Automatic Decentralized Clustering for Wireless Sensor Networks

Transcription

1 EURASIP Journal on Wreless Communcatons and Networkng 005:5, 97 c 005 C.-Y. Wen and W. A. Sethares Automatc Decentralzed Clusterng for Wreless Sensor Networks Chh-Yu Wen Department of Electrcal and Computer Engneerng, Unversty of Wsconsn-Madson, 5 Engneerng Drve, WI , USA Emal: wen@cae.wsc.edu Wllam A. Sethares Department of Electrcal and Computer Engneerng, Unversty of Wsconsn-Madson, 5 Engneerng Drve, WI , USA Emal: sethares@ece.wsc.edu Receved June 00; Revsed March 005 We propose a decentralzed algorthm for organzng an ad hoc sensor network nto clusters. Each sensor uses a random watng tmer and local crtera to determne whether to form a new cluster or to jon a current cluster. The algorthm operates wthout a centralzed controller, t operates asynchronously, and does not requre that the locaton of the sensors be known a pror. Smplfed models are used to estmate the number of clusters formed, and the energy requrements of the algorthm are nvestgated. The performance of the algorthm s descrbed analytcally and va smulaton. Keywords and phrases: wreless sensor networks, clusterng algorthm, random watng tmer.. INTRODUCTION Unlke wreless cellular systems wth a robust nfrastructure, sensors n an ad hoc network may be deployed wthout nfrastructure, whch requres them to be able to self-organze. Such sensor networks are self-confgurng dstrbuted systems and, for relablty, should also operate wthout centralzed control. In addton, because of hardware restrctons such as lmted power, drect transmsson may not be establshed across the complete network. In order to share nformaton between sensors whch cannot communcate drectly, communcaton may occur va ntermedares n a multhop fashon. Scalablty and the need to conserve energy lead to the dea of organzng the sensors herarchcally, whch can be accomplshed by gatherng collectons of sensors nto clusters. Clusterng sensors are advantageous because they () conserve lmted energy resources and mprove energy effcency, () aggregate nformaton from ndvdual sensors and abstract the characterstcs of network topology, () provde scalablty and robustness for the network. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Ths paper proposes a decentralzed algorthm for organzng an ad hoc sensor network nto clusters. Each sensor operates ndepently, montorng communcaton among others. Those sensors whch have many neghbors that are not already part of a cluster are lkely canddates for creatng a new cluster by declarng themselves to be a new clusterhead. The clusterng algorthm va watng tmer (CAWT) provdes a protocol whereby ths can be acheved and the processcontnuesuntlallsensorsarepartofacluster.because of the dffculty of the analyss, smplfed models are used to study and abstract ts performance. A smple formula for estmatng the number of clusters that wll be formed n an ad hoc network s derved based on the analyss, and the results are compared to the behavor of the algorthm n a number of settngs.. LITERATURE REVIEW Several clusterng algorthms have been proposed n recent years [,, 3,, 5,, 7,, 9,,, 3,, 5,, 7, ]. Many of the algorthms are heurstcs nted to mnmze the number of clusters. Some of the algorthms organze the sensors nto clusters whle mnmzng the energy consumpton needed to aggregate nformaton and communcate the nformaton to the base staton. Perhaps the earlest of the clusterng methods s the dentfer-based heurstc called the

2 Automatc Decentralzed Clusterng for Wreless Sensor Networks 7 lnked cluster algorthm (LCA) [5], whch elects sensor to be a clusterhead f the sensor has the hghest dentfcaton number among all sensors wthn one hop of ts neghbors. The connectvty-based heurstc of [, ] selects the sensors wth the maxmum number of -hop neghbors (.e., hghest degree) to be clusterheads. The weghted clusterng algorthm (WCA) [9] consders the number of neghbors, transmsson power, moblty, and battery usage n choosng clusters. It lmts the number of sensors n a cluster so that clusterheads can handle the load wthout degradaton n performance. These clusterng methods rely on synchronous clockng for the exchange of nformaton among sensors whch typcally lmts these algorthms to smaller networks []. The Max-Mn D-cluster algorthm [] generates D-hop clusters wth a complexty of O(D) wthout tme synchronzaton. It provdes load balancng among clusterheads n the network. Smulaton results suggest that ths heurstc s superor to the LCA and connectvty-based solutons. The low-energy adaptve clusterng herarchy (LEACH) of [] utlzes randomzed rotaton of clusterheads to balance the energy load among the sensors and uses localzed coordnaton to enable scalablty and robustness for cluster set-up and operaton. LEACH-C (centralzed) []usesa centralzed controller. The man drawbacks of ths algorthm are