. Thus cmplete ruting tables can be built and used t efficiently direct packets. In dynamic ad hc netwrks, ruting tables expire after a perid f time since the netwrk tplgy changes with time. If ruting tables are t be used with such mbile netwrks, hw ften shuld ne refresh them t update changes in netwrk tplgy? It is knwn that if ndes are fast mving, data in ruting tables will quickly becme bslete; the spatial lcatin f the ndes will change significantly in the time it takes fr a message t hp frm surce t destinatin. In cntrast, if the ndes are slw mving, ruting tables will persist fr sme amunt f time. In the extreme limit where the ndes are statinary, ruting tables will persist fr all times. But fast and slw are nt abslute quantities. They are defined nly relative t ther timescales f interest, such as the characteristic time fr message delivery. Thus n a netwrk with a shrt characteristic time (i.e., in which data exchanges ccur rapidly), airplanes can appear t be slw mving. On a netwrk with a lng characteristic time, peple can appear fast mving. T quantify the relatins amngst the varius timescales fr mbile, wireless, peert-peer netwrks we need t understand: (1) the density f ndes and f traffic in the netwrk; (2) the relative speed f the mbile devices; and (3) the relative speed f data flw acrss the netwrk. Quantifying the distinct timescales will be especially imprtant fr netwrks with hetergeneus clients, where different types f clients may mve with greatly varying velcities.

2 The aim f this paper is t define and measure the afrementined perfrmance metrics. These metrics are then used t cmpare netwrks built by a well knwn cnstant pwer level algrithm and thse built by a decentralized adaptive pwer algrithm. We cnsider ad hc netwrks made f a cllectin f identical ndes, distributed at randm with density ρ, in a tw-dimensinal space. We intrduce a framewrk fr analyzing the timescales fr flw f infrmatin, in a dense traffic regime, in the limit where the ndes are statinary, assuming that infrmatin diffuses n the netwrk. Thugh, in practice, an ptimized scheme fr exchanging data will be used, this simplificatin f diffusin makes it pssible t easily estimate varius timescales fr infrmatin flw. We are interested especially in the lngest timescale assciated with diffusive flw, τ, as it gives an estimate f the time required t learn the full netwrk tplgy (i.e., build a cmplete ruting table) via a sequence f queries and replies which diffuse alng the netwrk. This τ is the measure we use t define the characteristic time fr infrmatin t diffuse thrughut the netwrk. The distributin f τ is btained by studying many independent realizatins f netwrks with the same fixed user density. The feasibility f building such netwrks hinges n this distributin having a small enugh variance. Past wrk has assumed the diffusin f infrmatin ver mbile ad hc netwrks t derive results n their capacity[3]. Fr such mdels t be feasible, it must be shwn that the characteristic time t deliver messages des nt fluctuate drastically if the netwrk tplgy changes. Yet this issue f feasibility and the distributin f characteristic times, thugh extremely relevant, has nt been previusly addressed. As mentined, we als use the distributin f τ as ne f the perfrmance metrics fr quantitatively cmparing different schemes fr building up an ad hc netwrk frm a cllectin f initially islated devices. The well-studied cmmn pwer (CP) level mdel fr cnstructing an ad hc netwrk[4], [5] assumes all devices transmit at the same pwer level. We intrduce a an adaptive pwer (AP) level cnstructin, which uses directinal infrmatin, similar t [6]. This is an iterative scheme where each device sets its pwer individually and adaptively, using nly lcal infrmatin. We directly cmpare the perfrmance metrics f the CP and the AP schemes. Fr each instance, the AP cnstructin has a mre efficient tplgy (and hence a smaller value fr τ), smaller verall pwer cnsumptin fr bth the typical and extremal integrated pwer, and reduced interference. In additin, since each device sets its perating pwer level individually based n the lcal envirnment, the verall netwrk can quickly adapt t changes. Furthermre, by ptimizing based n gemetric cnnectivity, rather than minimizing pwer at each nde, sme ndes perate at higher pwer than by the CP scheme. Hwever such ndes intrduce shrtcut paths thrugh the netwrk. The latter tw issues, f adaptatin and shrtcuts, make the AP scheme especially well suited fr use alng with ruting algrithms, such as ad hc n demand distance vectr ruting[7], which require cntinual executin f rute discvery algrithms. This manuscript is rganized as fllws. Sectin II, is largely an extensin f past wrk, included t clarify the prblem frmulatin. We first describe the CP apprach, then determine the minimum pwer requirements fr full cnnectivity fr an ensemble f independent realizatins f netwrks, and extract the scaling behavir as the number f ndes is increased yet their spatial density, ρ, held cnstant. Sectin III intrduces the issue f timescales and the matrix frmalism used thrughut the remainder. A cnnectivity matrix that specifies which ndes are directly cnnected t which thers is cnstructed. Thse direct cnnectins define a static graph and we study a diffusive dynamics n that graph. The dynamics can be described by a state transitin matrix P. We are in particular interested in the eigen-spectrum f P. The eigenvalues determine the timescales fr infrmatin flw, and the crrespnding eigenvectrs can be used t identify simple bttlenecks. In Sec. IV we intrduce ur AP cnstructin, and cmpare it t the CP cnstructin using the perfrmance metrics discussed abve. II. CONNECTIVITY AND THE COMMON POWER LEVEL CONSTRUCTION We first determine the requirements fr building a fully cnnected netwrk f statinary ndes, that all transmit at a cmmn pwer (CP) level. Cnsider N devices initially distributed unifrmly at randm in a tw-dimensinal space f area L L (thus the spatial density f devices ρ = N/L 2 ). The crdinates f the ith device are dented by x i, and the spatial distance between the ith and jth devices, d ij = x i x j. The pwer level f a transmissin decreases n average with distance frm its