Quick convergecast in ZigBee beacon-enabled tree-based wireless sensor networks

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1 Aville online t Computer Communictions (8) Quick convergecst in ZigBee econ-enled tree-sed wireless sensor networks Meng-Shiun Pn *, Yu-Chee Tseng Deprtment of Computer Science, Ntionl Chio-Tung University, T Hsueh Rod, Hsin-Chu, Tiwn Aville online 5 Decemer 7 Astrct Convergecst is fundmentl opertion in wireless sensor networks. Existing convergecst solutions hve focused on reducing ltency nd energy consumption. However, good design should e complint to stndrds, in ddition to considering these fctors. Bsed on this oservtion, this pper defines minimum dely econ scheduling prolem for quick convergecst in ZigBee tree-sed wireless sensor networks nd proves tht this prolem is NP-complete. Our formultion is complint with the low-power design of IEEE We then propose optiml solutions for specil cses nd heuristic lgorithms for generl cses. Simultion results show tht the proposed lgorithms cn indeed chieve quick convergecst. Ó 7 Elsevier B.V. All rights reserved. Keywords: Convergecst; IEEE 8.5.4; Scheduling; Wireless sensor network; ZigBee. Introduction * Corresponding uthor. Tel.: E-mil ddresses: mspn@cs.nctu.edu.tw (M.-S. Pn), yctseng@cs. nctu.edu.tw (Y.-C. Tseng). The rpid progress of wireless communiction nd emedded micro-sensing MEMS technologies hs mde wireless sensor networks (WSNs) possile. A WSN consists of mny inexpensive wireless sensors cple of collecting, storing, processing environmentl informtion, nd communicting with neighoring nodes. Applictions of WSNs include wildlife monitoring [,4], oject trcking [6,8], nd dynmic pth finding [5,9]. Recently, severl WSN pltforms hve een developed, such s MICA [6] nd Dust Network []. For interoperility mong different systems, stndrds such s ZigBee [4] hve een developed. In the ZigBee protocol stck, physicl nd MAC lyer protocols re dopted from the IEEE stndrd []. ZigBee solves interoperility issues from the physicl lyer to the ppliction lyer. ZigBee supports three kinds of networks, nmely str, tree, nd mesh networks. A ZigBee coordintor is responsile for initilizing, mintining, nd controlling the network. A str network hs coordintor with devices directly connecting to the coordintor. For tree nd mesh networks, devices cn communicte with ech other in multihop fshion. The network is formed y one ZigBee coordintor nd multiple ZigBee routers. A device cn join network s n y the ssociting with the coordintor or router. In tree network, the coordintor nd routers cn nnounce econs. However, in mesh network, regulr econs re not llowed. Becons re n importnt mechnism to support power mngement. Therefore, the tree topology is preferred, especilly when energy sving is desirle feture. To support ZigBee econ-enled tree networks, the IEEE 8.5 WPAN Tsk Group 4 further defines revision of the IEEE [4] specifiction in 6. One of the mjor chnges is structure of superfrmes to support power mngement. On the contrry, to our understnding, power mngement is still impossile for mesh-sed ZigBee networks in the current specifiction. Therefore, we will focus on treesed, econ-enled ZigBee networks in this work. Considering tht dt gthering is mjor ppliction of WSNs, convergecst hs een investigted in severl 4-664/$ - see front mtter Ó 7 Elsevier B.V. All rights reserved. doi:.6/j.comcom.7..5

2 M.-S. Pn, Y.-C. Tseng / Computer Communictions (8) 999 works [8,9,,7,,]. With the gols of low ltency nd low energy consumption, Ref. [] shows how to connect sensors s lnced ing tree nd how to ssign CDMA codes to sensors to diminish interference mong sensors, thus chieving energy efficiency. The work [] ims to minimize the overll energy consumption under the constrint tht sensed dt should e ed within specified time. Dynmic progrmming lgorithms re proposed y ssuming tht sensors cn receive multiple pckets t the sme time. As cn e seen, oth [] nd [] re sed on quite strong ssumptions on communiction cpility of sensor nodes nd they do not fit into the