Relay Deployment and Power Control for Lifetime Elongation in Sensor Networks

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1 Relay Deployment an Power Control for Lifetime Elongation in Sensor Networks Yufeng Xin, Tuna Güven, Mark Shayman Institute of Avance Computer Stuies University of Marylan, College Park, MD {yxin, tguven, shayman}eng.um.eu Abstract In a sensor network, usually a large number of sensors transport ata messages to a limite number of sinks. Due to this multipoint-to-point communications pattern in general homogeneous sensor networks, the closer a sensor to the sink, the quicker it will eplete its battery. This unbalance energy epletion phenomenon has become the bottleneck problem to elongate the lifetime of sensor networks. In this paper, we consier the effects of joint relay noe eployment an transmission power control on network lifetime. Contrary to the intuition the relay noes consiere are even simpler evices than the sensor noes with limite capabilities. We show that the network lifetime can be extene significantly with the aition of relay noes to the network. In aition, for the same expecte network lifetime goal, the number of relay noes require can be reuce by employing efficient transmission power control while leaving the network connectivity level unchange. The solution suggests that it is sufficient to eploy relay noes only with a specific probabilistic istribution rather than the specifying the exact places. Furthermore, the solution oes not require any change on the protocols (such as routing) use in the network. I. INTRODUCTION Sensor noes are low-cost, low power evices equippe with sensing, communication an processing capabilities albeit with finite an nonrenewable battery energy. Naturally, energy efficiency is a major concern in sensor networks. As communication in sensor networks is generally multipoint-topoint (i.e., from sensor noes to an information sink noe), sensor noes in the proximity of the sink noe have to relay comparably excessive traffic especially when the number of noes in the network is large. For this reason, such noes inevitably eplete their energy rapily regarless of the routing algorithm utilize. This is the main energy bottleneck in a typical sensor network. To this en, Luo et al. propose to change the position of the information sink from time to time [9]. Since the sink mobility changes the set of sensor noes aroun the sink over time, the traffic loa, an consequently the energy epletion rate iscrepancy between the sensors becomes smaller. However, sink mobility is rather a complex solution with the necessity of carefully selecting the mobility parameters such as the trajectory an spee of the sink noe. It also assumes that the sink noe has no energy limitations as mobility comes with Partially supporte by AFOSR uner grant F an NSF uner grant CNS the cost of significant energy consumption. Furthermore, the sink mobility nees to be accompanie by a specific, complex routing algorithm to forwar ata from sensors to the mobile sink noe (See [9]). In another line of work, strategic placement of sensors in linear networks was stuie [8]. The necessary istance between neighboring sensors is obtaine in orer to achieve a specifie lifetime. The basic observation is that noes closer to the sink have shorter mutual istance. However, the geographical istribution of sensor networks is often ecie by the specifie sensory task in a certain area. Hence, strategic placement of sensors that is not complying with the sensory task naturally leas to a strictly suboptimal sensor noe istribution. This may incur an unacceptably high cost since the ensity of sensor noes woul be excessively high (as well as reunant from the sensory task point of view) aroun the sink. In this paper, we focus on the aforementione energy bottleneck issue an consier eploying aitional relay noes in orer to improve network lifetime of the multipoint-to-point ense sensor networks. Relay eployment is further couple with power control (i.e., intelligent assignment of transmission ranges for both sensor an relay noes). Contrary to the intuition, we moel relay noes as simple evices. Specifically, we assume that the relay noes are even simpler than the sensor noes in the sense that they have all the capabilities of a sensor noe with the exception that they o not possess any sensing component. Essentially, the relay noes exten the network lifetime by remeying the excessive traffic buren of a subset of sensor noes. Note that this moel iffers from the traitional moels in which relay noes have superior communication an processing