surce, s the magnitude at the surce determines the spatial range, R, ver which the signal strength will be distinguishable frm nise. The CP assumptin has analytic and practical advantages as discussed in [8]. In additin t reducing cllisins f transmissins, it ensures reciprcity if transmissins frm the ith device are perceptible by the jth, thse frm the jth are perceptible by the ith. Thus if d ij R, the devices are tw-way cnnected and exchange messages directly. If d ij >R, messages can be relayed between the tw ndes nly if there is a cnnected path f intermediaries. Ruting messages wuld be trivial if each nde bradcast at a large enugh pwer t cmmunicate directly with all ther ndes. Yet pwer is a limited resurce. Furthermre, the bradcast nature f wireless means a transmissin interferes with all ther simultaneus transmissins, having the greatest impact n thse in its range R. The desire t minimize interference and pwer cnsumptin means we want the transmissin range t be the smallest pssible while still ensuring full cnnectivity. We dente the value f this critical range by R c. With the CP scheme, all ndes bradcast at this crrespnding pwer level. This mdel was intrduced in the cntext f multihp cmmunicatin netwrks by Gilbert in 1961, and lse bunds n R c btained[4]. Recently strict bunds n R c have been btained in the asympttic limit,

3 R = 3.75 n = 1000, N = 78, ρ=0.1 Fig. 1. We illustrate a typical realizatin fr a netwrk cnsisting f N = 78 ndes at a spatial density ρ = 0.1 (hence L = 28). The ndes are initially distributed unifrmly at randm and are illustrated by the varius symbls scattered in the plane. Ndes which are cnnected, fr R =3.75 as shwn, frm a cluster and are represented by the same symbl, with slid lines illustrating the direct cnnectins. Fr this value f R, the system has c =6discnnected clusters. Frequency R c mean 90% 95% where the density ρ appraches infinite (and hence N, the number f devices, als appraches infinite)[5]. The prf uses techniques similar t thse fr perclatin thery and the thery f cverage prcesses (i.e., cvering a tw-dimensinal unit circle with disks f fixed size). We are interested, hwever, in the mre physically realistic regime where N is small and finite, and understanding hw R c scales with increasing N fr fixed ρ. We cnsider a range f finite values fr N, fr each f tw different densities, ρ =0.1and ρ =0.2, and generate n = 1000 independent realizatins f netwrks fr each set f parameters. Each realizatin is distinguished by the randm lcatins f the ndes. We determine R c fr each realizatin, as explained in detail in Sec. III-B; essentially we iteratively cnstruct a cnnectivity matrix fr a given value f R, and use the eigenvalues f the matrix t determine cnnectivity. We als determine the distributin f R c ver all these realizatins fr each {N,ρ}-pair studied. Fig. 1 illustrates a typical realizatin. R c =5.45 fr this realizatin. Figure 2 is a histgram displaying the frequencies f the different R c values bserved in the n = 1000 realizatins with N =78ndes and density ρ =0.1. The results fr ρ = 0.2 are similar, thugh rescaled, and thus nt included here. The dashed vertical lines mark respectively the mean (referred t frm here n as R c ), the 90th percentile and the 95th percentile. Nte the finite supprt indicated by the upper tail with rapidly decreasing density. This distributin in the nnasympttic regime had nt been previusly determined, yet it is imprtant. If ad hc netwrks are t be built they will cnsist f a finite number f ndes, initially starting with n the rder f tens f ndes. The infrmatin in such distributins culd be used as a starting pint fr building netwrks with prcessrs randmly distributed in space using the CP cnstructin. Mrever, having determined the distributin f R c fr a range f finite N s and different ρ s, we want t knw if Fig. 2. A histgram f the frequency with which a given value f R c ccurred fr n = 1000 realizatins f netwrks with N = 78prcessrs at density ρ =0.1. The vertical lines dente respectively the mean, R c, the 90th percentile and the 95th percentile. This gives us a clear idea f the distributin f R c in the nn-asympttic regime. this has predictive pwer: if it is pssible t extract a scaling functin fr hw the average value fr a given number f ndes and density, R c (N,ρ), varies as we vary N. Building up cnnectivity is analgus t building the minimal spanning tree f a cllectin f ndes: the minimum range fr full cnnectivity R c is analgus t the length f the lngest edge necessary t cmplete a cnnected tree. It is well knwn that the length f the lngest edge decreases frm the asympttic value as ln(n)/n, see fr instance [9]. Assuming this frm, we d a ne parameter fit f ur data fr R c (N,ρ) t the functin R c (N,ρ) = R (ρ)[1 ln(n)/n ]. Here the estimated parameter is R (ρ), the asympttic value f R c as N appraches infinite fr a given ρ. In Fig. 3 we plt R c (N,ρ) /R (ρ) fr varius values f N, fr ρ =0.1 and ρ = 0.2. The dtted line is the curve y = 1 ln(n)/n, the theretically expected behavir. It adequately describes the empirical data, capturing the general trend in a simple way. III. QUERY TRANSMISSION IN THE HIGH TRAFFIC LIMIT In additin t spatial cnnectivity requirements, tempral nes are als relevant. Fr instance, hw lng wuld it take fr the ndes t determine if they are fully cnnected? Fr each realizatin studied abve we first determine R c (wh s distributin is summarized in Fig. 2). Recall the ndes are statinary, hence the netwrk tplgy fixed. Once cnnectivity is established we cnsider a dynamics fr the flw f infrmatin n this static netwrk. The largest timescale assciated with this flw estimates the time required t learn the full netwrk

