Energy Savings Achievable in Connection Preserving Energy Saving Algorithms

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1 Enegy Savings Achievable in Connection Peseving Enegy Saving Algoithms Seh Chun Ng School of Electical and Infomation Engineeing Univesity of Sydney National ICT Austalia Limited Sydney, Austalia Guoqiang Mao School of Electical and Infomation Engineeing Univesity of Sydney National ICT Austalia Limited Sydney, Austalia Bian D.O. Andeson Reseach School of Infomation Sciences and Engineeing Austalian National Univesity National ICT Austalia Limited Canbea, Austalia Abstact Enegy saving is an impotant design consideation in wieless senso netwoks. In this pape, we analyze the enegy savings that can be achieved in a senso netwok whee each senso is capable of educing its tansmission powe fom a maximum powe p m, compaed with that in a senso netwok whee each senso can only tansmit at a constant powe level p m. To achieve a fai compaison, we assume sensos in both types of senso netwoks ae connected to the same set of neighbos, i.e. no connection is lost as a esult of a senso educing its tansmission powe. We futhe assume that sensos ae distibuted in a given aea following a Poisson distibution with known node density and the adio popagation is descibed by a log-nomal model. Ignoing bounday effect, we establish analytically the pobability fo a senso to achieve an enegy saving of at least h db. We also obtain the expected pecentage of enegy savings which can be substantial. The eseach epoted in the pape helps to answe questions such as whethe the enegy savings achieved by using a senso with a vaiable-tansmission-powe (and the consequent extension of its lifetime justify the additional cost involved in manufactuing it. Index Tems senso netwok, log-nomal shadowing, tansmit powe, enegy savings I. INTRODUCTION Wieless senso netwoks have been widely used fo monitoing envionments and detecting events. A wieless senso netwok consists of a set of sensos equipped with sensing hadwae, and able to communicate with each othe via adio links. Sensos ae nomally battey-opeated. In wieless senso netwoks, each senso plays two impotant oles. It sends data it has collected and it acts as a elay to othe sensos. Theefoe, failue of some sensos due to exhaustion of battey powe will cause topology change in a netwok, and in the wost case, the netwok will be disconnected pematuely. Since failed sensos ae nomally had to eplace, enegy saving becomes an impotant netwok design consideation in wieless senso netwoks. One of the common goals in designing wieless senso netwoks is to make sue sensing and communication tasks can be caied out and in the mean time the netwok lifetime is maximized. National ICT Austalia is funded by the Austalian Govenment Depatment of Communications, Infomation Technology and the Ats and the Austalian Reseach Council though Backing Austalia s Ability and the ICT Cente of Excellence Pogam. The appoaches taken so fa by vaious eseaches on impoving enegy efficiency have been vey divese []. The studies on enegy saving mechanisms cove almost all aspects of netwok design, including choices of hadwae. Thee ae also many enegy savings mechanisms that ae caied out in highe level aspects of netwok opeation. These include, but ae not limited to, outing potocols, scheduling mechanisms, and adjustment of tansmission powe. In [], the authos suvey the enegy-awae outing techniques which ae suitable fo wieless senso netwoks. These outing potocols use diffeent appoaches to meet the enegy constaint in wieless senso netwoks. Fo example, edundant data ae aggegated to educe the numbe of tansmissions. Thee is also an appoach that chooses diffeent paths at diffeent times to tansmit data, instead of using a single optimal path, to balance enegy consumption acoss all sensos. In [3], the authos eview 5 scheduling mechanisms in wieless senso netwoks. These scheduling mechanisms achieve the objective of minimizing enegy consumption by scheduling sensos into wok/sleep cycles. In this pape, we study the possible enegy savings achieved though adjustment of tansmission powe. Deciding a tansmission powe level is a complicated matte. It not only affects the amount of enegy consumed, but also detemines the establishment of diect links