Zig- Bee specifiction. In [7], the uthors propose n energyefficient nd low-ltency MAC, clled DMAC. Sensors re connected y tree nd sty in sleep stte for most of the time. When sensors chnge to ctive stte, they re first set to the receive mode nd then to the trnsmit mode. DMAC chieves low-ltency y stggering wke-up schedules of sensors t the time instnt when their children switch to the trnsmit mode. Similr to [7], Ref. [] rrnges wke-up schedule of sensors y tking trffic lods into ccount. Ech prent periodiclly rodcsts n dvertisement contining set of empty slots. Children nodes request empty slots ccording to their demnds. In [9], the uthors propose distriuted convergecst scheduling lgorithm. The sic concept is to connect nodes y spnning tree. Then the lgorithm reduces the tree to multiple lines. For ech line, the lgorithm schedules nodes trnsmission times in ottom-up mnner. Ref. [8] presents centrlized solution to convergecst. The lgorithm divides nodes into mny segments such tht the trnsmission of node in segment does not cuse interference to other trnsmissions in the sme segment. The im is to increse the degree of prllel trnsmissions to decrese ltencies. Although these results [8,9,,7] re designed for quick convergecst, the solutions re not complint to the ZigBee stndrd for the following two resons. Firstly, in these works, nodes wke/sleep times re dynmiclly chnged ccording to their schedules. However, in Zig- Bee econ-enled tree network, nodes wke/sleep times must e fixed in the wy tht ech router wkes up twice in ech cycle to receive its children s pckets nd to trnsmit pckets to its prent, respectively. The coordintor (resp., n end device) wkes up once to receive its children s pckets (resp., to trnsmit pckets to its prent). Secondly, the scheduling of [8,9,,7] is trnsmission-sed, while ours re receiving-sed. The impliction is tht the former my cuse router to e ctive multiple times per cycle. This is incomptile with the ZigBee specifiction. This pper ims t designing quick convergecst solutions for ZigBee tree-sed, econ-enled WSNs. This work is motivted y the following oservtions. First, we see tht most relted works re not complint to the ZigBee stndrd. Second, we elieve tht tree-sed topology is more suitle if power mngement is min concern in WSNs. The network scenrio is shown in Fig.. The network contins one sink (ZigBee coordintor), some ZigBee routers, nd some ZigBee. Ech ZigBee router is responsile for collecting sensed dt from end devices ssocited with it nd relying incoming dt to the sink. According to specifictions, ZigBee router cn nnounce econ to strt superfrme. Ech superfrme consists of n ctive portion followed y n inctive portion. On receiving its prent router s econ, n end device hs to wke up for n ctive portion to sense the environment nd communicte with its coordintor. However, to void collision with its neighors, router should shift its ctive portion y certin mount. Fig. shows possile lloction of ctive portions for routers A, B, C, nd D. The collected sensory dt of A in the k-th superfrme cn e sent to C in the sme superfrme. However, ecuse the ctive portion of B in the k-th superfrme ppers fter tht of C, the collected dt of B in the k-th superfrme cn only e relyed to C in the (k + )-th superfrme. The dely from B to C is lmost the length of one superfrme. The dely cn e eliminted if the ctive portion of B in the k-th superfrme ppers efore tht of C. The dely is not negligile ecuse of the low duty cycle design of IEEE For exmple, in.4 GHz PHY, with.56% duty cycle, superfrme cn e s long s s (with n ctive portion of.9 s). Clerly, for lrge-scle WSNs, the convergecst ltency could e significnt if the prolem is not crefully ddressed. The quick convergecst prolem is to schedule the econs of routers to minimize the convergecst ltency. We prove tht this prolem is NP-complete y reducing the -CNF-SAT prolem to it. We show two specil cses of this prolem where optiml solutions cn e found in polynomil time nd propose two heuristic lgorithms for generl cses. To the est of our knowledge, this