Energy Efficient TDMA Sleep Scheduling in Wireless Sensor Networks

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1 Energy Effiient TDMA Sheuling in Wireless Sensor Networks Junho M, Wei Lou Deprtment of Computing The Hong Kong Polytehni University Kowloon, Hong Kong {sjm, Ynwei Wu, Xing-Yng Li Deprtment of Computer Siene Illinois Institute of Tehnology Chigo, IL 60616, USA Guihi Chen Stte Key Lb of Novel Softwre Tehnology Nnjing University Nnjing, P. R. Chin Abstrt sheuling is wiely use mehnism in wireless sensor networks (WSNs) to reue the energy onsumption sine it n sve the energy wstge use by the ile listening stte. In tritionl sleep sheuling, however, sensors hve to strt up numerous times in perio, n thus onsume extr energy ue to the stte trnsitions. The objetive of this pper is to esign n energy effiient sleep sheuling for low t-rte WSNs, where sensors not only onsume ifferent mounts of energy in ifferent sttes (trnsmit, reeive, ile n sleep), but lso onsume energy for stte trnsitions. We use TDMA s the MAC lyer protool, beuse it hs the vntges of voiing ollisions, ile listening n overhering. We first propose novel interferene-free TDMA sleep sheuling problem lle ontiguous link sheuling, whih ssigns sensors with onseutive time slots to reue the frequeny of stte trnsitions. To tkle this problem, we then present effiient entrlize n istribute lgorithms tht use time slots t most onstnt ftor of the optimum. The simultion stuies orroborte the theoretil results, n show the effiieny of our propose lgorithms. Keywors: energy effiient lgorithms, sleep sheuling, wireless sensor networks. I. INTRODUCTION Wireless sensor networks (WSNs) onsist of lrge number of wireless sensor noes tht orgnize themselves into multihop rio networks. The sensor noes re typilly equippe by power-onstrine btteries, whih re often iffiult n expensive to be reple one the noes re eploye. Therefore, it is ritil onsiertion on reuing the power onsumption in the network esign. Previous work [1], [2] hs shown tht the ile listening stte is the mjor soure of energy wstge. In ft, it n onsume lmost the sme mount of energy s require for reeiving. Therefore, noes re generlly sheule to sleep when the rio moules re not in use [3]. After the sleep sheuling, noes oul operte in low uty yle moe tht they perioilly strt up to hek the hnnel for tivity. Keshvrzin et l. [4] nlyze ifferent sleep sheuling shemes n propose sheuling metho tht n erese the en-to-en overll ely. This metho i not, however, provie n interferene-free sheuling, in whih every noe n strt up n trnsmit or reeive its messges without interferene uring the ssigne time slots. One populr pproh to voi interferene is to opt the time ivision multiple ess (TDMA) MAC protools, whih n iretly support low uty yle opertions n hs the nturl vntges of hving no ontention-introue overhe n ollisions [1]. Moreover, TDMA n gurntee eterministi ely boun. Thus, we re intereste in esigning n effiient TDMA sleep sheuling for WSNs. TDMA protools ivie time into slots, whih re llote to sensor noes tht n turn on the rio uring the ssigne time slots, n turn off the rio when not trnsmitting or reeiving in the sleep sheuling. In orer to be interferenefree, simple pproh is to ssign eh ommunition link time slot, n thus, the number of time slots is equl to the number of ommunition links of the network. This sheme requires muh more time slots thn neessry, whih inreses the ely n reues the hnnel utiliztion signifintly. This is beuse multi-hop networks re ble to mke spe reuse in the shre hnnel, n multiple trnsmissions n be sheule in one time slot without ny interferene. TDMA link sheuling ttempts to minimize the number of time slots ssigne while prouing n interferene-free link sheuling, n it hs been shown tht the problem is NP-omplete [5], [6]. Severl pproximte lgorithms hve been propose in the link sheuling problem [7] [10]. However, if the TDMA link sheuling is use s the strtup mehnism in the sleep sheuling, noe my strt up numerous times to ommunite with its neighbors. Note tht the typil strtup time is on the orer of milliseons, while the trnsmission time my be less thn tht if the pkets re smll [11]. Consequently, the trnsient energy onsumption uring the strtup proess n be higher thn the energy uring the tul trnsmission. If sensor noe strts up too frequently, it not only nees extr time, but lso osts extr energy for the stte trnsition. Therefore, the stte trnsition, e.g., from the sleep stte to the tive stte, shoul be onsiere for n energy effiient TDMA sleep sheuling in WSNs. In this pper, we use new energy moel, where the energy onsumption of the stte trnsition is onsiere. We propose novel interferene-free TDMA sleep sheuling problem lle ontiguous link sheuling to reue the frequeny of stte trnsitions. In the sheuling, links inient to one noe re sheule together to obtin onseutive time slots so tht noes n only strt up one to monitor the hnnel /09/$ IEEE 630

