On the Throughput Capacity of Wireless Sensor Networks With Mobile Relays

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 4, MAY 2012 1801

On the Throughput Capacity of Wireless SensorNetworks With Mobile Relays

Wang Liu, Kejie Lu, Senior Member, IEEE, Jianping Wang, Liusheng Huang, andDapeng Oliver Wu, Senior Member, IEEE

Abstract—In wireless sensor networks (WSNs), it is difficult toachieve a large data collection rate because sensors usually havelimited energy and communication resources. Such an issue isbecoming increasingly more challenging with the emerging ofinformation-intensive applications that require high data collec-tion rate. To address this issue, in this paper, we investigate thethroughput capacity of WSNs, where multiple mobile relays aredeployed to collect data from static sensors and forward themto a static sink. To facilitate the discussion, we propose a newmobile-relay-assisted data collection (MRADC) model. Based onthis model, we analyze the achievable throughput capacity oflarge-scale WSNs using a constructive approach, which canachieve a certain throughput by choosing appropriate mobilityparameters. Our analysis illustrates that, if the number of relays isless than a threshold, then the throughput capacity can be linearlyincreased with more relays. On the other hand, if the number isgreater than the threshold, then the throughput capacity becomesa constant, and the capacity gain over a static WSN depends ontwo factors: 1) the transmission range and 2) the impact of inter-ference. To verify our analysis, we conduct extensive simulationexperiments, which validate the selection of mobility parametersand demonstrate the same throughput behaviors obtained byanalysis.

Index Terms—Data collection, mobile relays, throughputcapacity, wireless sensor networks (WSNs).

I. INTRODUCTION

W IRELESS sensor networks (WSNs) [1] are an importanttechnology that can enhance our capability of monitor-

ing and interacting with the physical world. A typical WSNconsists of a static sink (a.k.a. base station) and many staticsensors, where each sensor is battery powered. Usually, after

Manuscript received September 23, 2011; revised December 20, 2011;accepted February 1, 2012. Date of publication February 16, 2012; date ofcurrent version May 9, 2012. This work was supported in part by a Grant fromthe Research Grants Council of the Hong Kong Special Administrative Region,China, under Project CityU 121107, by the National Grand FundamentalResearch 973 Program of China under Grant 2011CB302-905, and by theNational Science Foundation under Award CNS-0922996. The work of W. Liuwas supported by the China Scholarship Council. The review of this paper wascoordinated by Dr. T. Taleb.

W. Liu is with the Department of Computer Science, University of Scienceand Technology of China, Hefei 230026, China, and also with the Departmentof Computer Science, City University of Hong Kong, Kowloon, Hong Kong.

K. Lu is with the Department of Electrical and Computer Engineering,University of Puerto Rico at Mayagüez, Mayagüez, PR 00681-9042 USA.

J. Wang is with the Department of Computer Science, City University ofHong Kong, Kowloon, Hong Kong.

L. Huang is with the Department of Computer Science, University of Scienceand Technology of China, Hefei 230026, China.

D. O. Wu is with the Department of Electrical and Computer Engineering,University of Florida, Gainesville, FL 32611 USA.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2012.2188145

collecting data from the environment, a sensor sends the data tothe sink using multihop transmissions. Although such a schemehas been widely deployed and can enable low-data-rate appli-cations, it is difficulty to support high-data-rate applications[2] because each sensor has limited radio resources and energysupply.

To improve the performance of data collection, several ap-proaches have been proposed, including data fusion [3], hetero-geneous architecture [4], and use of mobile devices [5]. Withdata fusion, correlated data obtained by neighboring sensorscan be compressed before being forwarded to the sink. In het-erogeneous architecture, powerful sensors with larger energycapacity and stronger communications capability are deployedto reduce the energy consumption of regular sensors and toincrease the data collection rate. Using mobile devices, datacan be relayed from a sensor to the sink using less number ofhops. In this paper, we will investigate the performance of datacollection in WSNs with mobile devices.

In the literature, different types of mobile devices have beenintroduced [6], including mobile sensors [7], mobile sinks[8]–[11], and mobile relays [12]–[18]. A mobile sensor isenabled with both mobility and sensing ability. A mobile sink,similar to the static sink, is the final destination of data collectedby sensors. A mobile relay stores data for a certain duration andwill forward these data to a sink at a later time. In this paper, wewill focus on mobile-relay-assisted data collection (MRADC)in WSNs.

Starting from [12], many MRADC models have been devel-oped and investigated. These studies can be classified accordingto their objectives: 1) to improve connectivity in sparse WSNs[6], [12], [19]; 2) to reduce the latency of each packet [15];3) to maximize the total amount of data gathered during thelifetime of a WSN [18]; and 4) to enhance the energy efficiency(and, thus, prolong the lifetime) of a WSN [13]–[16], [18].

