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Should I stay or should I go?Maximizing Lifetime with RelaysPeter Terlecky, Brian Phelan, Amotz Bar-Noy, Theodore Brown, Dror Rawitz

Abstract—We consider the problem of maximizing the lifetime of a wireless connection between a transmitter and a receiverusing mobile relays. Initially, all relays are positioned arbitrarily on the line between the transmitter and the receiver and havearbitrary battery capacities. Energy is consumed in proportion to the distance traveled for mobility and in proportion to anexponential function of the distance over which information is sent for communication. The objective is to find positions andthus transmission ranges for the nodes that maximize the lifetime of the network. We study two models, the single deploymentmodel and the multiple deployment model. We show how to compute an optimal solution for the case of no movement costfor both models. When there is a cost for movement we develop some structure for multiple deployment model. We considera discrete version of the single deployment model, in which relays are deployed on grid points. We provide two algorithms forthis case: a dynamic programming algorithm and a binary search algorithm. We prove that both algorithms are FPTASs for thenon-discrete problem, if batteries are not too small. We develop a number of heuristics for the multiple deployment model. Oursimulations demonstrate the benefit of moving over remaining at initial locations.

Index Terms—Deployment; Mobility; Network Lifetime; Relays; Sensor Networks.

F

1 INTRODUCTION

Keeping the network alive for as long as possibleis one of the main goals in any Wireless SensorNetwork (WSN). This paper uses recent advances innode mobility (see El-Moukaddem et al. [6] and thereferences therein) to maximize the lifetime of a WSNusing mobile-relay-based algorithms that move mo-bile relays to optimal locations along the line betweena transmitter and a receiver.

Several papers address the problem of optimalplacement of relays between a source and a des-tination to optimize other objective functions. Ap-puswamy et al. [1] optimize the capacity of the im-plied logical channel between the source and thedestination under some interference model. This pa-per allows only grid placements of relays with thesame, non-adjustable transmission range and does notconsider movement cost.

Goldenberg et al. [7] prove that, for equal batterylevels, the optimal locations for the relays on the lineare equidistant apart and present an algorithm formoving nodes to their optimal locations using infor-mation from 1-hop neighbors only. They prove that

• P. Terlecky is with the Mathematics Department, The Graduate Center,City University of New York, NY, 10016.E-mail: pterlecky@gc.cuny.edu

• B. Phelan, A. Bar-Noy, and T. Brown are with the Computer ScienceDepartment, The Graduate Center, City University of New York, NY,10016E-mail: {bphelan,tbrown}@gc.cuny.edu, amotz@sci.brooklyn.cuny.edu

• D. Rawitz is with the School of Electrical Engineering, Tel-Aviv, Isreal,69978E-mail: rawitz@eng.tau.ac.il

their algorithm preserves the connections between allthe relays.

Jiang et al. [9] develop a number of algorithms tospeed up the rate of convergence and, hence, mini-mize the number of iterations required to move eachof the nodes to their optimal locations. We extendthe above works by taking into consideration theremaining battery life in each node and the effectthat different mobility costs will have on the optimallocations of the nodes.

El-Moukaddem et al. [6] consider using mobilerelays to enhance the lifetimes of existing networkroutes between static nodes in a 2D plane. Each relaycan assist a single link between two nodes. Theyinclude realistic costs for movement and transmissionand take into consideration the battery power remain-ing in each of the nodes.

In this paper we study the problem of maximiz-ing the lifetime of a wireless connection between astationary transmitter and a stationary receiver usingmobile relays that are initially positioned arbitrarilyon the line between the transmitter and the receiverand have arbitrary battery levels. For example, therelays may be initially located at the transmitter thatdeploys them to assist it with a transmission. Weconsider several variants of the problem dependingon the deployment method. In many applications, [7],[9], [17], relays are allowed to move only at time zeroin order to decrease the amount of communicationoverhead between relays. It may also be the case thatrelays are initially deployed using an external agent,and cannot move by themselves. We refer to thismodel as the single deployment model. In the multipledeployment setting relays can move at any point in

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time.

Our contribution. We consider two network models,the first model corresponds to the case where relaysare allowed to be set once, while the second modelcorresponds to the case where relays can be adjustedmultiple times. In the first model relays can be de-ployed only at time 0 in an order preserving manner.The lifetime of the system is determined by the weak-est link, namely by the lifetime of the relay whosebattery is depleted first. This notion of Lifetime of FirstDeath was consider by El-Moukaddem et al. [6]. Werepresent this problem as a nonlinear program. Thesecond model allows multiple deployments. Relaysreadjust their transmission ranges after a deployment.In this case we wish to maximize the length of timethe transmitter can communicate with the receiver,that is the Transmission Lifetime.

We show how to compute relay locations for bothmodels when there is no cost for movement. It turnsout that for this case there exists a solution whichis optimal with respect to both models, namely thereexists an optimal single deployment order preservingstrategy that maximizes both notions of lifetime.

For non-zero movement cost, we develop somestructure for the optimal solution. In particular, weshow that there is no justification for movement un-less a relay becomes dies and that in any optimalsolution the transmitter must be the last node to die.

We consider a discrete version of the single de-ployment, in which relays must be deployed on gridpoints. Given any ε > 0, we show that if batteriesare not too small, there exists a grid density forwhich the discrete optimal lifetime of first death iswithin a factor of (1 + ε) from the optimal lifetimeof first death. We provide two algorithms for thediscrete problem. The first algorithm is a dynamicprogramming algorithm that computes an optimalsolution, while the second conducts a binary searchfor the optimum. For the case where the batteries arenot too small, both algorithms are FPTASs for thenon-discrete problem, since their running times arepolynomial in the size of the input and in 1/ε. We ex-perimentally test whether multiple deployments canassist in maximizing transmission lifetime. We findthat just a single deployment is necessary in maximiz-ing transmission lifetime. We develop heuristics formaximizing transmission lifetime based on this resultand based on our theoretical findings and show thatthey are well-performing in simulation. Finally, weconsider the natural case where relays with differentbattery levels are to be deployed from the location ofthe transmitter (base station). We experimentally findthat in a maximal lifetime of first death solution relaysshould be deployed so that relays with larger batteriestravel farther.

Related work. The numerous uses of mobility inWSNs are discussed in Francesco et al. [5] along

Fig. 1: n relays on the line between the transmitterand the receiver.

with the challenges that arise when mobile nodesare introduced to a network such as maintainingconnectivity. Along with mobile relays, other uses ofmobility include mobile sinks and data MULEs.

