Characterizing the Scaling Capacity of MultipleGateway Access in Wireless Sensor Networks

Panlong Yang, Chao Dong, Hai Wang, IEEE MemberInstitute of Communication Engineering

P.L.A University of Science and Technologyplyang@computer.org

Guihai Chen, IEEE MemberState Key Laboratory for Novel Software Technology

Dept. of Computer Science, Nanjing Universitygchen@nju.edu.cn

Abstract— We study the scaling capacity of wireless sensornetworks in case of multiple gateways are presented. In sucha heterogeneous network, many gateways are needed to gatherdata from the sensor nodes in network. Different from thepreviously studied single-gateway model, our proposed gatewayaccess model allows multiple gateway nodes being seamlesslyconnected with high speed wire-line or wide-band radios. Oncethe data from the sensor node is sent to any gateway node, itcan be collaboratively sent to the appropriate sink node amongmultiple gateways, which could effectively avoid throughputreduction due to multi-hop routing among the sensor node. Thismechanism is called “Destination-hub”, which could ensure thebounded time delay and network capacity. However under sucha cooperative scheme, two research issues need to be tackled.One is the number of nodes in network and the second is thescaling capacity for each node when multiple gateway nodes arepresented. Our proposed model is a fundamental research workfor scaling capacity, which could also provide an instructiveproposal for future gateway deployment. In this paper, wecharacterize the scaling capacity between gateways and sensornodes under different gateway access model. If gateway nodes arerandomly deployed, the number is Ω(log n), and there are at least4 log l gateway nodes are needed in each squarelet region so as tokeep the gateway connectivity, where r is the transmitting rangeof sensor nodes, and l is the side length of the deployed squareregion. And we can conclude that, the gateway nodes can betightly bounded by Θ(log n). For random placement, the scalingcapacity for such a scheme is Θ(n). And we have also proposedan optimized scheme for reducing the number of gateway nodes.Considering the interference Voronoi cell, the reduced gatewaynodes can be 2 ln l, and still bounded by Θ(log n).

I. INTRODUCTION

Capacity scaling technology is one of the most fundamentalresearch issues in wireless networks, which would promotethe efficiency of large-scale self-organizing wireless networkdeployment. The fundamental problem of capacity scaling isthe limit of network capacity as number of nodes in networkn trends to infinity. In order to evaluate the throughput ina randomly deployed large scale network, a lot of seminarworks on asymptotic capacity for scaling network have beenproposed. X.Y-Li et al’s work [4] unify the previous capacitybounds on unicast by Gupta and Kumar. Also, the capacitybounds of broadcast is presented in [1] [6]. At the sametime, store-carry-forward (SCF) mechanism is proposed andanalyzed in [2], which assumes infinite buffers on each node,and the end-to-end packet delay could be intolerable as numberof nodes trends infinity. Although the SCF mechanism could

improve network throughput in case of nodes’ mobility [2],in real applications for gateway access, few of them couldtolerant to infinite end-to-end delay due to limited buffer sizeand time critical applications. Indeed, almost no applicationcould be tolerable to infinite time delay.

We believe that, with the advances on wireless technology,emerging applications such as infrastructure monitoring andhealth-care surveillance systems need more gateway nodesfor bounded time delay and improved network throughput.Multiple gateway system could successfully avoid the prob-lem of single-point-failure; however, problems naturally ariseas to compute the throughput between gateway nodes andconventional nodes in network. The first is, how to build anefficient gateway access model. In a heterogeneous wirelessnetwork with multiple gateways, nodes intending to send datawould inevitably determine which gateway should be selectedfor transmission. And gateway access selection criteria andthe related methods would affect network accessibility andthroughput dramatically. The second problem we have to dealwith is to accurately estimate the available network capacity.Although in heterogeneous wireless network, multiple gate-ways would effectively balance between gateway nodes, it isnot the case when the data traffic load or number of users isovercrowded. The network access efficiency would possiblydrop dramatically. Conventional gateway access mode alwaysassumes one gateway which would suffer from the single pointfailure, and would very likely lead to network congestion.