to the neaby nodes, and hence affects the netwok topology and the paths that can be used fo outing to the destination. In this pape, we investigate the possible enegy savings that can be achieved in a senso netwok whee sensos ae allowed to educe thei tansmission powe without losing any connections to thei neighbo nodes, via some connection peseving algoithms. Moe specifically, we analyze the enegy savings that can be achieved in a senso netwok whee each senso is capable of educing its tansmission powe fom a maximum powe p m, compaed with that in a senso netwok whee each senso can only tansmit at a constant powe level p m. To achieve a fai compaison, we assume sensos in both types of senso netwoks ae connected to the same set of neighbos, i.e. no connection is lost as a esult of a senso educing its tansmission powe. Reducing tansmission powe may have othe complicated effects on the

2 netwok. Fo example, on the one hand, it educes intefeence to the tansmission of othe senso pais which is beneficial; on the othe hand, it deceases SNR which consequently inceases packet eo ate. We do not conside these effects in the pape. We futhe assume that sensos ae distibuted in a given aea following a Poisson distibution with known node density and the adio popagation is descibed by a log-nomal model. Ignoing bounday effect, we establish analytically the pobability fo a senso to achieve an enegy saving of at least h db. We also obtain the expected pecentage of enegy savings. The eseach epoted in the pape helps to answe questions such as whethe the enegy savings achieved by using a senso with a vaiable-tansmission-powe (and the consequent extension of its lifetime justify the additional cost involved in manufactuing it. The est of this pape is oganized as follows: Section II intoduces elated wok on enegy savings. Section III defines the system model consideed in this pape, followed by section IV which pesents the basic theoetical analysis and deivations. Section V pesents futhe analysis and numeical data, which isolates key aspects of the aveage pecentage of enegy savings obtained fom section IV. Finally, section VI concludes this pape and discusses possible futue wok. II. RELATED WORK Enegy saving elated topics have attacted many studies ove the yeas. Reseach in the aea has been done fom diffeent pespectives []. Among them, the widely adopted way of achieving enegy saving is via minimizing the tansmission powe. Deng et al. [4] identify the optimal tansmission ange that maximizes the atio of pe-hop tansmission distance to the enegy consumption. In thei pape, nodes ae assumed to be Poissonly distibuted ove a given egion. Ignoing bounday effect, the authos identify the aveage distance pogess fo each hop, which is defined to be the diffeence between the befoe-hop distance (distance between cuent node and the destination node and afte-hop distance (distance between next node in the path and the destination node. With taking into account the enegy equied to both tansmit and eceive packets, the authos find the optimal tansmission ange that maximizes the atio of distance pogess to enegy consumption. In [5], Clementi et al. study the poblem of minimizing the tansmission powe while meeting the equiement that any pai of nodes in the netwok is at most k hops away. In the pape, a finite set of nodes ae allowed to tansmit using diffeent tansmission anges. A node is diectly connected to anothe node if thei Euclidean distance is less than the tansmission ange. A deteminstic path loss model in an ideal envionment (path loss exponent has been used to elate the tansmission ange to the tansmission powe. The authos analyze the poblem of finding the best set of tansmission anges whee the sum of the tansmission powe of all nodes is minimum, at the same time ensuing that any two nodes in the netwok ae at most k hops away. In [6], the pefomance of diffeent dynamic tansmissionpowe-contol algoithms ae epoted. These algoithms show that dynamic tansmission-powe contol will educe powe consumption. The studies ae diffeent fom ou wok in a seveal aspects: a The algoithms may incease o decease the tansmission powe to meet some pedefined bound fo the numbe of neighbos of each node; b In some of the studies, the establishment of links between two nodes is decided