Controlled sink mobility is considered by Basagni etal. [2], while predictable sink mobility is considered byChakrabarti et al. [3], Chandra et al. [4] and Mhatre etal. [13]. Another approach considered in the literatureis allowing the sink to move randomly as in Juang etal. [10] and Kim et al. [12].

Data MULEs were introduced by Shah et al. [15]and are further explored by Jain et al. [8].

Song et al. [16] build a prototype mobile node todemonstrate that mobile nodes are feasible. Kansal etal. [11] proposes using controlled mobility to optimizethe power usage of a WSN and discusses a numberof potential network features, for example tracks, thatwould make adding controlled mobility cost effective.

Paper organization. Formal descriptions of the prob-lem, deployment methods, and network lifetimes arein given in Section 2. The case where movementis free of cost is analyzed in Section 3. Section 4gives some structural results. Section 5 considers thediscrete version of the single deployment model. Wepresent some heuristics for maximizing transmissionlifetime in Section 6. Our simulations are given inSection 7. Finally, Section 8 discusses areas for futureresearch.

2 MODEL AND PRELIMINARIES

In this section we define the models and introduce thenotation which will be used throughout the paper.

A transmitter would like to transmit information toa receiver. To aid in the transmission of information,n mobile relays are distributed on the line of commu-nication between the transmitter and the receiver (asin Fig. 1). Each relay has the task of passing alongthe information and thus maintaining the chain ofcommunication. The distance between the transmitterand the receiver is denoted by D. We assume w.l.o.g.that the transmitter is located at 0, while the receiveris located at D. The initial locations of the nodes aredenoted by x0, x1, . . . , xn, where x0 = 0 is the locationof the transmitter.

The initial battery power of node i ∈ {0, 1, . . . , n}, isdenoted by Bi. Communicating costs energy and theenergy required depends entirely on the distance overwhich the information must be transmitted. Followingthe model of Moscibroda et al. [14], a node transmit-ting data over a distance d invests P (d) = dα energy

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per time unit, where α > 1 is a constant. The costof mobility, or friction, is proportional to the distancetraveled, k, as in Wang et al. [18], i.e. M(k) = ak,where a is a constant.

We allow relays to occupy the same location. If twoor more nodes are located at the same point, then oneof them (say the node with the highest index) musttransmit to the next live node which is not at the samelocation, while the other nodes do not consume anypower.

Using the above definitions, we represent the net-work by the vector

N = 〈D,B0, . . . , Bn, x1, . . . , xn, α, a〉,

which gives the distance from transmitter to receiver,the initial battery powers and positions of the nodes,and the cost parameters, α and a. When a = 0 we saythat there is no friction.

2.1 Single Deployment and Lifetime of First Death

Our first model corresponds to the case where relaysare allowed to be set once. Relays are to be deployedat time 0 after which transmission may commence.The lifetime of the network is determined by its weak-est link, namely by the lifetime of the relay whosebattery is depleted first, thus breaking the chain ofcommunication.

We have n relays initially located at x1, x2, . . . , xnbetween 0 and D on the line. The relays are to bedeployed to some locations y1, y2, . . . , yn. Moving re-lay i from location xi to location yi decreases relay i’sbattery by a|xi−yi|. Let B′i be the new battery level ofrelay i after the movement, namely B′i = Bi−a|xi−yi|.(Clearly B′0 = B0.) Let righti denote the left-most nodeto the right of i, after the relays are deployed. A nodei ∈ {0, . . . , n} must transmit to righti. Let di be thetransmission range of node i, for every i ∈ {0, . . . , n},namely di = yrighti−yi, for every i ∈ {0, . . . , n}, whereyrighti = D if the receiver is the rightmost node to theleft of node i.

Once the relays arrive at locationsy = (y1, y2, . . . , yn), they may begin transmittingand must transmit from their respective locationsfor the duration of their lifetimes. The lifetime ofrelay i is defined as Li(yi) = B′i/d

αi . The lifetime of

the system given the deployment to y is the time inwhich the first relay dies, and it is defined as

LF (N, y) = min0≤i≤n

Li(yi) . (1)

We shall refer to this lifetime as Lifetime of First Death.This notion of network lifetime was considered byEl-Moukaddem et al. [6] for a max data mobile relayconfiguration.

The MAXIMUM LIFETIME OF FIRST DEATH problem(MAXFD) is the problem of finding a deployment thatmaximizes the lifetime of first death.

In this paper we focus on the no swapping case,where the relays are to be deployed to some locationsy1, y2, . . . , yn preserving the initial order between re-lays, i.e. xi ≤ xj implies yi ≤ yj , for every i 6= j. Inthis case, di = yi+1 − yi for i ∈ {0, . . . , n − 1} anddn = D− yn. This problem can be represented by thefollowing nonlinear (and non-convex) program:

maxy1,...,yn

min{B0

yα1, B1−a|x1−y1|

(y2−y1)α , . . . , Bn−a|xn−yn|(D−yn)α

}(2)

s.t. 0 ≤ y1 ≤ · · · ≤ yn ≤ D

The reason for focusing on this case is threefold.First, there are applications in which swapping isdisallowed. For example, Kansal et al. [11] suggestusing relays on a track. Second, in some scenariosadding this no swapping restriction gives additionalstructure that may be used for solving the problemwithout affecting the solution space. For instance,when n identical relays are initially located at thesame point. Finally, we show (in Section 3) that swap-ping is unnecessary in the non-friction case.

Another nice implication of the no-swapping as-sumption is multiple deployments do not help. Morespecifically, we can show that any solution whererelays redeploy after time 0 can be replaced by a singledeployment solution whose lifetime is not worse, seecorollary 5.

Furthermore, we assume that all relays must partic-ipate in communication. This may have a drawbackin that if any relay has a very small initial battery anda sizable transmission range, the lifetime of first deathwould be short. Note that if Bi ≥ amax{xi, D − xi},for all i, then a relay may effectively deactivateby moving arbitrarily close to the next relay. Thus,deactivation is encompassed by the model given ifBi ≥ amax{xi, D − xi}, for all i. We also note thatour FPTASs for MAXFD are based on a strongerassumption (see Section 5).

2.2 Multiple Deployment and Transmission Life-time

In our second model we assume relays can be de-ployed multiple times and can readjust their trans-mission ranges after a deployment or after the deathof a relay. In this model, we wish to maximize thelength of time the transmitter can communicate withthe receiver. We call such a lifetime the TransmissionLifetime.