2 in one sheuling perio T. Espeilly, if the topology is tree, noe only nees to strt up twie in perio, one for reeiving t from its hilren noes n one for trnsmitting its t to its prent noe. The min ontributions of this pper re summrize s follows: (1) We ress the sheuling problem in new energy moel, whih is loser to relisti sensors. (2) We propose the ontiguous link sheuling problem in WSNs, n prove it to be NP-omplete. (3) We present entrlize n istribute lgorithms tht hve theoretil performne boun to the optimum of the problem. (4) We evelop simultions to show the effiieny of the propose lgorithms. The reminer of this pper is orgnize s follows. Setion II reviews the relte works. Setion III esribes the system moel n then formultes the ontiguous link sheuling problem. Setion IV presents the entrlize lgorithm n Setion V presents the istribute lgorithm for the ontiguous link sheuling. Setion VI esribes n nlyzes the simultion results for the propose lgorithms. Setion VII onlues the pper. II. RELATED WORK Severl pproximte lgorithms hve been propose in the TDMA sheuling problem, inluing brost sheuling [12] [14] n link sheuling [7] [10]. Brost n link sheuling re time slot ssignments to noes n links, respetively. Rmswmi n Prhi [12] presente n effiient n interferene-free entrlize n istribute brost sheuling in multi-hop pket rio network. In [13], the brost sheuling problem ws moele s istne-2 oloring problem, n Krumke et l. propose pproximtion heuristi lgorithms for vrious geometri grphs. In [14], Ngo et l. presente entrlize geneti-fix lgorithm to reue the serh spe bse on within-two-hop mtrix. However, the performne of brost sheuling is worse thn link sheuling in WSNs, espeilly in terms of energy onservtion. In the brost sheuling, when noe wnts to trnsmit, ll the neighbors hve to turn on their rio n strt up, no mtter whether they re the intene reeiver or not. In ontrst, only the intene reeiver nees to strt up in the link sheuling. Rmnthn n Lloy [7] onsiere both the tree networks n rbitrry networks, n the performne of the propose lgorithms is boune by the thikness of network. In [8], Gnhm et l. propose link sheuling lgorithm involving two phses. In the first phse, vli ege oloring is obtine in istribution fshion. In the seon phse, eh olor is mppe to unique time slot, n the hien terminl n the expose terminl problems re voie by ssigning eh ege iretion of trnsmission. The overll sheuling requires t most 2(δ + 1) time slots, when the topologies re yli. In [9], Wng et l. propose both entrlize n istribute lgorithms with performne gurntee to obtin goo interferene-free link sheuling to mximize the throughput of the network. In the lgorithms, the sensors re sheule iniviully in preefine orer without onseutive ssignment of time slots, n eh noe is ssigne the best possible time slot to trnsmit or reeive without using interferene to the lrey-sheule sensors. Djuki n Vlee [10] propose n effiient min-mx ely sheuling metho to fin sheules with minimum roun trip ely in the link sheuling. The previous stuies in the TDMA sheuling i not onsier the energy onsumption of rio in the stte trnsition. Compre to brost sheuling n link sheuling, the ontiguous link sheuling oul reue the frequeny tht sensor noes strt up, n thus hieve better energy effiieny. III. SYSTEM MODEL AND PROBLEM FORMULATION In this setion, we present the system moel onsisting of network moel, n interferene moel n n energy moel, then we formulte the ontiguous link sheuling problem n prove it to be NP-omplete. A. System Desription Network Moel. We ssume tht WSN hs n stti sensor noes, whih re ll equippe with single omni-iretionl ntenns, n there exists sink noe to ollet the t from other sensor noes. With the ssumption tht ll the sensors hve the sme ommunition rnge r, the network n be represente s ommunition grph G = (V, E), where V = {v 1, v 2,, v n } enotes the set of noes, n E enotes the set of eges referre to ll the ommunition links. If {v i, v j } V, the ege e = (v i, v j ) E if n only if v j is lote within the trnsmission rnge of v i. In irete grph, the ege e is lle inient from v i, n inient to v j. Two types of network topologies for t olletion n ggregtion re isusse in this pper, t gthering tree n irete yli grph (DAG). A t gthering tree is tree route t sink noe, where eh intermeite noe ollets the t from its hilren noes n then forwrs the t to its prent noe. A DAG is grph with no irete yles, tht is, there is no pth tht strts n ens t the sme vertex. The epth of vertex in DAG is the length of the longest pth from tht vertex to sink. Interferene Moel. In wireless networks, the pkets trnsmitte by noe my be reeive by ll the noes within its trnsmission rnge. Therefore, interferenes my our mong these noes ue to the brost nture of the wireless meium. There re two types of interferenes: primry interferene n seonry interferene [7]. The primry interferene ours when noe hs more thn one ommunition tsk in single time slot. Typil exmples re sening n reeiving t the sme time n reeiving from two ifferent trnsmitters. The seonry interferene ours when noe tune to prtiulr trnsmitter is lso within the trnsmission rnge of nother trnsmission intene for other noes. Both primry interferene n seonry interferene re onsiere in this pper. The interferene between two links in the network epens on the interferene moel, n we use the protool moel [15], [16] in this pper. In the protool moel, eh noe v i hs 631

