Oblivious Routing with Mobile Fusion Centers over a Sensor Network

Transcription

1 Oblivious Routig with Mobile Fusio Ceters over a Sesor Network Devavrat Shah Departmet of Electrical Eg. ad Computer Sciece Massachusetts Istitute of Techology devavrat@mit.edu Sajay Shakkottai ECE Departmet The Uiversity of Texas at Austi shakkott@ece.utexas.edu Abstract We cosider the problem of aggregatig data at a mobile fusio ceter (fusor) (eg. a PDA or a cellular phoe) movig withi a spatial regio over which a wireless sesor etwork (eg., fixed motes) has bee deployed. Each sesor ode geerates packets destied to the fusor, ad our objective is to develop strategies that ca route the packets to the mobile fusor. For a arbitrary (possibly radom) fusor mobility patter over ay coected subset of the sesor deploymet area, we first derive upper bouds o the aggregatio data rate (i.e., the uiform rate regio from each sesor ode to the mobile fusor), where we allow all sesor odes to have complete kowledge of the mobility patter of the fusor. We the cosider aggregatio data rates that ca be achieved whe the mobility patter of the fusor is ukow to the sesor odes. Surprisigly, we show that for a class of mobility patters (radom mobility over coected-compositios of covex sets of the deploymet regio, e.g. radom walks over piece-wise liear sets), we ca costruct uiversal mobility-oblivious routig strategies that achieve aggregatio data rates that are of the same order as the (mobility-aware) upper boud. I. INTRODUCTION Networks of sesor devices, which have the capabilities of collectig, (wirelessly) forwardig ad aggregatig (fusig) data are of icreasig importace i a wide rage of applicatios. I this paper, we cosider the problem of aggregatig data at a mobile fusor device, such as a hadheld PDA which ca commuicate with earby sesor odes located over some geographic regio. Applicatios ivolvig such mobile fusors over a sesor field are may-fold. For example, a forema movig aroud o a civil costructio site ca use a had-held device to determie the locatio/completio status of various objects ad costructio material (e.g., pipig spools or structural steel compoets) which are sesor-ode tagged [1]. Such just-itime (JIT) maagemet which asks for small waitig or ispectio times o job sites ca help to raise productivity levels [2]. I this sceario, from the etworkig stadpoit, status data eeds to be routed from sesor odes to the forema s hadheld. However, the critical problem here is that sesor odes do ot a-priori kow the locatio of the forema. I a military cotext, a mobile soldier would eed sesor-data from fixed (say, radomly air-dropped) sesor devices over the battlefield. Agai, as i the costructio example, the locatio of the soldier may ot be ecessarily kow to the sesor odes. A aive approach to addressig the problem of commuicatig with a ukow sik (the mobile fusor) is through floodig. While floodig esures that the mobile fusor receives the iformatio regardless of its positio, the data rate that ca be supported is very poor. To see this, cosider the followig back-of-the-evelope calculatio. Example 1 Cosider a wireless sesor etwork with odes radomly placed over a uit area, with each ode havig a wireless radio of rage r() (i.e., the ode ca commuicate with all eighbors that are withi a distace of r()). Now, it immediately follows that there ca be o more tha 1/(πr 2 ()) simultaeous trasmissios at ay time (because each radio trasmissio covers a area of πr 2 ()). O the other had, to flood a sigle packet from a fixed ode to all other odes i the etwork, there eeds to be at-least 1/(πr 2 ()) trasmissios (this is because the packet has to cover the etire area i order to reach all the odes). These two observatios imply that for all odes to support a data rate of λ() each, we eed to satisfy λ() 1. Observe that this data rate is extremely low for istace, with poit-to-poit routig with kow source ad destiatio locatios, we kow from [3] that λ() = Ω(1/ ) is achievable, which is a much larger rate. A. Mai Cotributios The above discussio motivates the mai questios we address i this paper: (a) What is the best we ca do if all sesor odes possibly kow the locatio ad mobility patter of the fusor? (b) More importatly, ca we costruct oblivious routig schemes that do ot have ay kowledge of the locatio of the fusor, ad still maage to achieve data rates that are close to the upper bouds? For istace, floodig is a oblivious scheme however, the data rate with this strategy is very poor. The results that we preset i this paper are listed below. (i) We first develop a upper boud i this paper for arbitrary mobility patters. This uses cut-set ideas (i.e., the amout of data flux that ca eter ay set S i the etwork) to costruct the boud. (ii) For fusors with good mobility patters (which iclude mobility alog lies, over covex sets, ad coected X/07/$ IEEE 1541

