GeoQuorum: Load Balancing and Energy Efficient Data Access in Wireless Sensor Networks Jun Luo Ying He Scool of Computer Engineering Nanyang Tecnological University (NTU), Singapore Email: {junluo,ye}@ntu.edu.sg Abstract Wen data productions and consumptions are eavily unbalanced and wen te origins of data queries are spatially and temporally distributed, te so called in-network data storage paradigm supersedes te conventional data collection paradigm in wireless sensor networks (WSNs). In tis paper, we first introduce geometric quorum systems (along wit teir metrics) to incarnate te idea of in-network data storage. Tese quorum systems are geometric because curves (rater tan discrete node sets) are used to form quorums. We ten propose GeoQuorum as a new quorum system, for wic te quorum forming curves are parameterized. Our proposal significantly expands te quorum design metodology, by endowing a system wit a great flexibility to fine-tune itself towards different application requirements. In particular, te tunability allows GeoQuorum to substantially improve te load balancing performance and to remain competitive in energy efficiency. Our simulation results confirm te performance enancement brougt by GeoQuorum. I. INTRODUCTION Since teir inception, wireless sensor networks (WSNs) bear te task of intensive data collection troug teir large scales and dense deployments, wic represents a significant improvement over traditional sensing systems [1]. However, te limited energy storage of a node eavily confines WSNs ability of accomplising teir missions. Te related researc proposals to cope wit tis issue mainly focus on two objectives: namely load balancing and energy efficiency [2]. A WSN is often supposed to gater data from a large set of nodes to a particular (often small) set of nodes. Te resulting convergecast type of data transmission pattern makes te above two objectives contradict eac oter. For example, energy efficient communication protocols may lead to very unbalanced load distribution [3]. Fortunately, we will demonstrate a better tradeoff between tese two objectives under anoter data access paradigm involving in-network data storage. Tis latter paradigm also endows a great flexibility to data access: it can be performed wenever and werever needed. Here we first provide an illustration to contrast te two data access paradigms in Fig. 1. It is clear tat, wereas te convergecast collects te data at a single point, te innetwork storage replicates data at various nodes, to wic a later data query is directed. In tis paper, we focus on a particular design metodology, quorum systems, under te in-network data storage paradigm. Tis work is supported in part by AcRF Tier 1 Grant RG 32/09 and 69/07. (a) Convergecast (b) In-network data storage Fig. 1. Comparison of two data access paradigms. Specifically, data produced by sensor nodes and queries generated by uman users are bot directed to certain quorums (subsets of nodes). As te intersection between quorums are guaranteed in te design pase, users may access te sensory data witout directly communicating wit te sources tat generate tose data. Altoug quorum systems exist in distributed systems [4], we are reviving tem in te sensor networking scenarios. Moreover, our design metod, namely, geometric quorum systems (GQS), leverages on te recent developments in conformal geometry [5], and we propose GeoQuorum were te quorums are formed by parameterized curves. Tuning te parameters tat determine te quorums allows us to flexibly identify desired tradeoffs between load balancing and energy efficiency. Troug bot analysis and simulations, we furter demonstrate tat our design outperforms te existing ones in terms of bot load balancing and energy efficiency. In summary, our main contributions are: A formal definition of GQS and te related metrics. A toroug analysis of te existing quorum system designs for WSNs against te defined metrics. A general conformal geometry based quorum design metodology tat applies to WSNs wit any sape of te network areas. A specific quorum system, GeoQuorum, formed