nonautomatc clusterhead selecton and the requrement that the poston of all sensors must be known. LEACH s stochastc algorthm s exted n [3] wth a determnstc clusterhead selecton. Smulaton results demonstrate that an ncrease of network lfetme can be acheved compared wth the orgnal LEACH protocol. In [], the clusterng s drven by mnmzng the energy spent n wreless sensor networks. The authors adopt the energy model n [] and use the subtractve clusterng algorthm and fuzzy C-mean (FCM) algorthm to form clusters. Although the above algorthms carefully consder the energy requred for clusterng, they are not extensvely analyzed (due to ther complexty) and there s no way of estmatng how many clusters wll form n a gven network. The ad hoc network desgn algorthm (ANDA) [5] maxmzes the network lfetme by determnng the optmal cluster sze and the optmal assgnment of sensors to clusterheads but requres a pror knowledge of the number of clusterheads, number of sensors n the network, and the locaton of all sensors. The dstrbuted algorthm n [3] groups sensors nto a herarchy of clusters whle mnmzng the energy consumpton n communcatng nformaton to the base staton. They use the results provded n [] to obtan optmal parameters of the algorthm and analyze the number of clusterheads at each level of clusterng. Most of these desgn approaches are determnstc protocols n whch each sensor must mantan knowledge of the complete network [, 5] or dentfy a subset of sensors wth a clusterhead to partton the network nto clusters n heurstc ways [,,, 5,, 7,, 9, ]. The algorthms proposed n [,, 3, ] focus on reducng the energy consumpton wthout explorng the number of clusters generated by the protocols, though [, 9] demonstrate the average number of clusterheads va smulatons. For most of the algorthms, no analyss of the number of clusters s avalable. The method of ths paper s a randomzed dstrbuted algorthm n whch each sensor uses a random watng tmer and local crtera to decde whether to be a clusterhead. The algorthm operates wthout a centralzed controller, t operates asynchronously and does not requre that the locaton of the sensors be known. Based on smplfed models, an estmate of the number of clusterheads and a smple predcton formula are derved to approxmate and descrbe the behavor of the proposed algorthm. To examne the energy usage of the algorthm, the result provded n [9] s used to nvestgate stuatons where the mnmum transmsson range ensures that the network have a strong connectvty. The performance of the algorthm s nvestgated both by smulaton and analyss. 3. THE CLUSTERING ALGORITHM VIA WAITING TIMER Ths secton descrbes a randomzed dstrbuted algorthm that forms clusters automatcally n an ad hoc network. The man assumptons are () all sensors are homogeneous wth the same transmsson range, () the sensors are n fxed but unknown locatons; the network topology does not change, () symmetrc communcaton channel: all lnks between sensors are bdrectonal, (v) there are no base statons to coordnate or supervse actvtes among sensors. Hence, the sensors must make all decsons wthout reference to a centralzed controller. Each actve sensor broadcasts ts presence va a Hello sgnal and lstens for ts neghbor s Hello. The sensors that hear many neghbors are good canddates for ntatng new clusters; those wth few neghbors should choose to wat. By adjustng randomzed watng tmers, the sensors can coordnate themselves nto sensble clusters, whch can then be used as a bass for further communcaton and data processng. After deployment, each sensor sets a random watng tmer. If the tmer expres, then the sensor declares tself to be a clusterhead, a focal pont of a new cluster. However, events may ntervene that cause a sensor to shorten or cancel ts tmer. For example, whenever the sensor detects a new neghbor, t shortens the tmer. On the other hand, f a neghbor declares tself to be a clusterhead, the sensor cancels ts own tmer and jons the neghbor s new cluster. Assume the ntal value of the watng tme of sensor,, s a sample from the dstrbuton C+α U(0, ), where C and α are postve numbers, and U(0, ) s a unform dstrbuton. In the clusterng phase of the network, each sensor broadcasts a Hello message at a random tme. Ths allows each sensor to estmate how many neghbors t has. A Hello message conssts of () the sensor ID of the sng sensor, and () the cluster ID of the sng sensor. At the begnnng, the cluster ID of each sensor s zero. Note that a sensor WT (0)

