Extremum Tracking in Sensor Fields with Spatio-temporal Correlation

Transcription

1 The Military Commnications Conference - Unclassified Program - Networking Protocols and Performance Track Extremm Tracking in Sensor Fields with Spatio-temporal Correlation Prithwish Bas Raytheon BBN Technologies Abhishek Nadamani Cornell University Lang Tong Cornell University Abstract Physical phenomena sch as temperatre, hmidity, and wind velocity often exhibit both spatial and temporal correlation. We consider the problem of tracking the extremm vale of a spatio-temporally correlated field sing a wireless sensor network. Determining the extremm at the fsion center after making all sensor nodes transmitting their measrements is not energy-efficient becase the spatio-temporal correlation of the field is not exploited. We present an optimal centralized algorithm that tilizes the aforementioned correlation to not only minimize the nmber of transmitting sensors bt also ensre low tracking error with respect to the actal extremm. We se recent order statistics bonds in the formlation of the cost fnction. Since the centralized algorithm has high time complexity, we propose a sboptimal distribted algorithm based on a modified cost fnction. Or simlations indicate that a small fraction of sensors is often sfficient to track the extremm, and that the centralized algorithm can achieve abot 7% energy savings with almost perfect tracking. Frthermore, the performance of the distribted algorithm is comparable to that of the centralized algorithm with p to % more energy expenditre. I. INTRODUCTION Recent advances in embedded sensing and wireless commnications and networking technologies has reslted in the proliferation of low power sensor devices that are capable of sharing sensed information with each other over a wireless medim and ths forming a wireless sensor network. A common application of wireless sensor networks is in the domain of environmental monitoring; in particlar, they are sefl for sensing and tracking variations in physical phenomena sch as temperatre, pressre, hmidity, wind velocity etc. over a geographical area. Wireless sensor nodes sally have low power RF transceivers that reglarly transmit sensed data either directly or over mltiple hops to the fsion center (FC) which spports ser qeries on gathered data samples. A major concern that plages sensor networks is the limited battery life on sensor nodes since RF commnication and idle listening have a significant drain on battery. Since sensors can rarely be recharged once deployed in a remote environment, extending battery life is of paramont importance. Therefore, it is prdent to develop energy efficient mechanisms for sing these sensors and ths increase the lifetime of the sensor network. In this paper we consider the problem of Prepared throgh collaborative participation in the Commnications and Networks Consortim sponsored by the U. S. Army Research Laboratory nder the Collaborative Technology Alliance Program, Cooperative Agreement DAAD9---. The U. S. Government is athorized to reprodce and distribte reprints for Government prposes notwithstanding any copyright notation thereon. accrately monitoring a spatio-temporally correlated physical phenomenon in an energy-efficient manner. We assme that there are a finite nmber of static sensor nodes that can sense the phenomenon at their locations we refer to this discretized sampling of a continos phenomenon as a sensor field. Specifically, we are interested in tracking the maximm vale in the sensor field over time. In this paper, we focs on the single hop network scenario where all sensors and fsion center (FC) are within transmission range of each other. This is interesting in monitoring applications that involve sensors spread across a geographical area, e.g., rooftop of a bilding or a field. Typically, FC is connected to a continos power sorce whereas the sensor nodes are battery-powered. If the maximm is to be determined at a particlar instant of time, then all sensors cold transmit their measrements to FC which wold then determine the maximm vale of the sensor field. However, if there exist spatio-temporal correlations in the sensed data (as seen in data obtained from a weather sensor network testbed at BBN), we show that it is possible to exploit sch correlations and come p with an optimal sensor selection policy that saves energy while accrately tracking the maximm vale. We first propose a centralized algorithm that is exected at FC. It determines which sensor nodes shold be transmitting their sensed measrements in the next epoch. In this algorithm, FC maintains recent history of readings from varios sensors and attempts to minimize an objective fnction that captres both energy consmption and the expected vale of deviation from maximm, if only a sbset of nodes were transmitting. Minimization of this objective fnction yields low tracking error even if only a small fraction of sensor nodes transmit their measrements. We also propose a distribted algorithm in which the transceivers of sensor nodes are either in ON or OFF state. When in ON state, they overhear measrements reported by a sbset of other transmitting nodes and then make a decision abot whether to transmit their measrements in the next epoch or to sleep. In OFF state, the nodes do not sense, transmit, or receive. Every node independently attempts to minimize an objective fnction that is similar to the one sed in the centralized algorithm. Related Work: Bolis et al. propose a heristic mechanism for performing energy-efficient aggregation in sensor networks []. Unlike or proposed techniqe, their scheme to determine the maximm does not take into accont the //$. IEEE

