Randomized Scheduling Algorithms for Wireless Sensor Networks

Transcription

1 Randoized Scheduling Algoriths for Wireless Sensor etworks abhendra Bisnik, eeraj Jaggi Rensselaer Polytechnic Institute Troy, Y 80 bisnin@rpiedu, jaggin@rpiedu I ITRODUCTIO A wireless sensor network WS consists of devices nodes, with sensing and wireless counication capabilities, that are spatially distributed over a sensor field to onitor a set of physical phenoena [], [3] The applications of WSs include environental onitoring [6], [6], habitat onitoring in reote areas [], [9], surveillance [7], [], otion tracking [5], [8] etc The sensor nodes in a WS are typically tiny battery powered devices with liited counication and coputational capabilities It is iportant to conserve energy of the sensor nodes in order to ensure the longevity of the WS The ost coon way of conserving energy is to schedule only a sall subset of nodes to be in active state at a given tie while other nodes reain in low energy sleep state The objective of the scheduling schee is to axiize the lifetie of the sensor network while covering ost of the sensor field at all ties At the sae tie, for proper operation of the sensor network, all the active sensors at any given tie ust be fully connected In recent years there has been an increased interest in rechargeable sensors [9], [8], [3], [4] In contrast to battery powered sensors, which becoe useless once they run out of battery, the battery of re-chargeable sensors ay be replenished So theoretically the lifetie of a re-chargeable sensor networks is infinite However, in ost scenarios recharging is a slow process and the tie in which a sensor discharges ay be uch saller than the tie it takes to recharge it So even in the case of rechargeable sensor networks, there is a need to carefully schedule the sensors between active, sleep and recharge states If all the charged sensors are allowed to be active at the sae tie then they ay all get discharged siultaneously, leaving the sensor field to be uncovered for a long period of tie As opposed to battery powered sensors, where the ai of scheduling is to axiize the lifetie, the ai of scheduling in the case of rechargeable sensors is to axiize the tie average utility in the network, where the utility function would typically reflect the quality of coverage provided In this paper the utility function considered is the event detection probability In the rest of this paper we refer to the sensor networks with rechargeable sensors as rechargeable sensor networks, while the sensor networks with non-rechargeable sensors are referred to as battery-driven sensor networks It should however be noted that the rechargeable sensors also run on soe kind of battery which ay be recharged using soe renewable source of energy, such as solar, wind, theral energy etc A distributed schee for scheduling nodes in a sensor network is desirable in order to ake it resilient against node failures Since counication incurs a large energy cost, it is desired that the distributed schee should require inial counication If a distributed schee requires nodes to exchange large nuber of essages then the whole purpose of node scheduling, which is to conserve energy, is lost Also because of battery exhaustion, the sensor network ay be divided into disconnected coponents aking it ipossible for nodes to exchange essages with other nodes The requireent of low essage exchange overhead akes the developent of efficient node scheduling algoriths in sensor networks a very challenging proble In this paper we study randoized node scheduling algorith for two kinds of sensor networks, naely batterydriven and rechargeable sensor networks The randoized algoriths considered in this paper are copletely distributed in nature and do not require the nodes to exchange any essages For the battery-driven sensor networks we evaluate the coverage and connectivity properties achieved by the randoized scheduling algoriths For the rechargeable sensor networks we exaine the tie-average utility achieved by the randoized scheduling algorith A Project Goals Concentration results for sensor coverage: Although the coverage properties of randoized scheduling algorith in battery powered sensor network has been already studied, we revisit the coverage proble Through a novel forulation we provide concentration results that bound the deviation of fraction of area covered at any tie fro its expected value Connectivity properties of randoized scheduling: The connectivity properties of the disjoint subsets of sensors constructed by randoized scheduling is not well understood Depending upon the application of WS, it is iportant that either the disjoint sets are fully connected or a sensor is able to counicate with a central sink node using a path that consists of sensors belonging only to its sensor set In soe applications such as surveillance and otion tracking it is necessary that the sensors are able to share the sensed inforation aong theselves or counicate the inforation to a central sink in a real tie anner Thus connectivity is a critical factor