by the packet eception ate; c Simulation esults ae epoted without analytic backing. Indeed, to the best of ou knowledge, thee is no analytical esult obtained on the possible enegy savings in the scenaio mentioned in section I. In this pape, we assume that wieless nodes ae unifomly, independently, and identically distibuted ove a twodimensional aea. This type of node distibution has been commonly used in many majo studies of some fundamental netwok popeties, such as connectivity [7], [8] and capacity [9], []. Howeve, instead of following the analysis of these papes based on a simplistic unit disk channel model, we conside channel model with cetain andomness. Consideation of channel andomness is necessay because the non-negligible impacts of channel andomness, like log-nomal shadowing and Rayleigh fading, to the netwok connectivity have been epoted in the ecent wok [], [], [3]. In this pape, the log-nomal model has been selected to cay out the analysis. Howeve, the wok in this pape can be eadily extended to othe types of stochastic channel models. III. SYSTEM MODEL In this pape we conside that wieless sensos ae andomly, independently and identically distibuted ove a twodimensional aea accoding to a Poisson point pocess. The node density is known to be λ. Let us denote by X i the position of node i. Then we denote by R ij X i X j the Euclidean distance between a pai of andomly chosen nodes i and j. Since R ij ae also independently and identically distibuted andom vaiables fo diffeent pais of (i, j, we dop the index ij in the est of the pape whee the indices ae not impotant. Node i can diectly communicate with node j if the eceived powe at node j, denoted by P, is above a cetain known theshold p th. The elationship between P and the Euclidean distance between nodes i and j, R, follows the log-nomal model [4], P p α log R d + N ( whee P is the eceived powe at node j, p is the powe at a efeence distance d, α is the path loss exponent, N is a Gaussian distibuted andom vaiable with zeo mean and vaiance σ. The andom vaiable N eflects the shadowing effect in the envionment. Both P and p ae in dbm unit. Initially, node i tansmits at the maximum tansmission powe p m, which esults in a efeence powe p at a efeence distance d. This leads to the establishment of a set of diect connections between node i and some othe

3 nodes, i.e. its neighbos. Afte that, node i is allowed to educe its tansmission powe continuously while maintaining connections with the same set of neighbos. The assumption that node i can adjust its powe continuously is mainly used to simplify the late analysis, which does not citically ely on the assumption. Given the distibution function of enegy savings, which will be deived late, the esults can be easily modified fo nodes having a set of discete tansmission powe levels. The diffeence between the maximum tansmission powe p m and the minimum tansmission powe equied fo maintaining connections with the same set of neighbos as those when node i tansmits at the maximum powe p m, is in fact the enegy that could be saved by node i without any sacifice in connections. The enegy savings that can be achieved fo a andomly chosen node is of couse a andom vaiable depending both on the distibution of distances between the node and its neighbos and on the andomness in the log-nomal model. Denote this andom vaiable by H; in the next section we shall deive the distibution of H. IV. ANALYSIS Select an abitay node, say node i. Let the maximum tansmission powe be p m such that it esults in a eceived powe of p (in the absence of shadowing effect at efeence distance d. Let be a sufficiently lage positive value such that the pobability that a node, whose Euclidean distance to node i is lage than, is connected to node i can be ignoed. So, we shall only conside nodes whose Euclidean distance to node i is within. Denote by R the andom vaiable epesenting the Euclidean distance between a andomly chosen node j i within the cicula aea A centeed at node i and with a adius, and node i. Assuming node j is at a known and fixed distance < to node i, i.e. R, the eceived powe P at node j fom node i is Gaussian distibuted with mean p α log and vaiance σ. Theefoe, the conditional pobability that node j eceives a powe fom node i, which tansmits