capabilities. Furthermore, we consier a flat network architecture as oppose to a hierarchical moel, where sensors an relay noes communicate with the information sink via possibly a multi-hop route that consists of several sensor an/or relay noes. Intuitively, relay noe ensity shoul increase more an more with ecreasing istances to the sink noe(s) since we nee more relays to share the increasing traffic (i.e., energy) loa. On the other han, with enser noe istribution, one can presumably ecrease the transmission ranges in the regions of higher noe ensity without affecting the connectivity properties of the overall network (e.g., k-connecteness). Hence, as the noe ensity increases with the aition of the relay noes, the /06/$0.00 (c) 006 IEEE

2 energy consumption of sensor noes reuces not only because the traffic loa that a noe is require to forwar ecreases but also the energy consumption per transmission is reuce as well. Therefore, relay ensity an transmission range shoul be ecie jointly in optimizing the network lifetime. An obvious an maybe simpler alternative solution one may consier is to eploy sensor noes with ifferent levels of battery energy in such a way that the initial battery energy is higher for the noes that are closer to the sink noe. This way the energy rainage ue to heavy traffic buren aroun the sink can be hanle by the aitional battery energy installe in sensor noes. However, it is easy to see that for the same given total energy expeniture, the joint relay eployment an power control metho promises a longer network lifetime compare to eploying sensors with heterogeneous battery energy levels. This is ue to the fact that ajusting transmission ranges results in aitional energy savings which cannot be realize otherwise. In aition, one shoul note that it is often not possible to equip sensors with extene battery life ue to harware constraints on the size of the batteries. Another avantage of eploying aitional relay noes is that once relay noes are carefully eploye, no further complex protocols, like mobility an energy-efficient routing protocols, are neee. Moreover, the solution oes not specify the exact locations of relay noes; rather it suggests that it is sufficient for relay noes to follow a specific probability istribution. This can be beneficial for certain applications where the exact locations of the noes eploye cannot be specifie (e.g., military battlefiel networks). Therefore, it is our belief that the propose solution is a soli alternative for the aforementione network lifetime elongation problem. The rest of the paper is organize as follows. In Section II, we escribe the network moel an state the relay placement problem. The analytical solutions an numerical results regaring relay ensity an transmission power control are presente in Section III. In Section IV we escribe the experimental setup use for evaluation of the propose solution an present the simulation results. Section V conclues the paper. II. NETWORK MODEL AND PROBLEM STATEMENT Consier a wireless sensor network with static sensor noes an one information sink that is responsible to collect all the ata generate by the sensor noes. Let S = {1,...,S} be the set of sensor noes. Assume that the sensor noe istribution follows a Poisson process with fixe ensity within a circle C of raius D an center O. Thevalueof is in general ecie by the specific sensory task uner consieration. Each sensor noe creates ata traffic at a constant rate λ an the sink noe is locate at the center O of the circle C. We consier eploying aitional relay noes within C to prolong the network lifetime. Let R = {1,...,R} be the set of relay noes to be ae. Relay noe ensity shoul increase with ecreasing istance (0 < D) to the sink noe as mentione in the previous section. Towar this en, we assume that relay noes are istribute using a nonhomogeneous S1 S D t() n β Fig. 1. Loa istribution moel ( t 0 ). S 1 an S are two isjoint parts of a sector whose two sies are tangent to a circle with center n an raius t() [9]. Poisson process [10] with ensity ρ r () in contrast with the sensor noes. Note that here we make use of the symmetry arising from the specific istribution of sensor noes as well as the location of the sink noe an efine ρ() as a function of. Hence, the overall noe istribution also follows a Poisson process with ensity ρ()= + ρ r (). Our objective is to fin the optimal relay ensity ρ r () achieving maximum expecte network lifetime when sensor noes have a fixe an nonrenewable battery energy B. We assume that all noes (sensors, relays an the sink) use a common frequency channel. Let t s D enote