A solution, or solution path, is defined as P :={x1(t), . . . , xn(t)}∞t=0, where xi(t) is the location ofrelay i at time t for i ∈ {1, . . . , n}. Let lefti(t) andrighti(t) denote the right-most node to the left of nodei and the left-most node to the right of i at time t. Anode i ∈ {0, . . . , n} must transmit to righti(t). If relayi dies at some time t then the node lefti(t) increases itstransmission range to transmit to righti(t). The trans-mitting range of a live node i at time t, denoted by

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di(t), is the distance between that node and righti(t),i.e., di(t) = xrighti(t)(t) − xi(t). The remaining batterypower of node i at time t is denoted by Bi(t).

Let Xi(t) be the total distance traveled by node iup to time t (note X0(t) = 0 ∀t). Using this notation,the lifetime of node i under solution path P , Li(P),satisfies:

Bi = aXi(Li(P)) +∫ Li(P)

0

di(t)αdt (3)

A solution path P is feasible if (i) xi(0) = xi for everyi ∈ {1, . . . , n}, and (ii) xi(t) = xi(Li(P)), for everyi ∈ {1, ..., n} and t ≥ Li(P).

We define the Transmission Lifetime for a solutionpath P and network N , denoted by LT (N,P), to bethe length of time the transmitter can send data tothe receiver in a solution path P for a given networkN . This is equivalent to the lifetime of the transmitterunder a solution path P , L0(P). Thus:

LT (N,P) = L0(P) . (4)

The MAXIMUM TRANSMISSION LIFETIME problem(abbreviated MAXTL) is the problem of finding a solu-tion path P that maximizes the transmission lifetimeof a given network N .

Observe that the maximum transmission lifetimeis never smaller than the maximum lifetime of firstdeath, namely

maxy

LF (N, y) ≤ maxP

LT (N,P)

for every network N , since LF (N, y) ≤ LT (N,Py), forevery deployment y, where Py is the solution paththat corresponds to y, i.e., Py satisfies xi(t) = yi forevery relay i.

3 MAXTL AND MAXFD WITHOUT FRICTION

In this section we consider the no friction case, namelythe case where a = 0. Goldenberg et al. [7] considerthe case where there is no friction and all nodes haveequal battery power. They show that the energy costfunction P (d(t)) is a non-decreasing convex function,and that the optimal positions of the relay nodes mustlie entirely on the line between the source and thedestination and must be evenly spaced along the line.

We can extend this result to the case of non-equalbattery powers. We show that the lifetime of the net-work is optimized when we choose the transmissionranges di(t)’s to be such that the lifetimes of all nodesare equal, i.e. Li = Lj , ∀i, j ∈ {0, . . . , n} and fixed forthe lifetime of the network: di(t) = di for the lifetime(both transmission and first death) of the network.In this case, relays only need to move once to theiroptimal locations and transmission lifetime is equalto lifetime of first death.

Lemma 1. The transmission lifetime and lifetime of firstdeath of a network with n relays where there is no frictionis maximized when Li = Lj , for every i, j ∈ {0, . . . , n}.

Proof: Let T be the transmission lifetime of anoptimal solution. Define di to be the time average ofdi(t) from 0 to T for each i ∈ {0, . . . , n}:

di =1

T

∫ T

0

di(t) dt .

We first show that∑ni=0 di = D:

n∑i=0

di =

n∑i=0

1

T

∫ T

0

di(t) dt

=1

T

∫ T

0

n∑i=0

di(t) dt =1

T

∫ T

0

Ddt = D .

A feasible placement in which i’s range is di can beobtained by placing i at

∑i−1j=0 dj , for every i. We now

show that for arbitrary i:∫ T

0

(di)α dt ≤

∫ T

0

di(t)α dt

This follows with the use of Jensen’s Inequality:∫ T

0

(di)α dt = T · (di)α ≤ T · di(t)α

= T · 1T

∫ T

0

di(t)α dt =

∫ T

0

di(t)α dt .

It thus suffices to consider the solutions in whichthe transmission ranges di(t), i ∈ {0, . . . , n} are fixedfor the duration of lifetime. Of all such solutions,the one in which all node lifetimes are equivalent,Li = Lj , ∀i, j ∈ {0, . . . , n}, maximizes the lifetime ofthe network.

Assume this is not the case. Consider the time Mwhen the transmitter dies. Note that if a relay diesbefore time M the solution does not have fixed trans-mission ranges. Consider the leftmost relay which isstill alive at time M . WLOG, assume it is relay k. Shiftthe first k relays to the left by an amount δ > 0, where

δ ≤ min{d0(M), ε} and ε = α

√BkM − dk, i.e., ε satisfies

Bk/(dk + ε)α = M . The shift decreases d0, whileleaving d1, d2, . . . , dk−1 unchanged. Consequently thelifetime of this solution is greater than the lifetime ofthe optimal, a contradiction.

If Li = Lj , ∀i, j ∈ {0, . . . , n}, lifetime of first deathequals transmission lifetime. Since lifetime of firstdeath is never larger than transmission lifetime, itfollows that lifetime of first death is maximized aswell.

Theorem 2. If a = 0, an optimal solution for MAXTLand MAXFD is obtained by placing relay i at

∑i−1j=0 dj ,

where

di = D ·α√Bi∑n

j=0α√Bj

(5)

for every i ∈ {1, . . . , n}. The corresponding lifetime isD−α(

∑nj=0

α√Bj)

α.

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Proof: Consider relay i. Due to Lemma 1 we knowthat Li = Lj for every j ∈ {0, . . . , n}. It follows thatdj = α

√Bj/Bi · di, for every j ∈ {0, . . . , n}. Since∑n

j=0 dj = D, we have that

diα√Bi

n∑j=0

α√Bj = D ,

which gives the result.

Corollary 3. If a = 0, there exists a single deploymentand no swapping optimal solution for MAXTL.

4 MAXTL WITHOUT SWAPPING

In this section, we consider MAXTL with friction.We provide some structure for this case by givingnecessary conditions for any optimal solution.

First, we note that Theorem 2 does not hold in thenon-zero friction case. Consider the case where wehave one relay located at 0.25 with B0 = B1 = 1 andD = 1. Also, assume that a = 4. This means that goingfrom 0.25 to 0.5 depletes the relay. However, locatedat [0.25,0.5) the relay dies before the transmitter.

We prove that in an optimal solution, relays do notneed to move unless some relay dies.

Lemma 4. Given a MAXTL instance, any solution wherea relay moves at a time that does not correspond to the deathof another relay can be replaced by a stationary solutionwith at least as good a lifetime.