2 Fig. 1. Two stage oblivious routig. A mobile fusor ode moves over a restricted set S to which packets have to be obliviously routed. The paths of differet packets are chose idepedetly. Note that havig two stages is critical to esure that the secod hop is uiformly radom. compositios of covex sets, ad is formally defied i Sectio II-B, see also Figures 3 ad 4), we costruct a oblivious routig strategy that achieves aggregatio data rates which are of the same order as the (mobility-aware) upper boud. The mai idea i our routig strategy is for each packet from a sesor ode (source) to be spread oly over some restricted part the etwork, such that this set cuts the mobile fusor s path. This is doe by routig alog lies to a radomly chose destiatio but alog two radom stages (see Figure 1). Two stage routig esures that the secod stage is radom eough over the etwork (ote that the first stage path is correlated for all outgoig packets from the sesor source). Our scheme esures that a sufficiet umber of secod-stage lies cross ay good mobility patter (ad hece, the ukow path of the fusor). Thus, by usig appropriate codig at the sesor source, we esure that the fusor ca recover the packets from all odes. B. Related Work There has bee much iterest i the throughput-capacity ad delay of large wireless etworks. Gupta ad Kumar [3] itroduced (radom) large-scale wireless etwork models for studyig throughput scalig i a static settig. They cosidered radomly placed odes over a regio of uit area, each with a wireless commuicatio radius that scaled 1 log() as Θ( ), ad, each ode has a associated radomly chose destiatio (i.e., radom source-destiatio pairs). They showed that for large values of, each source destiatio pair ca support a data rate (through a multi-hop relayig strategy) that scales 1 We use the followig otatio i this paper: (i) f() = O(g()) meas that there exists a costat c ad iteger N such that f() cg() for > N. (ii) f() = o(g()) meas that lim f()/g() = 0. (iii) f() = Ω(g()) meas that g() = O(f()), (iv) f() = ω(g()) meas that g() = o(f()). (v) f() = Θ(g()) meas that f() = O(g()); g() = O(f()). as Θ(1/ log ). Other related work i the cotext of static etworks with differet chael models ad routig strategies iclude [4] [8]. I the cotext of mobile odes, [9], [10] have show that a throughput capacity of Θ(1) ca be supported, by usig mobility of odes itelligetly to miimize iterferece due to packet trasmissios, however, at the cost of a large delay. The results i [3] ad [9] have motivated a large body of work that has studied the trade-off betwee throughput ad delay, both i the cotext of static ad mobile etworks [11] [18]. However, to the best of our kowledge, we are ot aware of throughput or delay results (either bouds or costructive strategies) that address routig toward a sigle mobile ode that moves oly over a restricted regio of the domai i a large wireless etwork. I the followig sectios, we provide the eeded defiitios ad summarize our mai results. I Sectio II, we formally describe the models, ad defie the problem. We the preset the mai results i Sectio II-D. Next, i Sectio III, we preset a upper boud o the aggregatio data rate at the fusor. The, we proceed to preset a algorithm (Sectio IV-A) i Sectio IV, ad show that the algorithm is (order) optimal. We fially coclude i Sectio VI. A. Network topology II. SETUP AND PROBLEM STATEMENT We are give sesor odes radomly placed over a uit area. For cocreteess we assume that the uit area is a torus, i.e. a uit square with pair of opposite sides idetified. The x-axis of the uit torus costitutes oe edge of the square, ad the y-axis of the uit torus correspods to the orthogoal directio (see Figure 3). Each sesor ode is capable of wireless trasmissio. We assume that a ode ca commuicate with ay other ode withi its trasmissio rage r(). The followig is a wellkow property for such etwork, which follows from a applicatio of Cheroff s boud ad Uio boud. Property 1 Give odes throw uiformly at radom o the uit torus, we partitio the uit torus by a grid of equal sized square tiles, with each square partitio (heceforth referred to as a cell) havig area 4 log()/. The, each cell i the torus has at least oe ode with high probability 2. Further, each cell cotais o more tha 10 log() ode w.h.p. Give the Property 1, it ca be show that the etwork is coected w.h.p. if r() = 5 log()/. Further each ode is ot coected to more tha 100 log() odes w.h.p. The results of this paper will qualitatively remai the same as log as Property 1 (ad hece its implicatio) are satisfied. However, for ease of explaatio, i the rest of this paper, we will restrict our aalysis to the radom uiform ode placemet o the torus with r() = 5 log()/. Let G = (V, E) deote the graph formed by these sesor odes, with V represetig vertices ad E represetig edges. 2 I this paper, with high probability (w.h.p.) refers to a probability at least 1 1/ 2, uless specified otherwise. 1542

3 Fig. 2. A mobile fusor ode moves over a restricted set S of the fixed sesor odes V. B. Model for Traffic, Commuicatio ad Mobile Fusor Traffic Model: All sesor odes geerate ew data at rate λ(). The data is geerated accordig to a idepedet exogeous process, which is assumed to be Poisso (however, it ca be ay friedly arrival process such as a reewal process with fiite secod momet). The data geerated by odes is to be collected by a mobile fusor. Data that is collected or absorbed by the mobile fusor disappears from the etwork (i.e. departure of data i the stadard queueig termiology). Mobility Model: We cosider a discrete-time setup (i.e., time is partitioed ito time-slots). At each time-slot, the mobile fusor ca commuicate with exactly oe sesor ode. Thus, we deote the positio of the mobile fusor by the idex of the sesor ode it ca commuicate with. We assume that the mobility patter of the mobile fusio ceter (fusor) is described by a radom walk (RW) over a subset of V, deoted by S (see Figure 2). We will assume that S is a coected subset (where two odes are coected with they are withi distace Θ( log /). The RW is arbitrary