the maximum transmission range a noe can use to transmit. As iscusse earlier, noes can presumably have shorter transmission ranges in the regions with a higher noe ensity while keeping the network connectivity the same. It is a consequence of this fact to efine the transmission ranges of noes also as a function of istance to O; i.e., t() t s. In the following analysis we will assume that t() =t 0 for t 0. This simplifying assumption merely states that one-hop neighbors of the sink noe use a common transmission range. Note that t() is a nonecreasing function of, i.e., t 0 represents the minimum transmission range use in the network. The value of t 0 is naturally a esign parameter to be optimize. We represent the original sensor network without the relays as G(t s, S) an we assume that is selecte such that G(t s, S) is k-connecte 1 with a high probability p 0 (e.g., p 0 = 0.999), i.e., P(G(t s, S) G k )=p 0, (1) where G k is the set of k-connecte graphs. Moreover, let G(t(), S R ) be the overall network graph obtaine with the aition of relay noes R = {1,...,R} an by employing transmission power control. 1 Agraphisk-connecte if an only if no set of k 1 noes exists whose removal woul isconnect the graph [4]. 346

3 As in [9], for simplicity we assume an ieal loa-balance shortest path routing (See [6] for a presentation of such protocols). We o not consier ata aggregation. For simplicity, we efine ε() as the energy consume to transmit a unit of ata by a noe at a istance from O an assume that energy consumption for processing an receiving tasks is negligible. Specifically, we let ε()=c(t()) a, where c is a constant an a represents the attenuation exponent. It is typical to assume a 4. We aopt the network lifetime efinition presente in [], which is the time span from the sensor eployment to the first loss of coverage. Specifically, we assume that each sensor has a limite sensing range. Let C ρs be the particular area in C that is sense by at least one sensor. Loss of coverage is efine as the time when some area within C ρs that is initially covere by the network is not sense by any active noe anymore. A relate system parameter is the energy loa L() which is efine as the expecte power spent by a noe at a istance from O. Note that the average power consumption level is equal for all noes that are equiistant from the sink ue to the inherent symmetry of the problem. An analytical expression for L() is obtaine by extening the geographical moel presente in [9] an [5]. As shown in Fig. 1, for a noe n at istance from the center O, the average loa taken by noe n is in proportion to (S 1 + S )/S where S 1 an S can be approximate as follows. S 1 β() (D ), S π t(), t 0 () S 1 = πd, S = πt(), < t 0 (3) where β()=arcsin(t()/). Then, the expecte energy loa of a noe at istance is L() (S 1 + S ) λε() (4) S ( + ρ r ()) ( ) β() (D )+ π t () λε() π t, t 0 (5) ()ρ() = D λε() t ()ρ(), < t 0 (6) We observe the variation of expecte loa with respect to istance in Fig. when there is no relay eployment or transmission power control (i.e., ρ r () =0, t() =t s for all 0 < D). The aforementione energy bottleneck phenomenon can be clearly seen as the noes in the proximity of the sink noe eplete their energy rapily an the sink noe has no way to collect ata any more even though a large part of the network is still alive [9]. Consequently, network lifetime is strictly limite by the energy epletion rate of these sensors that are close to the sink. Following the iscussion in [9], the expecte network lifetime maximization problem can be reformulate as a energy loa balancing problem. The corresponing optimization problem can be state as follows: min ρ(), t() max L() (7) 0< D Loa Original Loa Distribution Fig.. Original loa istribution without relay, D = 100, t s = 5, λ = 1, a = 4, an = 1/π s.t. P(G(t(), S R ) G k ) p 0, (8) t() t s, (9) III. OPTIMAL RELAY DISTRIBUTION Due to shortest path routing assumption, the sensor noes locate at the outmost layer of the circle C (i.e., sensor noes that are at D units away from O) forwar only their locally generate ata an o not relay any aitional ata generate by other sensor noes (i.e., L(D) =λε(d)). L(D) constitutes a lower boun on the loa for all other sensors. L() L(D)=λε(D), for all 0 < D. (10) Note that eployment of aitional relay noes to the network has no effect on the lifetime of the noes at istance D. Hence, for an initial battery energy level B, the expecte network B lifetime can be upper boune by λε(d) an it can only be achieve if (10) hols with equality, i.e., L ()=L(D), for all 0 < D, (11) where L () represents the expecte