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Energy-Throughput Trade-offs in a WirelessSensor Network with Mobile Relay

Guanghua Zhu, Linda M. Davis, Terence ChanInstitute for Telecommunications Research

University of South Australia, Adelaide, AustraliaGuanghua.Zhu, Linda.Davis, Terence.Chan@unisa.edu.au

Abstract—In this paper we analyze the trade-offs be-tween energy and throughput for links in a wirelesssensor network. Our application of interest is one inwhich a number of low-powered sensors need to wirelesslycommunicate their measurements to a communicationssink, or destination node, for communication to a centralprocessor. We focus on one particular sensor source, andconsider the case where the distance to the destinationis beyond the peak power of the source. A relay nodeis required. Transmission energy of the sensor and therelay can be adjusted to minimize the total energy fora given throughput of the connection from sensor sourceto destination. We introduce a bounded random walkmodel for movement of the relay between the sensor anddestination nodes, and characterize the total transmissionenergy and throughput performance using Markov steadystate analysis. Based on the trade-offs between total energyand throughput we propose a new time-sharing protocol toexploit the movement of the relay to reduce the total en-ergy. We demonstrate the effectiveness of time-sharing forminimizing the total energy consumption while achievingthe throughput requirement. We then show that the time-sharing scheme is more energy efficient than the popularsleep mode scheme.

Index Terms—wireless sensor networks, time-sharing,mobile relay, random walk, energy efficiency, throughput

I. INTRODUCTION

Wireless sensor networks (WSNs) have found wideapplication for delivering measurement observations inrural areas, battle fields, body area networks and othercomplex environments [1]. Research to date has beenfollowed three main dimensions: sensing (e.g. sensorsampling [2]), processing (e.g. data aggregation [3], [4]),and communication (e.g. routing [5] or data dissem-ination [6]). Energy efficiency is a key design issuefor all the three dimensions because most sensor nodesare battery powered. Batteries often have small limitedenergy capacity and may be difficult to replace. Whendesigning WSNs and the protocols, maximizing thenetwork lifetime becomes one of the most significantproblems [7].

The objective in designing an energy efficient trans-mission protocol for WSNs is to deliver data from thesensor node to the destination efficiently using minimalenergy, while still satisfying quality of service (QoS)constraints. The choice of energy scheme depends onthe application that the sensor network is intended tosupport. For real-time constant bit rate voice traffic orobjective tracking scenarios, energy might be traded offto meet stricter QoS requirements [8], while for delaytolerant applications, such as rural area data acquisitionsystems (e.g. soil moisture sensing in [9]), protocolsemphasizing network lifetime may be utilized.

One example of a wireless sensor network is the bio-logical information acquisition system proposed in [10].The so-called shared wireless infestation model (SWIM)was introduced to examine the whale data acquisitionproblems. Rather than the classic data collection modelin which each whale transmits directly to a base station,the SWIM model permits sharing of information whenwhales are close to each other. Then, any of thesewhales that come into the coverage area of a basestation can upload all the stored data, thus avoiding hightransmission energy consumption at each whale node.In a similar problem, [11] applied a WSN based systemto a warehouse management model. Mobile forklifts(nodes) in the warehouse are fitted with sensor nodeswhich can collect information, including data relatedto the usage of the forklift, bumping/collision history,battery status, and physical location. Information can becollected using IEEE 802.15.4 standard in an efficientevent-driven manner and utilized to optimize forkliftdispatching so as to minimize warehouse operationalcosts.

In [12], Grossglauser and Tse proved that mobilityof nodes in a network can greatly increase the networkcapacity or the per-user throughput. In fact, two-hoprouting (where data transverse to the sink node viaat most one relay) was proved sufficient to attain thethroughput capacity. It is worth noticing that in such

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a scenario, data will never be transmitted from a relayto another relay. Thus, cooperation among relays is notnecessary.

In this paper, our application of interest is one inwhich a number of low-powered sensors need to wire-lessly communicate their measurements to a communi-cations sink (a destination node or an infostation [10])labelled D. This sink node is connected to some cen-tral processor for the measurement information via abackbone network, possibly wired. We will consider aspecific (but not uncommon) scenario where the distancebetween the source sensor node S and the infostation Dis so large that the power required by S to wirelesslycommunicate directly to D is beyond the peak powerof S. Thus, a relay node R, most likely another sensornode, is necessary to assist the source to transmit datato the sink.

We introduce a bounded random walk model formovement of the relay node between the sensor anddestination nodes. Following a similar setup as in [12],limited “central coordinated scheduling” is allowed inour model. In particular, we assume that the sink (i.e.the infostation) is not a typical sensor node. Hence, it isnot power limited and can send an almost instantaneousACK to all sensor nodes to coordinate their transmis-sions. As we shall illustrate, this assumption can berelaxed in general. However, for analytical simplicity,it is assumed throughout the paper. While we only focuson a single two-hop transmission from the sensor node Sto the infostation D, our model can also be extended tomore general scenarios including the optimal two-hoprouting scenario [12]. Further details will be given inSection II-C.

Most existing work (e.g., [13][14]) on routing in delaytolerant networks use an “on/off model” for availabilityof an error-free link. As in [15], we note that the total en-ergy consumption is related to both physical (PHY) layerparameters, including signal-to-noise ratio (SNR), andmedium access control (MAC) layer parameters includ-ing packet length and transmission protocol parameters.One of the novelties in this paper is the incorporation ofthe error performance of the PHY layer in terms of thetransmission energy in our link model. This leads to asoft characterization of the finite state machine, that isone in which transitions are present and the probabilitydepends in a non-trivial way on the transmission energyand the hop distances between the source and relay,and the relay and destination. By understanding howthe network throughput and transmission delays willbe affected by the PHY and MAC layers parameters,we derive design guidelines for how to choose or even

optimize the PHY or MAC layers parameters in a delaytolerant networks.