2 8 Wind Speed ( m/s ) Sensor Sensor Wind Direction (in degrees) Sensor Sensor 8 8 Fig.. Wind velocity measrements at two sensors for one hor: (a) Wind Speed; (b) Wind Direction spatio-temporal correlations of the phenomenon. Fhrmann and Widmer address the problem of determining the maximm (or minimm) in a network with large mlticast grops [], []. Their objective is to minimize the nmber of transmissions for determining the maximm sing mlticast feedback. The data measred at nodes is assmed to be independent, and the algorithm terminates after it has compted the maximm vale. On the other hand, or proposed approach exploits the spatio-temporal correlation in the sensor field and continosly tracks the extremm. Many researchers have investigated exploiting spatial correlation for wireless sensor network applications [], [8]. Borrowing terminology of [], these applications correspond to snapshot aggregation and do not take into accont the temporal characteristics of the sensed data. Akyildiz et al. emphasize the importance of spatial and temporal correlation in designing MAC protocols for wireless sensor networks []. Bt they consider the spatial and temporal correlation independently and not together. Moreover, their analysis on how spatial and temporal correlation can be sed for MAC protocol design is mostly qalitative. Like or proposed schemes, Mergen et. al. [7] have also considered the case where the sensors commnicate directly to FC. Their analysis shows that the cost of listening can be a dominant factor in a dense network. They have proposed wake p schemes and have determined the capacity of sch systems from an information theoretic perspective. II. SPATIO-TEMPORAL CORRELATION IN SENSOR DATA In order to verify or intition abot spatio-temporal correlation in physical phenomena, we gathered weather data from a sensor network testbed recently deployed on the rooftops of varios bildings at BBN. Each node consists of an embedded PC, a 8.a/b/g interface, and varios sensors for monitoring weather conditions as well as air polltants. The sensors can detect weather measrements, sch as wind speed and direction, temperatre, air pressre, relative hmidity, and rainfall. We collected pressre, temperatre, hmidity, wind speed and direction data for two sensors in the testbed (located approximately 8 meters apart) over a period of hors.these sensors crrently transmit the monitored data to FC periodically. Since they are connected to AC power otlets, these sensors have abndant energy resorces, and energy efficiency is crrently not an isse in the testbed. However, in the foreseeable ftre, some of these nodes may be operating on battery while monitoring physical phenomena sch as maximm temperatre or maximm wind velocity. Therefore, if all sensors transmit their sensed data it wold reslt in nnecessary energy expenditre and wold redce the lifetime of the sensor network significantly. If however, the sensors cold learn the spatio-temporal statistics of the sensed data and adapt themselves sch that only certain sensors transmit, then we can determine and track the extremm vales of the nderlying fields with high accracy. Figre shows wind velocity measrements; the maximm vales cold be determined with sensor transmitting most of the time and sensor transmitting at certain times. Therefore, it makes sense to se the spatio-temporal correlation characteristics of the sensor field in order to make decisions abot which sensors shold transmit with the objective of saving energy while not sacrificing accracy significantly. Stochastic modeling of weather data is a reasonably matre discipline. In particlar, modelling wind velocity is of great importance in civil engineering from a strctral engineering point of view [], []. It is sed to forecast maximm wind speed to determine the worst case wind load for a strctre. It is also reqired in wind energy prodction systems for forecasts of power which are generally derived from forecasts of speed. Researchers have explored the possiblities of modelling wind speed data with a first order Markov chain model []. Others have proposed techniqes for forecasting wind speed, based on cross correlation at neighboring sites []. Accrate modeling of wind direction is important in coastal applications for determining worst case directional load on the strctre. We modeled the data collected from or testbed as a first order Spatio Temporal Ato Regressive (STAR) model. In particlar, the first order spatio temporal process can be written as X [n] = φ X [n ] + φ X [n ] + ɛ[n], where φ and φ are called the space-time partial atocorrelation fnctions. φ measres the extent of temporal correlation whereas φ measres spatial correlation. X i [n] is the measred