energy loa at istance when relay noes are optimally istribute to the network. From (4) we can obtain the optimal relay istribution ρ () in terms of the optimal transmission ranges t () ρ r () = S 1ε()+S (ε() ε(d)) S ε(d) (1) ρ r () = S 1(t ()) a + S ((t ()) a (t (D)) a ) S (t (D)) a. (13) Note that t (), the optimal transmission range to be use at a istance to O, is yet to be etermine using the connectivity constraints. It is possible to maintain such a loa balance by merely eploying aitional relay noes to the network as will be shown in the following section. However, the number of relay noes require can be reuce if relay eployment is couple with transmission power control, i.e., by optimizing t() for all 0 < D. In fact, the savings in require number of relays is rather significant which is presente in Section III-B. 3463

4 Raley ensity Relay ensity istribution Fig. 3. Relay ensity, D = 100, t s = 5, λ = 1, a=4, an = 1/π A. Optimal Relay Deployment with Homogeneous Transmission Ranges In this section, we consier relay eployment uner the conition that all sensors an relays have the same fixe transmission range, i.e., t()=t s for all 0 < D. As there is no transmission power control involve with the optimization, the optimal relay istribution ρ r () can simply be obtaine from (13) as ρ r () = = β()(d ) πt s > t s, (14) = D t s t s. (15) Fig. 3 shows the variation of relay ensity ρ r () with respect to. We observe that a large amount of relays are neee to balance energy loa. For instance, just in the the one-hop neighborhoo of the sink noe (an area of πt s ), the require number of relays is equal 10,000, which is equivalent to the total number of sensors in the network. This is a rather large amount, even though the require relays ecreases ramatically with increasing. Such a high number of relays is require partly because the energy loa of the sensors at the ege of the network (L(D)) is very low which causes a strict constraint for the sensor that are close to the sink. Moreover, here we i not exploit the potential benefits one can observe through transmission power control. In fact, in the following section we will show how the relay ensity can be reuce significantly by controlling the transmission ranges. B. Joint Relay Deployment an Power Control In the previous section we observe that a high relay noe ensity is observe especially in the vicinity of the sink noe. A high noe ensity of wireless noes can potentially incur severe interference. In this section, we will show that the relay ensity can be reuce efficiently by employing transmission power control. It is clear that a shorter transmission range reuces the energy consumption of a noe per unit ata transmitte (ε()). As a result, the require number of relays to share the loa can be reuce ue to the reuction of expecte loa (L()) for such noes. However, the transmission ranges cannot be reuce inefinitely. A certain level of connectivity shoul be maintaine for the network to operate successfully. A straightforwar treatment on the connectivity issue is to require the connectivity of the new network with the relays be no less than that of the original sensor network. This constitutes a constraint on the optimization problem as state in (8). In [3], it has been shown that a homogeneous a hoc network can be moele as a geometric ranom graph G where the following inequalities hol: P( G G k ) P( min k) (16) P( G is isconnecte) 1 P( min > 0) (17) Here P( min k) is the probability that each noe has at least k neighbors, i.e., the network has a minimum egree min k. For the case of Poisson istribute S noes with a ensity, transmission range t s this probability is equal to the following: P( min k)= ( 1 k 1 n=0 S ( πt s ) n e πt s ) (18) n! Moreover, it is shown in [3] that the boun given in (16) is tight when number of noes is high. Therefore, it is reasonable to relate the probability of network connectivity to probability on the minimum egree of the network. However, one shoul note that the connectivity results above are vali only uner a homogeneous Poisson noe istribution. Towar this en, we evenly ivie the circle C into a set of layers M = {1,...,M}, where the relay ensity in the m th layer is constant (ρ m ). We also assume that all noes in layer m have the same transmission range t m. The with of a layer is enote as δ an is equal to D/M. The istance from the m th layer to the sink position O of C, m, represents the istance between O an noe n that is locate in layer m an is equiistant from both bounaries of layer m. For simplicity we assume that ρ M = 0. Furthermore, it is convenient to ivie the