by parameterized curves, allowing a flexible tradeoff to be made between load balancing and energy efficiency. Te remaining of tis paper goes as follows. In Sec. II, we define quorum systems (in te traditional sense) and teir metrics, and we also briefly review te application of quorum systems in networked settings, in particular a recent geometrybased quorum system design. We focus on GQS in Sec. III. Starting wit te conformal geometry basics and network model in Sec. III-A, we formally define GQS in Sec. III-B, we ten propose GeoQuorum in Sec. III-C. We report te simulation results in Sec. IV and conclude our paper in Sec. V.
II. FUNDAMENTAL OF QUORUM SYSTEMS A. Basic Definitions Quorum systems represent a fundamental abstraction for coordination among te nodes of a distributed system (e.g., a set of networked nodes). In its traditional sense, a quorum system is defined upon a finite set (also termed universe) U = {u 1, u 2,, u n } of nodes. In particular, te following definition caracterizes a quorum system [4]. Definition 1 (Quorum System): A quorum system Q 2 U is a set of subsets of U suc tat every two subsets intersect. Eac Q Q is called a quorum. Given a quorum system Q, a node may coose to access a quorum by eiter writing to or reading from it. Tanks to te intersection property, a read access will find te desired data from some quorum tat stores te data written by anoter node. Taking into account te inerent asymmetry between read and write accesses, we may redefine te quorum system in a asymmetric fasion as follows [6], te earlier definition ence specifies symmetric quorum systems. Definition 2 (Asymmetric Quorum System): An asymmetric quorum system Q 2 U consists of two disjoint sets, Q R and Q W, of subsets of U, suc tat eac subset in Q R intersects every subset in Q W. Eac subset in Q R (resp. Q W ) is called a read (resp. write) quorum. B. Metrics on Quorum Systems We introduce two metrics to measure te performance of quorum systems, namely, load and robustness. 1) Load: Tis metric measures te computational load taken by individual nodes due to teir participation in various quorums. Obviously, it depends not only on ow a quorum system is constructed, but also on wat strategy individual nodes adopt to access te system. Definition 3 (Access Strategy): An access strategy S consists of an access rate λ S and a probability measure P S on Q, i.e., Q Q P S(Q) = 1. Te strategy is pure if P S (Q) = 1 for some Q Q; oterwise it is mixed. For asymmetric quorum systems, we replace Q by Q R or Q W, depending on wic access operation is under consideration. Definition 4 (Load): Te load induced by S on a node u i is l S (i) = λ S P S (Q). Q Q:u i Q Te system load induced by S on a quorum system Q is te maximal load induced by S on any node in U, i.e., IL S (Q) = max u i U l S(i). Intuitively, tis metric measures te evenness of load distribution witin te wole system: te lower te system load, te more balanced te load is distributed. 2) Robustness: As anoter important metric, robustness indicates te ability of a quorum system to cope wit node failures (viz. its fault tolerance). Many measures ave been proposed for tis metric, we coose te most straigtforward one: te size of te intersection between two quorums. Definition 5 (Robustness): Te robustness of a quorum system Q is te size of te minimum intersection between an arbitrary pair of quorums IR(Q) = min Q i Q j. Q i,q j Q For asymmetric quorum systems, Q i, Q j Q is ence replaced by Q i Q W and Q j Q R. It is straigtforward to see tat, if te system robustness is k, ten any node failures involving less tan k nodes will not affect te intersection property of te system. C. Related Work on Conventional Quorum Systems Traditional quorum systems are confined in 2D space, e.g., a grid-like design [7], [8]. Tese designs are often so rigid tat tey allow very little tunability tat adapts a system to various application requirements. To improve te flexibility of te quorum systems, probabilistic quorum systems [9] were introduced to relax te intersection rule (making it a random