3 EURASIP Journal on Wreless Communcatons and Networkng () Each sensor ntalzes a random watng tmer wth a value WT (0). () Each sensor transmts the Hello message at random tmes: draw a sample r from the dstrbuton λ WT (0) U(0, ), where 0 <λ< 0.5, wat r tme unts and then transmt the Hello. (3) Establsh and update the neghbor dentfcaton: f a sensor receves a message of assgnng a cluster ID at tme step k (a) jon the correspondng cluster, (b) draw a sample r from the dstrbuton WT (k) U(0, ), (c) wat r tme unts and then s an updated Hello message wth the new cluster ID, (d) stop the watng tmer. (Stop!) else collect neghborng nformaton. () Decrease the random watng tme accordng to (). (5) Clusterhead check: f WT = 0 and the neghborng sensors are not n another cluster (a) broadcast tself to be a clusterhead, (b) assgn the neghborng sensors to cluster ID. (Stop!) elsef WT = 0 and some of the neghborng sensors are n other clusters jon any nearby cluster after τ seconds, where τ s greater than any possble watng tme. (Stop!) else go to step (3). Algorthm : The CAWT: an algorthm for segmentng sensors nto clusters. ID s not needed to be unambguously assgned to each sensor before applyng the CAWT. The followng are two possble ways for each sensor to determne ts sensor ID: () each sensor can automatcally know an ID number (lke an IP address or an RFID tag), and () each sensor could pck a random number when t frst turns on, whch s a random ID assgnment. If the range of numbers s large compared to the number of sensors, then t s unlkely that two sensors (wthn rado range) would pck the same number. Sensors update ther neghbor nformaton (.e., a counter specfyng how many neghbors t has detected) and decrease the random watng tme based on each new Hello message receved. Ths encourages those sensors wth many neghbors to become clusterheads. The updatng formula for the random watng tme of sensor s WT (k+) = β WT (k), () where WT (k) s the watng tme of sensor at tme step k and 0 <β<. If both of the followng condtons apply, then sensor declares tself a clusterhead: () the random watng tmer expres, that s, WT = 0; () none of the neghborng sensors are already members of a cluster. If sensor satsfes the above condtons, t broadcasts a message proclamng that t s begnnng a new cluster; ths also serves to notfy ts neghbors that they are assgned to jon the new cluster wth ID. When a sensor jons the cluster, t ss an updated Hello message and stops ts watng tmer. The complete procedure of the ntalzaton phase s outlned n the CAWT of Algorthm. After applyng the CAWT, there are three dfferent knds of sensors: () the clusterheads, () sensors wth an assgned cluster ID, and (3) sensors whch are unassgned. These unassgned sensors may jon the nearest cluster later depng on the neghborng nformaton or the demand of specfc applcatons, such as sensor locaton estmaton problem. Thus, the topology of the ad hoc network s now represented by a herarchcal collecton of clusters.. SIMPLIFIED METHODS OF CLUSTERING Because of the complexty of the CAWT, t s dffcult to evaluate the algorthm drectly other than va smulaton. Snce the connectvty among sensors and the number of neghborng sensors play mportant roles n the CAWT, t s reasonable to nvestgate the performance from the perspectve of these parameters. Therefore, we abstract the behavor of the algorthm usng two smplfed models whch approxmate the desred global behavor and serve to analyze ts performance... The neghborng densty model The frst smplfed model s the neghborng densty model (NDM) whch s detaled n Algorthm. The basc dea of NDM s to suppose that the probablty of each sensor of beng a clusterhead, p, s proportonal to the number of the

4 Automatc Decentralzed Clusterng for Wreless Sensor Networks 9 (a) Assgn a probablty to sensor, p, proportonal to the number of the neghborng sensors, N.Thats,p N / n = N. (b) Let B be the set of neghborng sensors of sensor. I s the ndex set of clusterheads. (c) P (k), P (k),and P (k) are by n vectors to store the probablty dstrbuton at tme step k. (d) Assgn k = 0andP (0) = (p, p,..., p n ). whle sum(p (k) ) > 0 () Select a clusterhead f j = arg max {p (k) } j I, () Update the probablty dstrbuton (k) p = p (k) {/ Bj, B B j =, j=arg max {p (k), }} (k) p j = 0. (3) Normalze the updated probablty dstrbuton. f sum( P (k) ) > 0 p (k) = p (k) / sum( P (k) ). else P (k) = P (k). () Store the normalzed probablty dstrbuton. P (k) = P (k), set k = k +. Algorthm : The neghborng densty model: a procedure for analyzng the CAWT. neghborng sensors, N. That s, p N n= N. () If the sensor s not already chosen as a clusterhead and ts neghborng sensors are not already n other clusters, then the sensor wth the largest p s chosen to be a clusterhead and t assgns probablty 0 to ts neghbors. Thus, a sensor becomes a clusterhead f t has the hghest neghborng densty among all sensors whch have not yet become cluster members.moreover,fasensorsnotamemberofaclusterand some of ts neghbors have already become cluster members, ths sensor should choose to wat and jon the nearest cluster later. After normalzng the updated probablty dstrbuton of sensors, the procedure repeats untl all sensors are members of a cluster. The ratonale for ths choce s that, f the