3 Ative Ative Ative Trnsmit Reeive Trnsient Trnsient Trnsient () Ative (b) Trnsient Fig. 1. The energy moel: () Before tive time slots merge, (b) After tive time slots merge. fixe trnsmission rnge r n n interferene rnge R, where R > r. We enote the rtio between the trnsmission rnge n the interferene rnge s γ = R r. In prtie, 2 γ 4. A trnsmission from v i to v j is suessful if ny noe v k lote within istne R from v j is not trnsmitting. TABLE I TIME AND POWER CONSUMPTION IN THE STARTUP PROCESS FOR A MICA2 MOTE WITH A CC1000 TRANSCEIVER Opertion proess Time Power onsumption 90μW Rio initiliztion 0.35ms 18mW Turn on Rio 1.50ms 3mW Swith to RX/TX stte 0.25ms 45mW Reeive 1 byte 0.416ms 45mW Trnsmit 1 byte 0.416ms 60mW Energy Moel. In B-MAC [17], Polstre et l. presente the energy onsumption of smpling the hnnel in low-power listening (LPL) on Mi2 mote. The strtup proess from the sleep stte to the tive stte inlues rio initiliztion, rio n its osilltor strtup, n the swith of rio to reeive/trnsmit stte. The strtup proess is slow ue to the feebk loop in the phse-loke loop (PLL) [18], n typil setting time of the PLL-bse frequeny synthesizer is on the orer of milliseons. The strtup time n energy onsumption in the strtup proess n lso be foun in the hnnel polling in [19]. Tble I lists the time n power onsumption in the strtup proess for Mi2 mote with CC1000 trnseiver. We n see tht the time to tivte sensor is 2.1ms, n the energy onsumption is bout 22μJ. Our energy moel is similr to the one use in [20]. In our moel, we ssume tht eh noe opertes in three sttes: tive stte (trnsmit, reeive n listen), sleep stte, n trnsient stte (stte trnsition). The energy onsumption of sensor noes in the sleep stte is muh less thn the onsumption in the tive stte, n signifint energy sving n be hieve if the sleep stte is employe uring the perios of intivity. The trnsient stte omprises two proesses: strtup (from the sleep stte to the tive stte), n turnown (from the tive stte to the sleep stte). The energy moel is illustrte in Fig. 1 (), n there is signifint energy onsumption n time overhe when the sensor s rio powers on n off. time time e b f () g h Noe Noe b Noe Noe Noe e Noe f Noe g Noe h (b) Noe Noe b Noe Noe Noe e Noe f Noe g Noe h Fig. 2. Link sheuling n ontiguous link sheuling: () Network topology, (b) Link sheuling, () Contiguous link sheuling Fig. 1 (b) shows tht merging the sensor s tive time slots together n reue the strtup frequeny so s to sve both energy n time, whih benefits the uty yle network esign. B. Problem Formultion In TDMA sleep sheuling, eh link l ij is ssigne time slot, in whih both sener noe v i n reeiver noe v j shoul strt up to ommunite. After the llote time slot, noes v i n v j hnge to sleep. When using the tritionl link sheuling lgorithms (e.g. [9], lle egree-bse heuristi in this pper) whih sheule the ommunition links one by one, noe v j my strt up w j times to monitor the hnnel in perio T, where w j is the number of neighbors. As resse before, the frequent strtup woul onsume lrge mount of extr energy n time. A resonble esign is to ssign onseutive time slots to ll irete links inient to the sme noe, n then noe only nees to strt up one to reeive ll the pkets from its neighbors. We refer to suh n interferene-free sheuling s the ontiguous link sheuling. A ontiguous link sheuling is si to be vli if ll the links inient to one noe re ssigne onseutive time slots. Fig. 2 shows smple of the ontiguous link sheuling. In Fig. 2 (), the given network is t gthering tree route t noe, in whih ny two links interfere with eh other. Fig. 2 (b) shows n interferene-free link sheuling, where noe strts up numerous times in perio. Fig. 2 () shows the ontiguous link sheuling tht noe n strt up only one for reeiving t from its neighbors. Note tht the ontiguous link sheuling n be pplie to both tree topology n mesh topology, suh s DAG. Espeilly, if the network is t gthering tree where eh noe only hs one prent, noe just nees to strt up twie t most in perio: one for reeiving t from its hilren noes n one for trnsmitting its t to its prent noe. An intervl vertex oloring [21], [22] is n ssignment of set S (i) ofw i onseutive olors to eh noe v i in suh wy tht S (i) S ( j) = for ny two jent noes v i n v j.inthe following prt of this setion, we show tht vli ontiguous link sheuling n be obtine by the intervl vertex oloring in the merge onflit grph, n we prove tht the ontiguous link sheuling problem is NP-omplete. Merge Conflit Grph. Given n interferene moel, the interferene of the links in the ommunition grph G = (V, E) () 632