4 <R c >/R N x ρ=0.1 ρ=0.2 1 ln(n) N Fig. 3. A scaling functin fr R c(n, ρ) /R (ρ) versus N, fr tw values f ρ. The errr bars represent the standard errr ver all the independent realizatins, n. Fr the first five pints n = Fr the largest tw pints n = 100. We use this data t estimate the asympttic value R (ρ), and find R (ρ = 0.1) = 5.42 ± 0.04 and R (ρ = 0.2) = 3.85 ± The slid line is the theretically expected behavir, R c(n, ρ) /R (ρ) = 1 ln(n)/n. tplgy and thus build a cmplete ruting table (i.e., the time fr the final message carrying new infrmatin f the tplgy t be received). In additin, it sets a reference pint. Prcesses ccurring during time intervals much lnger than this can be treated as apprximately statinary during duratins f time less than r equal t this reference interval. We make the simplest apprximatin fr the dynamics: a data packet lcated n a nde may take a randm walk step frm that current nde t ne f its directly cnnected ndes. This assumptin simplifies the mdel cnsiderably and allws us t quickly and easily estimate timescales. It crrespnds t a diffusin prcess n the graph. When making this assumptin we need t understand the cnnectin between randm walks and bradcasts ver wireless channels. Bradcast cmmunicatin means a data packet culd cnceivably be cmmunicated t all neighbrs during ne transmissin. Yet ur apprximatin assumes that at any time at mst nly ne neighbr is in a state f being ready t receive the packet (the thers being ccupied exchanging messages with ther devices). And thus this crrespnds t a regime f dense netwrk traffic. In additin, wireless transmissin means simultaneus bradcasts interfere with ne anther. As discussed belw, we weight the transitin prbabilities fr the randm walk t reflect these additinal effects f interference, and thus mre accurately capture the essence f wireless data transmissin. The assumptin f packets taking randm hps t a cnnected neighbr is a wrst case scenari with regards t efficiently transmitting data. With an actual ad hc netwrk, we wuld use sme strategy fr efficiently exchanging messages. Thus we are establishing upper bunds n the time t send data in a regime with dense netwrk traffic. A. State transitin matrix We can mdel the randm walks f the data packets using a matrix frmalism, where the matrix specifies the transitin prbabilities fr the walkers. The eigen-spectrum f the that matrix tells us mdes f behavir and assciated timescales. As discussed belw, we must first establish a cnnectivity matrix, then adapt it t incrprate a simple mdel fr interference, t btain the state transitin matrix. The direct cnnectins between the ndes in the netwrk specify the elements f the cnnectivity matrix, M. If ndes i and j are directly cnnected matrix element M ij = 1, therwise M ij = 0. Nte the diagnal elements M ii = 1, s ndes are cnnected t themselves. We are cnsidering a discrete time randm walk prcess, executed synchrnusly acrss the netwrk. During each discrete update f the netwrk, each data packet will chse at randm amngst ne f these direct cnnectins (including the ne t itself) and accrdingly hp t an adjacent nde r remain statinary. Fr instance if M ij = 1, a data packet n nde i wuld have sme prbability P ij t hp t nde j during the next discrete update. We can easily mdify the cnnectivity matrix M t btain the state transitin matrix, P, specifying these P ij s. In the simplest case, the prbability t hp t any cnnected neighbr is equally weighted, as is the prbability t stay statinary: P ij = M ij / M ij = M ij /k i. (1) j Nte, k i = j M ij, is the number f direct cnnectins fr nde i (i.e., the edge degree f nde i, including the self link). S fr the equally weighted case, P ij =1/k i if M ij =1, and P ij =0if M ij =0. We intentinally allwed fr the randm walkers t remain statinary since this prvides a mechanism fr incrprating interference. We make the simple assumptin that if any nde in yur neighbrhd is transmitting, yu cannt transmit. Thus if a nde is cnnected t k i thers, it can nly transmit n average 1/k i f the time, at which pint it wuld send ut a data packet. S, viewed in terms f packets, the prbability fr a data packet lcated n a specific nde t remain statinary n that nde, P ii =(k i 1)/k i. (2) When the packet takes a step every 1/k i updates, it hps with equal prbability t any f the ther (k i 1) directly cnnected ndes; P ij,i j =1/ [k i (k i 1)]. (3) These transitin prbabilities are illustrated in Fig. 4, where we shw ne nde cnnected t k thers and the prbability t hp alng the varius links.

5 Fig. 4. Edge weighting t apprximate the effects f interference. We shw a nde f the netwrk which is cnnected t k ther ndes (including the self link). A data packet lcated n this nde wuld hp alng ne f the links during the next discrete update f the space. The prbabilities fr fllwing each link are shwn adjacent t the link. This uses the apprximatin that a nde cnnected t k ther ndes nly transmits 1/k f the time. This simple apprach f edge weighting apprximates t first rder the effects f interference. But it neglects transmissins that fail due t the hidden terminal prblem[10], and als the cnnectivity f the (k i 1) adjacent ndes. We verestimate the interference caused by heavily cnnected adjacent ndes, as they will transmit less ften than the estimated rate f 1/k i. And we underestimate the interference caused by sparsely cnnected adjacent ndes, which will transit mre frequently than estimated. Recall this is in the regime with dense traffic s ndes want t transmit as ften as pssible. A secnd rder apprximatin, taking int accunt lnger range interactins, shuld be mre accurate. We shuld still be able t use distance-based truncatin f interference effects. Evidence has been published elsewhere that such mdels with simple truncatin adequately reflect interference in wireless cmmunicatins systems[11]. We wuld like als t incrprate results fr interference frm multicasting n trees[12], and multiple antennae nise cancellatin schemes[13]. B. Timescales and mdes Given a set f nde lcatins and a value fr R, wecan cmpute the crrespnding cnnectivity matrix M, and thus als the transitin matrix P. Once P is established fr such a realizatin, we study its eigenvalues and eigenvectrs. We are interested in the mdes f behavir assciated with the dynamics described by P. By definitin if v i is an eigenvectr f P, with assciated eigenvalue λ i, P v i = λ i v i. (4) And applying the state transitin matrix t-times yields: P t v i =(λ i ) t v i. (5) Nte, we btained P by independently nrmalizing each rw f a symmetric matrix (s the dynamics described by P cnserves prbability, meaning n randm walker is created r destryed). S all the elements f P are real and less