Our work is also in contrast to the volume of researchfocused on the trade-offs in multiplexing and diversityfor cooperative nodes in a relay network (usually withfixed transmission power). Instead, we focus on theenergy-efficiency of communication and the trade-offsin energy consumption, throughput and relay mobility.Important questions include, what is the impact of therelay mobility on the link performance and what rangeof power levels are needed to meet the desired QoSparameters. One of the main novelties of our contributionis in formulation of the energy consumption using theMarkov analysis to derive insights into non-trivial energyinteractions in a simple relay link. Transmission energyof the sensor node and the relay can be adjusted tominimise the total energy for a given throughput of theconnection from sensor source to destination.

Whether the relay is mobile or fixed, one commonapproach to energy saving in WSNs is to use sleep mode[16]. Sleep mode exploits the fact that the sensor eventsare rare in some scenarios, and sensor nodes need tospend very little time in actual communication. Sleep andwake up schemes typically trade off energy consump-tion for end-to-end delay and network connectivity. Asleep mode based scheme called S-MAC was proposedin [17], where nodes are organized into small groupscalled virtual clusters. All the nodes in a virtual clusterhave their sleep-wake schedule synchronized. Multi-hopforwarding within the same virtual cluster, can takeplace without requiring to wait for the next-hop nodeto wake up. The disadvantage of using such a scheme isthat the co-ordination of the sleep-wake cycles requiressome communication between the nodes, which in turnamounts to additional protocol and energy overheads.

An additional issue for WSN communication whereretransmission is used is the size of the data pack-ets. Large packets reduce the probability of successfultransmission while small packets lead to unnecessaryoverhead. In [15], a variable packet length adaptive mod-ulation and coding scheme was proposed to minimize theenergy consumption for a transmit session over a directwireless link. Such a scheme is beyond the scope of thispaper, but we note that it could be applied to the mobilerelay system analyzed here. In particular, we note that inthe time-sharing scheme, packet lengths and modulationparameters could be adapted to suit the different power(i.e. SNRs) at the PHY layer.

Based on the trade-offs between total transmissionenergy and throughput performance, we propose a time-sharing scheme with a high transmit energy mode and a

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low transmit energy mode. We show that such a schemecan exploit the movement of the relay to reduce thetotal energy consumption. When packets are bufferedfor transmission, the time-sharing scheme can be moreenergy-efficient than a sleep mode scheme in meeting anaverage throughput or delay requirement.

The paper is organized as follows: in Section IIwe present our detailed system model for the mobilewireless relay network, formulate the problem for op-timization of energy consumption, and introduce thesteady-state Markov analysis for calculating the two-hop throughput. In Section II-C we discuss extensionsand generalizations for the model. Then, we presentnumerical results to characterize the throughput-energytrade-off in Section III for case study examples. Basedon observations from the optimisation in Section III-A,we propose our time-sharing protocol and compare thetime-sharing scheme with the popular sleep mode forenergy saving in Section III-B. We conclude the currentwork with our derived design guidelines in Section III-Cand highlight some future directions in Section IV.

II. ANALYTICAL MODEL

A. Synchronization and Mobility

In this section we present our system model andformulate the energy-throughput trade-off using a finitestate model and Markov chain analysis. We will focuson a particular sensor source S, and its connection to acommunications sink or destination node D via a relaynode R that acts as an intermediary between S andD. We assume that D is not within the transmissionrange of S, and hence communications between S andD must be assisted by the relay node. We assume that thetransmission hops S → R and R → D are independentand so do not interfere with each other. The destina-tion node is a data collection infostation connected toa backbone network in order to communicate sensormeasurement data from several sensor nodes to a centralprocessor. It is assumed to have sufficient power andhence can send an acknowledgement (ACK) to both Sand R directly, whenever it successfully decodes andreceives a packet. Since we are interested in minimizingthe energy consumed by the low-power devices S and Rwhich is dominated by packet transmission (longer timescale), we assume that the ACK from D (very short timescale) is effectively instantaneous.

In our model, S, R, and D do not need to besynchronized. Nevertheless, we introduce the notion of atime slot interval as a reference for the amount of energyused by the source and relay, as well as the amountof movement or relative speed of the mobile relay. The

communication protocol is divided into two phases. Thefirst phase is for transmission from S to R. In thisphase, the sensor S will transmit a packet of lengthL channel input symbols to the relay R using energyETxS L. The energy ETx

S describes the average energy persymbol used to transmit the L symbol packet during thenotional slot interval. It can represent a relatively highenergy transmission of uncoded data or a lower energytransmission including redundancy coding. The sensor Swill continue to transmit the same packet until it receivesan ACK from the destination node D.

The second phase begins when the relay R success-fully receives the packet sent by S. In this second phase,the relay R will transmit the packet to the destinationnode D. Note that in this phase, both S and R aretransmitting, i.e. S continues to transmit the packet evenwhen the relay R has already received it. For the samepacket of L symbols, the relay uses energy ETx

R L inenergy. We assume the transmissions from S and R to beorthogonal, i.e. interference is non-existent or negligible.This might be achieved through various multiple accessschemes such as time-division, frequency-division, code-division multiple access or through directive antennas. Infact, strict orthogonality is not required at all because Sand D are so far apart that D receives negligible signalfrom S, and sees only the signal from R.

Finally, when D does decode the packet, an ACK willbe broadcast to both S and R. Upon receiving the ACK,the relay will stop forwarding the packet and S will starta new transmission (entering the first phase again).