is the first result tht provides convergecst solutions in ZigBee econ-enled tree networks. The rest of this pper is orgnized s follows. Section riefly introduces IEEE nd ZigBee. The quick convergecst prolem is formlly defined in Section. Section 4 presents our scheduling solutions. Simultion results re given in Section 5. Finlly, Section 6 concludes this pper.. Overview of IEEE nd ZigBee stndrds IEEE [] specifies the physicl nd dt link protocols for low-rte wireless personl re networks (LR-WPAN). In the physicl lyer, there re three frequency nds with 7 rdio chnnels. Chnnel rnges from 868. to MHz, which provides dt rte of kps. Chnnels to work from 9. to 98. MHz nd ech chnnel provides dt rte of 4 kps. Chnnels to 6 re locted from.4 to.485 GHz, ech with dt rte of 5 kps. IEEE devices re expected to hve limited power, ut need to operte for longer period of time. Therefore, energy conservtion is criticl issue. Devices re clssified s full function devices (FFDs) nd reduced

3 M.-S. Pn, Y.-C. Tseng / Computer Communictions (8) 999 Sink D B C B C A A ZigBee router (FFD) ZigBee end device (RFD) Interference neighor dt from dt from Schedule of A dt from Schedule of B dt from dt from Schedule of C dt from dt from Schedule of D k-th superfrme to sink (k+)-th superfrme Fig.. An exmple of convergecst in ZigBee tree-sed network. function devices (RFDs). IEEE supports str nd peer-to-peer topologies. In ech PAN, one device is designted s the coordintor, which is responsile for mintining the network. A FFD hs the cpility of serving s coordintor or ssociting with n existing coordintor/ router nd ecoming router. A RFD cn only ssocite with coordintor/router nd cn not hve children. The ZigBee coordintor defines the superfrme structure of ZigBee network. As shown in Fig. (), the structure of superfrmes is controlled y two prmeters: econ order (BO) nd superfrme order (SO), which decide the lengths of superfrme nd its ctive potion, respectively. For econ-enled network, the setting of BO nd SO should stisfy the reltionship 6 SO 6 BO 6 4. (A non-econ-enled network should set BO ¼ SO ¼ 5 to indicte tht superfrmes do not exist.) Ech ctive portion consists of 6 equl-length slots, which cn e further prtitioned into contention ccess period () nd contention free period (CFP). The my contin the first i slots, nd the CFP contins the rest of the 6 i slots, where 6 i 6 6. Slotted CSMA/CA is used in. FFDs which require fixed trnsmission rtes cn sk for gurntee time slots (GTSs) from the coordintor. A CFP cn support multiple GTSs, nd ech GTS my contin multiple slots. Note tht only the coordintor cn llocte GTSs. After the ctive portion, devices cn go to sleep to sve energy. In econ-enled str network, device only needs to e ctive for ðbo SOÞ portion of the time. Chnging the vlue of ðbo SOÞ llows us to djust the on-duty time of devices. However, for econ-enled tree network, routers hve to choose different times to strt their ctive portions to void collision. Once the vlue of ðbo SOÞ is decided, ech router cn choose from BO SO slots s its ctive portion. In the revised version of IEEE [4], router cn select one ctive portion s its outgoing superfrme, nd sed on the ctive portion selected y its prent, the ctive portion is clled its incoming superfrme (s shown in Fig. ()). In n outgoing/incoming superfrme, router is expected to trnsmit/receive econ to/from its child routers/prent router. When choosing slot, neighoring routers ctive portions (i.e., outgoing superfrmes) should e shifted wy from ech other to void interference. This work is motivted y the oservtion tht the specifiction does not clerly define how to choose the loctions of routers ctive portions such tht the convergecst ltency cn e reduced. In our work, we consider two kinds of interference etween routers. Two routers hve direct interference if they cn her ech others econs. Two routers hve indirect interference if they hve t lest one common neighor. Both interferences should e voided when choosing routers ctive portions. Tle lists possile choices of ðbo SOÞ comintions.