at the maximum powe p m, highe than the theshold powe level p th, i.e. node j is connected to node i, given thei Euclidean distance equals to, is P {P p th R } p th p th p +α log ( P p + α log σ d d dp d ( Note that when deiving this equation, we have made an assumption that the powe eceived by a andomly chosen node when node i is tansmitting is independent of the powe eceived by anothe andomly chosen node when node i is tansmitting. Equivalently, the andom path losses ae assumed to be independent. These assumptions ae also used late. Since nodes ae assumed to be Poissonly distibuted, and node j is within the cicula aea A centeed at node i and with adius, the pobability density function of R is f R (, fo. Combining this with Eq. ( gives the pobability that an abitay node within A is connected to node i. That is, P {P p th } P {P p th R } d ( p th p +α log d d (3 Note that we can futhe simplify Eq. (3 by changing the ode of double integal. Fistly, the oiginal sample space: { [, ] x [p th p + α log d, is equivalent to a new sample space, consists of two aeas S and S (as shown in Fig. : { [, d S x p th +p α ] x (, p th p + α log d ] { [, ] S x [p th p + α log d, Then, we can change Eq. (3 to: P {P p th } d p th p +α log pth p +α log d exp + d d p th p +α log d ( exp pth p +α log ( d d exp + p th p +α log d x p th +p α d x p th +p d Similaly, the pobability that an abitaily chosen node in A is connected to node i with eceived powe at least p th + h

4 Fig.. An aea gaph displays the egion of integation fo Eq. (3. will be P {P p th + h} pth +h p +α log ( d d exp + p th +h p +α log d x p th h+p Following fom the pevious esults, we can deive the pobability that an abitay connected node is connected to node i with eceived powe geate than p th + h: P {P p th + h P p th } P {P p th + h} P {P p th } As, the above conditional pobability becomes P {P p th + h P p th } (d x p th h+p (d x p th +p exp ( d p th h+p exp σ log h ( d p th +p exp σ log ( x σ dx exp ( x σ dx whee log(. is natual logaithm. Clealy, if the abitaily chosen node is connected to node i and has a eceived powe geate than p th + h, node i is able to educe its tansmission powe by h without losing the connection to this specific node. Let N be the numbe of nodes in A that ae connected to node i. Fom Mukhejee and Avido s ecent wok [5], we know that N is a Poissonly distibuted andom vaiable with mean µ λπ P {P p th }, whee λ is the node density. This popety elies on ou pevious assumption that powe eceived at an abitay node is independent of powe eceived at anothe abitay node. So the mean value as agees ( σ µ λπd log exp + (p p th log (4 Given the pobability distibution function of andom vaiable N and the equations deived so fa, the pobability that all neighbos of node i, ae connected with eceived powe geate than p th + h, can finally be deived. This is the pobability distibution of andom vaiable H defined in section III. Among all the possible values fo N, a special consideation should be taken fo N. That is the case when node i is an isolated node (no neighbo nodes. We assume no enegy saving can be achieved by node i if it is an isolated node. P {H h} µ n exp ( µ (P {P p th + h P p th } n n! n (µ h n exp ( µ n! n [ ] exp ( µ exp (µ h (5 In the above equation, we conside that if a node has no neighbo (isolated node, it cannot delive any enegy savings at all. Theefoe in the above equation n stats fom, instead of. The pobability density function of H is then f H (h d P {H h} dh log h µ exp( µ exp(µ h (6 Fom the pobability density function of H, we can also obtain the aveage pecentage of enegy saving fo a andomly selected node i. Define p n p m h. That is, p n is the tansmission powe achievable by node i whee the connections to all its neighbos ae maintained. Denote by p m and p n the coesponding p m and p n in milliwatts instead of dbm. Denote by h the coesponding h in decimal units. Thei elationship can be explained by the following equation: p n p m h Then, we can deive the faction of enegy savings g(h as g(h p m p n p m h h Using Eq. (6, the expected faction of enegy savings is, E[g(h] ( h log exp(µ h dh h µ exp( µ ( y α µ exp( µ exp(µy dy exp( µ ( µ α exp( µ [Γ( α +, µ Γ(α + ] (7