Despite the importance of these objectives, we note that thereis lack of theoretical understanding on the throughput capacityof WSNs with the MRADC model, where the throughputcapacity is defined as the maximal achievable data collectionrate from each sensor. In fact, most existing work assume thatthe data rate from each sensor is fixed [12]–[15], [17]. Toaddress this issue, in this work, we will analyze the throughputcapacity of large-scale WSNs with mobile relays. To the bestof our knowledge, this paper is the first work to construct theachievable throughput capacity of WSNs with mobile relays.

To facilitate the investigation, we propose a novel MRADCmodel, in which one static sink, n static sensors, and k mobilerelays are deployed in the network area. We first group the n

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1802 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 4, MAY 2012

sensors into k + 1 clusters (C0, C1, C2, . . . , Ck). In cluster C0,sensors send their data to the sink using multihop transmissionswithout the help of mobile relays. For each cluster Ci (i > 0),a mobile relay is assigned, and this relay periodically travelsbetween two specified locations. The first location is chosensuch that the mobile relay can forward stored data to the sink inone hop. The second location is inside Ci, at which the mobilerelay can collect data from cache nodes that are sensors withinone hop to the second location. Moreover, the cache nodes canstore data from other sensors when they are not forwarding datato the mobile relay.

Based on the proposed MRADC model, we investigate thethroughput capacity of large-scale WSNs using a constructiveapproach, which can achieve a certain throughput by choosingappropriate mobility parameters, such as the traveling speed,traveling distance, and other timing parameters. Our analysisillustrates that, if k is less than threshold k, then the throughputcapacity can be linearly increased with the increase in k. On theother hand, if k > k, then the throughput capacity is a constant,and the capacity gain over a static WSN depends on two factors:1) the transmission range and 2) the impact of interference. Toverify our analysis, we conduct extensive simulation experi-ments, which validate the selection of mobility parameters inthe constructive approach and demonstrate the same throughputbehaviors obtained by analysis.

The reminder of this paper is organized as follows: We firstreview existing MRADC models and propose our MRADCmodel in Section II. We then elaborate on an analytical frame-work to investigate the throughput capacity of large-scaleWSNs with MRADC in Section III. In Section IV, we provethe throughput capacity of WSNs with our MRADC model. Tovalidate the analysis, numerical and simulation results will bepresented and discussed in Section V. Finally, we conclude thispaper in Section VI.

II. DATA COLLECTION MODEL

In this section, we first briefly review existing work onMRADC models. We then introduce our MRADC model.

A. Existing MRADC Models

As we have mentioned in the first section, there are manyexisting MRADC models in the literature. In general, thesemodels can be classified according to the following three mainaspects: The first aspect is the number of mobile relays. In thiscase, the models in [14]–[16] assume that there is only onemobile relay. On the other hand, the models in [12], [13], [17],and [18] assume that there are multiple mobile relays.

The second feature is the mobility model of the mobilerelays. For this issue, an early study in [12] is based on therandom walk model, whereas most recent studies [13]–[18] arebased on controlled mobility because controlled mobility canimprove the performance of data collection, in terms of energyconsumption, delay of packets, etc.

The third feature is the routing of data packets. In this regard,there are three existing approaches. In [12], data from the staticsensor are forwarded to a mobile relay in one hop, and storeddata in a mobile relay are delivered to the sink later in one

Fig. 1. System model of MRADC with k = 4.

hop. In [13], [16], and [18], data from the static sensor aretransmitted to a mobile relay in one hop, and stored data inthe mobile relays are sent to the sink later using multihoptransmission. In [14] and [15], data are first forwarded to cachenodes using multihop routing; then, they will be gathered by themobile relay from the cache nodes using one-hop transmission,and finally, the data are delivered to a static sink in one hop.

In this paper, we develop a general MRADC model wherewe consider multiple mobile relays with controlled mobility,and we utilize different routing schemes for sensors located indifferent regions (i.e., C0 and Ci, 0 < i ≤ k in Fig. 1), whichwill be explained next.

B. New MRADC Model

In our MRADC model, we consider one static sink, n staticsensors, and k mobile relays. Sensors are first grouped intok + 1 clusters (C0, C1, C2, . . . , Ck). Cluster C0 is locatedaround the sink, and sensors in C0 send their data to the sinkusing multihop transmissions without mobile relay.

For any other cluster Ci (0 < i ≤ k), a mobile relay isassigned, and this relay can travel between two specified loca-tions, i.e., a rendezvous point (RP) that is inside Ci and a dumppoint (DP) that is one hop away from the sink. To collect datain Ci, sensors first send their data to cache nodes, which aresensors within one hop to the RP. The cache nodes can senddata to the mobile relay when it stops at the RP. Finally, themobile relay can deliver data to the sink when it stops at the DP.Fig. 1 shows an example for the MRADC scenario with k = 4.

To simplify the mobility of the mobile relay, we considerthat a mobile relay travels between the RP and the DP with aconstant period T . Consequently, we can define four states of amobile relay within each period T .

1) Traveling state: In this state, the mobile relay is travelingbetween RP and the DP. The duration of this state dependson the average traveling speed of a relay and the distancebetween the RP and the DP.