than r equal t unity, likewise the eigenvalues are all real and less than r equal t unity. Each eigenvalue λ i has an assciated eigenvectr v i. If the netwrk were cmpsed f N ndes, v i wuld be a N-dimensinal vectr. The j-th cmpnent f v i indicates the state f the j-th nde with a real number that can be psitive r negative (which we call the amplitude f the j- th nde). An eigenvalue λ 1 =1, has an assciated eigenvectr v 1 which describes a steady-state slutin t the dynamics. In steady-state, the amplitude at each nde n lnger changes with subsequent evlutin under P. If the netwrk f ndes is fully cnnected, there is ne unique steady-state slutin, s nly ne λ i =1. The remaining eigenvalues are all less than unity, hence describe decaying mdes (i.e., initialized in a state described by such a mde, the amplitudes at each nde decay). If the netwrk is nt fully cnnected, and instead cnsists f independent sub-clusters, there will be a steady-state slutin fr each sub-cluster. Using this apprach, we determine R c ;it is the smallest spatial range, R, fr which nly ne eigenvalue is unity. Fr a discussin f state-transitin matrices and their eigen-spectrum see fr instance [14]. We are interested in the timescales assciated with the decaying mdes, and in particular with the mst slwly decaying mde. We define the relaxatin time, T, fr a mde, as is standard in the physics literature, as the time fr the amplitude f the mde t decay by a factr f 1/e. Thus P T v i = v i /e. Hence the lngest timescale in ur system, τ, is the relaxatin time fr the slwest decaying mde (dented by v p ): Equivalently, P τ v p =(λ p ) τ v p = v p /e. (6) τ = 1 ln(λ p ). (7) This lngest timescale is assciated with the secnd largest eigenvalue, λ p,(i.e., the penultimate eigenvalue, which is the ne clsest t yet less than ne). We can interpret the psitive and negative amplitudes as crrespnding t tw different types f viscus fluids, and the relaxatin time as the time required fr the fluids t mix. C. Simulatin Envirnment Applying this framewrk invlves extensive numerical simulatin and statistical analysis f results. We wanted a sftware envirnment that wuld allw fr rapid high-level prttyping, accessibility t a rich cllectin f statistical tls, gd visualizatin f results, and flexibility t easily extend the mdel in the future. The R language and envirnment[15], an Open Surce implementatin f the S language develped at Bell Labratries, is a natural envirnment fr this type f simulatin. It prvides nt nly a high-level, interpreted prgramming language, but a rich cllectin f mdern statistical and graphical methdlgy, and amngst ther features, the facility t interactively mnitr simulatins as they prgress. Beynd the specifics f the mdel discussed herein, we have develped a cmputatinal framewrk which can be readily extended t mre cmplex and realistic mdels with little effrt, and int which we can embed cnventinal netwrk traffic simulatrs.

6 τ [min] = 109 t τ [med] = 604 t τ [max] = 5314 t R = R = 4.61 R = Fig. 5. Of the n = 1000 realizatins f netwrks with N = 78ndes distributed with spatial density ρ = 0.1, we illustrate the realizatins with the minimum, the median, and the maximum relaxatin times. These examples are representative f the cmmn tplgy amngst realizatins with shrt, average, and lng relaxatin times. Superimpsed n the netwrks are markers indicating the initial cnditin with the lngest relaxatin time n that netwrk, v p (the eigenvectr crrespnding t λ p). Ndes with a psitive cmpnent in v p are marked with pluses. Thse with negative cmpnent are marked with circles. D. Extremal and median behavir As mentined in Sec. II, we generated n = 1000 realizatins f netwrks fr varius values f ρ and N. We fcus n the realizatins with N = 78 ndes unifrmly distributed at randm with spatial density ρ = 0.1, and thse with N = 80 ndes and ρ = 0.2. Results fr bth densities are very similar, s nly thse fr the frmer are shwn explicitly. The cnnectivity requirement, R c, fr the ρ =0.1 realizatins cnstitute the histgram shwn in Fig. 2. Our apprach invlves slving fr the eigenvalues f a N N matrix, s we fcus n these realizatins as they are fr the largest values f N fr which we culd cnveniently gather extensive statistics. Once R c is knwn we determine the penultimate eigenvalue λ p fr each realizatin, and hence the assciated timescale τ = 1/ ln(λ p ). In Fig. 5 we single ut three f the 1000 realizatins fr ρ =0.1: the ne with the shrtest, the median, and the lngest relaxatin times. The significance f the plus signs and circles is explained in the subsequent sectin. The values f τ and f R c are included abve each realizatin. Nte that we write the time in units f the discrete time increment t. In ur simulatins t crrespnds t ne cmplete synchrnus update f the netwrk (i.e., each randm walker is updated nce). In an actual wireless netwrk t is rughly the characteristic distance, R c, divided by the data link speed. The realizatins shwn in Fig. 5 are typical. Mst f the netwrks with shrt relaxatin times have a relatively large value fr R c, and hence are highly intercnnected. Tplgically, they tend t have a densely cnnected central regin, and ne r tw ndes lcated at a large distance frm any ther nde. Accmmdating these utliers means R c is larger than average and that ndes in lcally dense regins bradcast at much higher pwer than necessary fr minimal cnnectivity. Netwrks with lng relaxatin times are nt directly cr- τ 2 = 1092 t τ 3 = 157 t Fig. 6. Higher rder mdes fr the realizatin with the maximum relaxatin time, Fig. 5(c). Again the signs f the cmpnents in the crrespnding eigenvalue are shwn by the circles and pluses. related with either large r small R c, having instead a range f values. Hwever, mst f them have a tplgy similar t that shwn in Fig. 5(c): tw main cnnected sub-clusters with little crss-cnnectivity between them. E. Simple bttlenecks and higher rder mdes The eigenvalue, λ p, determines the lngest timescale. We can als gain infrmatin frm v p, the assciated eigenvectr. The eigenvectr v p describes the initial cnditin fr the amplitudes at each nde with the lngest relaxatin time, typically having tw distinct regins, ne f psitive and the ther f negative cmpnents. In Fig. 5 we verlay n each nde a marker indicating whether its amplitude given in v p is negative r psitive. Ndes with negative amplitude are shwn by the circles. Thse with psitive amplitude are shwn by the plus signs. As mentined earlier, we can think f the circles and pluses as crrespnding t tw different viscus fluids diffusing n the netwrk, and the relaxatin time as the time required fr the fluids t mix. The ndes lcated