random watng tme of each sensor s long enough (n the sense that each sensor s able to collect suffcent neghborng nformaton), then the model s lkely to closely approxmate the behavor of the CAWT on any gven ad hoc network. The close connecton between the model and the algorthm s explored va smulaton... The averaged model Ths subsecton models the CAWT by a smplfed averagng procedure. Assume that a sngle clusterhead and an average number of neghborng sensors E (k) [N ] are removed durng each teraton k. Assume that each sensor wll be removed wth probablty p rm (k) = r k /m k,wherer k s the number of sensors to be removed and m k s the number of sensors remanng at teraton k. Denote the collecton of sensors at teraton k by V k. Snce a clusterhead and ts neghborng sensors are removed at each teraton, the collecton of sensors at the next teraton, V k+, s smply a new and smaller network. Theorem can be appled to approxmate the dstrbuton of the number of clusterheads at teraton k by N (µ k, σk ), where µ k = m k, σk = m k ( p (k) ), m k s the number = p (k) = p (k) of sensors n V k, p (k) s the updated probablty dstrbuton of sensors at teraton k, I k,andi k s the ndex set of sensors at teraton k. Once the procedure termnates, the number of teratons s an estmate of the number of clusterheads formed n the network. A statement of the averaged model I s gven n Algorthm Analyss of the averaged model Ths secton analyzes the averaged model of Algorthm 3 and derves a smple expresson for the expected number of clusterheads n a gven network. Later sectons show va smulaton that ths s also a reasonable estmate of the number of clusterheads gven by the mplementable CAWT of Algorthm..3.. The Lndeberg theorem Ths secton revews the probablty that s used when analyzng the performance of the model. Readers may see [0] for a complete dscusson and proof of the theorem.

5 90 EURASIP Journal on Wreless Communcatons and Networkng (a) Let N (k) b be the sum of neghborng sensors at teraton k. N (k) b = m k = N (k). I k ; I k s the ndex set of sensors at teraton k. (b) Let E (k) [N ] be the average number of neghbors at teraton k. (c) Assgn the probablty p (k) to sensor, proportonal to the number of neghborng sensors, N (k).thats,p (k) N (k) /N (k) (d) Assgn k = 0, m 0 = n, r 0 = 0. whle (m k r k ) > 0 r k = E (k) [N ] +, m k+ = m k r k, k = k +. s the celng functon. b. Algorthm 3: Averaged model I: procedure for analyzng the CAWT. Suppose for each n that ( ) X, X,..., X r, ( ) X, X,..., X r,. ( ) Xn, X n,..., X nrn are ndepent random vectors. The probablty space may change wth n. PutS n = X n + + X nrn. In the network applcaton, r n = n, X n = X,0, and (3) scalledatrangular array of random varables. Let X take the values and 0 wth probablty p and q = p. We may nterpret X as an ndcator that sensor s chosen to be a clusterhead wth probablty p and S n s the number of clusters n the network. Denote Y = X p.hence, S Y n Y = X p = S n p, = = = = E [ ] [ ] Y = E X p = 0, σy = σx ( ) = p p, s n = σy = σ ( ) X = p p. = = = For our case, the Lndeberg condton [0] reduces to lm n s = n Y dp lm Y ɛs n n (3) () s = dp = 0, (5) n Y ɛs n whch holds because all the random varables are bounded by and [ Y ɛs n ] 0asn. Theorem. Suppose that Y s an ndepent sequence of random varables and satsfes E[Y ] = 0, σy = E[Y ], S Y n = n = Y,ands n = n = σy. If the Lndeberg condton (5) holds, then S Y n /s n N (0, ). By Theorem, the dstrbuton of the number of clusters can be approxmated by N ( n = p, s n) snce E[S n ] = E[S Y n ]+ n= p = n = p and n = σ X = n = σ Y = s n..3.. Specal case Assume that n sensors are deployed n a crcle and the dstance between each par of neghborng sensors s equal. In addton, because of the rado range, assume that each sensor can detect two neghborng sensors. Hence each sensor may be chosen as a clusterhead wth probablty p = /n. As mentoned before, let X be the ndcator that sensor s chosen to be a clusterhead wth probablty p and let S n be the number of clusterheads n the network. Based on these assumptons, the expectaton and varance of S n are E [ ] n ( S n = kp r Sn = k ) = np, k= () s n = σx ( ) = np p. =.3.3. Analyss Ths secton shows that, wth approprate smplfcaton, the averaged model (AM) can be used to make smple predcton of the behavor of the CAWT. To obtan the mean and varance of the number of clusterheads of each teraton, the probablty dstrbuton of these random varables must be updated. However, t s not smple to calculate p (k) at each teraton snce the process of selectng a clusterhead at each teraton s complex. The followng smplfed analyss restructures the connectvty of the network so that each sensor has the same average neghborng densty at each teraton. Therefore, we have E (k+)[ N ] = N (k) b r k E (k)[ N ] m k+. (7) Ths smplfed averaged model s summarzed n averaged model IIn Algorthm.