5 whee µ is as in Eq. (4, Γ(. is the Gamma function defined as Γ(a t a exp( t dt and Γ(a, z is the uppe incomplete Gamma function defined as Γ(a, z z t a exp( t dt Eq. (7 indicate that the expected enegy savings achievable by using a senso with vaiable tansmission powe is detemined by the path loss exponent and the aveage node degee. V. NUMERICAL EVALUATION In this section, we evaluate the impact of vaious paametes, i.e. aveage node degee µ, vaiance of shadowing σ and path loss exponent α, on the expected pecentage of enegy savings. The vaiance of shadowing σ exets its impact on expected pecentage of enegy savings by affecting the aveage node degee µ. Fig.. Impact of path loss exponent α and standad deviation of shadowing σ on aveage numbe of neighbos µ pe node density λ. Eq. (7 shows that the expected pecentage of enegy savings depends only on the aveage numbe of neighbos µ (also known as the aveage node degee and the path loss exponent α. Howeve, fom Eq. (4 we know that the aveage node degee is not only diectly popotional to the node density λ, but also elated to two envionment paametes: the path loss exponent α, and the shadowing vaiance σ. Fig. shows the impact of these envionment paametes on the aveage node degee. Since the aveage node degee µ is known to be diectly popotional to the node density λ, we choose to plot the atio µ/λ as z-axis to emove the impact of λ fom the plot. The esult shows that shadowing has a positive impact on the aveage node degee, i.e. a lage shadowing vaiance leads to a highe aveage node degee, wheeas a lage path loss exponent has a negative impact on it. The changes in the aveage node degee due to diffeent σ ae small, compaed with the significant impact fom α, especially fo lowe values of α. Fig. 3. Expected pecentage of enegy savings with diffeent µ, whee α between to 5. Refeence [4] suggests that α epesents fee space envionment, α [, 3] epesents diffeent indoo envionment, and α [3, 5] epesents shadowed uban aea. Consideing the pevious analysis in Eq. (7, it is obvious that diffeent values of α will significantly affect the expected pecentage of enegy savings of a wieless senso. Diffeent values of α epesents diffeent envionments that the wieless sensos ae deployed. Refeence [4] suggests that nomally α is used to epesent a fee space envionment, α [, 3] epesents diffeent indoo envionment, and α [3, 5] epesents shadowed uban aea. Fig. 3 demonstates the expected pecentage of enegy savings fo diffeent values of α coesponding to these envionments. It shows that sensos in a lage α value envionment geneally expeience highe expected pecentages of enegy savings. This suggests that highe enegy savings ae to be expected in the obstucted envionment. Fig. 3 shows the expected pecentage of enegy savings inceases with µ when the values of µ is small; it peaks at a cetain value of µ and the value of µ that maximizes the expected pecentage of enegy savings vaies slightly with α; finally the expected pecentage of enegy savings dops with µ when the value of µ is lage. The expected pecentage of enegy savings deceases with µ when the value of µ is lage because it is hade to gain enegy savings when each senso gets connected to a lage numbe of neighbos. On the othe hand, the expected pecentage of enegy savings also dops at a vey small value of node degee. This phenomenon is mainly caused by the incease in pobability that a node will become isolated in the netwok. Such pobability is given by P {a node is isolated} exp( µ. Theefoe, a decease in the aveage node degee will esult in an incease in pobability of a node becoming isolated. An isolated node will not yield any enegy saving. Anothe inteesting esult fom Fig. 3 is that thee is always an optimal point fo each cuve in the plot. That is, in each envionment which is chaacteized by the value of the path loss exponent α, thee is an optimal aveage node degee µ that leads to the maximum enegy savings. Fig. 4 and Fig. 5