variable) and to leave more freedom in trading load for robustness. Interested readers are referred to [10], [6] for later developments on probabilistic quorum systems in coping wit mobility. In general, as nodes in WSNs are often static, we advocate a deterministic design for quorum systems, wile relying on oter tecniques (rater tan pure randomization) to improve its flexibility. D. Quorum Systems in A Projective Space Recently, a new design metodology for (deterministic) quorum systems was proposed in [11]. Tis metod suggests using projective map to first lift te 2D network area onto a 3D surface, a spere, ten design quorum systems on te 3D surface, and finally project te designed system back to te 2D area. As te system design done in a 3D space allows more diversity in saping te quorums, more flexible system designs become possible. Te practicality of tis design approac is backed by efficient localization mecanisms (e.g. [12]), as well as te trajectory based forwarding [13], wic may perform data forwarding along a continuous curve. Given a certain data type, two designs are proposed in [11]: 1 (a) Symmetric quorum system (b) Asymmetric quorum system Fig. 2. Quorum systems designed in [11]. We use red (resp. blue) color to indicate quorums accessed by a write (resp. read) access. For quorums in red, te corresponding geograpical as location and its antipodal point are sown. We also use pentagrams to represent te intersection between quorums, and triangles to represent te nodes tat access a quorum. 1 In te original paper, a quorum system design is termed a double ruling sceme. Te two designs we discuss ere are named double rulings retrieval and distance-sensitive retrieval.
(a) (b) (c) (d) (e) Fig. 3. Taking an arbitrary simply connected region D and a node set inside it as input (a), our design starts wit a Delaunay triangulation [14] and a construction of a closed genus-0 surface D by double covering, i.e., replicate D to D, reverse D s orientation, and glue D and D along te common boundary (b). Ten D is conformally mapped to te unit spere, suc tat te original boundary D is te equator, te red circle in (c). Note tat te input region D and its copy D are mirror reflected wit respect to te equator. Tis design allows us to use various sperical curves to form quorums, suc as circles and spirals sown in (d) and teir inverse-map in D (e). Symmetric quorum systems wit eac quorum represented by a great circle, sown in Fig. 2(a). Te access strategy for a write access is pure as te corresponding great circle is fixed by two points: te node accesses a quorum and te geograpical as of te data type. Asymmetric quorum systems wit write quorums represented by great circles and read quorums by latitude circles, sown in Fig. 2(b). Wile te access strategy for a write access is te same as te first design, tat of a read access also becomes pure, as te circle of eac read quorum is also defined by te node access te quorum and te geograpical as of te data type. In terms of quorum system design, apart from presenting euristics, no rigorous definitions and metrics are provided for te quorum systems in [11], tus no formal analysis is given to evaluate te performance of te designed system. Also, only planar curves are used to represent quorums on te 3D surface, wic significantly confines te design flexibility. III. GEOMETRIC QUORUM SYSTEMS FOR DATA ACCESS In tis section, we first introduce te geometry background and define our network model, along wit te properties and metrics of geometric quorum systems (GQS). Ten we present our asymmetric quorum systems, GeoQuorum, tat makes use of spatial curves to substantially improve te flexibility in finetuning system performance. A. Background on Computational Conformal Geometry Computational conformal geometry (CCG) is an emerging researc field spanning computer science and pure matematics [5]. Intuitively speaking, a conformal map is a function tat preserves te angles. Here we briefly discuss te CCG tools tat we will use in tis paper, wic is also illustrated in Fig. 3. Given a simply connected sape D R 