2 for the WS to achieve its goals In the paper we evaluate the probability that the disjoint subsets created by the randoized algorith are fully connected The technique used ay be extended to find the probability of a sensor set being connected to a sink node We also provide an alternative forulation for investigating the connectivity of sensor set with a central sink node Scheduling of rechargeable sensors: We propose a randoized scheduling algorith for the rechargeable sensor networks that unlike existing schees does not require nodes to exchange any essages We provide analysis of the tie average utility achieved by the scheduling algorith For the analysis we only consider the independent lifetie odel, where lifetie of one sensor does not depend on that of other sensors that are active at the sae tie We use siulations in order to investigate the perforance of the correlated lifetie odel, where lifetie of a sensor depends on that of other sensors that are active at that tie B Paper Outline The rest of the paper is organized as follows In Section II we present a brief overview of related work In Section III we discuss the coverage and connectivity properties of the randoized scheduling algorith in battery-driven sensor networks The randoized scheduling algorith and its perforance for rechargeable sensor networks is presented in Section IV We discuss our results and conclude in Section V II PREVIOUS RESULTS We first discuss the previous results for the coverage proble for battery driven sensor networks followed by previous results for rechargeable sensor networks A Previous Results for Battery Driven Sensors etworks The issue of coverage has been studied extensively in the literature Area coverage, where the goal is to onitor a specified region, has been considered in [5], [9], [7] Target or point coverage has been studied in [3], [5], [4], [], [6] Coverage has also been studied fro the perspective of axial support or breach path in [0], [4] In [6], deterinistic sensor placeent allows for topologyaware placeent and role assignent, where nodes can either sense or serve as relay nodes Zhang and Hou [9] show that if the counication range of sensors is at least twice as large as their sensing range, then coverage iplies connectivity They also develop soe optiality conditions for sensor placeent and develop a distributed algorith to approxiate those conditions, given a rando placeent of sensors An iportant ethod for extending the network lifetie for the area coverage proble is to design a distributed and localized protocol that organizes the sensor nodes in sets The network activity is organized in rounds, with sensors in the active set perforing the area coverage, while all other sensors are put in the sleep ode Set foration is done based on the proble requireents, such as energy-efficiency, area onitoring, connectivity, etc Different techniques have been proposed in literature [7], [9] for deterining the eligibility rule, that is, to select which sensors will be active in the next round This notion of classifying the sensor nodes into disjoint sets, such that each set can independently ensure coverage and thus could be activated in succession, has been considered in [3], [5] Cardei and Du [3] show that the disjoint set cover proble is P-coplete and propose an efficient heuristic for set cover coputations using a ixed integer prograing forulation A siilar centralized heuristic for area coverage has been proposed in [5], where the region is divided into ultiple fields such that all points in one field are covered by the sae set of sensors Then, a ost-constrained leastconstraining coverage heuristic is developed which is epirically shown to perfor well In [5] the constraints for the set of sensors to be disjoint and for these sets to operate for equal tie intervals, are relaxed and two heuristics, one using linear prograing and the other using a greedy approach are proposed and verified using siulation results Randoized scheduling of sensor nodes in order to provide prolonged coverage in the sensor network has been considered in [], [7], [8] Two iportant otivations for considering a randoized scheduling algorith are: i The disjoint set cover proble is P-coplete and hence it is not possible to solve it optially using any centralized deterinistic algorith in polynoial tie, and ii Randoized scheduling algoriths could be easily ipleented in a distributed network environent, since they require the sensors to aintain inial or no global state inforation, and also avoid unnecessary counications overhead for decision aking aongst the sensor nodes The basic idea is to divide the sensor nodes into K disjoint sets, such that each of the sets is independently able to provide sufficient coverage in the network Then each of the sets is activated in succession in the network Such an activation schedule is desirable in the presence of redundant deployent of sensor nodes, and it prolongs the network lifetie, which is actually proportional to K, the nuber of such sets of sensors A randoized schedule puts a sensor s into one of the cover sets, with probability that s joins a particular cover set given by K A randoized scheduling of sensor nodes is analyzed in [7] and the expected values of coverage intensity achieved in the network through rando cover set forations is provided Given the