The distances and geometry between the nodes S, Rand D affect the performance of the two hops in the link.In general the three nodes form a triangle in a plane. Inthe case when interference is negligible, it is only thedistances that impact the model and so we can think ofthe nodes in a straight line for convenience, as shown inFigure 1. Let d be the distance between S and D. Wewill assume that at any time instance, the relay must beat a distance of dS = id/(N +1) from the sensor for aninteger i from 1 to N . Similarly, the distance betweenR and D is given by dR = d− dS = (1− i/(N + 1)) d.We will simply say that the relay is in position i.

S DR

d

1 2 3 NN-1N-2i

Figure 1. Link geometry for S = sensor, R = relay, and D =destination nodes. There are N possible positions for the relay, currentposition, i.

While both the sensor node S and the destination nodeD are fixed, the relay R is mobile. In particular, we

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assume that the motion of R follows a bounded randomwalk model [18]. Let P be a N × N matrix such thatits entry P [i, j] at the ith row and the jth column ofP is defined as the conditional probability that R is inposition j at the next step given that R is in positioni at the current step. We assume the random walk is“symmetric” such that

P [i, j] =

1− pmove if j = i

pmove/2 if |j − i| = 1, 1 < i < N

pmove if (j = 2 and i = 1)

or (j = N − 1 and i = N)

0 otherwise.(1)

Roughly speaking, pmove is the probability that therelay R will move in the slot period. Within each timeslot (the time in which the source uses ETx

S L to transmita packet, and the relay if transmitting, uses ETx

R L), therelay will move s steps. Hence, s is essentially themobility or speed parameter: the larger the value of s,the faster the relay is moving. Let P s be the sth powerof P whose entry at the ith row and the jth columnis denoted as P s[i, j]. Then P s[i, j] is the conditionalprobability that R is in position j after s steps giventhat R is in position i at the current step.

As we shall demonstrate, relay mobility can signifi-cantly reduce the energy required to transmit a packet(especially in the low throughput regime). To see theidea, consider the scenario where both the sensor nodeS and the relay node R have a very limited transmissionrange (and hence also consume extremely low power).Transmission from the sensor to the relay is successfulonly when the relay node moves very close to the sensornode. Even after receiving a packet from S, the relaynode R cannot successfully forward the packet to thedestination node D due to the long distance from Rto D. Instead, it wanders between S and D. Clearly, itwill get very close to the destination by chance aftera period of time. Then the packet being forwarded byR can successfully be received at the destination. Thisapproach may require a long time for a packet to besent from S to R and then from R to D. Despite its lowthroughput, our numerical results however show a hugereduction in packet transmission energy, that is, the totalenergy used by the source S to successfully transmit thepacket between S and D.

Now, recall that ETxS is the average symbol energy, that

is, the average amount of energy required to transmit onechannel input symbol. The level of ETx

S is selected withinthe range of 0 ≤ ETx

S ≤ Emax, where Emax dependson battery peak power. Let τ be the packet transmission

delay which is defined as the expected number of timeslots needed to successfully transmit a packet between Sand D. Then, Etotal

S = ETxS ×L× τ will be the expected

total packet transmission energy needed by the sensor tosuccessfully transmit a packet.

In this paper, we focus on the amount of transmissionenergy consumed by the sensor source in calculatinga total energy cost. The model can be modified andextended to include the energy of the relay node usedfor transmission, even mobility, as well as the energiesconsumed by the relay node R and the destination nodeD to receive a packet. We focus only on S to simplifythe exposition. We will discuss this further in SectionII-C.

To study the trade-offs among packet transmissionenergy, mobility and system throughput, we will usea Markov chain to model the whole communicationsprocess. The chain has 3N states which are labelled by

Ω , F i,R i,B i, i = 1, . . . , N. (2)

The state in which the communications system is independs on 1) which nodes (S only, or both S and R) aretransmitting, 2) whether it is a new packet transmissionor not, and 3) the position of the relay node. The precisedefinition of the states are given as follows.• State F i: The system is in state F i if (1) the relay R

is in position i, and (2) the sensor S is transmittinga new packet for the FIRST time.

• State R i: The system is in State R i if (1) therelay is in position i, and (2) only the sensor isRETRANSMITTING the packet (because the relayfails to decode the packet previously sent by thesensor S).

• State B i: The system is in State B i if (1) the relayis in position i, (2) BOTH S and R are transmittingthe packet.

Suppose the current state of the system is Fi. It impliesthat the destination node D has successfully received apacket from the relay R in the previous slot and hasbroadcast an ACK to the sensor node S and the relaynode R. Therefore, in this current time slot, the sensor(having received the ACK) will transmit a new packet.Suppose the position of the relay R is j in the nexttime slot. Then the system will enter the state R j ifthe relay failed to decode the new packet transmittedby the sensor (hence, only the sensor will retransmit thepacket again). On the other hand, if the relay can decodethe packet successfully, then the system will enter intothe state B j (where both the relay and the sensor willtransmit in the next time slot). Similarly, if the system isin state Ri currently, then its next state can only be Rj

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or Bj . For the destination D to successfully receive thepacket from R, the system must be in state Bi. When Dsuccessfully receives the packet sent by the relay, the thesystem will enter the state Fj in the next time slot. Onthe other hand, if the destination node cannot decode,then the system will enter the state Bj instead. Figure2 illustrates the relations between these three types ofstates.

F R

B

Figure 2. State transition diagram of Markov chain for two-hopwireless link. Each set of states, F, R, and B contains N states.