6 Automatc Decentralzed Clusterng for Wreless Sensor Networks 9 (a) Let N (k) b be the sum of neghborng sensors of sensors at teraton k. = m k = N (k). N (k) b I k ; I k s the ndex set of sensors at teraton k. (b) Let E (k) [N ] be the average number of neghbors at teraton k. E (0) [N ] = N (0) b /m 0. (c) Assgn the probablty p (k) to sensor, proportonal to the number of neghborng sensors, N (k).thats,p (k) (d) Assgn k = 0, m 0 = n, r 0 = 0. whle (m k r k ) > 0 m k+ = m k r k, N (k) /N (k) b. E (k+) [N ] = (N (k) b r k E (k) [N ])/m k+, r k+ = E (k+) [N ] +, k = k +. s the celng functon. Algorthm : Averaged model II: procedure for analyzng the CAWT. Thus, the dstrbuton of the number of clusterheads can be approxmated by N(µ ch, σch ), where N t N t m k µ ch = µ k = k= k= = N t σch = N t σk = k= m k k= = p (k) p (k), () ( p (k)), where N t s the number of teratons. Moreover, suppose that the expectaton of the number of neghborng sensors of each sensor n the network s used to approxmate the number of neghborng sensors that wll be removed at each teraton (.e., the sensors whch wll eventually jon the new cluster). Thus, Then E (k)[ N ] = E [ N ] = n N, k. (9) = r k = E [ N ] +, () and a smple formula for predctng the number of clusterheads s N ch = n E [ N ] +. () The comparson of the performance of the CAWT and the smplfed models wll be llustrated n Secton. 5. ANALYSIS OF ENERGY CONSUMPTION Ths secton consders the energy consumpton of the CAWT assumng homogenous sensors. The total power requrements nclude both the power requred to transmt messages and the power requred to receve (or process) messages. In the ntalzaton phase, each sensor broadcasts a Hello message to ts neghborng sensors. Therefore, the number of transmssons N Tx s equal to the number of sensors n the network, n, and the number of receptons N Rx s the sum of the neghborng sensors of each sensor. That s, N Tx = n, N Rx = N j. () j= As a sensor, say sensor, meets the condtons of beng a clusterhead, t broadcasts ths and assgns cluster ID to ts neghborng sensors. Its neghborng sensors then transmt a sgnal to ther neghbors to update cluster ID nformaton. Durng ths clusterng phase, (+N ) transmssons and (N + j C N j ) receptons are executed, where C s the ndex set of neghborng sensors of sensor. Ths procedure s appled to all clusterheads and ther cluster members. Now let NT c x and NR c x denote the number of transmssons and receptons for all clusters, respectvely. Hence, NT c x = ( ) +N, I NR c x = ( ) (3) N j + N, I j C where I s a ndex set of clusterheads. Therefore, the total number of transmssons N T and the number of receptons

7 9 EURASIP Journal on Wreless Communcatons and Networkng 0 0 (a) 0 0 (b) 0 0 (c) Fgure : Clusters are formed n a random network of 50 sensors wth (a) R/l = 0.5, (b) R/l = 0., and (c) R/l = 0.5. N R are N T = N Tx + NT c x = n + ( ) +N, I N R = N Rx + NR c x = N j + ( ) N j + N. j= I j C () Suppose that the energy needed to transmt s E T,whch deps on the transmttng range R, and the energy needed to receve s E R.From(), the total energy consumpton, E total, for cluster formaton n the wreless sensor network s E total = N T E T + N R E R. (5) Observe that the above analyss s sutable for any transmttng range. However, overly small transmsson ranges may result n solated clusters whereas overly large transmsson ranges may result n a sngle cluster. Therefore, n order to optmze energy consumpton and encourage lnkng between clusters, t s sensble to consder the mnmum transmsson power (or range R) whch wll result n a fully connected network. Ths range assgnment problem s nvestgated n [9], whch proposes lower boundson the magntude of R d n (wth respect to l), R d n O(l d ), and shows that R d n l d ln(l) may be a good ntal value for the search of optmzed range assgnment strateges to provde a hgh probablty of connectvty. As usual, n s the number of sensors and l s the length of sdes of a d-dmensonal cube. The performance of the total energy consumpton of the CAWT wth dfferent selectons of R s examned va smulaton.. SIMULATION RESULTS The smulatons of ths secton examne the performance of the CAWT and valdate the smplfed models for whch analytcal results have been derved. Assume that n sensors are unformly dstrbuted over a square regon n a two-dmensonal space. Parameters for the random watng tmer, number of sensors, and rato of transmttng range R to the sde length l of the square, R/l, are nvestgated to provde a smulaton-based study of the CAWT. Note that the entre experments are conducted n a square regon wth sde length l = 00 unt length. The frst set of experments examnes the varaton of the average number of clusterheads wth respect to the rato R/l. Wth random watng tme parameters C = 0, α =, and