4 M.-S. Pn, Y.-C. Tseng / Computer Communictions (8) 999 Becon CFP Becon GTS GTS GTS Inctive SD = BseSuperfrmeDurtion SO symols (Active) BI = BseSuperfrmeDurtion BO symols Received Becon Trnsmitted Becon Received Becon Inctive Inctive SD = BseSuperfrmeDurtion SO symols (Incoming superfrme) Strt Time >SD SD = BseSuperfrmeDurtion SO symols (Outgoing superfrme) BI = BseSuperfrmeDurtion BO symols Fig.. IEEE superfrme structure. Tle Reltionship of BO SO, duty cycle, nd the numer of ctive portions in superfrme BO SO P9 Duty cycle (%) Numer of ctive portions (slots) P5. The minimum dely econ scheduling (MDBS) prolem This section formlly defines the convergecst prolem in ZigBee networks. Given ZigBee network, we model it y grph G ¼ðV ; EÞ, where V contins ll routers nd the coordintor nd E contins ll symmetric communiction links etween nodes in V. The coordintor lso serves s the sink of the network. End devices cn only ssocite with routers, ut re not included in V. From G, we cn construct n interference grph G I ¼ðV ; E I Þ, where edge ði; jþ E I if there re direct/indirect interferences etween i nd j. There is duty cycle requirement for this network. From nd Tle, we cn determine the most pproprite vlue of BO SO. We denote y k ¼ BO SO the numer of ctive portions (or slots) per econ intervl. The econ scheduling prolem is to find slot ssignment sðiþ for ech router i V, where sðiþ is n integer nd sðiþ ½; k Š, such tht router i s ctive portion is in slot sðiþ nd sðiþ 6¼ sðjþ if ði; jþ E I. Here, the slot ssignment mens the position of the outgoing superfrme of ech router (the position of the incoming superfrme, s clrified erlier, is determined y the prent of the router). Motivted y Brook s theorem [], which proves tht n colors re sufficient to color ny grph with mximum degree of n, we would ssume tht k P D I, where D I is the mximum degree of G I. Given slot ssignment for G, the ltency from node i to node j, where ði; jþ E, is the numer of slots, denoted y d ij, tht node i hs to wit to rely its collected sensory dt to node j, i.e., d ij ¼ðsðjÞ sðiþþmod k: ðþ Note tht the ltency from node i to node j ðd ij Þ my not y equl to the ltency from node j to node i ðd ji Þ. Therefore, we cn convert G into weighted directed grph G D ¼ðV ; E D Þ such tht ech ði; jþ E is trnslted into two directed edges ði; jþ nd ðj; iþ such tht wðði; jþþ ¼ d ij nd wððj; iþþ ¼ d ji. The ltency for ech i V to the sink is the sum of ltencies of the links on the shortest pth from i to the sink in G D. The ltency of the convergecst, denoted s LðGÞ, is the mximum of ll nodes ltencies. Definition. Given G ¼ðV ; EÞ, G s interference grph G I ¼ðV ; E I Þ, nd k ville slots, the minimum dely econ scheduling (MDBS) prolem is to find n interference-free slot ssignment sðiþ for ech i V such tht the convergecst ltency LðGÞ is minimized. To prove tht the MDBS prolem is NP-complete, we define decision prolem s follows. Definition. Given G ¼ðV ; EÞ, G s interference grph G I ¼ðV ; E I Þ, k ville slots, nd dely constrint d,