total nuber of sensor nodes in the network, they also provide an upper bound on the total nuber of disjoint cover sets possible, to obtain a desired average or expected coverage intensity In [8], the authors suggest turning on extra nodes in a cover set in order to provide connectivity in the network, and evaluate the perforance of their schees using siulations [] allows cover sets to cover area eleents partially and provides bounds on the nuber of ties an area eleent is covered by the randoized algorith in coparison with the axiu nuber of ties it could be covered using a centralized deterinistic algorith The sensors are partitioned such that the expected coverage is within a factor of e of the optial They also provide soe guarantees on the nuber of ties the least covered area eleent is covered

3 3 The connectivity properties of the cover sets created by randoized algorith has not been studied previously Also sufficient concentration results on coverage achieved by the cover sets and a ethodology to choose good K, the nuber of disjoint sets of sensors to use, have not been provided previously Our odel and forulation is different fro previous work We provide concentration bounds on coverage achieved by the cover sets fored In addition, we study the effect of choosing various values of K on the coverage properties of the cover sets fored For characterizing the connectivity property of a cover set, we evaluate the probability with which any pair of sensors belonging to a cover set are connected and provide a forulation to study the connectivity of a cover set with a central sink node We also discuss the effect of the ratio of counication and sensing radius on the connectivity of the cover sets if K is chosen so as to provide good coverage B Previous Results for Rechargeable Sensors etworks ode activation schedules to provide a better quality of coverage in rechargeable sensor networks have been considered in [8], [3], [4], [0] In the case of rechargeable sensors, the network lifetie is no ore the desired etric to optiize, since the sensor nodes would regenerate theselves after soe tie Thus a perforance etric based upon the quality of coverage provided is forulated and is easured over tie to evaluate the perforance of node activation schedules Since the discharge and recharge processes at the sensor nodes could depend on the occurrence of rando events, which could be correlated in space and/or tie, the perforance of activation schedules could depend on the degree of correlation of the discharge/recharge tie intervals/processes, across the sensor nodes [3], [4] consider a restrictive sensor syste odel, wherein a sensor node could be activated only upon coplete recharge of its battery, and show that a siple threshold based node activation policy achieves perforance ore than 3 4 ties the axiu achievable perforance [0], [8] consider sensor systes where a sensor could be activated as long as it has sufficient energy partially rechargeable sensors, and show that a threshold based node activation policy achieves asyptotically optial perforance with respect to the sensor energy bucket size, under various correlation syste odels The previous work only considers threshold based policies in order to activate the sensors when they are available The threshold based schees require the sensors to continuously exchange inforation with each other in order to find out the nuber of sensors in active state So in threshold based schees the sensors are required to broadcast their state inforation throughout the area that they cover If coverage area is very large than the counication area then the counication cost of such a broadcast ay be prohibitive and ay consue large portion of capacity of the network As opposed to the threshold based policies, the randoized activation policy considered in this paper does not require the sensor nodes to exchange any inforation Hence the randoized activation schee neither consues any extra energy of the sensor nor wastes any capacity of the network III RADOMIZED SCHEDULIG FOR BATTERY-DRIVE SESOR ETWORKS The objective of the randoized scheduling algorith for the battery-driven sensor networks is to divide the sensors into K disjoint sets Each disjoint set is referred to as a cover set The union of all the cover sets ust contain all the deployed sensors while intersection of any two cover sets ust be a null set Each cover set is scheduled for activation in a round robin anner The lifetie of the sensor network using such a scheduling schee is roughly K-ties the lifetie of a single sensor The randoized algorith for partitioning the sensors into cover sets is the following: Each sensor joins a cover set fro aong K cover sets by independently and uniforly generating a rando integer between and K We first investigate the coverage properties of cover sets constructed by the randoized algorith entioned above, followed by connectivity properties of the cover sets A Coverage Properties Syste Model: The sensor field is a torus with area, which is divided into squares of unit area Each unit square is called an area eleent sensors are distributed uniforly at rando and independently over the sensor field An area eleent is said to be covered