In this paper, we propose a new access model named“Destination hub”, where nodes in network can distribute anypart of data segments to any gateway nodes, and the packetscan be reassembled at the gateway side intelligently. Due to thecoordination between gateway nodes on packet level, it worksas if gateways are connected via a ethernet hub. Under thismodel, we are focusing on characterizing our capacity scalingproblem for multi-gateway access. We consider the in-networkgateway placement, where gateway nodes can be placed inany place of the closed network area. Although methods oncharacterizing available capacity have been widely studied,computing achievable network throughput between gatewayand conventional nodes is different. Two problems need tobe properly dealt with. One is the number of gateway nodesneeded in achieving connected destination hub and guaranteedtime delay, and the other is the scaling capacity for each sensor

2009 Eighth IEEE International Conference on Embedded Computing; IEEE International Conference on Scalable Computing and Communications

978-0-7695-3825-9/09 $26.00 © 2009 IEEE

DOI 10.1109/EmbeddedCom-ScalCom.2009.80

415

2009 Eighth IEEE International Conference on Embedded Computing; IEEE International Conference on Scalable Computing and Communications

978-0-7695-3825-9/09 $26.00 © 2009 IEEE

DOI 10.1109/EmbeddedCom-ScalCom.2009.80

415

2009 Eighth IEEE International Conference on Embedded Computing; IEEE International Conference on Scalable Computing and Communications

978-0-7695-3825-9/09 $26.00 © 2009 IEEE

DOI 10.1109/EmbeddedCom-ScalCom.2009.80

415

International Conference on Scalable Computing and Communications; The Eighth International Conference on Embedded Computing

978-0-7695-3825-9/09 $26.00 © 2009 IEEE

DOI 10.1109/EmbeddedCom-ScalCom.2009.80

415

node.Our study differs from previous research works in two folds.

One is that, we propose a newly built gateway access model,which could efficiently improve network throughput with guar-anteed time delay. The other is that, we characterize the scalingcapacity in different gateway placement modes. To the best ofour knowledge, ours is the first work on scaling capacity asmultiple connected gateways are presented. Capacity scalingproblem is modeled and analyzed properly according to ournewly proposed gateway model. Contributions of this paperare listed as follows:

1) We present a newly built gateway access model, wheregateway nodes can be connected seamlessly, and sensornodes can access any gateway for data collection as ifthey were connected in a destination hub.

2) We have proved that, the gateway nodes deployed in aclosed square region is tightly bounded by Θ(log n).

3) For reducing the number of gateway nodes and improvenetwork capacity, we reduce the number of gatewaynodes in half, which is 2 ln n, and still tightly boundedby Θ(log n).

The remainders of this paper are organized as follows:In section II, we make a literature review on related worksfor capacity scaling technology. In section III, we formulateour problem in mathematical models, where the network sys-tem model, destination-hub mechanism and scaling capacityproblems are properly addressed. In section IV, we makea thorough analysis on capacity of such a heterogeneousnetwork. In section V, we conclude our paper and point outthe future directions.

II. RELATED WORK

Asymptotic capacity of scaling wireless networks is a hotresearch topic in recent years. The ground-breaking work ofGupta and Kumar [1] present the unicast capacity in twoplacement models. As the nodes are randomly deployed, andthe scaling network capacity for each node could achieve ashigh as Θ( W√

n log n), where n is the number of nodes, and W

is the full transmitting rate for each node. If the node can beoptimally placed and the transmission range can be chosenin need, the scaling network capacity can achieve as highas Θ( W√

n). These fundamental research works assume each

node in network is equal, while in hybrid networks, ordinarynodes and base stations are all presented for data collections.[8] propose a multicast throughput for hybrid networks ask − 1 nodes with transmit range r are randomly selected inan area with side length a. The total multicast capacity isO(

√n√

log(n)·√

mk ·W ), where k = O( n

log n ), k = O(m), k√m→

∞, and m. And the paper also concluded that, the minimummulticast capacity is upper bounded by O( r·n

a · √m · Wk ) and

is at least Ω(W ). When k, the multicast capacity is Θ(W ).As in real network applications, the gateway nodes could notbe randomly selected as needed and the number of gatewaynodes should be considered for application requirements aswell.