Proof: Assume that in an optimal solution norelay dies in the time interval (t0, t2). Furthermore,assume that in this optimal solution, relays move atan instance t1, where t0 < t1 < t2 and are stationaryotherwise.

Consider the time average location of relay i from t0to t2, which has moved from location xi(t0) to xi(t1)at time instance t1:

xi =xi(t0) · (t1 − t0) + xi(t1) · (t2 − t1)

t2 − t0.

We claim that a solution that places relay i at locationxi at time t0 and does not move i before t2 is atleast as good as the original solution. First, noticethat xi ≤ xi+1 for every i, namely relay ordering ismaintained by the new solution. Also, since locationxi, the time average, is between locations xi(t0) andxi(t1), the cost of movement from xi(t0) to xi to xi(t2)is at most the cost of the original movement fromxi(t0) to xi(t1) to xi(t2).

Let us now consider the cost of transmission if relayi is placed at location xi at time t0. Let di(t0) =xi+1(t0)−xi(t0) and di(t1) = xi+1(t1)−xi(t1). Define dito be the time average from t0 to t2 of the transmittingdistance between node i and node i+ 1:

di =di(t0) · (t1 − t0) + di(t1) · (t2 − t1)

t2 − t0.

Observe that

di =di(t0) · (t1 − t0) + di(t1) · (t2 − t1)

t2 − t0

=(xi+1(t0)− xi(t0))(t1 − t0)

t2 − t0

+(xi+1(t1)− xi(t1))(t2 − t1)

t2 − t0

=xi+1(t0) · (t1 − t0) + xi+1(t1) · (t2 − t1)

t2 − t0

− xi(t0) · (t1 − t0) + xi(t1) · (t2 − t1)t2 − t0

=xi+1 − xiso that it is feasible to place relays i and i+1 at theirtime-averaged locations and obtain the time-averagedtransmission distance between them.

By Jensen’s inequality,

(di)α ≤ dαi =

(t1 − t0)di(t0)α + (t2 − t1)di(t1)α

t2 − t0and thus,

(t2 − t0)(di)α ≤ dαi = (t1 − t0)di(t0)α + (t2 − t1)di(t1)α.

That is, the amount of battery depleted for relay i inthe time-average case is at most the amount of batterydepleted in the case when the relay moves at time t1.We thus have the desired result.

Corollary 5. One deployment suffices when maximizingthe lifetime of first death.

It also holds that in any optimal solution the trans-mitter is one of the last to die.

Lemma 6. Given a MAXTL instance, any solution inwhich the transmitter is not among the last to die issuboptimal.

Proof: Assume the transmitter dies at time M , butrelay k is the leftmost relay which is still alive in theoptimal solution, i.e. relays 1, . . . , k − 1 die prior toor at time M in the optimal solution. Assume relaysi1, i2, . . . , il are among the first k− 1 relays and die attime M with the transmitter.

Move relay k an amount φk to the left at time M−τ ,where τ is small enough such that no relay dies inthe time interval [M − τ,M). Since no relay dies, wecan assume by Lemma 4 that the relays are stationaryin this time interval. Let φk be an amount whichprecludes relay k from dying before time M and isless than min{d0(M − τ), di1(M − τ), . . . , dil(M − τ)}.Sequentially move relays j = il, il−1, . . . , i1 at timeM − τ , an amount φj to the left where φj < φj+1 andis such that

Bj − aφj(dj(M − τ)− (φj+1 − φj))α

=Bj

dj(M − τ)α.

Note that such a movement is always feasible: con-sider the function

J(x) =Bj − ax

(dj(M − τ)− (φj+1 − x))α.

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We have that J(0) = Bj/(dj(M − τ) − φj+1)α >

Bj/dj(M−τ)α. Furthermore, J(x) is a strictly decreas-ing function on the interval [0, φj+1] and

J(φj+1) =Bj − aφj+1

dj(M − τ)α<

Bjdj(M − τ)α

.

Since J(x) is continuous, there is some point φj ∈[0, φj+1] for which

Bj − aφj(dj(M − τ)− (φj+1 − φj))α

=Bj

dj(M − τ)α.

Note that since

φk < min{d0(M − τ), di1(M − τ), . . . , dil(M − τ)}

and φk > φil > · · · > φi1 , relays can never swaporder with this shift. This shift of the relays decreasesd0(M − τ), resulting in a solution path with a longertransmission lifetime, a contradiction.

5 MAXFD ON GRID POINTS

In this section we consider a discrete version ofMAXFD in which relays are deployed on grid points.More specifically, we assume that the final locationsy1, . . . , yn of the relays must be one of the pointsjD/m for a pre-determined m and 0 ≤ j ≤ m. Thatis, we partition the interval [0, D] into m sub-intervalseach of length σ = D/m and restrict the final locationsfrom being in the interior of any sub-interval.

Given any ε > 0, we show that if Bi ≥ aD(1 +1n ), for every i, there exists a grid density for whichthe discrete optimal lifetime of first death is within afactor of (1+ε) from the optimal lifetime of first death.We provide two algorithms for the discrete problemthat are in fact FPTASs for MAXFD, if Bi ≥ aD(1+ 1

n ),for every i.

We note that our algorithms for the discrete versionof MAXFD can be extended to deal with relay deacti-vations. However, we do not consider deactivations,since they can be ignored if Bi ≥ amax{xi, D−xi}, forevery i. (Recall that in this case a relay may effectivelydeactivate by moving arbitrarily close to the nextrelay.)

5.1 Discrete MAXFD vs. MAXFD

Let OPTF be the optimal lifetime of first death, andlet OPTmF be the optimal lifetime of first death for thediscrete version with m grid points.

We first show that OPTF and OPTmF may be far aparteven when m is very large. Let m be an odd integer.Consider the following instance with three relays, a >0 and D = 1. The relay locations are given by x =( 12−

σ2 ,

12 +

σ2 , 1), and relay batteries are B1 = B2 = aσ

2 ,and B0 = B3 = B, where B is very large. The optimaldeployment is y1 = y2 = y3 = 1

2 , and in this case OPTFis close to B

0.5α , while OPTmF is at most a2σα−1 .

Such scenarios may be avoided if the deployment atgrid points does not deplete the batteries. In the nextlemma we make an assumption that ensures this.

Lemma 7. Let ε ∈ (0, 1). If Bi ≥ aD(1+ 1n ), for every i,

and m =⌈(n+ 1)2/ε

⌉, then

OPTmF >OPTF

(1 + ε)α+1.