2) Waiting state: When the mobile relay arrives at an RP,it may wait until the cache nodes collect a sufficientamount of data from other sensors. The duration of thisphase is Tw.

3) Harvesting state: In this phase, a mobile relay stops atthe RP and is receiving data from the cache nodes. Theduration of this state is Th.

LIU et al.: ON THE THROUGHPUT CAPACITY OF WIRELESS SENSOR NETWORKS WITH MOBILE RELAYS 1803

4) Dumping state: In this phase, a mobile relay stays at theDP and is sending the stored data to the sink. The durationof this phase is Td.

It shall be noted that T is not the constraint of packet delay,which is the time duration for a packet to travel from its sourceto the sink. In this paper, we mainly focus on the throughputcapacity, and we will not investigate the packet delay of theMRADC model.

III. ANALYTICAL FRAMEWORK

In this section, we develop an analytical framework to in-vestigate the throughput capacity of large-scale WSNs withMRADC. We first address the system models, including thetransmission model, the network model, and other assump-tions. We then discuss the approach to construct the achievablethroughput capacity.

A. System Models

1) Transmission Model: In this paper, we consider a par-ticular protocol model.1 Specifically, following [20], we let rbe the one-hop transmission range, and let (1 + Δ)r be theinterference range. Then, a transmission from node ni to nodenj is successful if and only if

|ni − nj | ≤ r (1)|nq − nj | ≥ (1 + Δ)r (2)

where nq represents any node that is simultaneously sending. Inthis paper, we assume that r is a constant and that the successfultransmission data rate is fixed to W bits per second, which isalso a constant.

2) Network Model: In our network model, we consider thatn (n → ∞) static sensors are deployed in a unit circle area (theradius of which is 1/

√π) randomly following a Poisson point

process. A static sink is located at the center of the circle. Notethat we choose a unit circle area to simplify the analysis; inreality, sensors can be deployed in an arbitrary region.

We further divide the unit circle area into two parts, i.e., thecentral circle with radius r0 and a ring. Sensors located in thecentral circle are grouped into cluster C0. Since n → ∞, withhigh probability, the number of nodes in C0 is N0 = πr2

0n.The ring area is partitioned into k sectors with the same

shape, as shown in Fig. 1. Sensors in the same sector aregrouped into a cluster. The number of nodes in each ring clusteris the same Ni = (1/k)(n − N0) (i > 0).

We also consider that all DPs are on a circle whose center isthe sink and whose radius is r. Similarly, all RPs are on a circlewhose center is the sink and whose radius is r + l. Moreover,the DP and the RP of the same relay are on a line that includesthe center of the circle. Therefore, the traveling distance of eachmobile relay is l.

To collect data from sensors, we apply the time-divisionmultiple access (TDMA) schedule in [21] for communicationsbetween sensors. To avoid interference between mobile relaysand sensors, the locations of the RP and DP must satisfy thefollowing conditions.

1The main results can be easily extended to other protocol models.

1) The distance between a DP and the edge of the centralcircle is greater than (Δ + 1)r. In this manner, the trans-mission from a mobile relay to the sink does not interferewith the communications in ring clusters. We can thenderive a requirement for r0:

r0 ≥ (2 + Δ)r = δr (3)

where δ = 2 + Δ is used to simplify the notations.2) An RP must be located within a ring sector, and the

distance between the RP and the border of its sector isgreater than δr. In this manner, the transmission from anycache node to the mobile relay does not interfere with thecommunication in other clusters. With such a design, wecan obtain the following conditions:

l ≥ r0 + (δ − 1)r ≥ (2δ − 1)r (4)

l ≥(

δ

sin πk

− 1

)r (5)

l ≤ 1√π− (1 + δ)r (6)

where the first condition shall hold to avoid interferingwith C0, the second condition shall hold to avoid interfer-ing adjacent ring clusters, and the third condition is dueto the fact that the network is inside a unit circle.

3) Other Assumptions: To simplify our discussion, we makethe following assumptions, many of which are common for in-vestigating the performance of large-scale wireless networks.

1) We assume that r is small enough so that we can constructk + 1 clusters according to our previous discussions.

2) We assume that each node (sensor or mobile relay) isequipped with a half-duplex transceiver, which meansthat it cannot simultaneously send and receive. Since thetransmission data rate is fixed to W , we have Td = Th.

3) We also assume that all nodes transmit data on the samefrequency. Moreover, TDMA is used for medium access.

4) We assume that the traveling speed of any mobile relay isa constant v meters per second.

5) Since we are interested in the fundamental capability ofthe network, we do not apply data fusion in any of thenodes.

B. Analytical Approach

In this section, we first define the throughput capacity andexplain the main idea of our approach. We then elaborate onthe throughput bounds based on the proposed MRADC model.Finally, we discuss the physical constraints that define thesearch space in which we can find the maximal achievablethroughput.

Following existing studies [20] and [21], we define thethroughput capacity U as the maximal data rate, at which everysensor can send data to the sink. The main idea of our analyticalapproach can be summarized as follows.