over the set S ad assumed to have a statioary distributio give by π = (π(i)) i S o S. The fusor chages its locatio alog the edges of G accordig to the RW at discrete time (i this paper, time will be measured i uits of the steps of this RW). Let p(s) deote the perimeter 3 of set S. Commuicatio Model: The mobile fusor ca upload data from the sesor ode (i.e., extract data) at its curret locatio at a rate R(). Throughout the paper, we assume that R() = ω(1), that is it scales so that R(). The precise scalig of R() will affect λ() as we shall see later i the paper. The rate required for fusor-to-sesor commuicatio is higher tha that of sesor-to-sesor commuicatio, as the fusor has to collect (at each time-slot) all the packets from a large 3 Precisely, the perimeter of set S is the legth of the shortest coected piece-wise liear closed curve (i.e. loop) that is made of lie-segmets joiig vertices of S ad geographically eclosig all the poits of S. collectio of sesor odes that have aggregated at a sesor ode. Usually, S is a small subset of V (this is the case whe it is more realistic ad questio becomes iterestig). Hece sesor odes eed to route their data to S so as to be collected by the fusor. For routig data, odes eed to trasmit data to its eighbor so as to spread it via multi-hoppig. The trasmissio is doe over a commo wireless chael. For successful trasmissio, we cosider the followig well-kow disk model or Protocol model itroduced by Gupta ad Kumar [3]. Defiitio 1 (Protocol model) A trasmissio from ode i to ode j is successful if ad oly if the followig coditios are satisfied: (1) distace betwee i ad j is o more tha r(), ad (2) ay other simultaeously trasmittig ode k is at least distace 2r() away from ode j. Whe trasmissio is successful, i ca trasmit uit amout of data to ode j. I this paper, we will use r() = Θ( log /), which is the radius of coectivity whe odes are throw at radom. A immediate implicatio of this model is that each ode ca trasmit or receive data from other sesor odes 4 at most at the uit rate. A useful defiitio about the capacity of S is give below. Defiitio 2 (Cut capacity) Give a set of odes S V, defie the cut-capacity of S as the maximal sum rate at which odes i S ca exchage (trasmit or receive) data with odes i S c (i.e. odes ot i S). Deote this by ρ(s). We state the followig result that relates the cut-capacity of S with its perimeter. Lemma 1 Cosider set S with perimeter p(s). The, uder the Protocol model with r() = Θ( log /), we have ( ) ρ(s) = O p(s). log Proof: We eed to use the followig two facts that are direct implicatio of Protocol model: (1) two odes that are farther tha 3r() caot commuicate with each other, (2) amog odes withi distace r(), trasmissio ca happe at rate O(1). Now to prove the above claim, lay dow a square grid Grid- 1 of side 2r(). Now, make a copy of this grid, call it Grid- 2, shift this grid by r() to the left ad r() dowwards. Now, ay set with perimeter p(s) cuts at most O(p(S)/r()) squares of Grid-1 as well as these may squares of Grid- 2 for the followig simple geometric reaso: to cut more tha 9 square cells of side 2r() by ay coected curve, the legth of the curve must be at least r(). It is ot hard to see that umber of odes that are i S ad S c that ca commuicate uder Protocol model with radius r() must belog to the square cells of Grid-1 ad Grid-2 which are itersected by perimeter of S. Thus, the total area i which 4 However, recall that the sesor ode ca commuicate with the mobile fusor at a higher rate of R(). 1543

4 y axis y axis l y(t)() l y(t)() T T l x(t)() x axis l x(t)() x axis Fig. 3. Example of a good pickup set. The ratio of the perimeter to the larger projectio max{l x(t )(), l y(t )()} for a good pickup set is a Θ(1) quatity. Fig. 4. Example of a good pickup set. The ratio of the perimeter to the larger projectio max{l x(t )(), l y(t )()} for a good pickup set is a Θ(1) quatity. the odes that ca exchage data betwee S ad S c belog is at most O(p(S)r()). By the Protocol model, the data rate per area Θ(r 2 ()) is uit. That is, the total rate at which data trasfer ca happe betwee S ad S c is bouded above by O(p(S)/r()). For r() = Θ( log /) as selected earlier, we have this as O(p(S) / log ). This completes the proof of Lemma 1. Fially, we defie a otio of equivalece of sets i terms of capacity. Defiitio 3 (Good pickup set) A set S of odes visited by mobile fusor is called good pickup set if there exists a collectio of cells, say T = {T 1, T 2,...}, such that the followig coditios are satisfied: (i) Coectedess coditio: The cells of T form a sigle coected compoet. Here, we call cells T i ad T j coected if they are eighbors of each other. (ii) Iclusio coditio: Sesor ode v S if v T i, for some T i T, (iii) Projectio coditio: Let l x (T ) be the legth of the projectio of T o the x axis, ad l y (T ) be the legth of the projectio of T o the y axis. Defie l(t ) = max{l x (T ), l y (T )}. The, l(t ) = Θ(p(S)). The iclusio coditio simply states that the set T cosists of cells which costitute the odes i S. The projectio coditio states that the perimeter of the set S is of the same order as the loger projectio of the set T o the coordiate axes. For istace, a collectio of odes alog a lie segmet, a smooth curve, a thi ad log set or a covex set such as a circle or ellipse are good pickup sets (see Figures 3 ad 4). A set such as a comb (see Figure 5), which has a large perimeter compared to its projectio is ot a good pickup