Proof: Let (y1, . . . , yn) be an optimal deployment,namely a deployment whose lifetime of first death isOPTF . Also, let di = yi+1− yi, for every i ∈ {0, . . . , n}.Observe that there must exists at least one relay i forwhich di ≥ (n+1)σ

ε , since otherwise

D =

n∑i=0

di < (n+ 1)(n+ 1)σ

ε=

(n+ 1)2D

εm≤ D .

Let ` be such a relay.Let d′i = σ ·

⌊diσ

⌋, for every i 6= `, and let d′` =

D −∑i 6=` d

′i. Observe that di − σ < d′i ≤ di, for every

i 6= `, and that

d′` < d` + (n+ 1)σ ≤ d` + εd` = (1 + ε)d` .

We describe a discrete deployment y′ using the newdistances between relays: y′i =

∑k<i d

′k, for every i.

Observe that

|y′i − yi| ≤ nσ =nD

m≤ εnD

(n+ 1)2<εD

n,

for every sensor i.Let R and R′ are the remaining battery powers after

moving to y and y′, respectively. Since Ri ≥ aDn , for

every i, it follows that

R′i ≥ Ri−a|y′i−yi| > Ri−εaD

n≥ Ri−εRi = (1−ε)Ri ,

Putting it all together we get that

L′i =R′i

(d′i)α≥ (1− ε)Ri

(1 + ε)αdαi>

1

(1 + ε)α+1· Li ,

and we are done.

5.2 Dynamic Programming AlgorithmIn this section we present a dynamic programmingalgorithm that solves discrete MAXFD called Lifetime-DP, shown as Algorithm 1. The running time ispolynomial in n and in the number of grid points,m+ 1.

The idea behind this dynamic programming al-gorithm is to try solving all possible instances forcovering the prefix segment [0, . . . , jD/m] for any0 ≤ j ≤ m using the i relays closest to the transmitterfor 0 ≤ i ≤ n.

For 0 ≤ j ≤ m and 0 ≤ i ≤ n, let f(i, j) be thelifetime of first death of the solution with the i relaysthat are closest to the transmitter in which there isa relay with infinite size battery positioned at pointjD/m (this is equivalent to moving the receiver to

7

Algorithm 1 : Lifetime-DP1: for all 0 = 1 to m do2: f(0, j)← B0

(jD/m)α

3: end for4: for all i = 1 to n do5: for all j = 0 to m do6: f(i, j)← 07: for all h = 0 to j do8: Calculate LF (y(i− 1, h), j)9: if LF ((y(i− 1, h), hD/m), j) > f(i, j) then

10: f(i, j)← LF ((y(i− 1, h), hD/m), j)11: y(i, j)← 〈y(i− 1, h), hD

m〉

12: end if13: end for14: end for15: end for

point jD/m). The desired final output will be f(n,m).Initially, for i = 0 and any 0 ≤ j ≤ m, the lifetimeis the time it takes the transmitter to die when ittransmits to a distance jD/m. Also, let y(i, j) =(y1, . . . , yi) be the vector of optimal positions assignedto the i relays when transmitting to j. Namely, y(i, j)corresponds to f(i, j).

Before showing how to compute f we need the fol-lowing definition. For 0 ≤ j ≤ m, let LF (z1, . . . , zi, j)be the lifetime of first death, when there are onlyi relays, those who are the closest to the trans-mitter, whose final positions are at points z1, . . . , zirespectively, that need to cover the prefix segment[0, . . . , jD/m]. The cost of moving relays 1, . . . , i topositions z1, . . . , zi is taken into account in the solu-tion.

Then, recursively,

f(i, j) =

max0≤h≤j

LF ((y(i− 1, h), hD/m), j) i, j > 0,

∞ j = 0,B0

(jD/m)α i = 0,j > 0.

That is, look for the lifetime maximizing position forthe ith relay among the first j + 1 possible positions.When a position is examined for the ith relay, use theoptimal locations for the first i−1 relays transmittingto such a position to find the lifetime for that con-figuration. Take the maximum lifetime of all positionevaluations to be f(i, j).

Lemma 8. Lifetime-DP finds an optimal solution forMAXFD on grid points.

Proof: We proof by induction on i that f(i, j), forevery j, is the maximum lifetime of first death of thefirst i relays when the receiver is located at jD/m.

Note that the claim is true for i = 0. Assume theclaim is true for all i−1. We would like to show that itis true for i. In order to compute f(i, j) the algorithmplaces the ith relay at locations h = 0, . . . , jD/mand considers the lifetime of this placement with themaximal solution of i − 1 relays transmitting to h,

y(i−1, h). By the induction hypothesis, y(i−1, h), for0 ≤ h ≤ j, are the positions of maximum lifetime ofthe first i− 1 relays transmitting to h. Hence, the bestlifetime of first death when i is placed at hD/m andtransmits to j is the minimum between f(i−1, h) andthe lifetime of i, namely LF ((y(i − 1, h), hD/m), j).Since the dynamic programming solution takes themaximum over all h, f(i, j) is optimal for MAXFD ongrid points, for every j.

The running time of Lifetime-DP is O(nm2 · lF (n)),where lF (n) is the time to compute LF (N). lF (n) =O(n), since it requires finding the minimum of n+ 1node lifetimes. Thus, the running time of the dynamicprogramming algorithm is O(n2m2).

We note that the running time can be improved toO(nm2) at the expense of the space complexity usingstandard techniques.

Because of Lemma 7, Lifetime-DP can be used as anapproximation algorithm for MAXFD.

Theorem 9. Lifetime-DP is an FPTAS for the special caseof MAXFD in which Bi ≥ aD(1 + 1

n ), for every i.

5.3 Binary Search AlgorithmIn this subsection we devise a binary search algorithmcalled Lifetime Binary Search. We show that this algo-rithm is an FPTAS for discrete MAXFD. It follows thatit is an FPTAS for MAXFD provided that the batteriescomply with the conditions of Lemma 7.

Lemma 10. Let T be the optimal lifetime of first death.Given T ′ ≥ 0, there exists a polynomial-time algorithmthat determines whether T ′ ≤ T .

Proof: Initially set yn+1 = D. Going from i = n to1 we try to move relay i to the leftmost position forwhich relay i has lifetime T ′. If we succeed for all i,we check whether the transmitter has enough powerto transmit to relay 1 at least until T ′.