by a sensor if any point belonging to the area eleent is within the sensing range of the sensor Let r denote the sensing radius of each sensor and d j denote the nuber of area eleents covered by sensor j For large r and, that is when the diensions of the grid are sall copared to the diensions of sensing radius and sensor field define d πr, d i πr = d i This sensing odel is otivated by the fact that in any application the phenoenon being sensed exhibits large spatial correlation So if the grid is fine enough, then knowledge about the phenoenon in an area eleent ay be inferred by looking at only a part of the area eleent This odel is illustrated in Figure The correspondence between sensors and the area eleents covered by the ay be represented by a bipartite graph, as shown in Figure Here, the out-degree of each node on the left hand side of the graph, representing the sensors, is at least d We assue that d/ is an integer and d/ ote that, in the average case an area eleent is covered by sufficiently large nuber of sensors = d/ The coverage etric considered in this paper is the nuber fraction of area eleents covered by a cover set We will evaluate the expected nuber of area eleents covered by a cover set and use Chernoff s bound to provide concentration results Coverage Analysis: The coverage property of the randoized scheduling algorith would definitely depend upon the nuber of cover sets used Large nuber of cover sets would lead to higher network lifetie but at the sae tie lower fraction of area eleents would be covered by each We choose torus area in order to avoid the boundary conditions faced in a square sensor field Asyptotically the results would hold true for the square area as well

4 4 Area eleent P[z ik = i = j] = j 0 j 6 K r Since the events i = j, j [0] are utually exclusive and disjoint, using the total probability theore and the fact that K = d, we have = πr Fig Area eleents covered by a sensor The coverage odel 3 Sensors 3 Area eleents P[z ik = ] = = P[z ik = i = j]p[ i = j] 7 " = = = = «# j K K «j j!» j K! K j «! πr j! πr «πr «j «j πr «j «j πr «j «j πr «j K «j 8 Fig graph Representation of area eleents covered by sensors as a bipartite cover set For this analysis we choose K = d/ and later discuss the effect of increasing or decreasing K by a factor α on coverage and lifetie achieved Lea : Let z ik denote the indicator rando variable, indicating whether the area eleent i is covered by cover set k, then P[z ik = ] = i, k K and li P[z ik = ] = 3 e Proof: Let i denote the nuber of sensors lying within radius r of the area eleent i Since the sensors are distributed uniforly at rando and independently in the region, we have, P[ i = j] = j πr j j πr 0 j 4 Since the sensors join one of the K sets independently with probability K, Or, P[z ik = 0 i = j] = K j 0 j 5 The asyptotic result follows fro the fact that li = e Thus if the nuber of sensors deployed is very large and K is chosen to be equal to d, then the probability that an area eleent is not covered by a cover set is given by e In the rest of this section we will discuss only the asyptotic case when Corollary : Let O k denote the nuber of area eleents covered by the cover set k, then in the asyptotic case as, E[O k ] = e k K 9 ext, we derive a concentration result using Chernoff bounds for the nuber of area eleents covered by a cover set Lea : With probability at least δ 0 < δ <, the following inequality holds for O k, k : k K e + log δ, e log δ O k in 0 Proof: Since O k = i= z ik, E[O k ] = e Using Chernoff bounds, we have, P[ O k E[O k ] t] e t Put δ = e t to get, t = P [ Ok + e log δ Thus, ] log δ With probability at least δ, we have, δ

5 5 Or, O k + e log δ e log δ O k e + log δ 3 4 Also, O k, the total nuber of area eleents Hence the result Choosing δ = e e, we get e O k 5 with probability at least e e, which tends to for large Thus, the nuber of area eleents covered by a cover set fored using the randoized set selection is greater than e, whp ext we coent on the effect of the nuber of cover sets, K, on the coverage achieved by the randoized scheduling Effect of K on O k : Suppose that instead of choosing K = d, we choose K = α d P[z ik = ] = and equation 3 becoes Thus, Then, equation becoes 6 α li P[z ik = ] = 7 e /α E[O k ] = 8 e /α and the concentration result corresponding to equation 5 becoes e O /α k 9 with probability at least e e /α We observe that by increasing the nuber of cover sets, K, by a factor α, the probability that an area eleent is not covered by a cover set is increased by a factor of e α, while the network lifetie, which is proportional to K, is increased only by a factor α Thus, the quality of coverage of a cover set deteriorates exponentially with α, while the network lifetie only increases linearly with α For large, better concentration results ay be obtained by choosing α < For exaple if e 4 and we choose α = 05, then fro equation 9, at least 96 percent of the area eleents would be covered whp by each cover set B Connectivity Properties In this section we investigate the connectivity property of the cover sets returned by the randoized algorith Intuitively the bound on probability that a cover set is connected would depend on the counication