6 Fig. 4. Maximum expected pecentage of enegy savings (i.e. the optimal point of each line in Fig. 3 with α [, ]. Fig. 5. The coesponding aveage node degee whee the maximum expected pecentage of enegy savings can be obtained, fo α [, ]. show the maximum expected pecentage of enegy savings that can be achieved fo each value of α, and the combination of α and µ that lead to the maximum enegy savings espectively. Fig. 4 shows that on aveage, at least 3% of enegy savings is achievable fo sensos with vaiable-tansmission-powe at most envionmental conditions. This esult shows that sensos with vaiable-tansmission-powe will significantly outpefom sensos with constant-tansmission-powe in tems of enegy efficiency and senso lifetime. Fig. 5 shows that the ange of aveage node degee to achieve the maximum expected pecentage of enegy savings is naow (fom.8 to.4. Hence, keeping the aveage node degee appoximately at will gain nea to optimal pefomance on enegy savings. VI. CONCLUSION AND FUTURE WORK In this pape, we have studied the achievable enegy savings in a senso netwok whee the sensos ae capable of educing thei tansmission powe while maintaining the connections to thei neighbos. The wieless channel is modeled by the log-nomal model. In geneal, the sensos with vaiabletansmission-powe will gain significant enegy savings, especially in an envionment with a high path loss exponent. The esults obtained in this study ae of pactical value. Fistly, the analytical esults show that sensos with vaiabletansmission-powe can delive significant enegy savings, which possibly justifies thei use in a eal netwok. Secondly, the esults also suggest that a nea-to-optimal aveage enegy saving is obtainable by a senso netwok if the aveage node degee is kept appoximately to. Ou existing wok analyzes the achievable enegy savings unde the condition that no connection is lost as a esult of educing tansmission powe. That means that the netwok topology is maintained duing the couse of educing tansmission powe. Howeve, this does not necessaily need to be the case. In the futue, we will extend cuent wok to examine the achievable enegy savings whee the eduction of tansmission powe is allowed to change the netwok topology, as long as the netwok still peseves some netwok popeties, such as connectivity o k-connectivity. REFERENCES [] A. Ephemides, Enegy concens in wieless netwoks, IEEE Wieless Communications, vol. 9, no. 4, pp , August. [] J. N. Al-Kaaki and A. E. Kamal, Routing techniques in wieless senso netwoks: a suvey, IEEE Wieless Communications, vol., no. 6, pp. 6 8, Decembe 4. [3] L. Wang and Y. Xiao, A suvey of enegy-efficient scheduling mechanisms in senso netwoks, Mobile Netwoks and Applications, vol., no. 5, pp , Octobe 6. [4] J. Deng, Y. S. Han, P.-N. Chen, and P. K. Vashney, Optimal tansmission ange fo wieless ad hoc netwoks based on enegy efficiency, IEEE Tansactions on Communications, vol. 55, no. 9, Septembe 7. [5] A. E. F. Clementi, P. Penna, and R. Silvesti, On the powe assignment poblem in adio netwoks, Mobile Netwoks and Applications, vol. 9, no., pp. 5 4, Apil 4. [6] J. Jeong, D. Culle, and J.-H. Oh, Empiical analysis of tansmission powe contol algoithms fo wieless senso netwoks, Univesity of Califonia at Bekeley, Tech. Rep. UCB/EECS-5-6, Novembe 5. [7] M. Desai and D. Manjunath, On the connectivity in finite ad hoc netwoks, IEEE Communications Lettes, vol. 6, no., pp , Octobe. [8] P. Gupta and P. R. Kuma, Citical powe fo asymptotic connectivity in wieless netwoks, Stochastic Analysis, Contol, Optimization and Applications: A Volume in Hono of W.H. Fleming, W.M. McEneaney, G. Yin, and Q. Zhang (Eds., pp , 998. [9], The capacity of wieless netwoks, IEEE Tansactions on Infomation Theoy, vol. 46, no., pp , Mach. [] M. Gossglause and D. N. C. Tse, Mobility inceases the capacity of ad hoc wieless netwoks, IEEE/ACM Tansactions on Netwoking, vol., no. 4, pp , August. [] C. Bettstette and C. Hatmann, Connectivity of wieless multihop netwoks in a shadow fading envionment, Wieless Netwoks, vol., no. 5, pp , 5. [] D. Mioandi and E. Altman, Connectivity in one-dimensional ad hoc netwoks: a queueing theoetical appoach, Wieless Netwoks, vol., no. 5, pp , 6. [3] D. Mioandi, E. Altman, and G. Alfano, The impact of channel andomness on coveage and connectivity of ad hoc and senso netwoks, IEEE Tansactions on Wieless Communications, vol. 7, no. 3, pp. 6 7, Mach 8. [4] T. S. Rappapot, Wieless Communications: Pinciples and Pactice, nd ed. Pentice Hall PTR, Decembe. [5] S. Mukhejee and D. Avido, Connectivity and tansmit-enegy consideations between any pai of nodes in a wieless ad hoc netwok subject to fading, IEEE Tansactions on Vehicula Technology, vol. 57, no., pp. 6 4, Mach 8.