4 b l b l b l b L (1) (2) L b R-r r r L (1) (1) L v s v t v q r v j v i v k v p e f le lf () Communition grph (b) Conflit grph () Merge onflit grph Fig. 3. Communition grph n orresponing onflit grphs n be represente s onflit grph G [16]. Corresponing to eh irete link between v i n v j in G, the onflit grph ontins vertex (enote by l ij ), n there is n ege between verties l ij n l pq in the onflit grph if l ij interferes with l pq in the network. Notie tht the onflit grph in the protool moel is irete grph, sine the links my not interfere with eh other. In [9], the link sheuling problem is moele s the vertex oloring in the onflit grph. Different from the onflit grph, we propose merge onflit grph G m to moel the ontiguous link sheuling problem. The w i irete links inient to the sme noe in G orrespon to vertex in G m, n there is n ege between ny two verties in G m if n only if t lest one pir of the orresponing links in G interfere with eh other. In G m,weusel vi (w i )to enote the orresponing w i links inient to noe v i, n w i is the weight of vertex L vi (w i ). For exmple, the links l b n l b in Fig. 3 (b) orrespon to L b (2) in Fig. 3 (). For the smple ommunition grph shown in Figs 3 (), if the interferene rnge is two hops, the orresponing onflit grph n merge onflit grph re shown in Figs 3 (b) n 3 (), respetively. In Fig. 3 (), we n see tht the number of verties in G m is equl to the number of reeiving noes in G, n w i is the number of links irete to the sme noe. Obviously, we oul obtin vli ontiguous link sheuling by the intervl vertex oloring in the merge onflit grph. Theorem 1. The ontiguous link sheuling problem is NPomplete. Proof: The problem is lerly in NP sine n ssignment n be verifie in polynomil time. We trnsform the intervl vertex oloring to the ontiguous link sheuling, n the intervl vertex oloring problem is prove to be NP-omplete [22]. Suppose tht eh vertex v i in weighte grph G w hs weight of w i, n the number of jent verties of eh vertex v i is w i, whih is no less thn w i. We onstrut ommunition grph G by repling eh vertex v i in G w with noe v i, n eh noe v i rnomly hooses w i noes s jent noes (or neighbors) in the orresponing w i jent verties in G w. Then, eh noe v i in G hs w i inient links. The remining w i w i jent verties of v i in G w n be seen s interferene in the ommunition grph. This trnsformtion n be performe in polynomil time n we oul obtin Fig. 4. Interferene of links in the protool moel ontiguous link sheuling in G by n intervl vertex oloring in G w, therefore, the problem is NP-omplete. IV. CENTRALIZED ALGORITHMS In this setion, we propose the entrlize ontiguous link sheuling lgorithms, entrlize sheuling n entrlize sheuling with sptil reuse (reursive bktrking n minimum onflits heuristi). A. Centrlize Sheuling We first stuy entrlize lgorithm to the ontiguous link sheuling problem. In ste of sheuling time slot iniviully for eh ommunition link, eh noe v i will be ssigne w i onseutive time slots, where w i is the number of links inient to v i. Eh noe n only strt up one to reeive ll the t from its neighbors. In the entrlize sheuling, we olor the noes in the eresing orer of their weight. Sine eh noe v i is ssigne the smllest w i onseutive time slots whih re not ssigne to other noes tht interfere with v i in the merge onflit grph G m, our sheuling lgorithm is interferene-free. The entrlize sheuling is esribe in Algorithm 1. We ssume R/r = γ, where r n R re the trnsmission n interferene rnge of noe v i, respetively. We use D(v i, x) to enote the isk entere t noe v i with rius x, n L(v i ) L(v j ) to enote the istne between noes v i n v j. Lemm 1. Let I(e) be the links tht interfere with link e in the orresponing weighte onflit grph G m uner the protool moel, then t lest I(e) /C 1 time slots re neee to sheule ll links in I(e), where C 1 = 9(γ+1)2 n I(e) is (γ 1) 2 the number of links in I(e). Proof: If noe v j n ommunite with noe v i, then noe v j must be in D(v i, r). Fig. 4 shows tht, if link inient to v p interferes with l ji in G m, v p must be in D(v i, R + r), n the link must be in D(v i, R + 2r), beuse there is t lest one jent noe of v p in D(v i, R) to interfere with v i, suh s v q. We observe tht the istne between two noes trnsmitting simultneously without interferene in G m shoul be t lest R r. For exmple, if l ji n l ts re interferene-free, the istne between v i n v t is lrger 633

5 Algorithm 1 Centrlize sheuling (Centrlize) Input: A ommunition grph G = (V, E). Output: A vli ontiguous link sheuling. 1: Construt the merge onflit grph G m, n initilize n empty stk S. 2: Push the verties in G m in the non-eresing orer of weight w i to the stk S. 3: while S is not empty o 4: Pop the verties in S sequentilly n ssign eh vertex L vi (w i ) the smllest w i onseutive time slots for noe v i to reeive, whih re not yet ssigne to ny of its neighbors in G m. 5: Sheule the w i time slots for trnsmitting sequentilly for eh jent noe of noe v i in G. thn R, n the istne between noe v j n v t shoul be L(v i ) L(v t ) L(v i ) L(v j ) R r. There re t most C 1 = π[r+2r+0.5(r r)]2 = 9(γ+1)2 links with rius (R r)/2 π[0.5(r r)] 2 (γ 1) 2 tht n be ple in D(v i, R+2r). Thus, there exists link set with the size t lest I(e) /C 1 suh tht eh pir of links in the set interferes with eh other. Therefore, t lest I(e) /C 1 time slots re neee to sheule ll links in I(e). g i1 w i1 g i2 w i2 g i3 w i3... g il w il w i T Fig. 5. The time slots tht onflit with v i in G m when v i is sheule Theorem 2. The number of time slots use by Algorithm 1 is t most onstnt ftor of the optimum. Proof: Suppose tht v i is the sensor noe to be sheule in the lst w i time slots, n ll the other sensors hve lrey been sheule, s shown in Fig. 5. In the figure, w il (1 l L) is the number of onseutive time slots oupie by the links inient to noe v il tht interfere with the links inient to v i in the merge onflit grph G m, n g i1, g i2,, g il represent the gps