B. Non-threshold Communications Model

To compute the transition probabilities in the Markovchain, we need to know the transmission packet errorrate (PER) of the two hops S → R and R → D. Mostexisting work [13][14][19] on delay tolerant networksassume a “threshold or on/off communications model”where data can be sent from one node to another if thetwo nodes are within a transmission range. Clearly, thetransmission range depends on various factors includingtransmission power, coding and modulation scheme,channel gains, noise power and many others. Hence,each transmission scheme (and thus also the associatedtransmission range) governs how states in the Markovchain transit from one to another. In other words, theconnection among the states are “hardwired” accordingto the chosen transmission scheme.

Such an approach is usually simpler “topologically”in the underlying state transition graph. However, it isalso less flexible and is very difficult (if not infeasible) toevaluate how each parameter in a transmission schemeaffects the overall system performance (e.g., in terms ofthroughput and energy consumption). Our paper howeverassumes a “soft communications” model such that thechoice of transmission scheme will affect the transitionprobabilities between states. While the state transitionmatrix becomes more complicated, it now becomes

feasible to optimize the right parameter in a transmissionscheme to optimize the system performance.

For now, we keep a general form where the aver-age packet error rate (PER) is a function of the hoptransmission energy, ETx

` , for transmitter ` = S,R, themodulation and coding scheme (MCS), the packet lengthL, the hop distance, d`, and the one-sided additive whiteGaussian noise spectral density, N0, at the hop receiver.

PER` = f(ETx` ,MCS`, L, d`, N0

)(3)

Let M be the transition matrix of our proposedMarkov chain. For any states C,D ∈ Ω, we defineM(C → D) as the transition probability that the sys-tem is in state D in the next time slot, given that itscurrent state is C. Similarly, for any subsets δ, β ⊆Ω, M(δ → β) is the set of all transition probabilitiesM(C→ D) for C ∈ δ and D ∈ β. In many cases, itis instrumentally convenient to regard M(δ → β) as asubmatrix of M such that its entries in the Cth row andthe Dth column is M(C→ D).

Let F1:N be the set of states Fi, i = 1, . . . , N, andsimilarly we define R1:N and B1:N . Using our notationconvention, the transition matrix M can be written as in(4).

We would also like to point out the following assump-tion in our model. The probability that the relay R candecode a packet sent by the sensor S depends only on thedistance between S and R (or equivalently, the currentposition i of R). This probability is independent of theposition of the relay (j) in the next time slot. Likewise,the probability that D can decode a packet sent by Rdepends only on the distance between R and D.

The set of probabilities needed for the transitionmatrix is:

M(Fi → Rj

)= f(ETxS ,MCSS , L, dS , N0

)P s[i, j] (5)

M(Fi → Bj

)=(1− f

(ETxS ,MCSS , L, dS , N0

))P s[i, j] (6)

M(Ri → Rj

)= f(ETxS ,MCSS , L, dS , N0

)P s[i, j] (7)

M(Ri → Bj

)=(1− f

(ETxS ,MCSS , L, dS , N0

))P s[i, j] (8)

M(Bi → Bj

)= f(ETxR ,MCSR, L, dR, N0

)P s[i, j] (9)

M(Bi → Fj

)=(1− f

(ETxR ,MCSR, L, dR, N0

))P s[i, j]. (10)

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M =

0 M(F1:N → R1:N ) M(F1:N → B1:N )0 M(R1:N → R1:N ) M(R1:N → B1:N )

M(B1:N → F1:N ) 0 M(B1:N → B1:N )

(4)

Having modelled the communications process by aMarkov chain, we can calculate the limiting probabilityof each state by solving a system of linear equations [20].Specifically, let π = (πA,A ∈ Ω) be the vector oflimiting probabilities. Then

πM = π (11)

and∑

A∈Ω πA = 1. After computing these limiting prob-abilities1, we can then evaluate the system throughputand the average packet transmission energy. In particular,the throughput (denoted by π0) is given as follows:

π0 =

N∑i=1

πFi . (12)

Roughly speaking, if the system has been operated forT time slots (where T is a very large number), thenby Little’s law [21], the expected number of packetsthat have been transmitted successfully is given by Tπ0.Equivalently, the expected packet transmission delay τis 1/π0 slots. The expected packet transmission energyconsumed by the sensor source (denoted by Etotal

S ) isthus given by

EtotalS =

ETxS L

π0. (13)

Before we end this section, we would also like tohighlight that this paper’s objective is not to derive themost detailed and accurate model for wireless sensornetworks. Instead, we aim to derive insights and systemdesign criteria or principals from our analysis, realisingthat insights obtained from our simpler model will stillbe valid in more sophisticated models. As an example,one idea obtained and verified from our model is thattime-sharing can improve system performance (see Sec-tion III-B).

C. Extensions

The Markov chain formulation has been used toevaluate the energy-performance trade-off for a two-hopnetwork. This approach really is quite general and couldreadily be applied to more sophisticated network models,especially where the Markov chain and calculation of its

1 The complexity to determine the limiting probabilities is of orderO(M3), where M is the size of the Markov chain state-space. If onewould plot the performance curve with K grid points, then the overallcomplexity will be of order O(M3K).

transition matrix can be automated. Some possibilitiesinclude larger, two-dimensional networks; nodes withfinite buffers; multiple relays; multiple antennas for mul-tiplexing or beamforming; networks with interference;adaptive modulation and coding schemes; and, variablepacket length protocols. Through modification of thedefinition of state, the Markov chain can be used tomodel the behavior of the network in terms of a varietyof parameters. In addition to transmission energy andexpected throughput, the state could be defined so as toinvestigate latency, buffer overflow, battery depletion andso on. The Markov chain models similarly could be usedto simulate time evolving performance of such networks.