vale at sensor i at time n. We sed techniqes in [9] for estimating the parameters for the spatio-temporal process measred from the testbed. Specifically for the wind speed data, (φ,φ )=(.,.8) and for the wind direction data, (φ,φ )=(.9,.). From these correlation coefficients, we observe that wind speed has both spatial and temporal correlation bt wind direction is mostly temporally correlated with little spatial correlation. III. EXTREMUM TRACKING PROBLEM UNDER SPATIO-TEMPORAL CORRELATION We consider the scenario in which N sensors, distribted randomly over an area A, are attempting to sense a spatiotemporal field at discrete time instants. They transmit the reslts to FC, which then determines the extremm vale at each time instant n. The basic tradeoff here that shold be exploited is the following: if more sensors transmit, the tracking error wold

3 Symbol Description TABLE I MATHEMATICAL NOTATION n Discrete instant (or epoch) of time. We se this convention instead of the sal t becase the latter is typically sed for modeling continos time. X i (n) =V (x i,y i,n) Random variable that indicates the vale of the field being measred at node i at time instant n. (x i,y i ) is the location of sensor i. X[n] =[X (n),x (n),...x N (n)] Vector of measrements at the sensors at time n. U i Random variable associated with sensor i that indicates sensor selection; U i = if sensor i transmits; otherwise, U i =. U =[U,U,...U N ] Random variables for transmission policy ( =[,,... N ] is a specific realization). H[n] Historical information available at the fsion center FC at time instant n. This incldes information received by FC before time n. H[n] =[H[n ],X i [n] s.t. i =]. δ =Pr(U = H[n]) Conditional probability of a particlar transmission policy being selected given the historical measrements. The optimal policy cold be randomized, which is given by δ. Note that δ =. M[n] =max(x [n],x [n],...x N [n]) F [n] =max(x i [n] s.t. U i =) E k() Maximm vale of the measrements from all sensors. This refers to the best possible tracking. Maximm vale of the measrements received from the sensors who were tasked to transmit by the algorithm and is a fnction of X[n] and U. F [n] denotes the measred maximm vale if the sensors were following transmission policy. Energy consmed by a sensor when it transmits. For the sake of simplicity we assme eqal transmission power. It wold be straightforward to extend it to a case where the transmission power depends on the distance to FC. Nmber of ones in the vector (denotes the nmber of sensors transmitting) be minimized bt the energy consmption wold be high. On the contrary, if too few sensors transmit, the energy consmed wold be low, bt we wold rn the risk of erroneos tracking. Therefore, or goal is to develop an algorithm that minimizes both the tracking error and the energy sing the measred spatio-temporal characteristics of the sensed field. If we know the spatio-temporal correlation strctre, crrent measrements can be sed to predict ftre vales and then a decision can be made by the algorithm abot which sensors shold transmit. In this paper we consider both centralized and distribted versions of the problem. In the centralized version of the problem, FC keeps history of all measrements from all sensors that it has received so far. At time instant n, FC decides which of the N sensors need to transmit after examining the previosly received data, and tasks those selected sensors to transmit in the next epoch. Upon receiving the measrements from the chosen sensors in the next epoch, FC determines the maximm and repeats the process. In the distribted version of the problem, FC does not make a decision abot which sensor shold transmit; instead the sensor nodes themselves make a decision abot whether to transmit after overhearing transmissions from other nodes and exploiting the spatiotemporal correlations in the sensed field. We introdce some sefl notation in Table I for the description of or algorithms. IV. CENTRALIZED TRACKING ALGORITHM We assme that each sensor can transmit once in an information gathering epoch. The epochs are assmed to be eqal in time dration. At the beginning of each information gathering epoch n, FC ses the previosly received measrements (in the n st epoch) to make a decision abot which sensors shold transmit their measrements in the crrent epoch. Upon receiving this information, the reqisite sensors transmit to FC. We assme a slotted TDMA style MAC protocol in operation. FC ses a broadcast slot to commnicate the information abot which sensors transmit next to all sensors listening in that slot. Along with this information, FC also specifies a slot schedle for which sensors transmits when in the crrent epoch. Sch a contention-free scheme allows sensors to pt their transceivers to sleep at all times except when it is their trn to transmit. In addition to mitigation of packet losses de to collisions, this generally reslts in tremendos energy savings becase of the redction in idle listening. FC, however, listens to all ongoing transmissions in the network. For this analysis we assme a perfect channel with no transmission errors or data loss de to fading or time asynchrony de to clock skews. FC chooses