radius of the sensors and the nuber of cover sets used If the nuber of cover sets are chosen so as to satisfy the coverage quality eg K = d/ as used in the last section, then the connectivity of a cover set would depend upon the ratio of counication and coverage radius Let r c denote the counication radius of a sensor and let β denote the ratio of coverage radius to counication radius That is r = βr c 0 Also it follows that πr c = πr β = d β or the coverage area of a sensor is β ties its counication area Let i be the nuber of sensors in the cover set i, i K We know that K i = and i= E[ i ] = K i K 3 Like the last subsection, we first use K = d/ in order to get asyptotic results and later discuss how scaling K with a factor α effects the connectivity properties For K = d/, E[ i ] = /d In order to obtain asyptotic expected value of the probability that a cover set is connected we consider an average cover set with = /d sensors Let A denote the adjacency atrix of such a cover set Fro [], we know that a path of length l exists between sensors i and j belonging to the cover set if A k ij = Al ij is the i,j eleent of A l, where the product operator is the boolean AD operator instead of noral ultiplication Thus sensors i and j belonging to a cover set are connected if l= A l ij = 4 where all the products and sus are boolean AD and OR operators We su up to l = because if the size of cover set is, the length of path between any two nodes could be at ost Lea 3: Let p l ij denote the probability that nodes i and j belonging to a cover set are connected by a path of length l, then in the asyptotic case and p ij = d β 5 p l p l ij = ij e β l 6 for all i j Proof: ote that p l ij = P[Al ij = ] A ij = if sensors i and j lie within counication range of each other This happens if i lies within disk of radius r c centered at j Since the sensors are distributed uniforly at rando over the torus of area, probability that A ij = equals πr c Using πr c = d β we get 5 Thus atrix A ay

6 6 be viewed as the adjacency atrix of a rando graph where probability that an edge exists between nodes i and j is given d by β ow consider p l ij for l A l ij is given by A l ij = A l ik A kj k= ow probability that A l ik A kj = is given by P[A l ik A kj = ] = P[A l ik = ] P[A kj = ] = p l ik d β Since i, j, k are arbitrary sensors, due to the syetry of forulation, we have p l ik = p l ij Thus probability that P[A l ik A kj = 0] k equals p l d ij β A l ij = 0 corresponds to the event that Al ik A kj = 0 k This event occurs with probability p l d ij β This iplies that P[A l ij = ] = p l d ij β Substituting = d and taking the liit d we get the following asyptotic result p l ij = li p l d d pl ij d ij = e β β 7 where we use the standard liit li x a x x = e a This proves 6 The following corollary gives the probability that two sensors belonging to a cover set are connected Corollary : In the asyptotic case, the probability that the sensors i and j belonging to a cover set are connected, denoted by p ij, is given by p ij = l= d β k= p k ij exp β 8 Proof: Sensors i and j are connected if l= A l ij = ow P[ l= A l ij = 0] is given by P[ A l ij = 0] = d β p l ij ow p ij is given by = = d β d β l= l= exp p ij = P[ A l ij = 0] l= exp pl ij β l= p l ij β 9 which leads to 8 The probability that two nodes belonging to a cover are connected, p ij, depends on i d which is inversely proportional Connectivity Probability Connectivity Probability in a cover set = 000, = 3000, r c = β coverage radius/counication radius Fig 3 Plot of p ij versus the β = r It is observed that if r r c > r, c then two nodes belonging to a cover are connected with high probability The probability drops sharply when r c r to the expected nuber of sensors in a cover set; and ii β the square of the ratio of coverage radius to counication radius of sensors It is observed that p ij increases as d and β decrease As d decreases, the expected nuber of sensors in a cover set also increases, thus probability that a cover set is partitioned into disconnected coponents decreases The dependence of p ij on β is uch ore coplex The nuber of cover sets K is chosen such that the sensor field is covered with high probability If a cover set covers the entire area with high probability then this iplies that the union of disks corresponding to the coverage area of sensors belonging to a cover set equals the entire sensor field with high probability Thus if counication radius of sensors is uch larger than the coverage radius ie sall β then the union of disks corresponding to the counication area of sensors belonging to a cover set would also be equal to sensor field with high probability Thus for sall β the cover set is connected with high probability Figure 3 shows how p ij varies with β It is observed that if β <, a cover set is connected with probability close to one However p ij drops sharply around β = and becoes alost 0 for β > 3 Therefore if the nuber of cover sets is chosen based on coverage criterion only, the resulting cover sets have good connectivity only if ratio of coverage radius to counication radius is sall < Up to now we have discussed results for the case where K = d/ We will now briefly discuss the effect of scaling the nuber of cover sets by a factor α If K = αd/, then probability that two nodes belonging to a cover set are connected is given by the following Corollary Corollary 3: If nuber of cover sets, K, equals αd/ then p ij = d β 30