The most similar work to this paper is B. Liu et al’s work[7] and U. Kozat et al’s work [6]. The differences of our workfrom B. Liu’s work are two-folds. One is that, we do not de-liberately increase the gateway nodes for network throughputimprovement. The number of gateway nodes are sufficient forconnecting all gateway nodes into a destination hub. The otheris that, we increase number of gateway nodes for applicationlevel requirements, such as the minimum bounded throughput,minimum delay etc. Comparing with the studies of U. Kozatand L. Tassiulas, our work does not assume the connectivityof N nodes in ad hoc network. Moreover, for each gatewaynodes, we do not bound the number of accessing sensor nodes.

Mobility is another important factor affecting networkthroughput. Grossglauser and Tse’s work [2] show the capacityimprovements with mobility, which would achieve a constantnetwork throughput with the increasing number of nodes.However, such a network would possibly suffer from theinfinity time delay. Also, we do not use Gaussian model inthis paper, because in our study, each gateway node can onlyaccept limited number of nodes for access.

III. PROBLEM FORMATION

In this section, we introduce our system model mathemati-cally. We begin with an introduction on interference model andour proposed destination hub model, which lay a foundationfor our analytical work. Gateway deployment is another keyfactor need to be investigated and we formulate this problemconsidering different scenarios with placement constraints.

We suppose that there are m gateways deployed in networkregion Ω, where Ω is a square region with side length a.Each node vi in network is in node set V , where V =v1, v2, ..., vn and each gateway node gi is in gateway set W ,where W = g1, g2, ..., gm. As mentioned in X.Y-Li’s paper[], random network model are categorized into three types.The first type is of fixed Region with increasing number ofnodes; the second type is of fixed network density and thethird type is of fixed Transmission Range.

A. Interference Model

We adopt the protocol model [3] as our proposed interfer-ence model, because the RTS/CTS model is not applicablein case of heterogeneous network, where transmission powerlevels between nodes are different. Distance between nodes vi

and vj . For the bidirectional communication between nodes vi

and vj , the following conditions should be satisfied:

1) dij(t) < minRiT , Rj

T , where RiT is the transmission

range of node vi.2) dkj(t) > (1+∆)maxRi

T , RjT , which means that, the

distance between node vk and vj should be greater thanthe maximum transmission range of node vi and nodevj .

We mainly consider the protocol interference model in ournetwork model, if and only if the two conditions are allsatisfied. In this paper, we assume that, Ri

T

RjT

= 1, and ∆ = 0.

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Fig. 1. Destination Hub based gateway access model

B. Destination Hub

As shown in Fig. 1, sensor nodes are randomly deployed innetwork, and each node has at least one traffic flow towardsthe gateway nodes . As gateway nodes are connected viawired-internet, it looks as if they are connected on a hub, anddata packets on any gateway nodes would. According to thedestination hub character, the data segment from source nodeto any . Data segments can be on different paths, and on thegateway/destination side, data segments are resembled togetherin achieving an integrated data package. Under the gatewayhub mechanism, data transmission model Problem to be solvedin. Actually, gateways can also be connected by relative highspeed (Comparing to conventional nodes in network) wirelesschannels.

C. Network Capacity

Capacity Definition: We denote the throughput vector−→λ

as a collection of transmission rate of each node in network,and

−→λ = λ1, λ2, ..., λn. The scaling network capacity is

the maximum network throughput as the number of nodes innetwork trends to infinity. We claim that the capacity per nodeof a random networks is of order Θ(f(n)), if there are twoconstants c and c′, where c < c′, such that:

limPr(λ(n) = c · f(n) is feasible) = 1

limPr(λ(n) = c′ · f(n) is feasible) < 1

We use a square region with side length l for our capacityanalysis, and the network density is constant with the increas-ing number of nodes in network.

IV. CAPACITY ANALYSIS

A. On Random Placement of Gateway Nodes

We firstly make attempts on bounding the capacity of“In-network placement” scenario, where gateway nodes aredeployed in the inner parts of the closed network region Ω.