2 wit boundary D and a node set inside it, we construct D as a closed surface of genus 0 (as explained in Fig. 3). We ten compute a armonic function φ mapping D to te unit spere, i.e., φ : D S 2 suc tat φ = 0 were is te Laplace-Beltrami operator. Tis map as te following promising properties: φ is conformal, tus, tere is no angle distortion; φ(d) and φ(d ) are mirror reflected wit respect to te equator; Te map applies to any 2D simply connected region D. Here we sould empasize tat te armonic map based metod as mentioned above allows us to map arbitrary simply connected region to cover te wole spere, tus avoiding various issues involved in stereograpic projection [11], suc as mapping nort pole to infinity and ence making potential quorum intersections out of te network boundary. B. Network Model and Geometric Design Basics We represent a WSN by U, wit u i U being a sensor node. U also serves as te universe upon wic a quorum system can be defined. 1) Geometric Model of WSNs: We apply te tool discussed in Sec. III-A to map te network area to a spere of unit radius. For te reverse projection, any curve tat passes across te equator as its upper and lower sections projected separately to te two network areas. Ten two (projected) sections are combined to get te projection on te original network area, as sown in Fig. 3(e). Tis improved map allows us to perform geometric analysis on te wole spere surface. 2) Geometric Quorum Systems: We extend te conventional definitions for quorum systems (presented in Sec. II-A) to geometric quorums system (GQS). Definition 6 (Geometric Quorum System): A GQS Q is a set of curves in space A (U A), suc tat every two curves intersect. Eac curve in Q defines a quorum. Te definition for asymmetric quorum systems is omitted; one simply splits Q into Q W and Q R, and intersection is only required between te two sets. We keep using te same definition for access strategy (Definition 3), load (Definition 4), and robustness (Definition 5). Te load taken by a node is te energy consumption for it to transmit te data (for write) or queries (for read) to quorums in wic it involves. Unlike traditional distributed systems, te energy efficiency (or total energy consumption of te wole WSN) is also a major concern of WSNs. Let M(Q) = {u A u Q} be a measure of te total energy consumption of a quorum Q, we furter define a metric to measure tis performance aspect.
Definition 7 (): Te total load induced by S on a certain quorum Q is IL T (Q) = Q Q λ S P S (Q)M(Q). In general, eac node may take a different access strategy. To simplify te analysis, we only distinguis between two types of strategies, namely S R and S W for read and write respectively. C. GeoQuorum: System wit Spatial Quorums Our analysis in [15] sows tat and ave (i) limited robustness (IR(Q) = 1), (ii) unbalanced load distribution (large IL S (Q) due to te pure access strategy), and (iii) no flexibility to be fine-tuned, ence tey cannot adapt to different access rates. In tis section, we present GeoQuorum as a new design. GeoQuorum makes use of spatial curves to form quorums, ence allows a great deal of freedom in finetuning te system performance. GeoQuorum is an asymmetric quorum system, wit write and read quorums formed by different type of curves. Specifically, we ave write quorums are formed by circles wit adjustable radius R W tat can be tuned according to te access rate. read quorums are formed by a special sperical spiral, defined as follows: x = cos (θ + θ 0 ) cos φ, y = sin (θ + θ 0 ) cos φ, z = sin φ were φ = aθ and φ [ π 2, π 2 ]. Let α be te angle (wit respect to te spere center) between two consecutive loops ( θ = 2π), we ave α = 2aπ. Te parameter a is determined by R W and te required robustness. access strategy is mixed: a write quorum is randomly cosen among all circles passing troug te node tat executes a write access; a read quorum starts from te node tat executes a read access and ends at its antipodal point, wit a randomly cosen θ 0. We illustrate suc a quorum system in Fig. 4(a). Note tat u i 2a (a) GeoQuorum (b) Proof of Proposition 