1) We note that the throughput capacity is limited by the datacollection capability in different states of the mobile re-lay. Therefore, we first develop three throughput boundsunder our MRADC model, and we can obtain an objective


function that aims to maximize the minimum of the threebounds.

2) To obtain the throughput capacity, we also need toconsider the impact of the physical constraints for im-plementing the MRADC model, including interference,timing, scheduling, and mobility of nodes. These phys-ical constraints define a search space, in which we canmaximize the throughput capacity.

3) With the objective function previously mentioned and allthe constraints, we can see that finding the throughputcapacity becomes an optimization problem, which will besolved in the next section.

C. Throughput Bounds

To develop the throughput capacity of MRADC, in thissection, we discuss the main bounds of throughput capacity.First, the throughput capacity is limited by the amount of datathat sensors in C0 can send to the sink in T . Second, thethroughput capacity is constrained by the amount of data thatsensors in every ring cluster can send to cache nodes within T .Third, the throughput capacity is also bounded by the amountof data that cache nodes can send to the mobile relay in T . Wedenote these three bounds as Us, Ui, and Uh.

1) Capacity Bound in C0: According to [21], in cluster C0,nodes can send data to the static sink using multihop transmis-sion with achievable rate W/δ2N0. Let Ts be the amount oftime that mobile relays are not sending data to the sink in eachT . We can obtain

Us =WTs

δ2N0T. (7)

2) Capacity Bound in Ci: Similar to the preceding discus-sion, in cluster Ci (i > 0), each sensor can forward its data tothe cache nodes using multihop transmissions with data rateW/δ2Ni. Let Ti be the amount of time that cache nodes arenot sending data to the mobile relay in each T . We can derive

Ui =WTi

δ2NiT. (8)

3) Capacity Bound at Relay: Since Th is the amount of timethat a mobile relay can harvest data from cache nodes in T , wehave

Uh =WTh

NiT. (9)

According to the preceding analysis, we can see that thethroughput capacity of the WSN with our MRADC model is

U = max{

mini{Uh, Ui, Us}

}. (10)

D. Physical Constraints

In our MRADC model, we have several physical con-straints.

1) The central cluster area constraint: Since r0 ≥ δr, we canderive

A0 = πr20 ≥ πδ2r2 (11)

where A0 is the area of the central cluster. Furthermore,we have

N0 = nA0. (12)

2) The ring cluster area constraint: Since the distance be-tween an RP and the border of a ring cluster must begreater than δr, we have

Ai ≥ πδ2r2 (13)

where Ai is the area of cluster Ci. In addition, we assumethat each ring cluster has the same area; therefore

A1 = A2 = · · · = Ak =1 − A0

k(14)

N1 = N2 = · · · = Nk =n(1 − A0)

k. (15)

3) The total area constraint: Clearly, the area of all clustersis a unit. Therefore, we have

(k + 1)πδ2r2 < A0 + kAi = 1. (16)

4) Traveling distance constraints: These constraints areshown in (4)–(6).

5) Time constraint for cache nodes: In each T , cache nodesin a cluster collect data from other sensors in Ti andforward data to a mobile relay in Th. Therefore, we have

T = Th + Ti. (17)

Note that, during Ti, sensors in each ring cluster can usethe TDMA schedule developed in [21] to forward data tocache nodes.

6) Time constraint for each mobile relay: In each T , amobile relay can stay at the RP for Th + Tw (Tw ≥ 0),move to the DP in l/v, dump data to the sink in anotherTh, and move back to the RP in l/v. Thus, we can derive

T = Tw + 2

(Th +

lv

). (18)

7) Time constraint for the sink: In each T , the sink needs toreceive data from k mobile relays in kTh. In the rest ofthe time, which is Ts, sensors in C0 can send data to thesink in a multihop style using the TDMA scheme in [21].As a result, we have

T = Ts + kTh. (19)

Considering the aforementioned constraints, since Tw and vonly occur in (18), we can always obtain a v′ = (vl/vTw +l) ≤ v to make Tw = 0. Therefore, we can rewrite the preced-ing equations as the following:

Th =T

2− l

v′ (20)

Ti =T

2+

lv′ (21)

Ts =T − k

(T

2− l

v′

). (22)


IV. THROUGHPUT CAPACITY OF A WIRELESS

SENSOR NETWORK WITH THE PROPOSED

MOBILE-DELAY-ASSISTED DATA COLLECTION MODEL

In this section, we will analyze the throughput capacity of aWSN with respect to k, under our MRADC model. Specifically,we first further analyze the throughput bounds introduced in thelast section, in which we will define new parameters that canhelp to derive the throughput capacity. Based on the analysis,we will prove the achievable throughput capacity with respectto k. Finally, we summarize the parameters to implement theMRADC model, such as the location of the RP and the travelingspeed.

A. Further Analysis for the Throughput Bounds

To facilitate further discussions, we define two parameters.