7 7 and p l p l ij = ij e αβ l 3 p ij = d k= p k ij β exp αβ 3 for all sensors i j belonging to the sae cover set Proof: The change in the nuber of cover sets does not effect the probability that two sensors belonging to a cover set lie within each others counication range which depends only on the counication range Thus p ij reains unchanged and hence 30 is sae as 5 Since K = αd/, E[ ] = αd and hence equation 7 is odified to p l ij = li p l αd d pl ij d ij = e β αβ 33 which leads to 3 Following procedure as used in the proof of Corollary and plugging in 30 and 3, we get 3 Corollary 3 iplies that the exponential part of the probability of a pair of sensors belonging to a cover set not being connected is raised to the power of /α as the nuber of cover set is increased by a factor α Thus if K = 05d/, then p ij = d P k= p k ij β ot β exp surprisingly this result is siilar to that observed for coverage achieved by a cover set However the rate of decrease of p ij with decreases in α is saller In the above section we investigated the asyptotic probability that any two sensors of cover set are connected to each other In certain scenarios the connectivity of a cover set is not an iportant issue Instead it is iportant that each sensor is connected to a sink station The requireent of connectivity with a sink can be easily incorporated in the above analysis by including the sink node in every cover set We refer to the set obtained by adding the sink node to a cover set as augented cover set ow notice that if each sensor is connected to the sink node then the augented cover ust be fully connected, since any two sensors are now guaranteed to have a path through the sink node if a direct path did not exist between the in the cover set Thus the above analysis can be extended to the cases where only the connectivity of sensors to sink node is of interest by considering augented cover set instead of cover sets We now present an alternative forulation for investigating the connectivity property of a cover set when connectivity of sensors to the sink node is of interest The alternative forulation is not easy to analyze and hence we present only an outline on how to proceed with the analysis Alternative forulation for connectivity of sensors to a sink node: Consider a scenario where there is a sink node which all the active sensors need to be connected with A cover set is said to be connected if all the nodes belonging to the cover set are able to counicate with the sink node Consider a sensor, say sensor i A path fro sensor i to the sink ay be represented as a sequence of sensors such that the first sensor in the sequence is i, the last sensor is the sink s and each sensor in the sequence lies within counication range of the preceding sensor Let Pi be the collection of all disjoint paths in the sensor network when all sensors are active Two paths are said to be disjoint if there exists at least one node that is not coon to both paths Let M i Pi and let {i,x j,x j,,x ljj,s} denote the j th path j M i, where l j is the nuber of interediate nodes between sensor i and the sink node in the j th path Without loss of generality, we assue that sensor i belongs to cover set We define a rando variable Y j such that Y j = if sensor j belongs to cover set and 0 otherwise Thus for j i, {, wp Y j = K 0, wp 34 K Let Z i denote a rando variable indicating whether i is connected to the sink or not, ie Z i = if there exists a path coprising of sensors in the cover set that connects sensor i to the sink Z i can be written as Z i = M i l j j= k= Y x kj 35 where the product and suation in the above equation are boolean AD and OR operations respectively So finding P[Z i = ] boils down to finding the probability that the satisfiability proble represented by the rhs of equation 35 is satisfied by rando assignent of variables Y j according to equation 34 This would depend on M i, K and l j, which in turn would depend on the spatial distribution of the sensors and the counication radius of the sensors r c It is siple to see that l j lj P[ Y x kj = ] = j M i 36 K k= If all the paths are independent ie no two paths share a coon sensor, P[Z i = ] is given by M i lj P[Z i = ] = 37 k j= However since the values of each product on the rhs of equation 35 is not independent, therefore calculation of P[Z i = ] is non-trivial Also the dependence of M i, K and l j on network paraeters is not iediately obvious We leave the exact characterization of connectivity of cover sets with sink node as a part of future work IV RADOMIZED SCHEDULIG FOR RECHARGEABLE SESOR ETWORKS As shown in figure 4, a rechargeable sensor has three states: Active state: If a sensor is in active state and an event occurs then the sensor detects the event with probability p d When the total energy of the sensor is consued then the sensor akes transition into recharge state Recharge state: In this state the sensor does not sense any event, but only replenishes its power supply The