the maximm among the received vales and denotes it as the maximm of the entire field. FC also pdates its history H[n] with the received vales. Objective Fnction: At the beginning of every epoch, FC depending on the history gathered so far, chooses the sensors that ensre minimm deviation from an estimate of the actal maximm in the epoch. If all sensors transmit, the tracking error wold be zero bt we shold choose the minimm nmber of sensors possible for redcing energy consmption. We captre the competing goals of minimizing both energy consmption and the tracking accracy by sing the following linear cost fnction as the objective fnction for the optimization problem (λ is a mltiplier that weighs the relative importance of energy efficiency and tracking error): λ Energy consmed +( λ) Tracking Error Since FC may not have access to the measrements (X) before making a decision, we attempt to minimize the expected vale of the deviation to accont for all possible realizations of X. Also the optimal policy U may be probabilistic. So we take the expectation over U as well. Or optimization problem then becomes the following:

4 8 8. Actal Maximm Centralized Algoirthm 8 Nmber of sensors that transmit. Maximm Maximm.. Nmber of sensors transmitting 8 All sensors transmit Wind Speed Wind Direction Actal Maximm Centralized Algorithm. 8 8 Fig.. Tracking the Maximm sing the Centralized Algorithm: (a) Wind Speed; (b) Wind Direction; (c) Energy consmption min {λ E U,X (E total H[n]) δ +( λ) E U,X (M[n] F [n] H[n])} () sbject to constraint δ =,wheree total denotes the total energy consmed by all sensors and E U,X denotes the expectation over all possible realizations of U and X given the history H[n]. Theorem. The cost fnction given in Eqation is minimized by a deterministic policy given by δ =, if = opt =, otherwise () where opt = argmin {λ k() E +( λ) E U,X (M[n] F [n] H[n])} () F [n] is the maximm among the readings of the sensors which transmit as indicated by. Proof: Consider the first term λ E U,X (E total H[n]) = λ δ k() E E U,X (M[n] H[n]) is independent of U and hence does not depend on δ. E U,X (F [n] H[n]) = F [n] p(, x H[n]) dx x = F [n] p(x H[n]) δ( H[n]) dx x = δ E X (F [n] H[n]) This can be formlated as a linear programming problem of the form: Minimize δ C, sbject to δ =,where C = λ k() E+( λ) E(M[n] F [n] H[n]). Bythe Fndamental Theorem of Linear Programming, the soltion to this optimization problem is attained at the end points of the N simplex formed by the δ s []. Hence Eqation yields the optimal soltion. Exploiting the Correlation Model: We assme that the measred data are jointly zero-mean gassian r.v. s with the following correlation strctre: E(X i [n ]X j [n ]) = σ e B n n e Adij () where n and n denote discrete instants of time, d ij is the distance between sensor i and j, andσ is the variance. A and B control the degree of correlation between the measred data samples higher A implies low spatial correlation and vice-versa. Similarly, B controls temporal correlation. If the measred data is spatially ncorrelated, (e.g., wind direction), E(X i (n )X j (n )) = σ e B n n δ ij () where δ ij is the kronecker delta, then it is sfficient to keep the most recently received vales in the history. We se this recent history information to estimate the conditional mean and the variance of the next measrement. Since the measrements are spatially ncorrelated, any measrements at sensor (withot loss of generality) from other sensors do not affect the conditional mean or the variance at sensor. This can be shown easily. For crrent time instant n and historical time instants n >n >...n N,wehave: E(X [n] all history from all sensors) = E(X [n] X [n ],X [n ],...X [n N ]) () = E(X [n] X [n ]) (7) Eqation follows becase the measrements are spatially ncorrelated. Eqation 7 follows de to the fact that for each sensor, the temporal measrements form a -D Gass Markov Random Process (as given by correlation strctre of Eq. ). We now explain how we se historical information to affect the expected vale of the highest order statistic. Given two jointly gassian random variables X and Y we have: E(X Y) = Σ XY Σ YY Y (8) Var(X Y) = Σ XX Σ XY Σ YY Σ YX (9) where Σ XX, Σ XY, Σ YX and Σ YY are covariance matrices. The determination of the optimal policy reqires the evalation of the expected vale of the highest order statistic conditioned on the history. The bond on the expected order statistic can then be evalated by calclating the conditional mean and conditional variance of the random variables, conditioned on the history. We can determine this by sing Eqations 8 and 9, where Y = H[n] and X = X i [n] for all i. We se known