7 at the transitin between the regins f psitive and negative crrespnd rughly t the bttlenecks fr diffusin. We cnsider als higher rder mdes (thse with shrter relaxatin times). Fr illustrative purpses we single ut the realizatin shwn in Fig. 5(c), and shw, in Fig. 6, the mde assciated with the next tw largest eigenvalues. We label the timescale assciated with each f these mdes respectively as τ 2 and τ 3, and indicate the value abve the crrespnding figure. Fr each timescale, the netwrk divides int subclusters. Ndes within each sub-cluster wuld be able t cmmunicate within a time bunded by the crrespnding timescale. This gives sme indicatin f the range f cmmunicatin accessible within that timescale and the size with which t build a ruting table fr that particular realizatin. F. Distributin f timescales By lking at individual instances we can gain insight int ruting n particular netwrks. But we are mre interested in general principles fr ruting n ad hc netwrks with timevarying tplgies, and ultimately in ad hc netwrks made f mbile ndes. Here we are trying t establish limits in which we can treat the ndes as statinary, and mrever understand if behavirs f individual netwrks are similar t the average behavir. We want t cnnect ur study t ne f netwrks with tplgies that change in time. An imprtant questin t ask is t what extent des the lngest timescale fr diffusin f infrmatin vary ver different realizatins f netwrks? If the variance is large, and we find several instances with τ appraching infinite, building ad hc netwrks f mbile elements will nt be very feasible: as the ndes mve, the lngest timescale may jump frm finite t near infinite, meaning we may have t wait clse t an infinite amunt f time t receive ur data! One f the few previus studies f mbility in ad hc netwrks als relies n the diffusin f data[3]. They shw that the thrughput f a netwrk increases if the ndes are mbile, yet they d nt address this issue f timescales r the distributin f timescales, bth f which are critical fr determining the feasibility f their scheme. T understand the variance f timescales ver independent realizatins f netwrks, fr every realizatin discussed thus far (thse cntributing t the histgram in Fig. 2), we determine the lngest relaxatin time, τ = 1/ ln(λ p ).In Fig. 7 we shw the distributin fr the 1000 samples with ρ =0.1. (Results fr ρ =0.2 are very similar). Nte we are recrding the lngest timescale fr each netwrk at the value f R c fr that netwrk, hence many different values f R c cntribute t these plts. We are particularly interested in the upper tail f this distributin, understanding hw frequently we shuld expect large utliers t ccur. The density appears t decay expnentially, and we verlay n it an expnential density, p(x) = νe νx, where we set ν = τ (i.e., we use the empirically btained value f the average time, τ, as the parameter fr the expnential density). Alngside the histgram, we shw an expnential-quantile plt[16] cmparing the expnential density and the empirical density. The Density 0e00 2e 04 4e 04 6e 04 8e 04 1 ln(λ p ) and expnential density, ρ=0.1 q empirical q expnential ln(λ p ) Fig. 7. Histgram f the empirical distributin f lngest timescale, λ p, ver the n = 1000 independent realizatins, fr ρ = 0.1. The dtted line verlaying the histgram is the expnential density νe νx, where ν is determined by the empirical data, ν = 1/ ln(λ p). The accuracy with which the expnential distributin describes the empirical ne is shwn in the inlayed expnential-quantile plt. values f the quantiles fr the expnential density are pltted against the empirically determined quantiles. The dtted line with slpe f unity describes the situatin f exact agreement. This plt illustrates that the empirical distributin is accurately described by the crrespnding expnential ne. The departure in the tail (i.e., the highest values) is due t the difficulty in estimating tail prbabilities, and is well within the range f sampling variatin, and accunts fr less than ne-percent f all the n = 1000 realizatins studied. Since the empirical distributin is well described by an expnential ne, the mments f the distributin are small and finite. It wuld be extremely unlikely t bserve an instance f a netwrk with a value f τ τ. Furthermre this means the ntin f a timescale fr a specified ρ is a well-defined quantity, and as ndes mve in space we d nt expect majr changes in the value f τ prvided that ρ, the user density, des nt change cnsiderably. IV. DISTRIBUTED ALGORITHM FOR NETWORK CONSTRUCTION Up t nw, we have been cnsidering a scenari where all devices perate at the same pwer level and hence have the same transmissin range, R c. Thugh this assumptin has advantages and appraches ptimal in the asympttic limit[8], it is nt necessarily an efficient mdel t implement in practice. As is the case with sensr netwrks, ptimizing fr pwer usage may be the mst critical factr. Fr ther applicatins, it may be ptimizing the tplgy t increase