1 Fig. 4. Geometric quorum system designed using spatial curves in 3D. te current design is based on te assumption tat λ W > λ R ; oterwise we adopt a dual design were we swap te write and read quorums. We first sow te relation between R W and a by te following proposition. Proposition 1: If R W kaπ, a (0, 0.5), ten te robustness of GeoQuorum is at least 2k. An intuitive explanation is sketced in Fig. 4(b). Given a certain robustness requirement, we ave a one-to-one correspondence between R W and a: R W = kaπ, as coosing te 2R W u i smallest circle minimizes te incurred system and total load. Under te assumption tat λ W > λ R, we may coose to tune R W according to λ W /λ R (te iger te ratio te smaller R W is), ten we matc a to R W based on te required robustness. Due to te use of mixed access strategy and te parameterized design, GeoQuorum can be tailored to meet te application requirements, suc tat bot system load and total load can be reduced; wic we will sow in Sec. IV-B. Interestingly, our design includes and as special cases. We refer reader to [15] for detailed discussions. IV. SIMULATIONS We ereby use simulation results to confirm te advantages of GeoQuorum over te existing designs. A. Simulation Settings We randomly put nodes in a square area (scenarios wit irregular areas are omitted due to space limitation). Ten we use Delaunay triangulation to generate te connectivity grap. If a quorum (curve) passes troug a triangle, all te tree vertices are carged wit a unit of communication load. Tis stems from te broadcast nature of wireless communication and te need for local coordination in te trajectory based forwarding. We assume WSNs wit 5000 nodes. Tere is one data type, 500 nodes are contributing to it and 100 nodes may query it. We normalize te data query rate to 1 and vary te data production rate r to test te system performance. Note tat te actual write and read access rates (to a quorum system) are 500r and 100, respectively. Suc an asymmetry between data production and consumption is reasonable, as oterwise multiple convergecasts may lead to better performance. For eac value of r, we obtain simulation results for 10 WSNs and sow te mean value and, if necessary, te standard deviation. B. Comparing GeoQuorum wit Existing Designs We compare GeoQuorum wit and. To make tem comparable, we apply te respective quorum designs to our CCG design space. For GeoQuorum, we set R W = 0.2π and a = 0.2. We first compare te system load of te tree quorum systems in Fig. 5(a), ten teir total load in Fig. 5(b). We also illustrate te actual load distribution in Fig. 5(c)-(d); te load distribution of is omitted, as it differs from only by about 1% to 2%. Te following observations are immediate from tese figures: Compared wit GeoQuorum, te load distributions of and are very unbalanced, exactly due to te existence of a as location and its antipodal point. GeoQuorum incurs a muc lower total load compared wit all oter tree systems, due to its adaptivity to te asymmetry in data production and consumption. C. Tuning te Load and Robustness of GeoQuorum We sow te performance of our GeoQuorum under parameter fine-tuning in tis section. We first tune te spiral parameter a from 0.025 to 0.3 wile increasing R W proportionally to maintain te same robustness. Te results on system and total
System Load 10 3 >500 GeoQuorum 4 5 6 7 8 9 10 Data production rate r (a) 10 5.7 10 5.6 10 5.5 10 5.4 GeoQuorum 4 5 6 7 8 9 10 Data production rate r (b) Te robustness of GeoQuorum can be tuned by canging a but keeping R W constant. As sown by Proposition 1, te robustness is tuned at a granularity of 2 under current setting; toug fractional granularity can be acieved troug randomization. Of course, increasing robustness comes at a cost of an increased total load, and we sow te relation between robustness and total load by Fig. 6(c). We consider two cases were R W = 0.6π and R W = 0.3π. Wen we tune a to linearly increase te robustness from 2 to 10, te total load increases by (rougly) following a power law. 400 300 200 100 (c) (d) GeoQuorum Fig. 5. Comparing