5 order statistics bonds to compte the expected vale of the maximm []. Note that for the case where the measred data is spatiotemporally correlated, from a theoretical point of view, only the most recently received data is not sfficient for the history. However, from a practical point of view, we observed from or simlation experiments (presented later) that sing the most recent history alone yields good tracking performance. Tracking Algorithm at FC: The algorithm for tracking the maximm of a spatio-temporally correlated sensor field can broken down into the start-p phase and sbseqent phases. Initialization phase (n =): ) All sensors transmit their measrements to FC ) F () = max(x [n],x [n],...x N [n]) ) H[] = [X [],X [],...X N []] Sbseqent phases (n >): ) FC determines which sensors have to transmit according to the optimal policy opt given H[n] (Eqation ). ) F [n] =max(x i [n] s.t. i =) ) H[n +]=[H[n],X i [n] s.t. i =] Simlation Reslts: In the simlations of the centralized algorithms, sensors are distribted randomly in a 8m 8m sqare. They sense a zero-mean jointly Gassian field with the correlation strctre of Eqation. We assme that each sensor reqires nit energy to transmit to FC. The energy consmed in every epoch is therefore the nmber of sensors transmitting in that epoch. We sed the partial space time atocorrelation fnctions determined by the experimental data presented in Section II. φ is the correlation coefficient for a temporal lag of one nit and no spatial lag. φ is the correlation coefficient for a temporal lag of one nit and a spatial lag of one. Hence, from Eqation, we have: e BΔn = φ and e Ad e BΔn = φ, where Δn is the time difference between the samples and d is the distance between the two sensors. Wind speed data has both spatial and temporal correlation; we have determined that A =.7 and.9 respectively. We set λ =. and σ =for these simlations since we wanted to pt eqal imporance on energy conservation and tracking accracy. In Figre (a), the actal maximm, M[n], and the maximm determined by the centralized algorithm, F [n], is plotted against time. The centralized algorithm tracks the maximm with very little deviation. Or simlations showed energy savings of.7% over the time period of s in comparison with the policy of all sensors tranmsitting in every time epoch. For less spatially correlated data like wind direction, A =. and B =.9 were respectively derived from the space-time partial atocorrelation fnctions. In this simlation we chose λ =. and σ =. We plotted the actal maximm and the maximm as determined by the centralized algorithm in Figre (b). For this s simlation rn, we achieved an energy savings of nearly 79%. Higher energy savings can be expected when sensing highly temporally correlated data if a sensor has sensed the maximm at a particlar instant of time, it is likely that in the next epoch the same sensor wold contine to report the maximm vale. This is becase the measrement of this sensor is nlikely to be affected by that of the neighbor de to low spatial correlation. We now analyze the ability of the algorithm to select the appropriate sensors to transmit. We have sed the simlation data of the parameters obtained for the wind direction data. Figre (c) illstrates how the energy consmption of the centralized algorithm varies over time for tracking both wind speed and direction. We observe that, on average, only sensors (ot of ) need to transmit for tracking the maximm wind speed and even fewer nmber of sensors are needed to track maximm wind direction. This is becase the latter phenomenon has a very high degree of temporal correlation, whereas the former phenomenon has mch less temporal correlation althogh it does have some spatial correlation. V. DISTRIBUTED TRACKING ALGORITHM The centralized algorithm presented in Section IV sffers from the problem of exponential time complexity since it ses a brte force approach to optimization over N transmission policies. Hence, we propose a sboptimal distribted algorithm which is a simple extension of or centralized algorithm. In the distribted algorithm, FC does not make a decision abot the sensors that shold be transmitting. Instead, each sensor node makes that decision locally. The sensors overhear transmissions and pdate their local history. We assme that the time is slotted as in Section IV and FC distribtes transmission slot-schedles a priori to all nodes. Each node knows which slot (in each epoch) to transmit its measrements in. At every time step, each sensor tries to locally minimize the cost fnction according to the policy defined by Eqation. Each sensor S i follows the following rle: if last-received vales from certain other sensors S i,...,s ik in S i s history are greater than S i s vale, then S i assmes that S i,...,s ik will transmit. If the decision is to transmit, S i senses and transmits its measrement to FC. In this distribted algorithm, each sensor node keeps awake in other slots as well in order to overhear the transmissions from other sensors. If the decision is not to transmit, that sensor sleeps for that time epoch. However, if the sensor sleeps for a long period, it may lose track of the state of the field. Hence, we propose that all sensors wake p, listen, and transmit at apriorifixed periodic intervals of dration T. More formally, when n =or n mod T =: ) All sensors transmit to FC ) Each sensor pdates its history abot other sensors after hearing their transmissions to FC ) FC determines the maximm M[n] In sbseqent phases (n > and n mod T ), for every sensor i, the following steps are taken: ) H i [n] is the vector denoting history available at sensor i at time n. In formal terms, H i [n] =[Hi,H i,...,hn i ], where H j i is the last received vale from sensor j at sensor i.