1) We define ρ = l/v′T , which is the proportion of time inT that each mobile relay travels from the RP to the DP(or from the DP to the RP).

2) We denote ψ = k((1/2) − ρ), which is an auxiliary pa-rameter for obtaining the throughput capacity.

With the aforementioned parameters, we can rewrite the threethroughput bounds with triplet (k, ρ,A0) as

Us(k, ρ,A0) =W

n

(1 + kρ − k

2

)δ2A0 (23)

Ui(k, ρ,A0) =W

n

k2 + kρ

δ2(1 − A0)(24)

Uh(k, ρ,A0) =W

n

k2 − kρ

1 − A0. (25)

Moreover, we can also define

U ′s(ψ,A0) =

W

n

(1 − ψ)δ2A0

(26)

U ′h(ψ,A0) =

W

n

ψ

1 − A0. (27)

To guarantee Us(k, ρ,A0) ≥ 0 and Uh(k, ρ,A0) ≥ 0, wehave

12≥ ρ ≥ max

{0,

12− 1

k

}(28)

which specifies the optimization space of the throughput capac-ity, in which we can also derive

0 ≤ ψ ≤ 1 (29)

as shown in Fig. 2.Next, to further understand the optimization space, we com-

pare the three bounds pairwise and obtain thresholds for k, ρ,

Fig. 2. Optimization space for the throughput capacity.

ψ, and A0.1) If Ui(k, ρ,A0) = Uh(k, ρ,A0)

Ui(k, ρ,A0) ≥Uh(k, ρ,A0) if ρ ≥ ρ′ (30)

Ui(k, ρ,A0) < Uh(k, ρ,A0) if ρ < ρ′ (31)

where ρ′ = δ2 − 1/2(δ2 + 1). On the other hand, sinceρ ≥ (1/2) − (1/k), we have

ρ′ >12− 1

k, if k < k′ (32)

ρ′ ≤ 12− 1

k≤ ρ, if k ≥ k′ (33)

where k′ = δ2 + 1.2) If U ′

s(ψ,A0) = U ′h(ψ,A0), we have

A0 =A′0 =

1 − ψ

(δ2 − 1)ψ + 1(34)

U ′s(ψ,A0) >U ′

h(ψ,A0) if A0 < A′0 (35)

U ′s(ψ,A0) ≤U ′

h(ψ,A0) if A0 ≥ A′0. (36)

On the other hand, since A0 ≥ πδ2r2, we have

A′0 > πδ2r2, if 0 ≤ ψ < ψ′ (37)

A′0 > πδ2r2 ≤ A0, if ψ ≥ ψ′ (38)

where ψ′ = 1 − πδ2r2/1 + πδ2(δ2 − 1)r2. Here, we canfurther derive the following: 1) When ψ = ψ′, ρ increaseswith the increase in k, and 2) when ψ = ψ′ and ρ = ρ′, wehave

k = k =(δ2 + 1)(1 − πδ2r2)1 + πδ2(δ2 − 1)r2

. (39)

3) If Us(k, ρ,A0) = Ui(k, ρ,A0), we have

A0 = A′′0 =

2 + 2kρ − k

2 + 4kρ(40)

Us(k, ρ,A0) > Ui(k, ρ,A0), if A0 < A′0 (41)

Us(k, ρ,A0) > Ui(k, ρ,A0), if A0 < A′′0. (42)

Similar to the preceding discussions, we have

A′′0 ≥πδ2r2, if ρ ≥ ρ′′ (43)

A′′0 ≥πδ2r2, if max

{0,

12− 1

k

}< ρ < ρ′′ (44)


where ρ′′ = 2πδ2r2 + k − 2/2k − 4kπδ2r2. We alsonote that ρ′′ increases with the increase in k and ρ′ = ρ′′

when k = k.

B. Throughput Capacity

In this section, we will prove the following Theorem.Theorem 1: The throughput capacity of the WSN with the

proposed MRADC model is

U(k) =

{Wn

(δ2−1)k+δ2+1δ2(δ2+1) , if 1 ≤ k ≤ k

Wn

11+πδ2(δ2−1)r2 , if k ≥ k.

(45)

Proof: According to the analysis in the preceding section,we study the throughput capacity in six regions in the space ofρ and k, as shown in Fig. 2.

• Region I: ρ′ ≤ ρ ≤ (1/2), and 1 ≤ k ≤ k.• Region II: ρ′ ≤ ρ ≤ (1/2), k ≤ k, and 0 ≤ ψ ≤ ψ′.• Region III: ρ′ ≤ ρ ≤ (1/2), k ≤ k, and ψ′ ≤ ψ ≤ 1.• Region IV: ρ′′ ≤ ρ ≤ ρ′.• Region V: ρ ≤ ρ′′ ≤ ρ′, and 1 ≤ k ≤ k.• Region VI: ρ ≤ ρ′ ≤ ρ′′, and k ≤ k ≤ k′.