set. C. Problem Statemet Each sesor ode geerates ew data at rate λ(). The data is geerated accordig to a idepedet exogeous process, which is assumed to be Poisso (however, ca be ay friedly arrival process). The data that is absorbed by T l x(t)() x axis l y(t)() Fig. 5. Example of a bad pickup set. Cosider the top lie of legth Θ(1) (equivaletly, Θ( / log()) cells). The, we ca have / log() dowward teeth to the comb, each of Θ(1) legth. This implies that the projectio o each of the axes is a Θ(1) quatity. However, the legth of the perimeter of the set is Θ( / log()). We disallow such shapes i our aalysis. mobile fusor disappears from the etwork (i.e. departure of data i the stadard queueig termiology). The primary goal i this paper is to idetify the maximal supportable rate λ() (for all sesors) that ca be supported by the etwork ifrastructure ad the mobile fusor so that et uabsorbed (or udeparted) data i the etwork remais fiite with probability 1. We heceforth refer to this as the stability coditio. D. Mai Results The mai result of the paper is about characterizatio of the maximal rate λ() at which data ca be trasmitted to S i a oblivious maer, i.e. whe sesor odes do ot kow the locatio/values of S, π. Specifically, our iterest is i a routig strategy that is oblivious. The precise defiitio is as follows. Defiitio 4 (Oblivious routig) A routig scheme is said to be oblivious if sesor odes that lie i the set S c do ot have ay kowledge of the locatio (either curret positio or statistical distributio) of the fusor. y axis 1544

5 The precise results are stated below. Upper boud. The followig is a straight forward upper boud o λ(). Theorem 2 For ay set S such that S /2, ( ) ρ(s) λ() = O. To support rate λ() while keepig system stable, we eed R() = Ω(λ()). Further, there exists set S supportig the maximal rate λ() = Θ(ρ(S)/), for which it is required that R() = Ω(ρ(S)) = Ω( S / log ) for rederig the system stable. We ote that the upper boud places o restrictio o the fusor locatio kowledge at sesor odes. I other words, all sesor odes ca kow the locatio/values of S, π of the fusor. Lower boud. The more surprisig result is the followig matchig (i order) lower boud via a simple oblivious routig scheme. Thus, our scheme is order optimal both i terms of maximal supportable λ() ad R() requiremet i a oblivious maer. Theorem 3 There exists a (radomized) oblivious routig ad fusor pick-up scheme so that ( ) ρ(s) λ() = Ω, ad R() = O ( S / log ), ( if S is a good pickup set with π such that π i = Θ ( ) geerally, R() = O is sufficiet. 1 log (mi i π i) 1 S III. UPPER BOUND: PROOF OF THEOREM 2 ). More The proof of upper boud follows usig simple argumets: For data geerated by odes outside S, it must eter set S from S c. Now the umber of odes outside S, i.e. S c is at least /2. Hece, the data rate that eeds eter S from outside is Θ(λ()) for ay supportable λ(). By defiitio, the maximal rate at which data ca eter S is ρ(s). Hece, the above discussio immediately imply that λ() = O(ρ(S)). Equivaletly, maximal supportable λ() = O(ρ(S)/). Further, it immediately follows that we require R() = Ω(λ()). This is because i each time-slot, the sesor odes geerate data (o average) at a total rate of Θ(λ()). O the other-had, the fusor ca pick up data at each time-slot at a rate of (at most) R(). We ext show that there exists set S (with S /2) such that for supportig this maximal rate λ(), we require R() = Ω(ρ(S)) for rederig the system stable. Now, cosider a set S cosistig of a collectio of say some M square cells (each cell has side / log()) o a horizotal lie. The the umber of odes i S is Θ(M log ). The, ρ(s) = Θ(M). The scheme of Theorem 3 will imply that we ca support rate ( ) ( ) ρ(s) M λ() = Θ = Θ. Hece, we require that R() = Ω(λ()) = Ω(ρ(S)) = Ω(M) = Ω( S / log ). This completes the proof of Theorem 2. IV. LOWER BOUND: PROOF OF THEOREM 3 To establish the proof of Theorem 3, we first preset a algorithm which we show will achieve the rate λ() claimed i Theorem 3. A. Algorithm The algorithm has the followig parts: (1) Routig of data by odes, (2) Schedulig of trasmissio, ad (3) Pick up of data by mobile fusor. These parts are described separately as follows. Routig. Each ode geerates data at rate µ r () = Θ ( 1/ log ) accordig to a Poisso process. Let the ode be i cell, say C 0. The ode seds packet as follows: first it picks two cells C 1 ad C 2 idepedetly ad uiformly at radom from all possible Θ(/ log()) cells. The, the packet is first set from C 0 to C 1 alog the straight lie joiig C 0 ad C 1 via hoppig alog the cells joiig them. After reachig cell C 1, route the packet from C 1 to C 2 i the same maer alog the straight lie 5 (see Figure 1). Durig routig, if a packet passes through a cell which cotais a ode v S, the ode v stores a copy of this packet i its FCFS queue. Whe packet reaches the cell C 2, the packet is destroyed (however its copy is stored i odes v S that were o the packet s path). Schedulig. A o-trivial task ivolved alog with routig packets i the etwork is that of schedulig packet trasmissios at cells while observig the Protocol model. There are may ways i which this is usually dealt with [3], [13]. We cosider the followig well-kow TDMA protocol. Sice trasmissio of a ode from a cell iterferes with odes i Θ(1) cells, trasmissio of cells ca be scheduled so that each cell gets Θ(1) fractio of the time to trasmit. Further, the time allocated to a cell is uiformly divided amog all of its odes. Thus, each ode gets Θ(1/ log()) fractio of the time to trasmit. The above descriptio of schedulig is sufficiet for establishig the rate feasibility. However, whe packets are of fixed size ad we wish to have costat delay o average per hop 5 Strictly speakig, we route to odes i the etwork, ot cells. However, for ease of otatio ad discussio, we will say that we route to a cell C 1 if we are route a packet to ay ode v C