For example, in our proposed model, we assume nointeraction between sources and relays. Therefore, thesource will continue transmitting the data, even if therelay has already received the data packet. Suppose thatthere is only one relay and that the relay can send anACK back to the source node. Then the transmissionscheme will be more energy efficient if the sourcecan stop its transmission whenever an ACK has beenreceived from the relay. Our Markov model can easilybe adapted to take into account the changes. Anotherextension is by relaxing the constraint that the sink nodecan send an instantaneous ACK to all sensor and relaynodes. We can instead requiring the ACK be sent fromthe sink node to the source node via the two-hop link,or with some tangible delay.

The exposition for the energy-throughput trade-offpresented in this paper focuses on a total energy costonly for the sensor source transmission, Etotal

S . Anotherextension of our model is to change this total costfunction. So long as the total cost can be calculated interms of the expected amount of time the finite statemachine spends in any given state, a similar approach toanalysis can be taken.

While this paper considered only the single relayscenario, our proposed approach can also be naturallyextended to cases where there are multiple relays. Insuch a scenario, the infostation can receive data via oneof the relays. As before, we can obtain the Markovstate transition diagram in Figure 2 first and evaluatethe transition probabilities according to the chosen PHYand MAC model. Then, the trade-off between delays andthroughput can be evaluated by finding the stationary dis-tribution of the associated Markov chain. This approach

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however has a downside that the complexity to find thestationary probability of the Markov chain scales withthe number of relays. In the context of two-hop routing,some tricks can be employed to simplify the analysis.

Under two-hop routing, data are sent from the sourcesensor S to the sink D via multiple relays. As packetsare not sent among relays, the underlying network isessentially a set of parallel two-hop links connecting thesource to the sink. Furthermore, the “transmission pro-cess” of these two-hop links are essentially independent2.

Let η denote the random time required to send a datapacket successfully from the source S to the sink D viaa single two-hop link. Suppose there are m relays (orequivalently two-hop links). Let ηk be the time requiredto send a data packet from the source to the sink relayk, for k = 1, . . . ,m. Then η1, . . . , ηm is a collectionof i.i.d. random variables, each of which has the samedistribution as the expected delay, τ , for the single relaycase. From the sink node’s perspective, a packet has beenreceived at time t if and only if

t ≥ min(η1, . . . , ηm). (14)

By analyzing the distribution of η, we can also charac-terise the time required to transmit a packet via m relays.For example, let p(t) be the probability that Pr(η ≤ t)and m (i.e., the number of relays) be a poisson randomvariable with mean λ. Then

Pr(t ≥ min(η1, . . . , ηm)) = 1− e−λp(t). (15)

In addition to extension to two-hop routing networks,we can also extend by incorporating a more sophisticatedmobility model. For simplicity, our previous model as-sumed a “collinear mobility model” in which the relay Rmoves along the straight line between the source S andthe destination D. It is worth noticing that the geometrybetween S, R and D is not important in our model.Instead, only the relative distances among the nodesmatter. Therefore, it is in fact quite straightforward toextend our approach using a two-dimensional mobilitymodel. In the following, we will sketch the ideas how itcan be done.

We assume a Nx × Ny grid, with scale ∆ as shownin Figure 3. There are Nx columns and Ny rows. Theposition of the sensor S and the destination D arearbitrary, but fixed. The relay R can move about the

2 The independence assumption holds in the scenario being consid-ered when there is only a single source and a single destination, orwhen the relays have sufficiently large buffers. However, it does nothold in the general sense. When there are multiple sources and therelays have limited buffers, one will need to extend the Markov chainmodel to capture the interactions among nodes.

2D field. For position i = 1, . . . , NxNy , we have

xi = ((i− 1) mod Nx) + 1 (16)yi = di/Nxe (17)

indexx

index y

S

D

R

dS

dR

Δ

Δ

1 Nx

1

Ny

Figure 3. 2D link geometry for S = sensor, R = relay, and D =destination nodes. The relay can be in one of the NxNy possiblepositions indexed by i or equivalently (xi, yi).

For the relay in position i, the relevant distances are:

dS = ∆√

(xi − xS)2 + (yi − yS)2 (18)

dR = ∆√

(xi − xD)2 + (yi − yD)2 (19)

In this 2D grid, a symmetric random walk can beestablished. One possibility is to replace (1) with:

P [i, j] =

1− pmove xi = xj and yi = yjpmove/4 |xi − xj | = 1 xor |yi − yj | = 1

1 < xi < Nx, 1 < yi < Nypmove/3 |xi − xj | = 1 xor |yi − yj | = 1

xi ∈ 1, Nx xor yi ∈ 1, Nypmove/2 |xi − xj | = 1 xor |yi − yj | = 1

xi ∈ 1, Nx and yi ∈ 1, Ny0 otherwise.

(20)These equations are based on consideration of Figure 4.In this case, the state space is now dimensioned by N =NxNy and (5) – (10) apply as before.

III. ENERGY-THROUGHPUT TRADE-OFF

A. Trade-off Optimization

Using the Markov chain formulation described inSection II, we can now characterize the trade-off between

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i

i

i

i

i

i

i i

i

Figure 4. 2D random walk cases for the relay in position i.

the packet transmission energy (consumed by the sensornode) and the transmission throughput in our mobilerelay model for a numerical case study.

In [22], a similar trade-off between packet transmis-sion energy and throughput was investigated. The modelproposed here has two important differences. First, therelay R in [22] was stationary and did not move duringthe whole communication process. Second, in that earlierwork, the transmission energy consumption included theenergies consumed by both the sensor node S and therelay node R for the case where ETx

S = ETxR = ETx.