first layer into two parts as, i.e., 0 < t 0 an t 0 < δ as L() terms for these two segments are ifferent accoring to (4). We will enote this inner layer as the layer m = 0 in the following. Since the uniform assumptions on the noe ensity an transmission range within one layer still hol, we have the following for the overall network G(t(), S R ): P(G(t(), S R ) G k ) P( min k) ( M k 1 (( + ρ m )πt m 1 ) n Sm +ir m e (+ρ m )πt m ) (19) n! m=1 n=0 where S m an R m enote the number of sensor an relay noes locate in the m th layer respectively. Note that there is an inaccuracy involve in (19) ue to neglecting the bounary effects between two ajacent layers where a noe in one layer has its transmission range exten into the other layer. However, it is easy to see that this is The with δ is selecte such that it is larger than t s. 3464

5 Relay ensity istribution ρ m Transmission range t m (Distance from center O) = 0.5/π > k=1 = 1/π > k=5 = 1.5/π > k=11 (a) Relay ensity istribution vs. k=1 k=5 k= (Distance from center O) (b) Transmission range vs. Fig. 4. k-connectivity ( varies), D = 100, t s = 5, λ = 1, a = 3, an p 0 = rather a strict assumption ue to the nature of ata transport which is always along the irection from an outer layer to an inner layer. In fact, if a noe in the m th layer has neighbor noes at (m 1) th layer where the noe ensity is higher, it can only have more neighbors to sen ata than the case where the bounary effects are ignore. Therefore, the relay ensities to be foun using the above formulation will be an upper boun to the actual case. With this simplifying assumption, for a given layer m we can obtain the necessary relay ensity ρ m an the corresponing transmission range t m inepenent of other layer parameters by jointly satisfying (13) an (16) as follows: an ( ρ m = D t m a +t m (t m a t s a ) t m t s a, m = 0 (0) 1 = β( m)(d m )t m a + πt m (t m a t s a ) πt m t s a, m > 0 (1) k 1 n=0 P( m min > k) p o ((ρ m + )πt m ) n Sm R m e (ρ m+ )πt m ) p o () n! where min m represents the minimum noe egree in layer m. The joint solution (ρ m,t m ) can be obtaine from (0), (1) an () using an iterative algorithm. The necessary relay ensity an transmission ranges uner various initial sensor ensity ( ) are epicte in Fig. 4. Note that for each we have a ifferent network connectivity level. For instance, for = 1/π, P(G(t s, S) G 5 )= This changes the set of constraints of the optimization problem since the overall network is constraine to leave original connectivity conitions unchange. The results emonstrate the traeoffs between the relay ensity an transmission range with the istance. By jointly consiering the relay ensity istribution an the transmission range control (power control), we observe that much lower relay ensity can be achieve for the same expecte network lifetime B/L(D), comparing Fig. 4(a) with Fig. 3. Comparably, the number of relays require in the one-hop neighborhoo of the sink noe is 1 (relay ensity ρ is equal to 5.71 when = 1/π). Note that, uner this solution not only the noe ensity aroun the sink is lower, but also the transmission range in the vicinity of the sink is smaller as well (t s =5 vs. t 0 =1.08). To be fair with the earlier analysis, we shoul also calculate the number of relays that are in the circle with an area of πt s. Using the values from Fig.4, the number of relays neee in the area of πt s is 19 (1 in the one-hop neighborhoo an 198 noes in the area between the circles with raius = 1.08 to = t s = 5) compare to 10,000 when no transmission range control is use. Hence, the savings in the number of relays is significant making the propose solution a feasible alternative for maximizing the network lifetime. One question that may arise is the effect of the aitional noes on the effective throughput of the network. However, as the relay noes o not generate aitional traffic they o not constitute a egraation on the throughput capacity of the network. We consier the etails of this issue in [11]. IV. SIMULATION RESULTS In this section, we present simulation results to verify the performance of the propose relay eployment an transmission power control scheme. We compare performance of the propose solution in Section III-B with the original sensor network, i.e., without aition of relay noes an with a fixe transmission range t s. Using the values for relay ensity an transmission ranges, our aim is to observe the effects of approximations mae throughout the analysis. Every noe is assigne an initial energy of E = units. We fix the relate parameters the same as given in Fig. 4 with k = 5(kconnectivity level) an the