8 8 Fig 4 Active State Activation Discharge Ready State States of a rechargeable sensor Recharge State Recharge power supply is fully replenished when the sensor collects K ax quanta of energy after which it oves to ready state 3 Ready state: This state corresponds to the sleep state of the battery-powered sensors In this state the sensor consues negligible energy and waits to be activated As shown in [3] and [4], the coverage perforance depends on how a sensor decides to igrate fro ready state to active state Let and nt denote the nuber of sensors deployed and the nuber of active sensors at tie t respectively For siplicity we assue that the entire sensing field is within sensing range of every sensor The utility at tie t, Unt is equal to p d nt The utility function is siply the probability that an event occurring at tie t is sensed by at least one active sensor The tie average utility is given by T U = li Untdt 38 T 0 ote that if Unt is ergodic then U siply equals the E[Unt], the expected value of Unt The policy used for deciding the igration of sensors fro ready to active state would decide nt Threshold based activation policies are studied in [3] [4], where a sensor in ready state is activated at tie t if nt is less than soe threshold It is shown that under certain conditions there exists an optial threshold which axiizes U Such threshold based policy avoids siultaneous activation of all the sensors which ight lead to alternating periods of large nuber of activated sensors and very few activated sensors However the threshold based policies have few drawbacks They require essage exchanges in order to find out the nuber of active sensors They also require the counication radius to be at least twice as large as the sensing radius, in order for the sensors to counicate with their neighbors The randoized activation policy proposed in this paper does not have these drawbacks Before presenting the randoized policy and its analysis, we first present the lifetie odels Throughout this section we consider only discrete tie event odels, where tie is indexed by t = 0,,,, For the discrete tie case the tie average utility is reduced to U = li L L + L Unt 39 t=0 Lifetie Model: Three lifetie odels are considered in this paper: Independent Lifetie IL Model: In IL odel the discharge and recharge ties of all the sensors are independent While in active state, a sensor looses one quantu of energy with probability q a at each tie step Thus the expected nuber of tie steps spent by a sensor in active state equals K ax /q a In recharge state a sensor gains one quantu of energy with probability q r at each tie step and the expected tie spent by a sensor in recharge state equals K ax /q r Also q r q a, which ensures that recharge rate is uch slower than the discharge rate Deterinistic Lifetie DL Model: In DL odel, each sensor reains in active and recharge state for Q a and Q r tie steps respectively This odel reflects the scenarios where constant aount of energy is consued during the active state irrespective of event occurrence or counication and the recharging source supplies unifor source of power during recharge state 3 Correlated Lifetie CL Model: In CL odel the discharge and recharge ties of all the sensors are correlated A typical correlated lifetie odel is the following With probability q a all the sensors in active state consue one quantu of energy while with probability q r all the sensors in recharge states gain one quantu of energy This odel reflects scenarios where one quantu of energy is consued at all sensors and all sensors receive energy fro the sae source Randoized Activation Policy: In order to avoid counication requireent and restriction on counication radius, we propose the following randoized activation policy At each tie step each sensor in ready state akes a transition to active state with probability p t and reains in ready state with probability p t Figure 5 shows Markov chain representation of the randoized activation policy for various lifetie odels The hypothesis on which our work is based is that there exists an optial value of p t, denoted as p t, siilar to the optial activation threshold that axiizes U A Perforance of the Randoized Activation Policy by IL Model: Lea 4: For the IL odel the tie average utility is given U IL = K ax /q a p d 40 K ax /q a + K ax /q r + /p t Proof: Let p Ai t denote the probability that sensor i is in active state at tie t Solving the Discrete tie Markov chain shown in Figure 5a, we get that under steady state p Ai t = K ax /q a K ax /q a + K ax /q r + /p t 4