5 M.-S. Pn, Y.-C. Tseng / Computer Communictions (8) 999 the ounded dely econ scheduling (BDBS) prolem is to decide if there exists n interference-free slot ssignment sðiþ for ech i V such tht the convergecst ltency LðGÞ 6 d. Theorem. The BDBS prolem is NP-complete. Proof. First, given slot ssignments for nodes in V, we cn find the ltency of ech i V y running shortest pth lgorithm on G D. We cn then check if LðGÞ 6 d. Clerly, this tkes polynomil time. We then prove tht the BDBS prolem is NP-hrd y reducing the conjunctive norml form stisfiility (- CNF-SAT) prolem to specil cse of the BDBS prolem in polynomil time. Given ny -CNF formul C, we will construct the corresponding G nd G I. Then we show tht C is stisfile if nd only if there is slot ssignment for ech i V using no more thn k ¼ slots such tht LðGÞ 6 4 slots. Let C ¼ C ^ C ^^C m, where cluse C j ¼ x j; _x j; _ x j;, 6 j 6 m, x j;i fx ; X ;...; X n g, nd X i fx i ; x i g, where x i is inry vrile, 6 i 6 n. We first construct G from C s follows:. For ech cluse C j, j ¼ ; ;...; m, dd vertex C j in G.. For ech literl X i, i ¼ ; ;...; n, dd four vertices x i, x i, x i, nd x i in G.. Add vertex t s the sink of G. 4. Add edges ðt; x i Þ nd ðt; x i Þ to G, for i ¼ ; ;...; n. 5. Add edges ðx i ; x i Þ nd ðx i ; x i Þ to G, for i ¼ ; ;...; n. 6. For ech i ¼ ; ;...; n nd ech j ¼ ; ;...; m, ddn edge ðc j ; x i Þ (resp., ðc j ; x i Þ)toG if x i (resp., x i ) ppers in C j. Then we construct G I s follows.. Add ll vertices nd edges in G into G I.. Add edges ðx i ; x i Þ nd ðx i ; x i Þ to G I, for i ¼ ; ;...; n.. Add edges ðc j ; x i Þ nd ðc j ; x i Þ to G I, for i ¼ ; ;...; n nd j ¼ ; ;...; m. Then we uild one-to-one mpping from ech truth ssignment of C to slot ssignment of G. We estlish the following mpping:. Set sðtþ ¼.. Set sðc j Þ¼, j ¼ ; ;...; m.. Set sðx i Þ¼ nd sðx i Þ¼, i ¼ ; ;...; n, ifx i is true; otherwise, set sðx i Þ¼ nd sðx i Þ¼. 4. Set sðx i Þ¼ nd sðx i Þ¼, i ¼ ; ;...; n, ifx i is true; otherwise, set sðx i Þ¼ nd sðx i Þ¼. The ove reduction cn e computed in polynomil time. By the ove reduction, vertices x i or x i, i ¼ ; ;...; n, tht re ssigned to slot (resp., slot ) will hve ltency of (resp., 4) nd vertices x i or x i, i ¼ ; ;...; n, tht re ssigned to slot (resp., slot ) will hve ltency of (resp., ). Hence, for those x x C C C vertices x i, x i, x i, nd x i, i ¼ ; ;...; n, the longest ltency will e 4. To prove the if prt, we need to show tht if C is stisfile, there is slot ssignment such tht k ¼ nd LðGÞ 6 4. Since C stisfile, there must exist n ssignment such tht ech cluse C j, j ¼ ; ;...; m, is true. If cluse C j is true, t lest one vrile in C j is true. According to the reduction, C j cn lwys find n edge ðc j ; x i Þ or ðc j ; x i Þ with wððc j ; x i ÞÞ ¼ orwððc j ; x i ÞÞ ¼, where i ¼ ; ;...; n. Thus, when C is stisfile, the ing ltency for ech cluse is. This chieves LðGÞ ¼4. For the only if prt, if ech vertex C j, j ¼ ; ;...; m, cn find t lest n edge with weight to one of x i nd x i, for i ¼ ; ;...; n, to chieve ltency of, it must e tht ech cluse hs t lest one vrile to e true. So formul C is stisfile. Otherwise, the ltency of C j, j ¼ ; ;...; m, will e 6. h For exmple, given C ¼ðx _ x _ x Þ^ðx _ x _ x Þ^ ðx _ x _ x Þ, Fig. shows the corresponding G. The truth ssignment ðx ; x ; x Þ¼ðT ; F ; T Þ mkes C stisfile. According to the reduction nd the mpping in the ove proof, we cn otin the network G nd its slot ssignment s shown in Fig. such tht LðGÞ ¼4. 4. Algorithms for the MDBS prolem 4.. Optiml solutions for specil cses x x x x x Fig.. An exmple of reduction from the -CNF-SAT to the BDBS prolem. Optiml solutions cn e found for the MDBS prolem in polynomil time for regulr liner networks nd regulr ring networks, s illustrted in Fig. 4. In such networks, ech vertex is connected to one or two djcent vertices nd hs n interference reltion with ech neighor within h hops from it, where h P. In regulr liner network, we ssume tht the sink t is t one end of the network. Clerly, the mximum degree of G I is h. We will show tht n optiml solution cn e found if the numer of slots k P h þ. The slot ssignment cn e done in ottomup mnner. The ottom node is ssigned to slot. Then, t x x x x x