It was shown in that paper that the packet transmissionenergy and the network throughput depend highly onboth the transmission symbol energy level ETx and theposition of the stationary relay. In this paper, we focuson the trade-off between the packet transmission energyconsumed by the sensor node S and the throughput whenthe relay is mobile.

In the following, we will use our proposed Markovchain model to analyze an example to better understandthis throughput-energy trade-off. In our example, thedistance between S and D is d = 300m, which is thesame as that used in our earlier setup in [22]. For theaverage packet rate (PER) of equation (3), we adopt themodel proposed in [23] (based on [24]) for uncodedpacket over a Rayleigh fading channel:

PER =anr

αN0

rαN0 + gnETxGe−γpn(gn+

N0rα

ETxG)

+ (1− e−γpnN0r

α

ETxG ) (21)

where we have dropped the implied subscript ` desig-nating the transmitter S or R. The form of this equation

comes from the averaging over flat-fading Rayleighchannels, and an approximate exponential expression forthe PER for an additive white Gaussian noise channel ata given SNR. Details are in [23]. We use parametersfrom the model in [24] with b = 1 for BPSK, antennagain G = 1, and with parameters an = 67.7328,gn = 0.9819, rpn = 6.3281 dB, that depend on theMCS and the packet length, chosen to be L = 1080. Weassume that N = 10 (i.e. the relay can be in any one often possible positions). The level of symbol energy ETx

`

(for ` = S,R) can take any value from 1 × 10−7 J to2× 10−4 J. In addition, pmove is chosen to be 0.98.

Figure 5 depicts the throughput-energy trade-off forvarious mobility speeds s (i.e. the number of stepsthat a relay can move in any one time slot). For eachmobility speed s and a chosen symbol energy levelETxS = ETx

R = ETx, we can evaluate a throughput-energytuple (π0, E

totalS ) by computing the limiting probabilities

of the states in the Markov chain. By varying ETx, wethus obtain a throughput-energy curve for a given mobil-ity speed s. It is not difficult to see that if ETx increases,then π0 also increases. However, increasing ETx doesnot directly imply that the packet transmission energyEtotalS also increases. We see in Figure 5, especially for

s = 1, that EtotalS may decrease even if ETx increases.

Such a reduction in EtotalS is not hard to explain. While

the increase of ETx requires an increase in the amountof energy needed to transmit a packet in one time slot,it also increases the opportunities for a packet to bereceived successfully by the relay or by the destinationnode. Therefore, the average number of retransmissionsof a packet can also be reduced. Our simulation showsthat this reduction in the number of retransmissions canresult in a substantial reduction in the total energy neededto transmit a packet.

One important conclusion we can draw from themobility results in Figure 5 is that relay mobility canhelp reducing packet transmission energy. To justifyour claim, we will compare our obtained throughput-energy curves for various mobility speeds with theones for when the relay is stationary. In particular, weconsider two stationary relays scenarios. In the firstscenario (called the random stationary relay scenario),the position of the relay is randomly chosen before thecommunications process. Once the position is selected,the relay will stay in the same position during the wholeprocess. For fair comparison, the probability that thestationary relay will be at the position i will be the sameas the probability for that the mobile relay will be at thesame position.

In the second scenario (called the optimal stationary

8

Figure 5. Trade-off in total energy and throughput for mobile relay link.

relay scenario), the position of the relay will be selectedoptimally. Figure 6 shows the packet transmission energyEtotalS consumed by the sensor source node S and the

throughput of the network, π0 as a function of the relayposition for a fixed level of ETx. We can see that packettransmission energy and throughput are optimized whenthe relay is at the midpoint between S and D. This occursbecause of our assumption that both S and R use thesame transmission energy level. Therefore, in this secondscenario, we will always assume that the relay R is atthe midpoint between S and D. Under this assumption,we can show the throughput-energy curve by varying thesymbol energy level ETx. See Figure 7.

If we compare the throughput-energy curves witha mobile relay to the one with a random stationaryrelay, we can see from Figure 5 that relay mobilitydoes reduce energy consumption for the whole rangeof throughputs. In fact, the larger the mobility speeds, the lower the packet transmission energy Etotal

S . Wealso observe that the reduction in transmission energy

from mobility decreases as the throughput increases.This phenomenon can be explained as follows. Whenthe system is operated in the high throughput regime,packets must reach the destination in a fairly short periodof time. Clearly, within such a short time, the locationof the relay will not change significantly. Thus, the relaylooks as if it is stationary. Therefore, we do not expectthat mobility can greatly reduce transmission energy inthe high throughput regime.

However, when comparing with the throughput-energycurve in the optimal stationary relay scenario, we noticeseemingly contradictory outcomes that for a wide rangeof throughput levels, a stationary relay performs betterthan a mobile relay. The reason is largely due to thatthe position of the stationary relay is optimally chosen,while the position of mobile relay changes continuously.Therefore, in the high throughput regime (where mobil-ity does not help in general), an optimally positionedrelay performs better than a mobile relay in the senseof reducing packet transmission energy. However, if we

9

Figure 6. Energy transmission of S and R fixed at ETxS = ETx

R − ETx. Source energy consumption EtotalS as a function of relay position in

meters.

focus only on the low throughput regime, relay mobilitycan still reduce transmission energy, as illustrated fromFigure 5.

B. Enhancement by Time-Sharing

There are scenarios where energy is critically scarce(especially when it is essential to lengthen the lifespanof a battery-operated sensor network). The numericalexample in the earlier section showed a very interestingproperty: Suppose that the relay is stationary (or ismoving very slowly) and that the current level of symbolenergy ETx is very small. If ETx increases slightly,then the throughput may only increase slightly whilethe transmission energy of each packet may increasesignificantly. In fact, our example showed that it issometimes better to operate the system at an evenhigher symbol energy level to reduce packet transmissionenergy (while at the same time also increase the networkthroughput). In this section, we will propose a “time-

sharing” approach to further reduce packet transmissionenergy.