6 for packets, a sophisticated schedulig is required for packet trasmissios at the odes. Such a scheme was used i [14]. A aalogous scheme ca be used i our cotext as well. We refer reader to Sectio V ad Appedix for further details. Pick up. The mobile fusor, whe preset at ode i S, picks up the head-of-lie R() packets from the FCFS queue at ode i. The fusor makes sure that it does ot pick up the same packet multiple times. We assume a simple (low-overhead) mechaism i place to esure this, which ca be doe i may ways (for example, a hadshakig mechaism at the begiig of each time-slot where list of packet IDs ca be set by the sesor ode to the fusor, based o which packets ca be selected for upload to the fusor). B. Aalysis Our objective is to show that ) the effective rate achieved by the above scheme is Ω. We ote that uder the ( ρ(s) above described scheme, the geeratio rate µ r () of each ode remais the same irrespective of the S. However, the effective rate (i.e. umber of packets reachig successfully) depeds o the properties of S. Determiig this effective rate is sufficiet to show that our scheme achieves the desired data rate. This is because sources ca do appropriate codig i the followig sese. Suppose sources fid that oly a fractio p of their packets reach the mobile fusor successfully. As we shall soo see, the probability of each packet reachig is idetical ad idepedet uder our scheme. Hece, the etwork ca be see as a erasure chael betwee each source ad the mobile fusor with the probability of erasure beig 1 p. Usig appropriate codes for a erasure chael such as MDS codes [19] or low-complexity LDPC [20] codes ca yield that the source ca effectively trasmit data at rate p. We refer reader to Sectio V for some additioal related discussio. Hece, i what follows we will be iterested i determiig effective rate at which packet reach from source to destiatio. Effective rate computatio. The computatio of effective rate is divided ito three parts. (i) First, we establish that the routig scheme is feasible, i.e., the etwork load iduced by the routig scheme ca be supported by the Θ(1) badwidth that is available at each cell. (ii) Secod, we compute the effective rate at which data eters the set S which satisfies the hypothesis of Theorem 3 uder the above described scheme. (iii) Third, we show that uder the hypothesis of Theorem 3, the fusor ca be pick up data at sufficiet rate to reder the system stable. (i) Feasibility of Routig: Uder the routig scheme described above, each ode geerates data at rate Θ(µ r ()). Thus, each cell geerates data at rate Θ(µ r () log ) give Property 1. As described i Sectio IV-A, each packet is routed i two stages. The first stage cosists of the selectio of a radom cell C 1 uiformly from all Θ(/ log()) cells ad routig packet from C 0 to C 1. I a similar fashio, the secod stage ivolves selectio of aother radomly chose cell C 2, ad routig the packet from cell C 1 to C 2. Property 2 Uder the two stage routig for each packet, the straight route legth joiig ceters of C 0, C 1 ad C 1, C 2 is Θ(1) with probability at least 0.9. Sice each route legth is at most 2, we have that a packet travels Θ(1) distace o average durig the two-stage routig. The Property 2 implies that packet makes Θ( / log ) hops o average durig the routig i two stages sice each hop makes the packet travel a distace r() = Θ( log /). Now each ode geerates packets at rate µ r () = Θ(1/ log ). Sice each packet makes Θ( / log ) hops o average ad there are odes, we require that i uittime the total umber of hops (over all flows) made is Θ(µ r () / log ), which is Θ(/ log ). By Property 1, each cell has Θ(log ) odes. Subsequetly, the symmetry of the routig strategy implies that the hops are distributed uiformly (i order). There are total Θ(/ log ) cells. Hece, we have a requiremet of Θ(1) hops o average per cell. By selectio of a appropriate Θ(1) scalig costat, this meas that we ca support the routig algorithm for a rate of µ r () = Θ(1/ log ). This proves the feasibility of routig scheme. (ii) Effective Rate to S: Give the feasibility of routig, we ext compute the effective rate at which data eters the set S. Recall that i our scheme, if a packet traverses through set S, the data is picked up by (oe or more) odes i set S. Lemma 4 For a good set S, uder the two stage routig, a packet eters a ode i S with probability at least Ω(p(S)). Proof: From Defiitio 3 of a good pickup set S, there exists a collectio of cells 6 T such that these cells (ad their odes) are iside S; l(t ) = Θ(p(S)). l x (T ) 1/6 ad l x (T ) l y (T ). That is, the whole set T ca be covered i a square of side 1/6. Give T, we ca ow draw a coected curve C that passes through cells of T ad its projectio o x axis, l x (C) = Θ(p(S)) ad l y (C) l y (T ). Let C s ad C e deote the start poit ad the ed poit of C respectively (see Figure 6). Now, cover the curve withi a square box, Bx of side at most Θ(p(S)), which has side smaller tha 1/6 by above discussio. Sice torus does ot have edge effects, we ca shift (traslatio of origi) the coordiates such that the ceter of Bx is the same as ceter of torus. Now cosider two thi horizotal strip St u, St d each of height 1/12 ad width of uit legth: St u at vertical distace 1/12 from the top of Bx above it ad St d at vertical distace 1/12 from bottom of Bx below it. Now for ay poit P St u, coect it with the ed poits of C, i.e. C s ad C e ad 6 I this proof, for ease of discussio (ad without loss of geerality), we assume that l(t ) 1/6, ad that l x(t )() l y(t )(). For, if l(t )() > 1/6, we ca always work with a coected subset of T which has a projectio that is 1/