packet size is µ = 0.5. We aapte the following neighbor-base loa balancing routing protocol in the simulation. During the simulation, every noe maintains a neighbor list that contains noes within its transmission range that satisfy two conitions: (1) enough resiual energy; () equal or smaller istance to the sink. When there is a packet, either generate locally or relaye from another noe, it chooses the noe in its neighbor list that has the maximum resiual energy as its next-hop estination. Due to the broacast nature of wireless communication, this information can be obtaine when neighbours attach their resiual energy value to the regular ata or routing beacon packets. We assume a perfect MAC layer for simplicity. 3465

6 Resiual energy Energy consumption rate e e e e e+006 1e L 0 L 1 L L 3 L 4 L 5 L 6 L 7 L 8 L 9 L (a) Original sensor network without relays t Original sensor network Sensor network w/ relays (b) Sensor network with relays Fig. 5. Simulation result, D = 100, t s = 5, λ = 1, a = 4, an p 0 = The simulation results are epicte in Fig. 5(a) an Fig. 5(b). In Fig. 5(a), the X-axis represents the actual time in the network an the Y-axis represents the average resiual energy over all noes within a layer. We observe that the simulation results confirm the analytical results, suggesting the errors ue to approximations are negligible. For the original sensor network, the noes in the inner layer eplete their energy very fast an the network ies roughly at time t = 64. Fig. 5(b) shows the average energy consumption rate (per time unit) with respect to the istance for both original network an the one with relays an transmission range control. We observe that, in the later case, the energy of wireless noes over all layers is consume at a similar pace. Consequently, the lifetime of the network is prolonge ramatically after almost all noes in the network eplete their energy. This is a very esirable result in terms of the scalability because we make the lifetime of the sensor network above 100 times longer than the original network by aing only about the same number of cheap relays as the number of original sensors an ajusting the transmission ranges. V. CONCLUSION Due to the multipoint-to-point communications pattern observe in general homogeneous sensor networks, the closer a sensor to the sink, the quicker it will eplete its battery. This unbalance energy epletion phenomenon has become the bottleneck problem to elongate the lifetime of sensor networks. In this paper, we propose to eploy wireless relay noes in the sensor network as a simple metho to solve this problem. These relay noes are rather simple evices contrary to the common approach in the literature. This way a cost-effective solution to elongate the lifetime of sensor networks is propose. By transforming the original homogeneous wireless sensor network to a heterogeneous sensor an relay hybri network, the expecte energy loa is evenly istribute over all the wireless noes (sensors an relays) so that all noes will eplete batteries at the same time. We presente analytical results uner ifferent scenarios where the maximum transmission ranges can be either homogeneous or heterogeneous. In the heterogeneous transmission range case, we take the network connectivity into consieration. By jointly consiering the loa balance an network connectivity, the results showe that the maximum network lifetime coul be achieve by aing relay noes with relatively lower ensity, at the same time the maximum transmission ranges of both relay an sensor noes can be significantly reuce. In this paper, we consiere a common istribution for sensor noes an exploit the properties to obtain the solution. However, the results can easily be extene uner ifferent sensor istribution moels. There are more problems that are worth further investigation. We have intentionally ignore the receiving an ata processing energy consumption. In aition, we have consiere only a single information sink that is locate at the center of a circle that constitutes the network area an we mae use of the inherent symmetry cause by these assumptions. Even though it is shown in [9] that locating the sink noe at the center of a circular network is optimal in terms of energy efficient ata collection in a static homogeneous sensor network, it is of interest to generalize the problem to the case where the sink noe is not necessarily locate at the center as well as to the case of multiple sinks. REFERENCES [1] I. F. Akyiliz, W. Su, Y. Sankarasubramaniam, an E. Cayirci. A survey on sensor networks. 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