The idea is very simple. Instead of using only onesymbol energy level, we allow the use of two levels, ETx

lowand ETx

high. For each packet, when it is first transmitted,the sensor node will randomly select whether the packetshould be sent using the symbol energy level ETx

low(with a probability 1 − q) or ETx

high (with a probabilityq). Once the level is determined, the packet will betransmitted using the same symbol energy level for allsubsequent retransmissions. It turns out that this time-sharing approach can reduce packet transmission energyand also offer an opportunity to provide priority serviceto transmit delay-sensitive packets.

As before, we will model the whole communicationsprocess with a Markov chain, which consists of thefollowing states

Ω∗ ,Filow,Fihigh,Rilow,Rihigh,Bilow,Bihigh, i = 1, . . . , N

.

(22)

10

Figure 7. Relay node fixed at midpoint between S and D. Energy consumption ETxS as a function of transmission energy ETx.

The definition of the states are defined similarly asbefore. For example, the communication system is instate Fihigh if 1) the sensor node is going to transmit anew packet, 2) the position of the relay is i, and 3) thepacket will be transmitted by the sensor with a symbolenergy E high. Other states can also be defined similarlyas before. Figure 8 illustrates the relations between thesesix types of states.

Let π∗ = (π∗A,A ∈ Ω∗) be the vector of limiting prob-abilities. Note that these limiting probabilities depend onthe value of the time-sharing factor q. The transmissionthroughput π∗0 can be computed by

π∗0 =

N∑i=1

π∗Filow+

N∑i=1

π∗Fihigh. (23)

Average packet transmission energy can also be com-

puted similarly. Let

π∗low,0 =

N∑i=1

π∗Filow(24)

π∗low =

N∑i=1

π∗Filow+

N∑i=1

π∗Rilow+

N∑i=1

π∗Bilow(25)

π∗high,0 =

N∑i=1

π∗Fihigh(26)

π∗high =

N∑i=1

π∗Fihigh+

N∑i=1

π∗Rihigh+

N∑i=1

π∗Bihigh. (27)

Then the average packet transmission energy is givenby

E∗totalS = L

(ETx

low π∗low + ETx

high π∗high

π∗0

)(28)

We now evaluate the performance of our time-sharingapproach by considering a similar setup to Section III:

11

Flow Rlow

Blow

Fhigh Rhigh

Bhigh

q

1-q

q

1-q

Figure 8. State transition diagram of time-sharing scheme.

Let s = 1 and N = 10 so that the relay can be at onlyone of the 10 possible positions between S and D. Welet ETx

low = 1.0 × 10−7 J and ETxhigh = 1.8 × 10−5 J. For

each choice of the time-sharing parameter q (from 0 to1), we can compute the tuple (π∗0 , E

∗totalS ). By varying

q, we obtain the throughput-energy curve formed byconnecting all the tuples (π∗0 , E

∗total). As transmission

delay τ∗ is equal to the reciprocal of throughput, ifwe are interested, we can also connect all the tuples(1/π∗0 , E

∗totalS ) to form the delay-energy curve. In fact,

in some cases, it will be more convenient to examine thedelay-energy curve directly.

In Figure 9, we show the delay-energy curves obtainedfrom the time-sharing approach. And for comparison,we also plot the delay-energy curve of the system weoriginally proposed in Section III. If we compare thetwo curves, we can see that the time-sharing approachcan significantly reduce packet transmission energy. Thereason why a time-sharing scheme can reduce packettransmission energy is as follows. As we have seenearlier, there is a trade-off between transmission through-put (also the average transmission delay) and packettransmission energy. If we aim to minimize packet trans-mission energy, then the best option is to choose a verylow symbol energy level, as illustrated in Figure 7. Theprice to be paid however is the extremely low throughput.To address this problem, we can send a portion of thepackets by using a large symbol energy level to increasethe overall throughput. It turns out from our simulation

that this time-sharing approach can increase throughputwithout significantly increasing the packet transmissionenergy.

Remarks: The gain from time-sharing depends ona clever choice of ETx

low and ETxhigh. It also depends on

the moving speed of the relay. Usually, the gain is largerwhen the relay’s moving speed is low. Yet, even when therelay is moving fast, there is still a fairly large reductionin packet transmission energy in the low throughputregion.

A different operating characteristic could be achievedby introducing more than two choices for transmissionenergy level. Referring to Figures 5 and 9, time-sharingallows the system to operate on a curve that connects thetwo operating points corresponding to ETx

low and ETxhigh.

Careful choice of these operating points will ensure theminimum energy time-sharing curve. More transmissionenergy levels will result in an operating characteristicwith the same or higher total energy.

If we examine the delay-energy curve (obtained bytime-sharing) in Figure 9, we can notice that the curveis fairly close to a straight line. In the following, we aimto explain why. Using our Markov chain model, we caneasily prove that

q =π∗high,0

π∗0, and 1− q =

π∗low,0

π∗0(29)

So equation (28) can be rewritten as:

E∗total

=

(ETx

low ·π∗low

π∗low,0·π∗low,0

π∗0+ ETx

high ·π∗high

π∗high,0·π∗high,0

π∗0

)L

=

(ETx

low ·π∗low

π∗low,0· (1− q) + ETx

high ·π∗high

π∗high,0· q

)L (30)

Let

τ∗low =π∗low

π∗low,0(31)

τ∗high =π∗high

π∗high,0(32)

Then τ∗low (or τ∗high) is the average delay for a packet tobe transmitted using the symbol energy level ETx

low (orETx

high). Similarly, we can also prove that

1

π∗0=

π∗low

π∗low,0· (1− q) +

π∗high

π∗high,0· q. (33)

As the relay is moving, τ∗low and τ∗high are not constant, butare functions of the time-sharing parameter q. In Figure10, we plot the values of τ∗low and τ∗high in terms of q. Itturns out that the two values are fairly insensitive to the

12

Figure 9. Total energy consumption for sensor node versus average packet delay (1/throughput).

value of q. Together with (30) and (33) this explains whythe energy-delay curve in Figure 9 resembles a straightline.