8 thrughput. We ultimately want t build up netwrk tplgy in a distributed manner, using a lcal algrithm that, when cmpared with the CP algrithm, reduces the average per nde pwer requirement, and increases the efficiency f the tplgy. Fr prcessrs distributed at randm, in practice the spatial density f prcessrs varies lcally. Each realizatin has dense patches f ndes and ther patches with just ne r tw ndes, as illustrated in Fig. 5. Nte that ndes in dense regins end up ver-cnnected. They culd perate at much lwer pwer, be cnnected mre sparsely, and yet still be cnnected t the entire remainder f the netwrk. The sparser cnnectivity wuld als result in less interference between simultaneus transmissins. The adaptive pwer algrithm discussed in [6] is a distributed cnstructin fr lcally setting pwer t be the minimum necessary at each nde, while still ensuring full netwrk cnnectivity. Hwever, instead f ptimizing with respect t pwer usage as in [6], we want t ptimize the efficiency f the tplgy with respect t τ, the lngest timescale. We actually want sme ndes t bradcast at higher pwer than necessary and hence have mre cnnectins than the minimum necessary. If certain ndes n the edges f the cluster bradcast at higher pwer, previusly disjint sub-clusters wuld cnnect up, allwing new paths thrugh the netwrk, which can intrduce shrtcuts and eliminate bttlenecks, resulting in mre efficient netwrk tplgies. See [8] fr a brief review f existing appraches fr tplgy cntrl via distributed adaptive pwer cnstructins. We describe ur adaptive pwer algrithm, then evaluate its perfrmance relative t the cmmn pwer level scheme using the metrics defined thus far. We shw that this adaptive apprach reduces interference and pwer cnsumptin, and generates netwrks with mre efficient tplgies. In additin t these benefits, since the ndes set their pwer levels using infrmatin f the lcal envirnment, they can adjust these levels dynamically in respnse t changes in the lcal envirnment. In cntrast, the cmmn pwer level is nt as rbust t changes: if the ndes mved in space the value f R c wuld vary. Each new value f R c wuld first have t be determined and then bradcast acrss the system. Building a fully cnnected netwrk using nly lcal infrmatin requires mre than just specifying a required number f cnnectins (as prpsed in [17]), r using maximum nearest neighbr distance. Our scheme relies upn directinal infrmatin, and is similar t [6]. It hinges upn the bservatin that ndes in lcally dense regins tend t have cnnectins distributed istrpically in all directins f space, yet ndes n the perimeter f clusters tend t have cnnectins emanating frm a small cnvex hull f cnnectivity, as can be seen in the netwrks illustrated in Fig. 5. The adaptive pwer cnstructin prceeds fr each nde independently. Each initially islated nde begins by transmitting at lw pwer, gradually ramping up until either satisfying a gemetric cnstraint n cnnectivity, as described belw and illustrated in Fig. 8, r reaching a prespecified maximum allwed pwer level. As the nde ramps up pwer it first Fig. 8. The gemetric cnstraint. If a nde is cnnected t m neighbrs, the vectrs frm the central nde t the m neighbrs divide a unit circle arund the central nde int m disjint sectrs. If the angle f each sectr is less than r equal t π, the cnstraint is satisfied. establishes a link with its clsest neighbr, then with its next nearest neighbr (prvided neither f these neighbring ndes is lcated further away than the maximum allwed range). With each new cnnectin made, the gemetric infrmatin is assessed. In general, when a nde is cnnected t m neighbrs, the vectrs frm the central nde t the m neighbrs divide a unit circle arund the central nde int m disjint sectrs. If the angle f each sectr is less than r equal t π, the cnstraint is satisfied and the nde sets its perating pwer at the current value. If any angle is greater than π the cnstructin cntinues until either the central nde makes a new cnnectin, at which pint the sectr angles wuld be retabulated and the cnstraint rechecked, r the maximum perating pwer level is reached (in which case the pwer is lwered t the level where the last cnnectin was established). The cnstructin in [6] is very similar. Hwever they use the value 2π/3, which we believe is t cnservative fr ur realizatins. In additin, we explicitly set the maximum allwed perating pwer t be higher than the minimum necessary fr cnnectivity. As the cnstructin prgresses we build a cnnectivity matrix, M, as defined in Sec. III-A. Since each nde cnstructs its cnnectivity independently f the ther ndes, we ccasinally give up reciprcity and cnstruct uni-directinal links. Thus the matrix M is nt symmetric. Fr a discussin f the benefits and tradeffs invlved with uni-directinal links see [18], and fr a prtcl level abstractin which deals with them see [19]. In the future we plan t cmpare current tplgies t thse which result when nly bidirectinal links are accepted. The maximum range fr cnnectivity was set t be 1.35 R c (where R c fr each realizatin was the ne determined by the CP cnstructin). This range was chsen as it was fund t be the smallest pssible range fr which every realizatin resulted in a fully cnnected netwrk. Nte we needed t ensure cnnectivity f all ndes fr bth transmissin and receptin f messages. This was cnfirmed by defining tw cnnectivity matrices, ne fr transmissin, M T and ne fr receptin M R as fllws, Mi,j T = M i,j i>j ; Mi,j T = M j,i i<j; Mi,j R = M j,i i>j ; Mi,j R = M i,j i<j. (8)

9 τ [min] = 135 t τ [med] = 316 t τ [max] = 561 t Y Y Y X X X Fig. 9. Netwrks resulting frm the AP cnstructin. The nde lcatins are identical t thse fr the realizatins shwn in Fig. 5. Hwever the netwrk tplgy results frm the AP instead f the CP cnstructin. The darkly drawn links are bi-directinal, the lightly drawn nes are unidirectinal. Nte the mre balanced lad sharing, and the cnnectins between sub-clusters that were almst disjint with the CP cnstructin. Such cnnectins are shrtcut paths thrugh the netwrk. In ther wrds, M T is the lwer triangular prtin f the matrix M, mirrred acrss the diagnal. M R is the upper triangular prtin f M mirrred acrss the diagnal. Using these tw cnnectivity matrices, we can define the state transmissin matrices, P T and P R, as befre and check that each f these have nly a single eigenvalue that equals ne, ensuring the netwrk is cmpsed f ne fully cnnected cluster thus we have a test fr cnnectivity. Nte, this test requires glbal infrmatin (the full cnnectivity matrix). In Fig. 9, we shw the netwrks which resulted fr ndes with the same lcatins as shwn in Fig 5, yet cnstructed with the AP instead f the CP scheme. The darkly drawn links are bi-directinal, the lightly drawn nes are uni-directinal. Nte the AP cnstructins in general have mre balanced lad sharing, and many shrtcut paths cnnecting tgether subclusters that were almst disjint with the CP cnstructin. We can quantitatively cmpare the alternate cnstructins directly with tw different perfrmance