6 4 M.-S. Pn, Y.-C. Tseng / Computer Communictions (8) 999 t t r t r size: size: l left group right group r l r left group right group Fig. 4. Exmples of optiml slot ssignments for regulr liner nd ring networks ðh ¼ Þ. Dotted lines men interference reltions. for ech vertex v, sðvþ ¼ðkþÞmodk, where k is the slot ssigned to v s child. Theorem. For regulr liner network, if k P h þ, the ove slot ssignment chieves ltency of j V j, which is optiml. Proof. Clerly, the slot ssignment is interference-free. Also the ltency of j V j is clerly the lower ound. h For regulr ring network, we first prtition vertices excluding t into left nd right groups s illustrted in Fig. 4() such tht the left group consists of the sink node t nd c other nodes counting counter clockwise from t, nd the right group consists of those d e nodes counting clockwise from t. Now we consider the ring s spnning tree with t s the root nd left nd right groups s two liner pths. Assuming tht c P h nd k P h, the slot ssignment works s follows:. The ottom node in the left group is ssigned to slot.. All other nodes in the left group re ssigned with slots in ottom-up mnner. For ech node i in the left group, we let sðiþ ¼ðj þ Þmod k, where j is the slot of i s child.. Nodes in the right group re ssigned with slots in topdown mnner. For ech node i in the right group, we let sðiþ ¼ðj cþmodk, where j is the slot ssigned to i s prent nd c is the smllest constnt ( 6 c 6 k) tht ensures tht sðiþ is not used y ny of its interference neighors tht hve een ssigned with slots. It is not hrd to prove the slot ssignment is interference-free ecuse nodes receives slots sequentilly nd we hve voided using the sme slots mong interfering neighors. Although this is greedy pproch, we show tht c is equl to in step in most of the cses except when two specil nodes re visited. This gives n symptoticlly optiml lgorithm, s proved in the following theorem. Theorem. For regulr ring network, ssuming tht k P h nd c P h, the ove slot ssignment chieves ltency LðGÞ ¼ cþh, which is optiml within fctor of.5. Proof. We first identify three nodes on the ring (refer to Fig. 4()): l : the ottom node in the left group. r : the first node in the right group. r : the node tht is h hops from l clockwise. counting counter The ltency of ech node cn e nlyzed s follows. The prent of node x is denoted y prðxþ. A. For ech node i in the left group except the sink t, the ltency from i to prðiþ is. A. The ltency from r to t is h. A. For ech node i next to r in the right group ut efore r (counting clockwise), the ltency from i to prðiþ is. A4. The ltency from r to prðr Þ is if the ring size is even; otherwise, the ltency is. A5. For ech node i in the right group tht is descendnt of r, the ltency from i to prðiþ is. It is not hrd to prove tht A, A, nd A re true. To see A4 nd A5, we mke the following oservtions. The function pr i ðxþ is to pply i times the prðþ function on node x. Note tht pr ðxþ mens x itself.