oupie by other non-onfliting links. The links re sheule in the non-inresing orer of weight, so w il (1 l L) is not smller thn w i. Sine eh vertex v i is ssigne the smllest w i onseutive time slots, whih o not interfere with v i, g il (1 l L) issmllerthnw i. The totl number of time slots use by our sheuling is T = g il + w il + w i < L w i + w il + w i 2( w il + w i ) The number of links tht interfere with the links inient to v i in G m is L w il + w i. We enote T opt s the minimum number of time slots use by ny sheuling, i.e., the minimum sheuling perio. From Lemm 1, we know tht L w il +w i T opt C 1. Then, we get T 2C 1 T opt = C T opt, where C = 2C 1. b e f () g h b e f (b) Fig. 6. Contiguous link sheuling: () Centrlize sheuling, (b) Centrlize sheuling with sptil reuse B. Centrlize Sheuling with Sptil Reuse In the ontiguous link sheuling, it is possible to ssign the verties L vi (w i ) n L v j (w j ) the sme time slots when L vi (w i ) interferes with L v j (w j )ing m, beuse not every link in L vi (w i ) interferes with every link in L v j (w j ). A smple is illustrte in Fig. 6. The given network is irete yli grph (DAG), n the interferene rnge of eh noe is two hops. Sine L (2), L b (3) n L (3) interfere with eh other in the merge onflit grph, n the time slots ssigne using Algorithm 1 re shown in Fig. 6 (). We oul lso get vli ontiguous link sheuling shown in Fig. 6 (b), whih uses fewer time slots. Bse on the observtion, we propose the entrlize sheuling with sptil reuse to improve the entrlize sheuling by re-rrnging the time slots in n interferene mtrix. Definition 1. An interferene mtrix of noe v i is n m n mtrix M = (m j,k ) m n (1 j m, 1 k n) tht shows whether time slot oul be ssigne to link inient to v i without interferene. In the mtrix M, m j,k = : link l k oul not use time slot t j (inter f erene) 0:link l k oul use time slot t j (inter f erene- free) 1:link l k selets time slot t j (seletion) where enotes interferene with the lrey-sheule links, 0 enotes interferene-free n 1 enotes the selete (or ssigne) time slot for the link (seletion). Note tht the number of olumns n is equl to the number of links inient to noe v i, n the number of rows m is the number of time slots ssigne in the sheuling. A smple of the interferene mtrix is shown below. In the mtrix M 0, noe v i hs 5 inient links {l 1, l 2, l 3, l 4, l 5 }, n the time slots whih hve been use by other links tht interfere with links l 1, l 2, l 3, l 4, l 5 re {t 1, t 2, t 6 }, {t 2, t 4, t 7 }, {t 1, t 2, t 5 }, {t 2, t 3, t 6 } n {t 2 }, respetively. l 1 l 2 l 3 l 4 l 5 t t 2 t M 0 = t t t t g h 634

6 Definition 2. An ssignment in the interferene mtrix M = (m j,k ) m n of noe v i is si to be vli if there is only one 1 in eh row n eh olumn, n there re n onseutive rows tht hve 1 in eh row in the mtrix. The links inient to v i oul be sheule onseutive time slots by obtining vli ssignment in the interferene mtrix. Definition 3. An interferene submtrix is n n n mtrix M = (m j,k ) n n, whih onsists of n onseutive rows in the interferene mtrix M. Algorithm 2 Centrlize sheuling with sptil reuse Input: A ommunition grph G = (V, E). Output: A vli ontiguous link sheuling. 1: Construt the merge onflit grph G m, onstrut n interferene mtrix for eh vertex L vi (w i ), n initilize n empty stk S. 2: Push the verties in G m in the non-eresing orer of weight w i to the stk S. 3: while S is not empty o 4: Pop the verties in S sequentilly n ssign eh vertex L vi (w i ) the smllest w i onseutive time slots, using Algorithm 3 or Algorithm 4 in the interferene mtrix M to fin solution. 5: v i brosts the time slot ssignments to the noes whose inient links interfere with the links inient to v i, then the noes upte their interferene mtrixes. Algorithm 2 esribes the entrlize sheuling with sptil reuse. In orer to reue the time slots ssigne, the reursive bktrking lgorithm is use to mke sure tht noe is ssigne the smllest vilble onseutive time slots. In the lgorithm, when noe v i is ssigne time slots, it will brost the informtion to ll the noes whose inient links interfere with the links inient to v i. Then eh noe in the network woul hve n interferene mtrix to inite the time slots tht its inient links oul not use. The lgorithm first fins the smllest n onseutive rows tht hve 0 in eh row in the interferene mtrix M, n onstruts n interferene submtrix M whih onsists of the n rows. Then it strts the first seletion in the first row, n the seon seletion in the seon row without interferene to the seletion in the first row. The lgorithm ontinues the seletion to the next row until vli ssignment is foun. During the proess of eh seletion in row, there my be severl nites 0, n the selete nite is referre to s preeessor, n the nites 0 in the next row re referre to s suessors of the preeessor. If suessor fils in the seletion, it then exeutes the bktrking proeure: the lgorithm heks whether the next suessor of the preeessor stisfies the onition tht there is only one 1 in eh olumn. If the suessors re exhuste, the lgorithm bktrks to the previous preeessor n tries the next suessor of the previous preeessor. If there re no more preeessors, the lgorithm s new time slot interferene-free to ll the links inient to v i, n the orresponing interferene mtrix s zero row vetor (0) 1 n in the lst row. The etils of the reursive bktrking lgorithm re shown in Algorithm 3. Algorithm 3 Reursive bktrking Input: An interferene mtrix M = (m j,k ) m n. Output: A vli ssignment in M. 1: Construt n interferene submtrix M onsisting of the smllest n onseutive rows tht hve 0 in eh row. 2: Strt the first seletion in the first row. 3: Continue the seletion in the next row stisfying the onition tht there is only one 1 in eh olumn, until vli ssignment is obtine. 