6 Centralized Algorithm Distribted Algorithm Distribted Algorithm No periodic wake p Centralized Algorithm Distribted Algorithm Distribted Algorithm periodic wake p of s 8 Nmber of sensors that transmit All sensors Tx Centralized Alg Dist. No periodic wake p Dist. periodic wake p s Maximm Maximm Nmber of sensors Fig.. Tracking the Maximm sing the Distribted Algorithm: (a) No periodic wake p; (b) Periodic wakep (s); (c) Energy consmption ) Let j =if H j i Hi i ) Define i =[,..., i,, i+,..., N ] and i = [,..., i,, i+,..., N ] ) Every sensor i calclates: λ k( i ) E+( λ) E(M[n] F i H i [n]) () λ k( i ) E+( λ) E(M[n] F H i[n]) () i ) If expession > expression, then sensor i transmits. ) If it transmits, it also listens to the transmissions of the other sensors and pdates H i [n +], else it sleeps ntil the beginning of the next time epoch. The above distribted algorithm involves comptation of only two vales (Expressions and ) at each sensor; therefore, it is mch more comptationally efficient in comparision with its centralized conterpart which has to make N comptations at each time epoch. Simlation Reslts: We simlated the distribted algorithm with the same data (wind direction) that was sed for the centralized algorithm, and compared their performance. From Figre, we can observe that the distribted algorithm performs almost as well as the centralized algorithm althogh the total energy consmed is % more than the latter (for periodic wakep of T =s). The mean sqared error is. which is lower in comparison with.77 for the centralized algorithm becase more sensors transmit in the case of the former. The distribted algorithm is efficient in its operation since it jst involves the comptation of two vales before making a decision abot whether to transmit. With minimal extra energy expenditre, we obtain very good tracking performance. We observe from Figre (c) that the distribted algorithm is able to conserve energy almost as significantly as its centralized conterpart. However, the exact energy efficiency depends on the vale of wakep period T. For the case where there is no wakep at all, the energy consmption is qite low (lower than centralized) at the cost of moderate tracking error. For wake p period T =s, we observe that energy consmption shoots p occasionally bt is low overall. Therefore, we conclde that a distribted scheme with a low freqency periodic wakep has decent energy efficiency and good extremm tracking performance. VI. CONCLUSION In this paper we show that a single hop wireless sensor network that is continosly tracking the extrema of time-varying phenomena can intelligently exploit the nderlying spatiotemporal correlations in order to save precios battery energy and ths extend network lifetime. In particlar, we proposed both centralized and distribted versions of an optimization algorithm that simltaneosly attempt to minimize the energy consmption as well as the tracking error. We showed that both the centralized and the distribted algorithms cold track the maximm of a spatio-temporally correlated field remarkably well over varying degrees of correlation while spending little energy. The algorithms were also adaptive to detect sdden changes in the phenomena. REFERENCES [] I. F. Akyildiz, M. C. Vran, and O. B. Akan, On Exploiting Spatial and Temporal Correlation in Wireless Sensor Networks, Proc. WiOpt,. [] M. C. Alexiadis, P. S. Dokopolos and H. S. Sahsamanoglo, Wind speed and power forecasting based on spatial correlation models, IEEE Transactions on Energy Conversion, September 999. [] D. Berstimas, K. Natarajan and C. Teo, Tight bonds on Expected Order Statistics, Probability in the Engineering and Informational Sciences,. [] A. Bolis, S. Ganeriwal and M. 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