7 1/12 1/12 St u C s T P l (T)() x St (P) ud Fig. 6. Illustratio of the costructio of the regio ST ud (P ) used i Lemma 4. exted these lies till they cross the strip St d completely. Call the regio i St d betwee these two lies itersectio as St ud (P ). From geometry, it is ot hard to see that the area of St ud (P ) = Θ(p(S)) for ay poit P. Next, we will use this to determie the probability of packet eterig S as Θ(p(S)). To this ed, ote that selectig C 1 ad C 2 for secod stage is same as choosig two poits, P, Q uiformly at radom i the torus ad the selectig the cells C 1 ad C 2, where P, Q belog to C 1 ad C 2 respectively. Goig with this probabilistic iterpretatio, if we choose P i St u ad Q i St ud (P ), the the lie joiig them will itersect the curve C sice uder our routig scheme, we are routig alog the shortest path joiig C 1 ad C 2 ; by costructio the shortest path must cross the C. If the lie cuts C, the it must cut a cell of T. That is, the packet travelig alog that lie must eter S. Thus, the probability of eterig S is at least the probability that first poit P is i St u ad Q is i St ud (P ). This is Θ(1) times Θ(p(S)), which is Θ(p(S)). This completes the proof of Lemma. Give Lemma 4, clearly the probability of each packet reachig S is at least Θ(p(S)). Sice each ode geerates data at rate µ r (), we have effective rate give as ( ) p(s) λ eff = Θ(µ r ()p(s)) = Θ. log By Lemma 1, we kow that ρ(s) = O(p(S) log /). From this ad above equality, we have C e St d 1/12 1/12 λ eff = Ω(ρ(S)/). (1) Now, by Theorem 2, we kow that λ eff ca ot be larger tha O(ρ(S)/). Thus, we have λ eff = Θ(ρ(S)/). (2) (iii) Pickup Rate of Fusor: The above two steps establish that the data ca be pushed i the set S at desired effective rate. The remaiig task is to show that it ca be picked up by the mobile fusor at a appropriate rate from the odes i S. To this ed, we compute the effective rate at which data is comig i a cell of S. The we show that mobile fusor, uder the hypothesis of Theorem 3 ca pick up data at rate higher tha this effective rate. This will complete the proof of the desired claim. Cosider a cell that cotais a ode i S. Uder the two stage routig scheme, we ow compute the effective rate at which data is passig through this cell. From the computatio doe to establish feasibility of the routig scheme, it ca be see that the et data comes ito a cell at rate O(1). Now, uder statioary distributio π such that π i = Θ(1/ S ), the mobile fusor speds Θ(log / S ) fractio of time i each cell. If we ow choose R() = Ω( S / log ), the the fusor will pick up data from each cell at effective rate of ( log R eff = Ω S ) = Ω(1). S log I this case, each cell is served at required rate ad hece we have a stable system. This completes the proof of Theorem 3. V. DISCUSSION: CODING & SCHEDULING Here we preset some more details related to the codig ad schedulig schemes that were metio i the descriptio of oblivious scheme described i Sectio IV. First, codig. As remarked earlier, codig is required because a packet set by each source reaches S (ad subsequetly is picked up by mobile fusor) with probability p idepedetly. Thus, the effective chael betwee the source ad mobile fusor ca be see as a erasure chael with erasure probability 1 p (erasures happeig idepedetly). Such a chael has bee well-studied ad may codes with excellet performace are kow. Specifically, it has bee show that by usig codes like appropriate LDPC codes [20] (ad it recet modificatios like [21], [22]), with block legth of some large N, the source ad mobile fusor ca commuicate at rate p ε N, where ε N < exp( αn) for some positive costat α. The codig ad decodig is extremely simple (roughly Θ(N log(n)) operatios). For this reaso, by computig the value of p implies the rate as stated i Sectio IV. We also refer to recet work by [23] o the use of codig i the cotext of the throughput-delay trade-off for ad hoc etworks. Now, schedulig. This essetially deals with the questio of which packet to trasmit whe a cell gets a trasmissioopportuity. The discussio i Sectio IV establishes that there exists (such as Time-Divisio-Multiple-Access (TDMA)) schedulig schemes uder which the routig scheme is supportable. However, if oe wishes to miimize (i order) the average delay experieced by packets, we eed more clever schemes. Such schemes were desiged i [14]. We describe a adaptatio of such a scheme i the Appedix. VI. CONCLUSION I this paper, we have studied the problem of aggregatig data at a mobile fusor movig withi a restricted geographic 1547