Fig. 2. Illustration on the grid placement for each squarelet

Firstly, we assume that, the number of gateway nodes is notlimited. A question naturally arises that, how many gatewaynodes are needed in creating a connected gateway destinationhub, and how about the scaling capacity? Before answeringthis question, we give the following lemmas.

Lemma 1: Assume that there are n nodes distributed uni-formly and independently at random in R = [0, l]d, each withtransmitting range r, and we also assume that rdn = kld ln l.For some constant k > 0, with r = r(l) ¿ l and n = n(l) Àl. If k > d · kd or k = d · kd and r = r(l) À l, then the

communication graph is connected w.h.p., where kd = 2d ·dd/2Proof: The proof is referenced from et al’s work [].

Lemma 1 provides a necessary condition on a connectedgraph where nodes are connected in a d-dimension region. Inthis paper, the region is 2-dimension, and obviously d = 2.

Lemma 2: In a two dimensional region network, if a com-munication graph is connected, the number of nodes in net-work is at least 16

(lr

)2ln l.

Proof: Proof: As the sensor nodes are deployed in a plainregion, let d = 2, we can easily get this result according tolemma 1.

There are totally M =⌈

l2

( r√2)2

⌉squarelets in network

region, on average each squarelets has 16 ln l gateway nodes.For each squarelet, w.h.p, the number of nodes can connect tothe gateway node without relay.

Lemma 3: If m balls are thrown into b bins independently,and uniformly at random, then

Pr(any bin has >2m

bballs) ≤ b · exp(−m

3b)

Lemma 4: Chernoff Lower Tail Bound. Let X1, X2, ..., Xn

be independent Poisson trials, where Pr[Xi = 1] = p. Let

X =n∑

i=1

Xi, then for 0 < β < 1. We have

Pr(X ≤ (1− β)E[X]) ≤ exp(−β2

2E[X])

According to lemma 3 and 4, we can easily find that, W.h.pthe number of gateways in a squarelet is at least 4 ln l.

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Fig. 3. Connected dominate set for arbitrary gateway placement

As shown in Fig. 2, in each squarelet, there are gatewaynodes labeled with the black color, which is lower boundedby 8

(lr

)2ln l. And the sensor nodes randomly deployed in

each squarelet would access the gateway at the same squarelet.As the number of gateway nodes for each squarelet is lowerbounded, the throughput for such a network is subjected bythe number of sensors in each squarelet.

If we set β = 12 in lemma 4, we have Pr(X ≤ 1

2E[X]) ≤exp(− 1

8E[X]). With the increasing number of nodes in net-work, the side length l of the deployed square would increasealso, and the side length l →∞.

B. A Tight Bound For Random Gateway Assignment

As we have the lower bound on the number of gatewaynodes in network, the contending sensor nodes would affectnetwork throughput and the number of sensor nodes shouldalso be bounded. According to our network model, there aren nodes randomly deployed in a square with M squarelets,where M =

⌈l2

( r√2)2

⌉. The expected number of nodes in a

squarelet is nM . According to lemma 4, w.h.p, the number of

sensor nodes in a squarelet is lower bounded by n2·M . We

have assumed that, the network density is constant with theincreasing number of sensor nodes. We can easily concludethat, the upper bound of sensor nodes in each squarelet is Ω(1).Considering the lemma 4, for each squarelet, the expectednumber of sensor nodes could also be bounded by O(1).

Based on these results, we can achieve the lower bound ofnetwork capacity in each squarelet. As each node can directlycommunicate with the gateway nodes, and the nodes in eachsquarelet is O(1), we can achieve the throughput lower bound,which is Ω(W ). Since the channel bandwidth is upper boundedby Wbps, the throughput for each contending area is upperbounded by O(W ). And we could achieve the tight bound foreach squarelet, which is Θ(W ).

Although the scaling network capacity could be as highas Θ(n). We can find that, such a scheme would not bean efficient scheme due to the unnecessary consumption andredundancy. Large amount of gateway nodes deployed are notfully used due to the contending sensor nodes in the same

Fig. 4. Voronoi Placement for gateway nodes

squarelet region. We are to present an improved gatewayassignment scheme and optimized network throughput withfewer gateway nodes.