metrics, the first based n timescales, the secnd based n expected pwer cnsumptin. We d this fr all n = 1000 instances with ρ = 0.1 discussed thus far (nte, as with all ther results presented, the cmparisns fr the instances with ρ = 0.2 are extremely similar, s we chse nt t reprduce them here). Fig. 10 shws tw different cmparisns f the timescales. Fig. 10(a) is a histgram f the values f τ which result using the AP as ppsed t the CP cnstructin. The verlayed dashed line is the same expnential density pltted in Fig. 7, which accurately described the envelpe f the histgram fr the CP cnstructin. Nte the significant shift tward shrter timescales, which reflects nt nly the shrtcut paths, but als reduced interference. Fig. 10(b) is a scatterplt directly cmparing τ fr each individual realizatin generated first by the CP and then the AP cnstructin. In almst every instance except ne, the AP timescale was cnsiderably shrter. And nte that the extreme cases, with the largest values f τ under the CP cnstructin, have small values f τ with the AP. S fr these extreme instances the adaptive algrithm is especially superir under this measure. We als cmpare the relative pwer cnsumptin f the netwrks generated with the alternate cnstructins. We assume that all ndes are transmitting all the time. Furthermre, as a rule f thumb, we assume that the pwer falls ff with distance R as, P 1/R 2.5. Since we knw the value f the transmissin range fr each nde, we can thus calculate its pwer cnsumptin. In Fig. 11 we shw a scatterplt cmparing average pwer cnsumptin fr the 1000 instances with ρ = 0.1. The hrizntal axis dentes the pwer cnsumptin with the CP scheme. The vertical axis dentes pwer cnsumptin with the AP scheme. The slpe, δ =0.41. Thus fr each unit f pwer increase with the CP scheme, we expect nly a 0.41 unit increase with the AP scheme. V. DISCUSSION AND CONCLUSIONS In this manuscript we have attempted t accurately define the perfrmance metrics relevant t ad hc netwrks. They are pwer cnsumptin, interference, and the characteristic time fr message delivery. We als discuss an adaptive pwer algrithm fr netwrk cnstructin, and using these metrics, assess its perfrmance relative t a mre standard algrithm. The characteristic time, τ, required fr a message t be delivered via peer-t-peer cmmunicatin in an ad hc netwrk is a fundamental quantity. Nt nly des this timescale ptentially cnstrain the feasibility f building such netwrks, it gives an estimate f the time required t build up a cmplete ruting table fr the netwrk. Furthermre it serves as a

10 Adaptive pwer 1 ln(λ), cmpared t expnential density f Cmmn Pwer Scatterplt f timescales cmmn versus adaptive pwer Density Adaptive pwer ln(λ) Cmmn pwer Fig. 10. (a) A histgram f the values fr τ fr the same n = 1000 realizatins with ρ =0.1 discussed thus far, hwever cnstructed with the AP instead f CP schemes. The dashed line is the same expnential density pltted in Fig. 7. (b) A scatterplt cmparing each f the 1000 instances. The hrizntal axis dentes the value f τ resulting frm a CP cnstructin, the vertical axis, frm the AP cnstructin. There is ne bvius utlier where the value f τ was nt lwered by the AP cnstructin. In all ther case τ is lwered, which is especially significant fr thse realizatins with the largest values f τ under the CP cnstructin. perfrmance metric fr evaluating alternate netwrk cnstructin schemes. Yet, despite its relevance, n previus wrk has quantitatively discussed the relevance nr attempted t quantify the characteristic time. We intrduce a framewrk based n the assumptin that messages diffuse alng the netwrk. Diffusin means n strategy is used t efficiently exchange data. If any strategy fr ruting messages were used we wuld expect the value f τ t decrease. Thus the time btained by ur methd is an upper bund n the actual time. We are als interested in the distributin f this time acrss many independent realizatins f netwrks with similar user densities. This distributin gives insight int the feasibility f cmmunicating efficiently with ad hc netwrks with timevarying tplgies. In particular, if the distributin has a large variance, we wuld expect the time t exhibit large fluctuatins as the underlying netwrk tplgy changes. Instead we find the empirical distributin is well described by an expnential distributin. Hence the fluctuatins n average will nt be large and the timescale will nt change drastically if the underlying tplgy changes while the lcal user density remains fixed. We als intrduce a decentralized algrithm fr netwrk cnstructin, which is a variant f [6]. Instead f ptimizing with respect t minimal pwer, we ptimize with respect t minimizing τ, the largest timescale. Our adaptive pwer (AP) algrithm lets each device set its pwer level individually t ptimize its wn cnnectivity, using nly infrmatin f the current state f its lcal envirnment. When cmpared t netwrks generated with the standard cmmn pwer (CP) algrithm the netwrks resulting frm the AP cnstructin have mre efficient netwrk tplgies and imprved perfrmance by all three metrics (pwer cnsumptin, interference, and timescales). In additin, since the AP scheme uses nly infrmatin lcal t each device, the tplgy f the netwrk can change rapidly in respnse t envirnmental changes, such as mving users r time-varying wireless channels; the cnstructin can be iterated lcally as necessary. In fact many ruting algrithms rely upn cntinually executing rute discvery algrithms, such as ad hc n demand distance vectr ruting[7]. Using the AP cnstructin fr tplgy and rute discvery, shrtcut paths thrugh the netwrk are fund which wuld nt be fund with the CP apprach. Thus the AP netwrks shuld have a higher thrughput than the CP nes. With regards t the pwer cnsumptin metric, we assumed that all devices were transmitting at all times (i.e., a high traffic density limit). Mre accurately we culd instead estimate usage and use this t determine pwer cnsumptin. In additin, past algrithms fr adaptive pwer during usage can be verlayed nt the netwrks we cnstructed. Such algrithms are based n devices transmitting at the minimum pwer necessary t reach a specific neighbr, nt the minimum pwer fr full netwrk cnnectivity[20]. Much future wrk which fits naturally int ur framewrk invlves quantifying distinct regimes fr ruting. Fr instance, answering the questin n hw large t build ruting tables and hw ften t refresh them. We expect such answers t depend n a cmbinatin f factrs described herein, such as the density f ndes and traffic, and the relative speed f the devices with regards t the instantaneus value f the characteristic time. We nw have a way t quantify the characteristic time. Als, depending n the distance between the surce and destinatin, we expect that different strategies