4: if seletion fils to obtin vli ssignment then 5: Exeute the bktrking proeure. 6: if the reursive bktrking fils then 7: A zero row vetor in the lst row of M, n elete the first row in the n onseutive rows. Upte M, n repet the reursive bktrking in M. The reursive bktrking lgorithm is brute-fore serh lgorithm, whih is too omplex for sensor networks. Therefore, we present fst lgorithm lle minimum onflits heuristi, s shown in Algorithm 4. Definition 4. A onflit mtrix is n n n mtrix M C = ( j,k ) n n tht esribes the number of onflits in the interferene submtrix M = (m j,k ) n n, n eh element j,k in the mtrix is the number of onflits, whih is the sum of the seletions 1 in the row n olumn tht inlue j,k. Tht is, j,k = n m l,k + n m j,l m j,k,ifm j,k. The minimum onflits heuristi lgorithm first onstruts n interferene submtrix M whih onsists of n onseutive rows, n then strts with rnom initil onfigurtion in M, e.g., with one seletion per olumn or per row. One initil onfigurtion of mtrix M 0 is shown in the mtrix M 1.The orresponing onflit mtrix of mtrix M 1 is shown in the mtrix M C1. The lgorithm then uses heuristi to etermine how to reue the interferene by moving the seletion 1 with the lrgest number of onflits to the position in the sme olumn where the number of onflits is minimum. The first step of improvement is shown in the mtrix M 2, n the orresponing onflit mtrix of mtrix M 2 is M C 2. It ontinues to reue the interferene until there is no interferene or the initil onfigurtion fils. If the initil onfigurtion fils, it will new time slot n ontinue. Though the lgorithm onverges muh fster, the minimum onflits heuristi my get stuk on lol optimum, thus, it oes not gurntee solution to fin the onseutive time slots whih tully exist. M 1 = , M C1 =

7 M 2 = , M C2 = Algorithm 4 Minimum onflits heuristi Input: An interferene mtrix M = (m j,k ) m n. Output: A vli ssignment in M. 1: Construt n interferene submtrix M onsisting of the smllest n onseutive rows tht hve 0 in eh row. 2: Initilize the mtrix M with rnom onfigurtion. 3: Count the number of onflits in M, n obtin the onflit mtrix M C. 4: Use heuristi to reue the interferene, moving the seletion 1 with the lrgest number of onflits to the position in the sme olumn, where the number of onflits is minimum. 5: if the minimum onflits heuristi fils then 6: A zero row vetor in the lst row of M, n elete the first row in the n onseutive rows. Upte M, n repet the minimum onflits heuristi in M. As Algorithm 2 reues the number of time slots ompre to Algorithm 1, it lso follows onstnt boun performne gurntee of Algorithm 1. Note tht two verties tht onflit with eh other in the merge onflit grph my overlp some time slots in Algorithm 2. Corollry 1. The number of time slots use by Algorithm 2 is t most onstnt ftor of the optimum. V. DISTRIBUTED ALGORITHMS Wireless sensor networks re self-orgnize n istribute, entrlize lgorithms oul not be use without preefine leer. Therefore, it is neessry to esign effiient istribute lgorithms. In this setion, we propose two istribute lgorithms, istribute sheuling n istribute sheuling with effiient ely. A. Distribute Sheuling In the istribute sheuling, we use rnom orer rther thn globl eresing orer of the weight, n we ssume tht there is ontention-bse MAC (e.g. S-MAC [1]) vilble for noe to ompete the hnnel n to obtin n interferene-free ontiguous link sheuling. The istribute sheuling is simple n effiient so tht eh sensor noe n run the sheuling without extr omputtion. The istribute sheuling is shown in Algorithm 5. Theorem 3. The number of time slots use by Algorithm 5 is t most onstnt ftor of the optimum. Proof: Suppose tht noe v i is sheule in the lst w i time slots, n ll the other noes hve lrey been sheule. We enote K = w i w min, where w min = min w i l. 1 l L Algorithm 5 Distribute sheuling (Distribute) 1: Eh noe v i monitors n ompetes for the hnnel. 2: if noe v i obtins the hnnel then 3: v i ssigns the smllest w i onseutive time slots sequentilly to its inient links whih o not interfere with the links tht hve lrey been sheule. 4: v i brosts the informtion to the noes tht re in the interferene rnge of v i, n these noes oul not trnsmit in the time slots ue to the interferene. 