C. An Improved Gateway Assignment – With Fewer Nodes

As stated in the previous subsection, if the gateway nodesare randomly deployed, there are large number of gatewaynodes are wasted due to the densely deployed patterns. Ifwe can arbitrary place the gateway nodes and keep theirconnectivity for “Destination-hub” character, the necessarynumber of gateway would be fewer, and the gateway accessefficiency would improve also.

As shown in Fig. 3, there are two difficulties in findinga minimum connectivity dominant set (CDS), where all thegateway nodes can cover every sensor nodes. The first is thedifficulty in finding an optimum CDS. The second difficultyis to ensure covering all the sensor nodes. Due the hardnessin finding an optimized CDS gateway placement, we turn tothe voronoi cell for a sub-optimal result.

Voronoi tessellation for a given region is formed by a setof “center” points on this region. Each construction pointidentifies a unique Voronoi cell and all the remaining pointson the region are partitioned into disjoint Voronoi cells byassigning each point to the Voronoi cell that has the closestconstruction point to its own position. We apply our problemto Voronoi cell problem. As shown in Fig. 3, to any node inVoronoi cell, its’ distance to gateway node in is smaller thanthat of to gateway nodes in any other Voronoi cell. Voronoitessellation of a unit square As shown in Fig. 3, gateway nodesseparate totally conventional nodes. We call such a method theVoronoi based gateway access.

Lemma 5: There exist a circle with radius ε, which can beenclosed by a voronoi cell, and a circle with radius 3ε canenclose this voronoi cell also.

This lemma has been proved in [6]. As shown in Fig. 5,the Voronoi cell is “sandwiched” by two circles with radiu εand 3ε respectively. We will firstly study the scaling capacityfor Voronoi based gateway access scheme and also study thebounding techniques for gateway reduction secondly. In each

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Fig. 5. Demonstration on the existence of ε

Fig. 6. Voronoi based gateway access

Voronoi cell, there is one gateway node. Each sensor nodewould select the nearest gateway for accessing. According tolemma 5, the longest distance between sensor node to gatewaynode at the same voronoi tessellation unit would be at most3ε. Since ε < r, we could bound the distance between sensornode and gateway node to 3r. As shown in Fig. 6, if thesensor node could adjust transmission power, and send datawith radius 3ε, the interference zone would possibly includemany gateway nodes.

Lemma 6: The longest edge in the Euclidian MinimumSpanning Tree (EMST) tree connected by gateways is at most√

log n+βn·π · a.

Lemma 7: Considering the interfered voronoi cell, the re-duced gateway nodes can be 2 ln l, and still in the order ofΘ(log n).

Proof: The overlapping area shown in Fig. 6 is of size9 2

3π−√

32 ε2. As nodes are randomly deployed, each squarlet

is of size 2r2, and the nodes deployed in a squarelet is atmost 4 ln l. According to lemma 6, is ε is upper bounded bythe longest EMST, so that, ε is of order O(

√log n). For the

interference area, the gateways are not needed for duplicatedeployment. And we can conclude that, the reduced gatewaynodes can be 2 ln l, and still in the order of Θ(log n).

V. CONCLUSIONS AND FUTURE WORK

The gateway access scheme is important to network deploy-ment and capacity in a scaling network as nodes increasinglytend to infinity. If gateway nodes are randomly deployed, thegateway nodes are tightly bounded by Θ(log n). For randomplacement, the scaling capacity for such a scheme is Θ(n).And we have also proposed an optimized scheme for reducingthe number of gateway nodes. Considering the interferencevoronoi cell, the reduced gateway nodes can be 2 ln l, andstill in the order of Θ(log n).

The future research directions include the multi-hop accessmode for load balancing and need less gateway nodes. More-over, we need to study the scaling capacity in case of GaussianChannels.

ACKNOWLEDGMENTS

This work is supported in part by the National BasicResearch Program of China (973 Program) under grant No.2006CB303004, No. 2009CB3020402. China National 863Project under grant No. 2008AA01Z216. China NationalScience Fund under grant No. 60673154, No.60672080 andChina Jiangsu Provincial High Technology under grant No.BG2007039.

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