11 Adaptive pwer scheme, Padp Overall pwer requirements, cmmn versus adaptive pwer slpe, δ= Cmmn pwer scheme, Pfixed Fig. 11. A scatterplt cmparing average pwer cnsumptin fr the 1000 instances, where we assume all ndes are transmitting at all times, and that the pwer decays as a functin f distance R as P 1/R 2.5. The hrizntal axis dentes the pwer cnsumptin with the CP scheme. The vertical axis dentes pwer cnsumptin with the AP scheme. The slpe, δ =0.41. Thus fr each unit f pwer increase with the CP scheme, we expect nly a 0.41 unit increase with the AP scheme. fr ruting wuld be better. Cnsider the netwrk shw in Fig. 5(c) and Fig. 6. We have illustrated the sub-clustering that ccurs fr different timescales. Clearly, based n timescale cnstraints, different ruting strategies wuld be necessary fr efficient intra-cluster and inter-cluster cmmunicatin. There are many challenges still nt addressed in ad hc netwrking. As shwn recently, the assumptin intrinsic t almst all existing mdels f wireless netwrk tplgy, that received pwer falls ff mntnically with distance frm the surce, is nt necessarily valid[21]. We are currently develping simple mdels t generate realistic wireless ftprints which can then be cupled t the CP and AP mdels. Ruting strategies need t be specifically tailred t the underlying netwrk building algrithm. Our adaptive pwer algrithm wuld require that the devices have either directinal antennae r ther means f directinal sensing[22]. Design issues fr such systems need t be cnsidered. Ad hc netwrks als prvide us with pprtunities fr intelligent nise cancellatin schemes, which have yet t be intrduced and studied. Furthermre, actual ad hc netwrks will experience nise and scattering frm the envirnment which need t be understd. Finally, recent wrk has fcused n hybrid netwrks with sme ad hc and sme base-statin cmmunicatin, alng with hetergeneus clients[23]. As mentined in Sec. I, quantifying timescales fr ad hc netwrks with hetergeneus clients is particularly pertinent. ACKNOWLEDGMENTS We wuld like t thank Suman Das, Piyush Gupta, and Albert Cerpa fr helpful cnversatins and feedback. R.M.D. thanks the Institute fr Pure and Applied Math at UCLA fr supprt and hspitality during the wrkshp n Large Scale Cmmunicatins Netwrks. REFERENCES [1] Embedded, Everywhere: A Research Agenda fr Netwrked Systems f Embedded Cmputers. Cmputer Science and Telecmmunicatins Bard (CSTB) Reprt, [2] T. S. Rappaprt, Wireless cmmunicatins : principles and practice. Upper Saddle River, N.J.: Prentice Hall, [3] M. Grssglauser and D. Tse, Mbility increases the capacity f ad-hc wireless netwrks, in Prceedings f INFOCOM, [4] E. N. Gilbert, Randm plane netwrks, J. SIAM, vl. 9, n. 4, pp , [5] P. Gupta and P. R. Kumar, Critical pwer fr asympttic cnnectivity in wireless netwrks, in Stchastic Analysis, Cntrl, Optimizatin and Applicatins, W. M. McEneaney, G. Yin, and Q. Zhang, Eds. Bstn: Birkhäuser, [6] R. Wattenhfer, L. Li, P. Bahl, and Y. Wang, Distributed tplgy cntrl fr pwer efficient peratin in multihp wireless ad hc netwrks, in Prceedings f IEEE INFOCOM, [7] C. Perkins and E. Ryer, Ad-hc n-demand distance vectr ruting, in MILCOM 97 panel n Ad Hc Netwrks, [Online]. Available: citeseer.nj.nec.cm/ html [8] S. Narayanaswamy, V. Kawadia, R. S. Sreenivas, and P. R. Kumar, Pwer cntrl in ad-hc netwrks: Thery, architecture, algrithm and implementatin f the COMPOW prtcl, in Prceedings f Eurpean Wireless Cnference, [9] M. D. Penrse, A strng law fr the lngest edge f the minimal spanning tree, Ann. Prbab., vl. 27, n. 1, pp , [10] P. Karn, MACA - a new channel access methd fr packet radi, in AARU/CRRL Amateur Radi 9th Cmputer Netwrking Cnference, 1990, pp [11] M. Liljenstam and R. Ayani, Interference radius in PCS radi resurce management simulatins, in Prceedings f the 1998 Winter Simulatin Cnference, 1998, pp [12] K. Ramanan, A. Sengupta, I. Ziedins, and P. Mitra, Markv randm field mdels f multicasting in tree netwrks, T appear in J. Appl. Prb. [13] G. Zysman et. al., Technlgy evlutin fr mbile and persnal cmmunicatins, Bell Labs Technical Jurnal, pp , January March, [14] J. R. Nrris, Markv Chains. Cambridge, U.K.: Cambridge University Press, [15] R. Ihaka and R. Gentleman, R: A language fr data analysis and graphics, Jurnal f Cmputatinal and Graphical Statistics, vl. 5, n. 3, pp , [16] S. Ktz, N. L. Jhnsn, and C. B. Read, Eds., Encyclpedia f Statistical Sciences. Wiley, [17] R. Ramanathan and R. Rsales-Hain, Tplgy cntrl f multihp wireless netwrks using transmit pwer adjustment, in Prceedings f IEEE INFOCOM, 2000, pp [18] M. Marina and S. Das, Ruting perfrmance in the presence f unidirectinal links in multihp wireless netwrks, in Prceedings f MOBIHOC, June [19] V. Ramasubramanian, R. Chandra, and D. Mssé, SRL: Prviding a bidirectinal abstractin fr unidirectinal adhc netwrks, in Prceedings f IEEE INFOCOM, June [20] B. Narendran, J. Sienicki, S. Yajnik, and P. Agrawal, Evaluatin f and adaptive pwer and errr cntrl algrithm fr wireless systems, in IEEE Internatinal Cnference n Cmmunicatins (ICC 97), [21] D. Ganesan, B. Krishnamachari, A. W, D. Culler, D. Estrin, and S. Wicker, Cmplex behavir at scale: An experimental study f lwpwer wireless sensr netwrks, [22] L. Gird and D. Estrin, Rbust range estimatin using acustic and multimdal sensing, in Prceedings f the IEEE/RSJ Intl Cnf n Intelligent Rbts and Systems (IROS), [23] T. Phan, G. Zrpas, and R. Bagrdia, An extensible and scalable cntent adaptatin pipeline architecture t supprt hetergeneus clients, T appear at The 22nd Internatinal Cnference n Distributed Cmputing Systems, July 2-5, 2002, and P. Gupta, private cmmunicatin.