5: else 6: v i wits for rnom time, then goes to step 1. Cse 1: If K is onstnt, the number of time slots use is t most onstnt ftor of the optimum, sine T = g il + w il + w i < K w il + (K + 1)C 1 T opt w i + w il + w i w il + w i (K + 1)( w il + w i ) Cse 2: If K is not onstnt, we oul ivie the time slots ssigne before v i is sheule into severl onseutive groups, n k g = w g i w gmin is onstnt in eh group, where w gi is the number of time slots ssigne to the lst noe in group g, n w gmin is the minimum number of time slots ssigne to noe in group g. LetT gopt be the optimum number of time slots in group g, n it must be tht T gopt T opt. Then the number of time slots use in eh group is T g (k g + 1)C 1 T gopt (k g + 1)C 1 T opt. Thus, the totl number of time slots use is T (K g +1)N g C 1 T opt, where N g is the number of groups, n K g is the lrgest k g mong the groups. Note tht the weight of the noes in the ltter group is lwys higher orer infinite ompre to the noes in the former group, i.e., the rtio of the weight of ny noe in the former group to the weight of ny noe in the ltter group is 0. Otherwise, we oul merge the two groups into new group g, n the rtio between w g i n w g min in group g is still onstnt. Sine the sheuling is in rnom orer, the number of groups N g is finite. We efine C = (K g + 1)N g C 1, then we get T C T opt. B. Distribute Sheuling with Effiient Dely e 4 e 3 e 2 e 1 v 5 v 4 v 2 v 1 v 3 () Line topology of 5 noes e 1 e 2 e 3 e 4 e 1 e 2 e 3 e 4 e 1 e 2 e 3 e 4 e 1 e 2 e 3 e 4 0 T 2T 3T 4T (b) Time ely Fig. 7. TDMA sheuling ely 636

8 In the TDMA sleep sheuling, noe stys in the sleep stte for most time, n perioilly strts up to hek for tivity. As forwring noe hs to wit until its nexthop neighbor strts up n is rey to reeive, the messge elivery ely will inrese. When pkets re forwre from n inoming link to n outgoing link, they oul only be forwre to the outgoing link in the next perio T if the inoming link is sheule to be tive fter the outgoing link. This kin of ely will umulte t every hop in the network, whih my le to long lteny. A smple is illustrte in Fig. 7. The network topology is line s shown in Fig. 7 (), n the trnsmission sequene is e 1 e 2 e 3 e 4 s shown in Fig. 7 (b). Uner this sitution, the time ely for pket trnsmitting from v 5 to v 1 is lmost 3T. However, if the trnsmission is e 4 e 3 e 2 e 1, v 5 oul trnsmit the t to v 1 in one perio T. In orer to reue this ely, we sheule the links from bottom to top, tht is, noe with higher epth shoul be sheule erlier. Hene, noe v i n only be sheule until ll the hilren noes of v i re lrey sheule. The lgorithm is esribe in Algorithm 6. Algorithm 6 Distribute sheuling with effiient ely (Distribute-ely) 1: Eh noe v i monitors n ompetes for the hnnel if ll the hilren noes of v i re lrey sheule. 2: if noe v i obtins the hnnel then 3: v i ssigns the smllest w i onseutive time slots sequentilly to its inient links whih o not interfere with the links tht hve lrey been sheule. 4: v i brosts the informtion to the noes tht re in the interferene rnge of v i, n these noes oul not trnsmit in the time slots ue to the interferene. 5: else 6: v i wits for rnom time, then goes to step 1. As the links re still sheule in rnom orer, the lgorithm follows onstnt boun performne gurntee of Algorithm 5, tht is, the number of time slots use by Algorithm 6 is t most onstnt ftor of the optimum. Corollry 2. The number of time slots use by Algorithm 6 is t most onstnt ftor of the optimum. VI. SIMULATION RESULTS In this setion, we stuy the verge-se performne of the propose entrlize n istribute lgorithms for the ontiguous link sheuling using simultor built in C++, n we lso ompre our lgorithms with the egreebse heuristi in [9]. The performne metris use in the evlution re the number of stte trnsitions, the number of time slots ssigne, n time ely. In the simultions, noes with trnsmission rnge of 15m n n interferene rnge of 30m re eploye in squre re of 100m 100m. We test the networks when the number of noes vries from 200 to 400 in steps of 50. We onstrut breth first serh (BFS) tree n irete yli grph () BFS tree Fig. 8. () BFS tree Fig. 9. (b) DAG grph Mximum number of stte trnsitions (b) DAG grph Averge number of stte trnsitions (DAG) roote t the sink noe s the topologies of the network. For eh se, 50 networks re rnomly generte, n the verge performne over ll of these rnomly smple networks is reporte. Fig. 8 n Fig. 9 show the mximum n verge number of stte trnsitions of the following shemes: entrlize, reursive bktrking, minimum onflits heuristi, istribute, istribute-ely, n egree-bse heuristi (egree-bse). In the BFS tree moel, the mximum number of stte trnsitions is two in the shemes of the ontiguous link sheuling, while mny noes nee to strt up numerous times uner the egree-bse sheme, whih is proportionl to the totl time slots require by this noe. In the DAG moel, the number of stte trnsitions of the ontiguous link sheuling shemes is muh less thn the number of the egree-bse sheme, n the trnsient energy ost n be reue. The verge number of time slots ssigne in the sheuling is shown in Fig. 10. In both the BFS tree n DAG moel, the number of time slots ssigne inreses s the number of noes inreses, for the number of interferene links inreses when the number of noes inreses. In the ontiguous link sheuling, the links inient to one noe re sheule together to obtin onseutive time slots to voi frequent stte trnsitions, n severl gps re forme mong the ssigne time slots (seen in Fig. 5), whih ereses the hnnel utiliztion n requires more time slots. Fig. 10 shows tht the overhe is not high, n the reursive bktrking sheuling sheme hs performne omprble to the egree-bse sheme. Although the minimum onflits heuristi my get stuk on lol optimum, it lmost hs the sme performne ompre to reursive bktrking. If the entrlize sheuling with sptil reuse is not use, the results woul be little worse, 637