8 Automatc Decentralzed Clusterng for Wreless Sensor Networks 93 Average number of clusterheads R/l n = 5 n = 50 n = 75 n = 0 Fgure : Average number of clusterheads as a functon of the rato R/l. β = 0.9, Fgure depcts typcal runs of the algorthm based on the same network topology but wth dfferent R/l ratos. The results show that each cluster s a collecton of sensors whch are up to hops away from a clusterhead. Fgure shows the relatonshp between the average number of clusterheads and the R/l rato wth varyng the number of sensors. The average number of clusterheads n each case s the sample mean of the results of 00 typcal runs. Observe that the average number of clusterheads decreases as the rato R/l ncreases (.e., the transmsson power ncreases). Snce larger transmsson power allows larger rado coverage, a clusterhead has more cluster members, whch reduces the number of clusters n the network. Fgure also shows that when the transmsson range s small, the network wth a lower sensor densty wll have a larger percentage of solated sensors whch eventually become clusterheads n ther own rght. Ths s because the network s only weakly connected wth these values. On the other hand, when the transmsson power s large enough to ensure strong connectvty of the network, the average number of clusterheads stablzes as the number of sensors ncreases. The second set of experments n Fgure 3 evaluates the performance of the neghborng densty model (NDM), whch compares cluster formaton when usng the NDM and the CAWT. The outputs of the two methods are not dentcal due to the randomness of the watng tmer. Nonetheless, both these clusterng structures are qualtatvely smlar gven the same network settngs, suggestng that the NDM provdes a good approxmaton to the CAWT. The thrd set of experments compares the estmates of the number of clusterheads when applyng the CAWT, the neghborng densty model (NMD), the averaged model (AM), and the predcton formula. In each method, the results of 00 typcal runs are merged. For the CAWT, the NDM, and the predcton formula cases, the estmates of the number of clusterheads are gven by the sample mean and sample varance of the results of typcal runs. For the AM case, the estmates of mean and varance of the number of clusterheads are generated n each typcal run, whch means the best estmate may not be obtaned by averagng the typcal runs. The covarance ntersecton (CI) method of [] provdes the best estmate gven the nformaton avalable. The CI algorthm takes a convex combnaton of mean and covarance estmates that are represented n nformaton space. Snce these typcal runs are ndepent, the crosscorrelatons between these estmates are 0. Therefore, the general form s P cc = ω P a a + + ω n P a na n, P cc c = ω P a a a + + ω n P a na n a n, () where n = ω =, n>, a s the estmate of the mean from avalable nformaton, P aa s the estmate of the varance from avalable nformaton, c s the new estmate of the mean, and P cc s the new estmate of the varance. We choose to weght each typcal run equally. In order to compare the CAWT and the smplfed models, Fgures a and b show the standard devaton of the mean number of clusterheads. The plots vary the number of sensors n and the transmsson power R/l. Also shown n Fgures c and d are the confdence ntervals for the mean number of clusterheads at a 90% confdence level. The graphs suggest that the NDM approxmates the CAWT somewhat better than the AM. Ths s reasonable because the NDM retans global connectvty nformaton whle the AM uses only the average densty nformaton. Though the NDM outperforms AM, these results provde evdence that the AM provdes a way to roughly predct the performance of the CAWT. The fourth set of experments consders the total energy consumpton of the CAWT. Assume that the communcaton channel s error-free. Snce each sensor does not need to retransmt any data, two transmssons are executed, one for broadcastng the exstence and the other for assgnng a cluster ID to ts cluster members or updatng the cluster ID nformaton of ts neghbors. Hence, the total number of transmssons s n. Under these crcumstances, sensor wll receve N messages. Then, the total number of receptons s n = N. Fgures 5 and show the average number of transmssons and receptons of random networks after applyng the proposed algorthm. Fgure also shows that the number of receptons ts to ncrease as the raton R/l ncreases. Ths mples that energy consumpton s hgher for the network wth larger transmsson power. Ths can be attrbuted to the fact that larger transmsson power allows sensors to detect more neghbors, whch ncreases the number of receptons when assgnng cluster ID or updatng cluster ID nformaton. Therefore, n order to mnmze energy use and keep strong connectvty n the network, an approprate selecton of the transmsson range R s essental. In [9], the authors