8 regio over which a wireless sesor etwork has bee deployed, ad where each sesor ode geerates packets destied to the fusor. First, we have characterized the maximum data rate that ca supported from the sesor odes to the fusor. For a arbitrary fusor mobility patter over ay coected subset of the sesor deploymet area, we have derived a upper boud o the aggregatio data rate that ca be supported at the fusor, where we allow all sesor odes to have complete kowledge of the mobility patter of the fusor. Next, we have developed a oblivious routig strategy based o multi-hop routig over two radom segmets. Usig this strategy, we have show that for a class of mobility patters, we ca achieve (order) optimal aggregatio data rates. VII. ACKNOWLEDGMENTS D. Shah was supported by NSF CAREER grat ad DARPA ITMANET Program. S. Shakkottai was supported by NSF Grats CNS , CNS ad CNS , ad the DARPA ITMANET Program W911NF REFERENCES [1] E. Jaselskis, T. Elmisalami, ad B. Stepha, Radio frequecy idetificatio taggig: Applicatios for the costructio idustry, report to the Costructio Idustry Istitute, Austi, TX, [2] L. S. Pheg ad C. J. Chua, Just-i-time maagemet of precast cocrete compoets, Joural of Costructio Egieerig ad Maagemet, vol. 127, o. 6, pp , [3] P. Gupta ad P. R. Kumar, The capacity of wireless etworks, IEEE Tras. Iform. Theory, vol. 46, o. 2, pp , [4] S. R. Kulkari ad P. Viswaath, A determiistic approach to throughput scalig i wireless etworks, IEEE Tras. Iform. Theory, vol. 50, o. 6, pp , Jue [5] L. Xie ad P. R. Kumar, A etwork iformatio theory for wireless commuicatio: Scalig laws ad optimal operatio, IEEE Tras. Iform. Theory, vol. 50, o. 5, pp , May [6] O. Leveque ad E. Telatar, Iformatio theoretic upper bouds o the capacity of large exteded ad hoc wireless etworks, IEEE Tras. Iform. Theory, vol. 51, o. 3, pp , March [7] A. Jovicic, P. Viswaath, ad S. R. Kulkari, Upper bouds to trasport capacity of wireless etworks, IEEE Tras. Iform. Theory, vol. 50, o. 11, pp , November [8] S. Subramaia ad S. Shakkottai, Geographic routig with limited iformatio i sesor etworks, i Proceedigs of Iformatio Processig i Sesor Networks, April 2005, pp [9] M. Grossglauser ad D. Tse, Mobility icreases the capacity of adhoc wireless etworks, i Proc. IEEE INFOCOM, Achorage, Alaska, 2001, pp [10] S. N. Diggavi, M. Grossglauser, ad D. Tse, Eve oe-dimesioal mobility icreases ad hoc wireless capacity, i Proc. IEEE ISIT, Laussae, Switzerlad, July [11] N. Basal ad Z. Liu, Capacity, mobility ad delay i wireless ad hoc etworks, i Proc. IEEE INFOCOM, [12] M. J. Neely ad E. Modiao, Capacity ad delay tradeoffs for ad-hoc mobile etworks, IEEE Tras. Iform. Theory, Jue [13] A. E. Gamal, J. Mamme, B. Prabhakar, ad D. Shah, Throughputdelay trade-off i wireless etworks part I: The fluid model, IEEE Tras. o Iformatio Theory, Special Issue o Networkig ad Iformatio Theory, [14], Throughput-delay trade-off i wireless etworks part II: Costat packet size, To appear i IEEE Tras. o Iformatio Theory, [15], Optimal throughput-delay trade-off i eergy costraied wireless etworks, i Proc. IEEE ISIT, Chicago, 2004, p [16] G. Sharma, R. Mazumdar, ad N. Shroff, Delay ad capacity trade-offs i mobile ad hoc etworks: A global perspective, i IEEE Ifocom, Barceloa, Spai, April Fig. 7. The torus o the left which has 16 cells ad each cell cotais at least oe ode. The circled ode i each cell acts as a relay. The correspodig queuig etwork of 16 cell-odes (servers), with each server correspodig to a cell i the wireless etwork, is show o the right [14]. [17] X. Li, G. Sharma, R. Mazumdar, ad N. Shroff, Degeerate delay/capacity tradeoffs i ad hoc etworks with browia mobility, IEEE Tras. o Iformatio Theory, Special Issue o Networkig ad Iformatio Theory, [18] R. Mada ad D. Shah, Capacity-delay scalig i arbitrary wireless etworks, Allerto Coferece o Commuicatio, Cotrol, ad Computig, [19] J. V. Lit, Itroductio to Codig Theory. Spriger, [20] R. G. Gallager, Low Desity Parity Check Codes. MIT Press, [21] M. G. Luby, M. Mitzemacher, M. A. Shokrollahi, D. A. Spielma, ad V. Stema, Efficiet erasure codes. [Olie]. Available: citeseer.ist.psu.edu/ html [22] Richardso ad Urbake, The capacity of low-desity paritycheck codes uder message-passig decodig, IEEETIT: IEEE Trasactios o Iformatio Theory, vol. 47, [Olie]. Available: citeseer.ist.psu.edu/richardso98capacity.html [23] L. Yig, S. Yag, ad R. Srikat, Codig achieves the optimal delaythroughput trade-offs i mobile ad-hoc etworks, preprit, November, [24] F. Kelly, Reversibility ad Stochastic Networks. Wiley, Chichester, APPENDIX SCHEDULING SCHEME Now, we describe the schedule scheme that decides which packets to trasmit for a cell (this is a direct adaptatio of a scheme preseted i [14]). To this ed, we ca view the etwork of cells as made of Θ(/ log ) odes, oe each correspodig to a grid-cell i the origial etwork. Call this etwork N D of N = Θ(/ log ) cell-odes. A example of such a etwork is give i Figure 7. Stadard argumets for Protocol model imply that each cell-ode i N D gets to trasmit oce i Θ(1) time-slots. For simplicity, we ormalize a time-slot so that each ode gets to trasmit twice i each (ormalized) time-slot: oe for the first stage of routig, which is used oly to trasmit packets that are beig routed i the first stage, ad the other for secod stage of routig which is used oly for trasmissio of packets which are i the secod stage of their routig. I etwork N D, each ode has its ow queue ad it is coected to four of its eighbors. Uder the routig scheme of Sectio IV, packets travel from cell odes correspodig to C 0 to C 1 (resp. C 1 to C 2 ) via multi-hops through cell odes o the straight-lie joiig them i the first stage (resp. secod stage). I what follows, we describe schedulig decisio for 1548