9 () BFS tree Fig. 10. () BFS tree (b) DAG grph Averge number of time slots ssigne Fig. 11. Averge time ely (b) DAG grph s shown in the entrlize sheuling. The two istribute lgorithms hve the worst performne, ue to the ft tht they o not hve the globl informtion. The number of time slots in the istribute lgorithms is bout 1.5 times of the number of time slots in the reursive bktrking sheuling. Fig. 11 shows the verge time ely, n the ely inreses s the number of noes inreses in both BFS tree moel n DAG moel. The istribute sheuling with effiient ely sheme hs the best performne, for the links re sheule from the bottom to the top in the sheuling, whih is helpful to reue the ely. We summrize observtions from the simultion results s follows: (1) The entrlize n istribute lgorithms propose n reue the number of stte trnsitions, n thus hieve better energy effiieny. If the topology is tree, the noes n only strt up twie in perio. (2) Our propose istribute lgorithms n hieve performne omprble to the entrlize lgorithms. (3) The istribute sheuling with effiient ely sheme n reue the network ely. VII. CONCLUSION In this pper, we propose new interferene-free TDMA sleep sheuling problem in WSNs, lle ontiguous link sheuling. In the sheuling, sensor noe only strts up one to reeive ll the t from its neighbors, n thus n reue the energy ost n time overhe in the stte trnsition. Espeilly, if the topology is tree, the noes n only strt up twie in one sheuling perio. We lso propose entrlize n istribute lgorithms tht use time slots t most onstnt ftor of the optimum. The simultion results orroborte the theoretil nlysis, n show the effiieny of our lgorithms in terms of the number of stte trnsitions, the number of time slots ssigne, n time ely. ACKNOWLEDGMENT This work ws supporte in prt by grnts A-PH12, PolyU 5236/06E, PolyU 5232/07E, PolyU 5243/08E, HKUST 6169/07E, HKBU 2104/06E, NSF CNS , NSF CCF , Ntionl Bsi Reserh Progrm of Chin (2006CB30300), Ntionl High Tehnology Reserh n Development Progrm of Chin (2007AA01Z180), n Chin NSF grnts ( , , , ). REFERENCES [1] W. Ye, J. Heiemnn, n D. Estrin, An energy-effiient MAC protool for wireless sensor networks, in Pro. of IEEE INFOCOM, [2] T. Dm n K. Lngenoen, An ptive energy-effiient MAC protool for wireless sensor networks, in Pro. of the First ACM Conferene on Embee Networke Sensor Systems (SenSys), [3] Y. Sun, S. Du, O. Gurewitz, n D. B. Johnson, DW-MAC: low lteny, energy effiient emn-wkeup MAC protool for wireless sensor networks, in Pro. of ACM MobiHo, [4] A. Keshvrzin, H. Lee, n L. Venktrmn, Wkeup sheuling in wireless sensor networks, in Pro. of ACM MobiHo, [5] E. Arikn, Some omplexity results bout pket rio networks, IEEE Trnstions on Informtion Theory, vol. 30, no. 4, pp , [6] A. Ephremeis n T. Truong, Sheuling brosts in multihop rio networks, IEEE Trnstions on Communitions, vol. 38, no. 4, pp , [7] S. Rmnthn n E. L. Lloy, Sheuling lgorithms for multihop rio networks, IEEE/ACM Trnstions on Networking, vol. 1, no. 2, pp , [8] S. Gnhm, M. Dwne, n R. Prksh, Link sheuling in sensor networks: Distribute ege oloring revisite, in Pro. of IEEE INFO- COM, [9] W. Wng, Y. Wng, X. Y. Li, W. Z. Song, n O. Frieer, Effiient interferene-wre TDMA link sheuling for stti wireless networks, in Pro. of ACM MobiCom, [10] P. Djuki n S. Vlee, Link sheuling for minimum ely in sptil re-use TDMA, in Pro. of IEEE INFOCOM, [11] A. Wng, S. Cho, C. Soini, n A. Chnrksn, Energy effiient moultion n MAC for symmetri RF mirosensor systems, in Pro. of the 2001 Interntionl Symposium on Low Power Eletronis n Design (ISLPED), [12] R. Rmswmi n K. Prhi, Distribute sheuling of brosts in rio network, in Pro. of IEEE INFOCOM, [13] S. Krumke, M. Mrthe, n S. Rvi, Moels n pproximtion lgorithms for hnnel ssignment in rio networks, Wireless Networks, vol. 7, no. 6, pp , [14] C. Y. Ngo n V. O. K. Li, Centrlize brost sheuling in pket rio networks vi geneti-fix lgorithms, IEEE Trnstions on Communitions, vol. 51, no. 9, pp , [15] P. Gupt n P. R. Kumr, The pity of wireless networks, IEEE Trnstions on Informtion Theory, vol. 46, no. 2, pp , [16] K. Jin, J. Phye, V. N. Pmnbhn, n L. Qiu, Impt of interferene on multi-hop wireless network performne, in Pro. of ACM MobiCom, [17] J. Polstre, J. Hill, n D. Culler, Verstile low power mei ess for wireless sensor networks, in Pro. of the Seon ACM Conferene on Embee Networke Sensor Systems (SenSys), [18] R. E. Best, Phse-loke Loops: Design, Simultion n Applitions. MGrw-Hill, [19] W. Ye, F. Silv, n J. Heiemnn, Ultr-low uty yle MAC with sheule hnnel polling, in Pro. of the 4th ACM Conferene on Embee Networke Sensor Systems (SenSys), [20] S. Cui, A. J. Golsmith, n A. Bhi, Energy-onstrine moultion optimiztion, IEEE Trnstions on Wireless Communitions, vol. 4, no. 5, pp , [21] D. e Werr n A. Hertz, Conseutive olorings of grphs, Mthemtil Methos of Opertions Reserh, vol. 32, no. 1, pp , [22] M. Kuble, Intervl vertex-oloring of grph with forbien olors, Disrete Mthemtis, vol. 74, no. 1-2, pp ,