Clearly, in Regions I, II, and III, since ρ′ ≤ ρ ≤ (1/2), wehave Ui(k, ρ,A0) ≥ Uh(k, ρ,A0). Consequently, the through-put bound is

U(k) = max {min {U ′h(ψ,A0), U ′

s(ψ,A0)}} .

On the other hand, in Regions IV, V, and VI, the throughputbound is

U(k) = max {min {Ui(k, ρ,A0), Us(k, ρ,A0)}} .

Next, we discuss the achievable throughput in differentregions.

Region I: In this region, we can observe that 0 ≤ ψ ≤ ψ′, andtherefore, we have A′

0 ≥ πδ2r2. Since U ′s(ψ,A0) decreases

and U ′h(ψ) increases with the increase in A0, we can obtain

the maximum throughput capacity when A0 = A′0, i.e.,

U(k)=U ′h (ψ,A′

0)=U ′s (ψ,A′

0)=W

n

(δ2−1)ψ+1δ2

. (46)

Hence, to maximize the throughput capacity, we have tomaximize ψ = k((1/2) − ρ). Since ρ ≥ ρ′ and 1 ≤ k < k,the maximum throughput is

U(k) =W

n

(δ2 − 1)k + δ2 + 1δ2(δ2 + 1)

(47)

which is achieved when ρ = ρ′, and which is the first equa-tion in Theorem 1.Region II: Similar to Region I, we also have U(k) =Wn

(δ2−1)ψ+1δ2 when A0 = A′

0. Since the maximum ψ in thisregion is ψ′, the maximum throughput becomes

U(k) =W

n

11 + πδ2(δ2 − 1)r2

(48)

which is the second equation in (45).

Region III: In this region, as ψ ≥ ψ′, A0 ≥ πδ2r2 ≥ A′0, and

the maximum throughput is

U(k) = U ′s(ψ,A0) =

W

n

(1 − ψ)δ2A0

. (49)

To maximize U(k), we have to minimize both ψ and A0.Letting ψ = ψ′ and A0 = πδ2r2, we can derive the samethroughput as shown in (48).Region IV: Since ρ′′ ≤ ρ ≤ ρ′ in this region, we have A′′

0 ≥πδ2r2. Therefore, based on (23) and (24), we let A0 = A′′

0 toachieve the maximum throughput

Ui (k, ρ,A′′0) = Us(k, ρ,A′′

0) =W

n

1 + 2kρ

δ2. (50)

Clearly, the preceding equation can be maximized when ρ =ρ′, i.e.,

U(k) = U (k, ρ′, A′′0) =

W

n

(δ2 − 1)k + δ2 + 1δ2(δ2 + 1)

(51)

which is the same as (47).Region V: In this region, since ρ ≤ ρ′′, A0 ≥ πδ2r2 ≥ A′′

0,and the maximum throughput is

U(k) = Us(k, ρ,A0) =W

n

(1 + kρ − k

2

)δ2A0

. (52)

To maximize U(k), we shall maximize ρ and minimize A0.By letting ρ = ρ′′ and A0 = πδ2r2, we can obtain

U(k) = Us(k, ρ′′, πδ2r2) =W

n

k − 1δ2(1 − 2πδ2r2)

. (53)

Here, we note that, when k < k, the achievable U(k) in thisregion is smaller than (51).Region VI: Similar to Region V, the maximal throughput islimited by Us(k, ρ,A0). The only difference is that we shallchoose ρ = ρ′ to maximize the throughput, which is

U(k) = Us(k, ρ′, πδ2r2) =W

n

δ2 + 1 − k

πδ4(δ2 + 1)r2. (54)

Comparing (48) and (54), we observe that when k ≥ k, themaximum throughput capacity is (48). �

Corollary 1: When k ≥ k, the throughput capacity is

Umax =δ2

1 + πδ2(δ2 − 1)r2U . (55)

where U = W/nδ2 is the throughput capacity of a static WSNin [21].

C. How to Achieve the Throughput Capacity

In this section, we discuss how to choose appropriate mo-bility parameters to achieve the throughput capacity proved inTheorem 1. Note that bold symbols, such as ρ, will be used torepresent the optimal parameters.


1) Optimal ρ: Based on the proof of Theorem 1, to achievethe throughput capacity, we have to choose ρ, depending on therange of k. Specifically, when k < k, the maximum throughputcapacity can be achieved (in Regions I and VI) if ρ = ρ′; whenk ≥ k, ψ = ψ′ (in Regions II and III), and thus, ρ = (1/2) −(ψ′/k). To summarize, we have

ρ =

{δ2−1

2(δ2+1) , if k < k12 − 1−πδ2r2

k+kπδ2(δ2−1)r2 , if k ≥ k.(56)

2) Optimal r0: We now consider parameter r0, which isthe radius of the central cluster. In the proof of Theorem 1, thearea of the central cluster is A0. In addition, to achieve thecapacity, we have two regions of k. Specifically, when k ≥ k,we have ψ = ψ′, and A0 = πδ2r2. In this case, we shall chooser0 = δr.