9 9 EURASIP Journal on Wreless Communcatons and Networkng 0 0 (a) 0 0 (b) 0 0 (c) 0 0 (d) Fgure 3: Cluster formaton n a random network wth 0 sensors and (a) the CAWT wth R/l = 0.5, (b) the NDM algorthm wth R/l = 0.5, (c) the CAWT wth R/l = 0., and (d) the NDM algorthm wth R/l = 0.. suggest that log l R l d (7) n may be a good choce for the ntal range assgnment for sensors n the d-dmensonal space. Hence, f l = 00 m and n = 0, then R 73. m. Ths means that for R/l 0.73, t may lead to a strongly connected network and energy conservaton. The fnal set of experments compares the cluster formaton when usng the Max-Mn D-cluster formaton algorthm [] and the new decentralzed clusterng algorthm wth random watng tmer. The Max-Mn heurstc generalzes the clusterng heurstcs so that a sensor s ether a clusterhead or at most D hops away from a clusterhead. Ths heurstc has complexty of O(D) rounds whch s better than most clusterng algorthms n the lterature (see [5,, 7,, ]) wth tme complexty of O(n), where n s the number of sensors n the network. In the proposed CAWT, each sensor ntates rounds of local floodng to ts -hop neghborng sensors, one for broadcastng sensor ID and the other for broadcastng cluster ID, to select clusterheads and form -hop clusters. Hence, the tme complexty s O() rounds. Ths mples that the CAWT and the Max-Mn heurstc wth D = have the same tme complexty O(). Thus the Max-Mn heurstc wth D = provdes a good way to benchmark the performance of the CAWT. As shown n Fgure and by the fgures n [], load balancng may not be acheved wthout an approprate transmsson range snce ths may lead to ether too large or too small cluster szes. Hence, the cluster formaton s examned wth respect to the R/l rato and network densty suggested n (7) when usng both the CAWT and the Max-Mn heurstc. Fgures 7 and show that both the average number of the CAWT clusterheads and the Max- Mn clusterheads ncrease approxmately lnearly wth ncreased network densty though the Max-Mn heurstc has more clusterheads and slghtly smaller cluster szes than the CAWT. Fgure also demonstrates that a good selecton of transmsson range may lead to a mnmal varaton of the cluster sze wth ncreased network densty. Ths

10 Automatc Decentralzed Clusterng for Wreless Sensor Networks R/l = 0.75 R/l = 0.5 R/l = 0.75 R/l = R/l = 0.75 R/l = 0.5 R/l = 0.75 R/l = (a) (b) 5 R/l = R/l = R/l = 0.5 R/l = 0.75 R/l = R/l = 0.5 R/l = 0.75 R/l = (c) (d) Fgure : The number of clusterheads formed n a random network usng () the CAWT, () NDM, (3) AM, and () the predcton formula, respectvely, wth varyng R/l ratos. Parts (a) n = 50 and (b) n = 0 show the standard devaton over 00 runs. Parts (c) n = 50 and (d) n = 0 show the confdence ntervals at the 90% level. may help to acheve the load balance among the clusterheads. The above set of experments mply that the CAWT s compettve wth the Max-Mn heurstc n terms of tme complexty and cluster formaton. The authors n [] show that the Max-Mn heurstc may fal to provde a good cluster formaton n some network confguratons and more study s needed to determne approprate tmes to trgger the Max- Mn heurstc. In comparson, the CAWT may be relably appled to any network topology and network densty. 7. CONCLUSION Ths paper has presented a randomzed, decentralzed algorthm for organzng the sensors of an ad hoc network nto clusters. A random watng tmer and a neghbor-based crtera were used to form clusters automatcally. Two smplfed models are ntroduced for the purpose of understandng the performance of the CAWT. Smulaton results ndcated that the smplfed models agree well wth the behavor of the algorthm. Under the assumpton of fxed transmsson power and homogenous sensors, the energy requrements of the method were determned. There are several ways ths work may be generalzed. For a fxed clusterhead selecton scheme, a clusterhead wth constraned energy may dran ts battery quckly due to heavy utlzaton. In order to spread the energy usage over the network and acheve a better load balancng among clusterheads, reselecton of the clusterheads may be a useful