If xi < yi+1, move relay i to the leftmost positionyi in the closed interval [xi−1, yi+1], for which relay ihas lifetime Li ≥ T ′, if such a location exists. Findingyi can be done by solving the following equations:

Bi − api(di + pi)α

= T ′ , (6)

if Li ≥ T ′, and

Bi + api(di + pi)α

= T ′ , (7)

if Li < T ′, where pi = xi − yi. Note that yi exists ifLi ≥ T ′. In addition it exists if Li < T ′ and Bi−adi >0. If yi exists, update di−1 ← di−1 + (yi − xi) anddi ← di + (xi − yi). Otherwise output NO.

If xi = yi+1 = · · · = yi+k, for some k ≥ 1, proceedas follows. Move relays i, . . . , i+ k to the left as longas i, . . . , i + k − 1 have battery power and i + k haslifetime at least T ′ and until reaching xi−1. Computingthe maximum distance i + k may travel can be done

8

as explained in the case of xi < yi+1. The maximumdistance relay i + j, for j < k, may travel is B′i+j/awhere B′i+j is the remaining battery power of relayi + j. If all relays have enough battery power to bemoved to xi−1 with relay i+ k still having lifetime atleast T ′, update yi+j ← xi−1, for j ∈ {0, . . . , k} andmove to iteration i − 1. Otherwise, the move to theleft was stopped at x ≤ xi by relay j: either j < k andthe battery of relay i + j was depleted, or j = k andthe lifetime of relay i+k is exactly lifetime T ′. Assignyi+` ← x, for every ` ∈ {j, . . . , k}, and continue theprocess of moving i with k = j − 1.

There are n+1 relays, each may cause O(n) move-ments. In each such movement we check O(n) relaysand solve either (6) or (7).

In the discrete MAXFD case, instead of solving theequations, we find the leftmost grid point that canbe reached using the battery. This is O(m) time inthe worst case. Equation (7) is monotone decreasingand hence the requested grid point can be found inO(logm) time using binary search.

Let Ta denote the lifetime for movement cost a.Lifetime Binary Search performs a binary search forT in the interval [T∞, T0].

Theorem 11. Lifetime Binary Search is an FPTAS fordiscrete MAXFD.

Proof: Observe that Ta ≥ Tb if a ≤ b. It followsthat Ta ∈ [T∞, T0]. By Theorem 2 we have that

T0 =(∑nj=0

α√Bj)

α

Dα≤ (n α

√Bmax)

α

Dα=nαBmax

Dα,

where Bmax = maxiBi. On the other hand,

T∞ = mini

Bi(xi+1 − xi)α

≥ Bmin

Dα,

where Bmin = miniBi. Hence the number of iterationsof the binary search is bounded by

log(

T0

εT∞

)≤ log

(nαBmax

εBmin

)= α log n+log Bmax

Bmin+log 1

ε ,

which is polynomial in the input size and 1/ε.Due to Lemma 7 Lifetime Binary Search can be used

to approximate MAXFD.

Theorem 12. Lifetime Binary Search is an FPTAS for thespecial case of MAXFD where Bi ≥ aD(1+ 1

n ), for everyi.

6 HEURISTICS FOR MAXTL

In this section we develop heuristics for single deploy-ment MAXTL, when a > 0. The reason for consideringsingle deployment MAXTL is due to experimental evi-dence that suggests that any optimal MAXTL solutionrequires just one deployment, see section 7.2.

6.1 Lifetime-DP and NLP as Heuristics for MAXTL

If instead of evaluating the function LF (N), Lifetime-DP evaluates LT (N) in each iteration, it can be usedas a heuristic for single deployment MAXTL. LT (N)can be computed by summing over successive com-putations of the lifetime of first death until the deathof the transmitter. Since the lifetime of first death on knodes can be computed in time O(k), LT (N) for n+1nodes can be computed in worst-case time O(n2). Therunning time of this dynamic programming heuristicwhich we call DP is O(nm2 · lT (n)), where lT (n) isthe time to compute LT (N). Since lT (n) is O(n2), therunning time of DP is O(n3m2).

Single deployment MAXTL can also be formulatedas a nonlinear program if LT (N) is used as theobjective function in (2). We implement a nonlinearprogramming algorithm NLPalgo for this formulationusing the MATLAB function FMINCON which weset to use the medium-scale line-search method. Thismethod finds local maxima and is not guaranteed tofind the global maximum.

6.2 Other Algorithms and Heuristics for MAXTL

We now describe the MtoOpt algorithm as shownin Algorithm 2. Since we know the set of optimallocations when there is no friction, Opta=0, we wouldlike to move all the relays a fixed percentage towardsthese optimal locations under the assumption thatthe optimal location for a relay in the presence offriction is somewhere between the relay’s initial lo-cation and Opta=0. The algorithm evaluates each ofthe solutions where the relays move 1%, 2% . . . , 100%towards Opta=0 and outputs the solution resulting inthe longest transmission lifetime. This algorithm runsin O(n2) time as lT (n) is of order O(n2) and a constantnumber of points are evaluated.

We also implement an exponential algorithm, EXP.This algorithm does not use any heuristic but insteaduses brute force to evaluate all possible

(m+1n

)combi-

nations for the n relays over the m+1 possible, equallyspaced, grid locations between the transmitter and thereceiver. The algorithm preserves order among therelays. The EXP runs in O(mn) time and, hence, isonly practical for small values of n when over 100grid points are considered.

Algorithm 2 The MtoOpt algorithmINPUT: NCalculate the set of optimal locations, Optstat, using Eqn. 5For Each Percentage i do:

1) Li ←− Lifetime(InitLoc+ (Optstat − InitLoc) ∗ i)2) If L(i) > BestLifetime then

a) BestLifetime←− L(i)b) BestLocs←− InitLoc+ (Optstat − InitLoc) ∗ i)

OUTPUT: Final locations of the relays.

9

7 SIMULATIONS AND RESULTS

This section summarizes the results of the compre-hensive simulation study that was performed. Thesimulations were performed using MATLAB 6.5 R13on a machine running Windows XP 2002 SP3 with anIntel Pentium D CPU 2.8GHz and 2GB RAM. For afixed unit length network with one transmitter andone receiver we varied each of the parameters n, α,a and m, where n is the number of nodes in thenetwork, α is the path loss exponent in the communi-cation model, a is the coefficient of friction and m isthe number of discrete points that are considered aspossible final locations in single deployment MAXTLwith grid relay locations. For each range of values ofthese parameters we evaluated their effect by runningsimulations with all the mobile relays being initiallylocated at the transmitter, at the midway point be-tween the transmitter and the receiver, and at thereceiver.

We also simulated the relays being randomly dis-tributed along the line according to both the uniformdistribution and the normal distribution. In the ran-dom initial location cases we used Monte Carlo sim-ulations and present the results with 95% confidenceintervals.