Before we end this section, we would like to compareour proposed time-sharing approach to a relaying schemewith a sleep mode. The idea of sleep mode is based onthe fact that a node should only be awakened to receiveor transmit information when necessary [16]. In manydata gathering scenarios, it would also be reasonableto turn off the radio and only keep the sensor on formost of the time [4]. At a first glance, it seems thatthe two schemes should perform similarly (in particularwhen ETx

low is very small in our time-sharing scheme).However, as we shall see, the two schemes are in factquite different.

To evaluate the performance of a transmission schemewith a sleep mode, we will specify formally the schemeusing our Markov chain model: Assume that after eachsuccessful packet transmission, there is a probabilitypsleep that the sensor will go to the sleep mode. Boththe sensor S and the relay R will not transmit dur-ing the sleep mode. However, the mobile relay willcontinue to move during the sleep mode. The durationof when the sensor and the relay will remain in thissleep mode follows the geometric distribution with mean1/(1 − psleep) − 1. Once the system is awakened fromthe sleep mode, it will start transmission of a new packetagain.

If we examine the protocol carefully, we can see rightaway that a sleep mode enabled scheme cannot reducepacket transmission energy. It can only lengthen thelifespan of a node by not using all its energy. Whenthe system is awakened and is transmitting a packet, itconsumes as much energy as one without a sleep mode.

Figure 11 shows the energy-delay curves obtainedfrom our proposed time-sharing scheme and from thesleep mode enabled relaying scheme. For reference,we also re-plot one of the throughput-energy curves inFigure 5 as the energy-delay curve. For our-time sharingscheme, the system parameters are exactly as in SectionIII. For the scheme with sleep mode, symbol energy ETx

is chosen to be 1.8×10−5 J, which is the same as ETxhigh

in our time-sharing scheme.As expected, it turns out that transmission scheme

with a sleep mode does not help reducing packet trans-mission energy. For a transmission scheme with a sleepmode, the packet transmission energy needed is essen-tially the same as that without using the sleep mode.

C. Design Principles

In this section, we summarize the application oflessons learned from our study of the energy andthroughput trade-off for two-hop networks with eithera fixed or mobile relay. We form a set of operatingguidelines or application notes.

13

Figure 10. Packet delay low energy and high energy in terms of q.

We see from our analysis in this paper that havingaccess to a relay with mobility provides us with theopportunity to access the low total energy (very lowthroughput) operating region for the two-hop network,and furthermore, opens up the possibility of using atime-sharing scheme. Transmitting with very low powerenables the sensor to significantly extend its batterylifetime so long as retransmissions do not dominate theenergy consumption. Mobility alleviates the high needfor retransmissions at low transmission energies, andallows for successful communication despite a very tightconstraint on peak power from the battery.

We found that if we can only adjust the transmissionenergy level and the mobility of the relay node, the net-work throughput at low transmissions energies dependsa lot on the relay mobility. If the transmission energy isfixed, we can only improve the throughput by increasingthe speed of the mobile relay. If the mobility of the relayis fixed, we can select the proper transmission energy tofulfil the requirement of the network throughput.

If the relay mobility is fixed under the time-sharingscheme, we can adjust the selection of time-share energylevels to ensure the delay of the network. If the relaymobility is not fixed, we need to find the optimal time-sharing energy levels to make sure the time scheme isenergy efficient in terms of the delay and energy trade-off.

The time-sharing scheme also offers the opportunity

to offer different levels of service (i.e. priority or pref-erential treatment) for different packets the network.For example, in many applications, communication ofa changed condition is of far greater priority than com-munication that the status is unchanged. A time-sharingscheme can easily accommodate the two priority levels.Sleep mode can be used to extend the lifetime of thenetwork. However, it does not achieve the energy savingsper packet that the time-sharing scheme can achieve.

IV. SUMMARY AND FUTURE DIRECTION

In this paper, we have modeled a simple two-hopwireless network with a mobile relay node where trans-mission energy can be adjusted to improve the perfor-mance of the whole network. We introduced a boundedrandom walk model for the mobile relay node andcharacterized the energy and throughput performanceusing steady-state analysis of the Markov chain modelwe formulated. Based on observations, we proposed atime-sharing scheme for the low throughput operatingregion, extending our formulation and analysis for thisscenario. We demonstrated the effectiveness of time-sharing in significantly reducing the sensor node en-ergy consumption while achieving the same averagethroughput and packet delay as a single transmit en-ergy system. The results indicate that time-sharing isan attractive operating mode for the two-hop networkwith mobile relay. While the models presented in this

14

Figure 11. Performance comparison of single transmit energy, time-sharing and sleep mode in terms of total energy per packet versus averagedelay

paper are simple, the analysis approach developed isbroadly applicable. We have highlighted and discussedpossible extensions and generalizations. Nevertheless,even the simple models were demonstrated to be usefulin deriving design principles and operating guidelines forenergy conservation in a wireless sensor network.

ACKNOWLEDGEMENT

The authors would like to thank the reviewers and theeditor for their comments and suggestions, which greatlyhelp improving the quality of the paper.

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