11 9 EURASIP Journal on Wreless Communcatons and Networkng Average number of transmssons R/l n = 5 n = 0 n = 75 n = 50 n = 5 Fgure 5: The number of transmssons n random networks as a functon of the number of sensors and R/l rato. 000 Average number of clusterheads Number of sensors CAWT NDM AM Max-Mn Fgure 7: The average number of clusterheads as a functon of the number of sensors and R/l rato usng the CAWT (and the two smplfed models) and the Max-Mn heurstc. n = 0, R/l = 0.73; n = 00, R/l = 0.5; n = 300, R/l = 0.; n = 00, R/l = 0.07; n = 500, R/l = Average number of receptons R/l n = 5 n = 50 n = 75 n = 0 Fgure :Thenumberofreceptonsnrandomnetworksasafuncton of the number of sensors and R/l rato. strategy. Also, f the sensors are movng slowly, then the algorthm s flexble and cheap enough to be appled teratvely as the network confguraton changes. Ths can be acheved by modfyng the condtons under whch the random tmng counter s ncremented or decremented. From an adaptve cross-layer desgn perspectve, the random tmer may be adjusted usng current channel condtons (sgnal-to-nterference-and-nose rato (SINR), lnk connectvty, etc.) and energy constrants (energy level of neghborng sensors) from the physcal layer. Moreover, the random tmer may adapt based on the moblty of the sensor and the constrants from the MAC layer to acheve network robustness and scalablty. Therefore, such adaptve clusterng pro- Average cluster sze Number of sensors CAWT NDM AM Max-Mn Fgure : The average cluster sze wth the same network settngs as n Fgure 7. tocols may provde a relable method of cluster organzaton for wreless ad hoc sensor networks. REFERENCES [] A. D. Ams, R. Prakash, T. H. P. Vuong, and D. T. Huynh, Max-mn d-cluster formaton n wreless ad hoc networks, n Proc. 9th IEEE Annual Jont Conference Computer and Communcatons Socetes (INFOCOM 00), vol., pp. 3, Tel Avv, Israel, March 000. [] A. D. Ams and R. Prakash, Load-balancng clusters n wreless ad hoc networks, n Proc. 3rd IEEE Symposum on

12 Automatc Decentralzed Clusterng for Wreless Sensor Networks 97 Applcaton-Specfc Systems and Software Engneerng Technology, pp. 5 3, Rchardson, Tex, USA, March 000. [3] S.Bandyopadhyay and E.J.Coyle, An energy effcent herarchcal clusterng algorthm for wreless sensor networks, n Proc. nd IEEE Annual Jont Conference of the IEEE Computer and Communcatons Socetes ( INFOCOM 03), vol. 3, pp , San Francsco, Calf, USA, March Aprl 003. [] S. Basagn, Dstrbuted clusterng for ad hoc networks, n Proc. th Internatonal Symposum on Parallel Archtectures, Algorthms, and Networks (I-SPAN 99), pp. 3 35, Perth/Fremantle, WA, Australa, June 999. [5] D. Baker and A. Ephremdes, The archtectural organzaton of a moble rado network va a dstrbuted algorthm, IEEE Trans. Commun., vol. 9, no., pp. 9 70, 9. [] M. Gerla and J. T. C. Tsa, Multcluster, moble, multmeda rado networks, Wreless Networks, vol., no. 3, pp. 55 5, 995. [7] A. Ephremdes, J. E. Weselther, and D. J. 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IEEE Internatonal Conference on Communcatons (ICC 97), vol., pp , Montreal, Que., Canada, June 997. Chh-Yu Wen receved the B.S.E.E. and M.S.E.E. degrees wth hgh honors from the Natonal Cheng Kung Unversty n Tawan n 995 and 997, respectvely. He receved an M.S.E.E. degree from the Unversty of Wsconsn-Madson n 00 and wll complete the Ph.D. n electrcal engneerng from the Unversty of Wsconsn-Madson n 005. He has worked on cellular moble systems emphaszng the capacty of wreless channels and networks, modulaton, and error control codng. Current research nterests nclude wreless communcatons, adaptve sgnal processng, software-defned rado, and adaptve dstrbuted algorthms for wreless ad hoc sensor networks. Wllam A. Sethares receved the B.A. degree n mathematcs from Brandes Unversty, Waltham, Mass, and the M.S. and Ph.D. degrees n electrcal engneerng from Cornell Unversty, Ithaca, NY. He has worked at the Raytheon Company as a Systems Engneer and s currently a Professor n the Department of Electrcal and Computer Engneerng at the Unversty of Wsconsn - Madson. Hs research nterests nclude adaptaton and learnng n sgnal processng, communcatons, and acoustcs, and s the author of Tunng, Tmbre, Spectrum, Scale (Sprnger, 99) and the coauthor of Telecommuncaton Breakdown (Prentce-Hall, 00).