When 1 ≤ k ≤ k, we have

ψ = k

(12− ρ′

)=

k

δ2 + 1

A0 =A′0 =

δ2 + 1 − k

(δ2 − 1)k + δ2 + 1.

Hence, we shall choose

r0 =

√A′

0

π=

√δ2 + 1 − k

π(δ2 − 1)k + πδ2 + π.

3) Optimal l: Next, we consider l as the distance betweenan RP and a DP. According to (6), we know that 1/

√π − (1 +

δ)r ≥ l, and we also have the relationship between l and r0.Based on the preceding discussions, we have

l ≥

⎧⎨⎩max{

(2δ − 1)r, δrsin π

k− r

}, if k ≥ k

max{

r0 + (δ − 1)r, δrsin π

k− r

}, if k < k.

(57)

In large-scale WSNs, due to the density of sensors, the posi-tions of RPs will not affect the throughput capacity. However,for any WSN with finite sensors, the throughput capacity canbe affected by the locations of RPs. In this paper, to analyze theimpact of RPs’ locations, we study two cases.

1) RPs are located at the closest positions to the sink, whichimplies that we can choose the minimal value in (57).

2) RPs are located at the centers of ring clusters, whichmeans that the radius of the RPs is (1/2)(1/

√π + r0).

Hence, the traveling distance l=(1/2)(1/√

π+r0)−r.

4) Optimal v′: Finally, since ρ = (l/v′T ), we can easilyobtain the optimal v′, depending on the selection of l; thus,v′ = l/ρT .

V. SIMULATION RESULTS

In this section, we develop simulations to evaluate the per-formance of our MRADC model and to verify our theoreticalanalysis. We first introduce the simulation settings and ourmethodology. We then analyze the maximum throughput ca-pacity under a given number of mobile relays k.

A. Settings

In our simulation, n = 9000 sensors are uniformly deployedin a unit circle area, the radius of which is 600 m. Thetransmission range is r∗ (r∗ ∈ {20, 30} m). Since the radius ofthe unit circle is 1/

√π units, we have r = r∗/600

√π units.

To guarantee the connectivity of the network, we can chooser∗ such that the expected number of neighbors for each nodeis at least 10 [22]. For the mobility of mobile nodes, we letT = 1200 s, and we let the traveling speed v be from 0.01 to2.5 m/s.

B. Methodology

Based on the MRADC scheme previously described, weimplement our simulation in the following four steps:

In Step One, we randomly deploy n nodes in the unit circlearea. We then build a graph, with the maximum length of eachedge being δr. With such a graph, we identify a number Nx,which is the minimum number to color all the nodes such thattwo adjacent nodes have different colors.

In Step Two, we specify an r0 in range δr ≤ r0 ≤ 1/√

π −2δr and a ρ in range max(0, (1/2) − (1/k)) < ρ < (1/2).Following (6), we calculate l and the associated v. We can alsocalculate Th = (T/2) − ρT and Ts = T − k((T/2) − ρT ).

In Step Three, we partition the network into two parts, i.e., acentral circle with radius r0 and a ring. For the ring, we furthercreate k sectors and specify k RPs based on our discussionsin the previous sections. According to our MRADC model, weselect all nodes that are one hop from an RP as cache nodes ofthe particular RP.

In Step Four, for the central cluster, we build a balanced tree,with the sink being the root of the tree using Algorithm 1 in[11]. Similarly, for each ring cluster, we build a balanced treefor all sensors with the root being the RP using Algorithm 1 in[11] as well.

Next, we calculate the throughput capacity of the network.For the central cluster, we have

Us(k, ρ,A0) = minj

{W (T − kTh)

N j0TNx

}(58)

where N j0 is the number of sensors in the subtree of sensor j in

cluster C0.To calculate the throughput bound for cache nodes, we use

Ui(k, ρ,A0) = minm,j

{W (T − Th)N j

mTNx

}(59)

where N jm is the number of sensors in cluster Cm (m > 0) that

are in the subtree of sensor j, which is not a cache node.For the throughput bound for mobile relay, we have

Ur(k, ρ,A0) = min0<i≤k

{WTh

TNi

}. (60)

As we want to find the optimal r0 and ρ, we have

U(k)= maxρ,A0

{min

i

{Us(k, ρ,A0),Ui(k, ρ,A0),Ur(k, ρ,A0)

}}.

(61)


Fig. 3. U(k) and U(k) versus k under different r’s and Δ’s.

Fig. 4. Optimal ρ versus k under different Δ’s, where r = 1/30√

π.

C. Results Analysis

In our experiments, for a given WSN setting with specifiedk, Δ, r, we simulate 50 randomly generated topologies. Toachieve the maximum throughput capacity, we increase r0

with steps of 0.1r from δr to 1/√

π − 2δr, and we increaseρ with steps of 0.01 from max(0, (1/2) − (1/k)) + 0.01 to(1/2) − 0.01.

Fig. 3 first shows both the simulation and theoretical resultsof throughput capacity U(k) and U(k) versus k under ourMRADC. We can observe the similarity between the simulationresults and theoretical results. In particular, our experimentsshow that the simulation results are about 80% of the numericalresults with the same setting.