9 cell odes for the trasmissios correspodig to first stage routig. The schedulig for secod stage will follow i the same fashio (with a little differece described at the ed). I the first stage, each ode geerates data for all other N 1 cells with rate Θ(µ r ()/N). Also as each cell has precisely Θ(log ) odes, we have that betwee ay two cells, data is geerated at rate Θ(log µ r ()/N). Thus, we have effectively N 2 routes, with each havig rate Θ(log µ r ()/N). Sice the origial process is Poisso ad the cell C 1 selectio is at radom, we have that the arrival process for each route is Poisso as well. As argued before, the etwork is stable, i.e. each cell ode ca serve all arrivig data (with appropriate choice of costats i µ r ()). Now, if we had a cotiuous time etwork ad service disciplie beig Pre-emptive Lasti-First-Out, deoted as LIFO (PL), the sice each packet has determiistic service time (uit ormalized time-slot), the queue-size distributio at each ode would have bee product form 7 with average queue-size beig O(1) due to the stability coditio of the etwork. But this etwork is discrete ad hece we eed a differet policy which is described ext. For this, we eed some defiitios. Queuig etwork N C : Cosider a cotiuous-time ope etwork, with the same topology, routig policy, exogeous arrival process ad service requiremet for packets, as that of N D. The oly differece is that N C operates i cotiuous time ulike N D, i which though the arrival process is cotiuous time, the service happes oly at the discrete times. We have Preemptive LIFO (PL) queue maagemet at each server i N C (see [24] for more details). As argued above, this etwork has all the desirable properties: product form distributio, O(1) average queue-size at each ode ad hece O(1) average delay at each ode for a packet ad Poisso departure process. Next, will use the simulatio of such a etwork to iduce schedule for N D. Packet Schedulig i N D usig N C : We caot use the PL policy of N C directly i N D because of the followig reasos: 1) Due to the discrete time ature of the etwork N D, a packet that is geerated at time t becomes eligible for service (i.e. ext hop trasmissio) oly at time t. 2) A complete packet has to be trasmitted i a time-slot, i.e. fractios of the packets caot be trasmitted. This meas that a preemptive type of service like PL is ot allowed. To address these problems for N D, we preset a cetralized schedulig policy derived from emulatig i parallel, the cotiuous-time etwork N C with PL queue maagemet at each server. The exogeous arrivals i both N C ad N D are the same. Let a packet arrive i N C at some server at time a C ad i N D at the same server at time a D. The it is served i N D usig a LIFO policy with the arrival time cosidered to be a C istead of a D. Clearly such a schedulig policy ca be implemeted if ad oly if each packet arrives before its scheduled departure time. Accordig to our schedulig policy, the scheduled departure time ca be o earlier tha a C, whereas the actual arrival time is a D. Hece for this schedulig policy to be feasible, it is sufficiet to show that a D a C for every packet at each server. Let d C ad d D be the departure times of a packet from some server i N C ad N D respectively. Sice the departure time at a server is the arrival time at the ext server o the packet s route, it is sufficiet to show that d D d C for each packet i every busy cycle of each server i N C. I what follows, we show that for all packets i ay busy cycle of ay server, the departures i N D occur at or before the departures i N C. Lemma 5 ( [14]) Let a packet depart i N C from some server at time d C ad i N D at time d D, the d D d C. Proof: Fix a server ad a particular busy cycle of N C. Let it cosist of packets umbered 1,..., k with arrivals at times a 1... a k ad departures at times d 1,..., d k. Let the arrival times of these packets i N D be A 1,..., A k ad departures be at times D 1,..., D k. By assumig that A i a i for i = 1,..., k, we eed to show that D i d i for i = 1,..., k. Clearly this holds for k = 1 sice D 1 = A a i + 1 = d 1. Now suppose it holds for all busy cycles of legth k ad cosider ay busy cycle of k + 1 packets. If a 1 < a 2, the because of the LIFO policy i N D based o times a i, we have D 1 = a a 1 + k + 1 = d 1. The last equality holds sice i N C, the PL service policy dictates that the first packet of the busy cycle is the last to depart. Also, the remaiig packets would have departure times as if they are from a busy cycle of legth k. Otherwise if a 1 = a 2 the the LIFO policy i N D based o arrival times a i results i D 1 = a 1 + k + 1 = d 1 ad the packets umbered 2,..., k depart exactly as if they belog to a busy cycle of legth k. This completes the proof by iductio. Lemma 5 ad the already discussed property of N C establishes that the average delay per hop for a packet is O(1). Each packet travels at most O( N) = O( / log ) hops. Thus, the average delay is O( / log ). Fially, schedulig for secod stage routig at each ode is doe i the same way but with the followig differece. The packets, after they reach their resp. cell C 1, wait i a special queue till they come out i the N C etwork correspodig to the first stage routig. The, they are ijected back i the etwork for secod stage routig. This will make sure that the arrival process for secod stage routig is also Poisso ad hece same scheme will work. 7 This is due to results o product form solutios for quasi-reversible queuig etworks. See book by Kelly [24] for details. 1549