GeoQuorum wit and. load are plotted in Fig. 6(a) and 6(b), respectively. We only sow mean values, as te standard deviations are too small to be discerned (partially due to te load balancing effect brougt by GeoQuorum). System Load 10 2.8 10 2.7 10 2.6 10 2.5 10 2.4 r = 4 r = 6 r = 8 r = 10 (a) 0 0.05 0.1 0.15 0.2 0.25 0.3 Spiral parameter a 9 x 105 8 7 6 5 4 3 2 1 Fig. 6. R W = 0.6π R W = 0.3π (c) 10 5.7 10 5.6 10 5.5 10 5.4 0 2 3 4 5 6 7 8 9 10 Robustness (b) Tuning te system performance. r = 4 r = 6 r = 8 r = 10 0 0.05 0.1 0.15 0.2 0.25 0.3 Spiral parameter a In general, one always as to make a tradeoff between load balancing and energy efficiency. Te tunability of GeoQuorum allows us to make different tradeoffs upon different application requirements. For example, wen te data production rate is low (r = 4), a (0.75, 1.5) appears to acieves a balanced performance in bot system and total load. Tis region sifts towards smaller values wit an increasing r. For r = 10, a is better to be around 0.05. Te flexibility of freely tuning te system performance is one of te major advantages of GeoQuorum over te existing designs. V. CONCLUSION We ave investigated te issue of data access in WSNs, aiming at balancing (communication) load distribution wile maintaining energy efficiency. Specifically, we ave revived te application of quorum systems in WSNs, and proposed te concept of geometric quorum systems based on a new development in combining computational conformal geometry wit sensor networking. In particular, we ave proposed GeoQuorum tat makes use of parameterized spatial curves to form quorums, suc tat te system performance can be finetuned to meet different application requirements. Troug bot analysis and simulations, we ave confirmed te advantages of GeoQuorum over existing proposals. REFERENCES [1] I. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, A Survey on Sensor Networks, IEEE Communication Mag., vol. 40, no. 8, pp. 104 112, 2002. [2] J. An and B. Krisnamacari, Fundamental Scaling Laws for Energy Efficient Storage and Querying in Wireless Sensor Networks, in Proc. of te 7t ACM MobiHoc, 2006. [3] J. Luo and J.-P. Hubaux, Joint Sink Mobility and Routing to Increase te Lifetime of Wireless Sensor Networks: Te Case of Constrained Mobility, IEEE/ACM Trans. on Networking, vol. 18, no. 3, pp. 871 884, 2010. [4] D. Malki and M. Reiter, Byzantine Quorum System, Springer Distributed Computing, vol. 11, no. 4, pp. 569 578, 1998. [5] X. Gu, F. Luo, and S.-T. Yau, Recent Advances in Computational Conformal Geometry, Comm. Inf. Sys., vol. 9, no. 2, pp. 163 196, 2009. [6] J. Luo, P. Eugster, and J.-P. Hubaux, Pilot: Probabilistic Ligtweigt Group Communication System for Ad Hoc Networks, IEEE Trans. on Mobile Computing, vol. 3, no. 2, pp. 164 179, 2004. [7] M. Naor and A. Wool, Te Load, Capacity and Availability of Quorum Systems, SIAM J. on Computing, vol. 27, no. 2, pp. 423 447, 1998. [8] Z. Haas and B. Liang, Ad Hoc Mobility Management wit Uniform Quorum Systems, IEEE/ACM Trans. on Networking, vol. 7, no. 2, pp. 228 240, 1999. [9] D. Malki, M. Reiter, and A. Wool, Probabilistic Quorum Systems, Elsevier Info. and Comp., vol. 170, no. 2, pp. 184 206, 2001. [10] J. Luo, J.-P. Hubaux, and P. Eugster, PAN: Providing Reliable Storage in Mobile Ad Hoc Networks wit Probabilistic Quorum Systems, in Proc. of te 4t ACM MobiHoc, 2003. [11] R. Sarkar, X. Zu, and J. Gao, Double Rulings for Information Brokerage in Sensor Networks, in Proc. of te 12t ACM MobiCom, 2006. [12] J. Luo, H. Sukla, and J.-P. Hubaux, Non-Interactive Location Surveying for Sensor Networks wit Mobility-Differentiated ToA, in Proc. of te 25t IEEE INFOCOM, 2006. [13] D. Niculescu and B. Nat, Trajectory Based Forwarding and its Applications, in Proc. of te 9t ACM MobiCom, 2003. [14] J. Sewcuk, Triangle: Engineering a 2D Quality Mes Generator and Delaunay Triangulator, in Springer LNCS 1148, 1996, pp. 203 222. [15] J. Luo and Y. He, GeoQuorum: Load Balancing and Energy Efficient Data Access in Wireless Sensor Networks, 2010. [Online]. Available: ttp://arxiv.org/abs/1101.0892