9 9 q a q a q a q a q r q r q r A A A R R R K K -q a -q -q a -q r a -q r -q r q r q r = 00 q r = 005 q r = 0 p t Ready -p t a The IL odel Tie Average Utility A A A R R R K K 03 0 p t Ready p t -p t b The DL odel Fig 6 Siulation results showing how U CL varies with p t for various q r For these siulations q a = 05 and p d = 08 Fig 5 q a q a q a q a q r q r q r A A A R R R K K -q a -q a -q a p t Ready -p t c The CL odel -q r -q r -q r q r Markov chain representation of various lifetie odels for all i Let A t denote the nuber sensors in the active state at tie t, then P[ A t = j] = p j Ai t j p Ai t j 4 for all 0 j Thus utility at tie t, U A t, is given by U A t = p d At 43 The expected value of U A t is given by E[U A t] = p d j P[ A t = j] = p d j j = j «p Ai t j p Ai t j «p Ai t p d j p Ai t j = p Ai tp d «K ax/q a = p d 44 K ax/q a + K ax/q r + /p t Since the Markov chain representing the states of the sensors is ergodic, the tie average utility equals the expected value of utility, ie U IL = li L L + which leads to 40 L U IL t = E[U A t] 45 t=0 DL Model: Lea 5: For the DL odel, the tie average utility is given by Q a U DL = p d 46 Q a + Q r + /p t The proof of the above Lea is siilar to that of Lea 4 3 CL Model: For the CL odel we were not able to generate any closed for expressions for the tie average utility function So we perfored siulations in order to evaluate the dependence of U CL on p t Figure 6 shows how U CL varies with p t 4 Discussions : The results of Lea 4 and 5 are obtained by steady state analysis of Markov chains shown in figure 5 and by using the fact that Unt is an ergodic process It should be noted that U IL, U DL and U CL are all strictly increasing functions of p t This iplies that contrary to our hypothesis, there exists no optial p t and it is advantageous to always go fro recharge state to the active state, with probability This corresponds to the threshold activation policy with threshold equal to However this activation policy is not optial as shown in [3] and [4] Hence avoiding counication overheads leads to a loss of perforance in a rechargable sensor network V COCLUSIOS AD FUTURE WORK In this paper we investigated the perforance of the randoized scheduling algoriths in wireless sensor networks In particular we considered the coverage and connectivity properties of cover sets fored using randoized algoriths for partitioning sensors into cover sets and randoized activation algoriths for rechargeable sensor networks We derived asyptotic values of the probability with which a cover set covers an area eleent and expected nuber of area eleents covered by a cover set Using Chernoff bound, we also present concentration results for the nuber of area eleents covered by each cover set We discuss the effect of scaling the nuber of cover sets on the coverage property and effective lifetie

10 0 of the sensor network It is found that if the nuber of cover sets is scaled by a factor α then the probability that an area eleent is not covered by a cover set is scaled by e /α ext we investigated the connectivity property of a cover set In order to characterize the connectivity of a cover set we use the adjacency atrix of the cover set A and use the fact that if the cover set is fully connected then all the off-diagonal eleents of A = l= Al equal when all the suations and products are binary AD and OR operators respectively Using this we obtain asyptotic value of P[A ij = ] ie probability that any two sensors of a cover set are connected We found that if β is ratio between coverage radius and counication radius and the cover sets are chosen so as to satisfy coverage with high probability then P[A ij = 0] varies as e /β ie if counication radius is uch larger than the coverage radius, the cover sets are connected with high probability The effect of scaling the nuber of cover sets on the connectivity of a cover set is siilar to the effect on coverage of a cover set We started investigating the randoized activation policy for rechargeable sensor networks based on the hypothesis that there exists an optial randoized activation policy that axiizes the life-long utility of the rechargeable sensor network We evaluated its perforance using discrete tie Markov chain analysis and siulations It is observed that the optial randoized activation policy is a trivial one, which activates the sensors as soon as they are recharged There is still a lot of scope for future work related to scheduling algoriths for wireless sensor networks All the results presented in this paper assue an initial unifor distribution of sensor nodes in the sensor field When the initial distribution is unifor and the nuber of cover sets is carefully chosen, then randoized scheduling algorith achieves very good coverage and connectivity However a real deployent of a sensor network ay not be uniforly distributed and ight consist of regions of high and low sensor density For such a scenario the randoized algorith ay lead to coverage holes and disconnected coponents In such cases it is iportant that a sensor uses soe local inforation, such as local density or location inforation, in order to decide which cover set to join Cover set ebership policies that allow counication and cooperation aong one hop neighbors is also of uch interest For cases of uneven sensor distributions, the cover set ebership policies should also allow a sensor to join ultiple cover sets in order to avoid coverage holes or disconnected coponents This is in contrast to the randoized schees that allow a sensor to join only a single cover set REFERECES [] Jaes San Jacinto Mountain Reserve jaesreserveedu [] Z Abras, A Goel, and S Plotkin Set k-cover algoriths for energy efficient onitoring in wireless sensor networks In IPS 04: Proceedings of the third international syposiu on Inforation processing in sensor 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