Here, we note that the simulation results are obtained whenthe RPs are located at the centers of ring clusters. If the RPs areplaced to closest possible locations to the sink, the achievablethroughput will be about 60% of the theoretical results.

Fig. 3 also shows the impact of r and Δ. Comparing theresults under different r’s, we can observe that the throughputcapacity decreases when r increases. On the other hand, we findthat the throughput capacity decreases when Δ increases.

In Figs. 4 and 5, we compare the optimal conditions forboth r0 and ρ to achieve the maximum throughput capacityin both simulation and theory. These two figures show thatthe simulation results are almost the same as the theoreticalresults, which validates the selection of optimal parameters inthe theoretical analysis.

Fig. 5. Optimal r0 versus k under different Δ’s, where r = 1/30√

π.

Fig. 6. Optimal v versus k under different Δ’s, r = 1/30√

π.

Finally, in Fig. 6, we compare the optimal speed versus kthat can achieve the maximal throughput in both simulationand analysis. According to our previous discussions, in thesimulation, the RPs are located at the centers of ring clusters.We can observe that the analysis results match well with thesimulation results. We can also note that, in both scenarios, theoptimal v decreases with the increase in k.

VI. CONCLUSION

In this paper, we have investigated the throughput behaviorsof WSNs with mobile relays. We have first proposed a newMRADC model, in which we have considered multiple relayswith controlled mobility. Based on this model, we analyzed theachievable throughput capacity of large-scale WSNs using aconstructive approach, which can achieve a certain throughputby choosing appropriate mobility parameters. Our analysisillustrates that, if the number of relays is less than a threshold,then the throughput capacity can be increased with more relays.On the other hand, if the number is greater than the thres-hold, then the throughput capacity is a constant, and the ca-pacity gain over a static WSN depends on two factors: 1) thetransmission range and 2) the impact of interference. To verifyour analysis, we have conducted extensive simulation exper-iments, which validate the selection of mobility parameters


and demonstrate the same throughput behaviors obtained byanalysis.

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Wang Liu received the B.S. degree in computerscience from Harbin Institute of Technology, Harbin,China, in 2007. He is currently working toward thePh.D. degree under the Joint Ph.D. degree programwith both the Department of Computer Science,University of Science and Technology of China,Hefei, China, and the Department of ComputerScience, City University of Hong Kong, Kowloon,Hong Kong.

His research interests include throughput, delay,lifetime in wireless networks, and mobility-assisted

wireless networks.

Kejie Lu (S’01–M’04–SM’07) received the B.Sc.and M.Sc. degrees in telecommunications engineer-ing from Beijing University of Posts and Telecom-munications, Beijing, China, in 1994 and 1997,respectively, and the Ph.D. degree in electrical en-gineering from the University of Texas at Dallas,Richardson, in 2003.

In 2004 and 2005, he was a Postdoctoral Re-search Associate with the Department of Electricaland Computer Engineering, University of Florida,Gainesville. In July 2005, he joined the Depart-

ment of Electrical and Computer Engineering, University of Puerto Rico atMayagüez, where he is currently an Associate Professor. His research interestsinclude architecture and protocols design for computer and communication net-works, performance analysis, network security, and wireless communications.

Jianping Wang received the B.Sc. and M.Sc. de-grees from Nankai University, Tianjin, China, in1996 and 1999, respectively, and the Ph.D. degreefrom the University of Texas at Dallas, Richardson,in 2003.

She is currently an Assistant Professor with theDepartment of Computer Science, City Universityof Hong Kong, Kowloon, Hong Kong. Her researchinterests include dependable networking, optical net-working, service-oriented wireless sensors, and adhoc networking.

Liusheng Huang received the M.Sc. degree in com-puter science from University of Science and Tech-nology of China, Hefei, China, in 1988.

He is currently a Professor and Ph.D. Supervisorwith the School of Computer Science and Tech-nology, University of Science and Technology ofChina. He has published six books and more than200 papers. His research interests are in the ar-eas of wireless sensor networks, information secu-rity, distributed computing, and high performancealgorithms.

Dapeng Oliver Wu (S’98–M’04–SM’06) receivedthe B.E. degree in electrical engineering fromHuazhong University of Science and Technology,Wuhan, China, in 1990, the M.E. degree in electricalengineering from Beijing University of Posts andTelecommunications, Beijing, China, in 1997, andthe Ph.D. degree in electrical and computer engineer-ing from Carnegie Mellon University, Pittsburgh, PA,in 2003.

Since 2003, he has been with the faculty of theDepartment of Electrical and Computer Engineering,

University of Florida, Gainesville, where he was an Assistant Professor from2003 to 2008, an Associate Professor from 2008 to 2011, and is currentlya Professor. His research interests are networking, communications, signalprocessing, computer vision, and machine learning.

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On the Throughput Capacity of Wireless Sensor Networks With Mobile Relays