We evaluate each of the heuristic algorithms inthe following ways: i) how they perform against theupper bound lifetime achievable when there is nofriction and relays can be located at any point, ii)how they perform against the lower bound givenby the case where mobility has infinite cost and thenodes must remain in their initial locations, and iii)the time cost of each of the algorithms. In general,i) illustrates the cost of mobility and the effect ofincreasing the number of relays in a network whileii) demonstrates the benefit that mobile nodes bringto a WSN and how, even with relatively high friction,they can extend the lifetime of a network.

We define a unit of friction to be that which allowsa relay with one unit of battery power to travel theunit distance from the transmitter to the relay. Inother words, with friction set to 1 a relay with fullbattery will use up all of its power moving from thetransmitter to the receiver.

7.1 Performance of Algorithms for MAXFDThe performance of all the algorithms and heuristicsfrom the preceding two sections was tested on theMAXFD problem. The effects of varying the initiallocations of the relays, their initial battery levels andthe cost of movement were simulated. Also, for thediscrete algorithms, the effect of varying the numberof grid points available was tested. On the MAXFDproblem, the Lifetime Binary Search, the Lifetime-DPand the EXP algorithms had very similar perfor-mances with over 50 gridpoints although they weresensitive to the exact number of points used. Above

200 gridpoints, the algorithms performed nearly iden-tically to the NLPalgo-FD.

7.2 Super Exponential AlgorithmWe conducted extensive simulations to decidewhether allowing the relays to move more than onceduring the lifetime of the network results in an in-crease of network lifetime in the MAXTL problem.To this end, we created a super-exponential algorithmthat pre-computes the entire lifetime of the networkfor each possible configuration of the n relays on them possible locations. For each of the

(m+1n

)configura-

tions the SuperEXP algorithm with the fitness functionLifetime of First Death was re-run after the first nodedied on the

(m+1n−1

)remaining reconfigurations. This

was repeated for the n − 2, n − 3, . . . 1 relays or untilthe transmitter died. Also, we evaluated exponentialversions of the DP and MtoOpt algorithms where wereconfigure the network after each relay death and useLifetime of First Death to evaluate the optimal config-uration at each iteration. However, as the SuperEXPoutperforms both DPexp and MtoOpt-exp for large m,we will only present our results from simulationsusing this algorithm.

Our simulations show that while the SuperEXPoften outperforms the EXP and DP algorithms forlevels of friction close to 0, the EXP and DP are justas good as SuperEXP for all other levels of friction.Furthermore, the super EXP algorithm never outper-forms the NLPalgo. Our empirical observations leadus to the following conjecture:

Conjecture 13. In an optimal MAXTL solution, the relaysneed only be deployed once.

Based on this conjecture, in the remaining simula-tions we evaluate the algorithms on a move once basiswith the remaining relays adjusting their transmissionranges after the death of a relay to accommodate forthe gap in communication.

Another important observation from analyzing theexponential versions of the heuristic algorithms is thatthey perform just as well when the battery levels arenot equal across the nodes in the network. After thedeath of the first relay, the remaining nodes will havealready moved different amounts and communicatedover different distances and so, when the heuristicsare applied to the reduced set of nodes in the seconditeration the battery levels across the nodes in thissub-problem will be unequal. Therefore, in evaluatingthe performance of our algorithms we set the initialbattery levels to be one unit across all nodes in thesimulations that follow.

7.3 How many grid points do we need?By varying the value of m between 5 and 200 insteps of 5, while holding all the other parametersconstant, we evaluated the effect that increasing the

10

number of possible final locations for the relays in theEXP and DP algorithms had on the optimal locationsfound by these algorithms. We observed that theoptimal lifetimes found by the heuristic algorithmsincreased asymptotically towards the upper boundwith m. Increasing m from 5 to 65 results in anincrease of the worst lifetime found by the algorithmsby 28% in absolute terms, while increasing m from100 to 200 results in an increase of just 1%. How-ever, the EXP algorithm takes 340.422 seconds torun when m = 200 and only 41.437 seconds whenm = 100. The DP, by contrast, takes 3.859 and 15.578seconds when m = 100 and m = 200, respectively.Since using large values of m was impractical forour simulations, especially those requiring the montecarlo method, we set m = 100 when evaluating theEXP algorithm against other parameters except whereexplicitly stated otherwise. The Lifetime Binary Searchalgorithm was extremely fast and could easily accom-modate high values of m but was highly inconsistentat values < 100.

Note that the EXP algorithm outperforms the DPfor many values of m, this is due to the algorithms be-ing evaluated using Transmission Lifetime as opposedto the Lifetime of First Death.

0 1 2 3 4 5 6 7 8 9 1020%

30%

40%

50%

60%

70%

80%

90%

100%

Friction

Lifetim

e

Lifetime Binary Search

MtoOpt

DP − b=200

NLPalgo

EXP − m=200

Fig. 2: Performance of the heuristics compared to theoptimal non-friction solution

7.4 The Effect of α on Network Lifetime

Using the values 1 ≤ α ≤ 6, Bi = 1, n = 4 and varyingthe cost of mobility between 0 and 10 we investigatedthe effect path loss exponent has on the performanceof the algorithms and the resulting optimal lifetimes.We found that the parameter α has little effect onthe respective algorithms for values greater than 1other than to consistently reduce the optimal lifetimesfound. Consequently, we chose to set α = 2 in thesimulations to follow.

7.5 Increasing the Relay Count: How do the algo-rithms scale?

Using the values 2 ≤ n ≤ 30, Bi = 1, α = 2 andvarying the cost of mobility between 0 and 10 weevaluated the effect the number of relays has on thealgorithms. We found that increasing the number ofmobile relays in the network has the expected effect ofincreasing the lifetime of the network. However, theincrease is consistent across the algorithms and, hence,the number of relays in the network did not affectthe performance of our algorithms. For the belowanalysis, we fix the number of relays at n = 4 sothat we can compare results across all experimentsincluding those requiring monte carlo simulations.

One important observation from our simulations isthat the NLPalgo does not scale well: for values of nover 40 the DP algorithm with m = 200 finds moreoptimal solutions when a > 0.8.

Note, we were unable to test the EXP and SuperEXPalgorithms for values of n > 11 with m > 100 asMATLAB was unable to fully compute the possiblepermutations.

7.6 The Effect of Friction

Using the values α = 2, Bi = 1, n = 4, m = 200 andsetting the cost of mobility, a, between 0 and 10 we getthe graph in Fig. 2. Out of all the algorithms we evalu-ated, the NLPalgo algorithm gave the longest MAXTLlifetimes across the entire range of parameter settings.The dynamic programming approach, DP, performsas well as the EXP for levels of friction below 2.5but its performance quickly drops off above this level.MtoOpt gives surprisingly good results considering itssimplicity and for a ≥ 2.5 it outperforms the DP. TheLifetime Binary Search also performs optimally at lowlevels of friction but also drops off above a = 2.5. Notethat it outperforms the DP at higher levels of frictionwhich is surprising considering that it is designed forthe MAXFD problem.

All of the algorithms we evaluated passed the sensechecks that for zero friction the relays should go to thelocations described by Eqn. 5 and for infinite frictionthe relays should remain in their initial locations.More importantly, all of the algorithms performedat least as well as the case where the relays mustremain in their initial locations and from this we canconclude that it always pays to move unless frictionis effectively infinite or the relays are initially locatedat their optimal positions.

7.7 Random Locations

To fully understand the effect of the initial locationsof the relays on the maximum lifetime achievable by anetwork we ran monte carlo simulations on networklifetime using each of the heuristic algorithms with theinitial locations of the relays randomly chosen from

11

the uniform and normal distributions while varyingthe level of friction and keeping all other parametersconstant. We implemented the Control Variate tech-nique (using the theoretical optimal locations) andAntithetic Variate technique to reduce the standarddeviation in the mean value of the simulations by afactor of 3.

We present the results from the uniform distributionsimulations with 50, 000 iterations of the NLPalgo al-gorithm and 95% confidence interval error bars in Fig.3. Clearly, we can see that increasing the friction hasthe combined effect of decreasing the average lifetimein the networks and increasing the standard deviation.This means that the initial locations of the relays has agreater impact on the achievable lifetime of a networkas the level of friction is increased. Also, this impactgrows rapidly: increasing the level of friction from 0to 2 results in an increase in standard deviation from0 to 4.96% resulting in error bars of ±9.72% aroundthe mean.

7.8 Time cost of algorithms

The heuristic algorithms were fast relative to theexponential algorithms and especially the SuperEXPalgorithm. For a = 10, m = 200, Bi = 1, n = 4 andα = 2 the Lifetime Binary Search algorithm ran thefastest by far at 0.0023s, the MtoOpt algorithm wasnext fastest at 0.047s, the NLPalgo and DP algorithmswere comparable at 16.56s and 15.94s, respectively.The exponential algorithms on the same set of param-eters all took between 475s and 500s.

To give an idea of how slow the SuperEXP is, onthe same set of parameters as before but with n = 3the SuperEXP took 481.9s while the EXP took 5.7s.

The monte carlo simulations were only run onthe Lifetime Binary Search, MtoOpt, NLPalgo and DPalgorithms with n ≤ 4 and a ≤ 2 with the simulationstaking up to 7 hours for the larger parameter values.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

60%

65%

70%

75%

80%

85%

90%

95%

100%

Friction

Life

tim

e

Fig. 3: MC simulations of lifetimes for random startinglocations with 95% Error Bars

1.6 1.8 2 2.2 2.4 2.610

20

30

40

50

60

70

80

x

Lifetim

e

small0

big0

a=0

a=10

a=20

Fig. 4: Instance of simulation in experimental verifi-cation of Conjecture 14

7.9 Deployment from Base Station

Assume n relays with battery powers B1, . . . , Bn areinitially positioned with the transmitter at 0. In whatorder should the relays be deployed to maximize thelifetime of first death? We can verify experimentallythat relays with larger batteries should be deployedto positions farther than relays with smaller batteries.

Conjecture 14. If all relays have initial locations at 0,it is optimal for them to be ordered after deployment inincreasing order of initial battery power.

Assume that in an optimal solution there are twoadjacent relays s and t with initial battery powersBs < Bt but ys > yt after deployment. Note that ifrelays are not ordered in increasing order of initialbattery power there is always such a pair. Assumerelay m, located at ym is the relay which relay stransmits to after deployment.

Let y = yt. Consider the interval I = (y, ym]. Let B′sand B′t denote the battery levels of relays s and t aftertraveling from 0 to location y. It must be that B′s < B′t.Consider case 1 where relay t remains at y and relay smoves to location y+x in interval I . For this case thelifetime of relay t is B′t/xα and the lifetime of relay sis (B′s − ax)/(ym − x)α. Consider case 2 where relays remains at y and relay t moves to location y + x ininterval I . For this case the lifetime of relay s is B′s/xα

and the lifetime of relay t is (B′t − ax)/(ym − x)α.Assume a is large enough so that (B′s − ax)/(ym −

x)α does not intersect with B′t/xα in case 1. Since

(B′t − ax)/(ym − x)α > (B′s − ax)/(ym − x)α, and byassumption (B′s − ax)/(ym − x)α < B′t/x

α ∀x ∈ I ,the lifetime of first death of case 2 is greater than thelifetime of first death of case 1, contradicting the initialassumption of optimality.

For the a for which (B′s − ax)/(ym − x)α intersectswith B′t/x

α (intersection point is maximum lifetime

12

of first death), we verify that lifetime of first deathof case 2 is greater than the lifetime of first death ofcase 1 via simulation. An instance of the simulationis shown in Fig 4.

In Fig. 4, B′t = 150, B′s = 100, α = 2 and I =(0, 4]. Big0 is the function 150/x2 and small0 is thefunction 100/x2. There are also the functions whichdepend on a: f150(a, x) = (150 − ax)/(4 − x)α andf100(a, x) = (100 − ax)/(4 − x)α. Keep in mind that(150−ax)/(4−x)α > (100−ax)/(4−x)α for a given aand ∀x ∈ I when viewing Fig. 4. That is, for a givena, f150(a, x) will be the larger function.

When a = 0, f150(0, x) and f100(0, x) intersectsmall0 and big0 respectively at exactly the same life-time. Note that when a = 10, f150(10, x) intersectssmall0 at a larger lifetime than does f100(10, x) inter-sect big0. When a = 20, the difference in lifetimesof the intersection points is even more drastic. Thisbehavior is common across all experiments. Thus,there is strong experimental evidence supporting theconjecture.

8 FUTURE RESEARCHFirst, we list several open problems:• Formally proving conjectures 13 and 14• We conjecture that MAXTL is NP-Hard when

there is friction.Finally, there are a number of possible natural gener-alizations for our problem:• Consider the model which allows swapping

and/or activation/deactivation.• Consider the model where the initial and final

locations of the relays can be anywhere in theplane.

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