7 M.-S. Pn, Y.-C. Tseng / Computer Communictions (8) O. When the ring size is even, the equlity sðpr i ðl ÞÞ ¼ sðpr i ðr ÞÞ holds for i ¼ ; ;...; c h. More specificlly, this mens tht (i) l nd prðr Þ will receive the sme slot, (ii) prðl Þ nd pr ðr Þ will receive the sme slot, etc. This cn e proved y induction y showing tht the i-th descendnt of t in the right group will e ssigned the sme slot s the ðh þ i Þ-th descendnt of t in the left group (the induction cn go in top-down mnner). This property implies tht when ssigning slot to r in step, c ¼ in cse tht the ring size is even. Further, r nd its descendnts will e sequentilly ssigned to slots k, k,..., k h, which implies tht c ¼ when doing the ssignments in step. So properties A4 nd A5 hold for the cse of n even ring. O. When the ring size is odd, the equlity sðpr i ðl ÞÞ ¼ sðpr i ðr ÞÞ holds for i ¼ ; ;...; c h. This mens tht (i) prðl Þ nd prðr Þ will receive the sme slot, nd (ii) pr ðl Þ nd pr ðr Þ will receive the sme slot, etc. Agin, this cn e proved y induction s in O. This property implies tht c ¼ when ssigning slot to r in step, nd c ¼ when ssigning slots to descendnts of r.so properties A4 nd A5 hold for the cse of n odd ring. The equlity of slot ssignments pointed out in O nd O is illustrted in Fig. 4() y those numers in gry nodes. In summry, the ltency of the left group is c. When the ring size is even, the ltency of the right group is the numer of nodes in this group, jv j plus the extr ltency h incurred t r. So LðGÞ ¼ jv j j þ h ¼jV cþh. When the ring size is odd, the ltency of right group is the numer of nodes in this group,, plus the extr ltency h incurred t r nd the extr ltency incurred t r.so LðGÞ ¼ cþh. A lower ound on the ltency of this prolem is the mximum numer of nodes in ech group excluding t. Applying c s lower ound nd using the fct tht c P h, LðGÞ will e smller thn :5 c, which implies the lgorithm is optiml within fctor of.5. Note tht the condition c P h is to gurntee tht t will not locte within h hops from r. Otherwise, the oservtion O will not hold. h 4.. A centrlized tree-sed ssignment scheme Given G ¼ðV ; EÞ, G I ¼ðV ; E I Þ, nd k, we propose centrlized slot ssignment heuristic lgorithm. Our lgorithm is composed of the following three phses: phse. From G, we first construct BFS tree T rooted t sink t., phse. We trverse vertices of T in ottom-up mnner. For these vertices in depth d, we first sort them ccording to their degrees in G I in descending order. Then we sequentilly trverse these vertices in tht order. For ech vertex v in depth d visited, we compute temporry slot numer tðvþ for v s follows.. If v is lef node, we set tðvþ to the miniml non-negtive integer l such tht for ech vertex u tht hs een visited nd ðu; vþ E I,(tðuÞmodk) 6¼ l.. If v is n in-tree node, let m e the mximum of the numers tht hve een ssigned to v s children, i.e., m ¼ mxftðchildðvþþg, where childðvþ is the set of v s children. We then set tðvþ to the miniml non-negtive integer l > m such tht for ech vertex u tht hs een visited nd ðu; vþ E I,(tðuÞmodk) 6¼ ðlmodkþ. After every vertex v is visited, we mke the ssignment sðvþ ¼ tðvþ mod k. phse. In this phse, vertices re trversed sequentilly from t in top-down mnner. When ech vertex v is visited, we try to greedily find new slot l such tht (sðprðvþþ l)modk < ðsðprðvþþ sðvþþ mod k, such tht l 6¼ sðuþ for ech ðu; vþ E I, if possile. Then we ressign sðvþ ¼l. Note tht in phse, node with higher degree mens tht it hs more interference neighors, implying tht it hs less slots to use. Therefore, it hs to e ssigned to slot erlier. Also note tht, the numer tðvþ is not modulus numer. However, in step of phse, we did check tht if tðvþ is converted to slot numer, no interference will occur. Intuitively, this is temporry slot ssignment tht will incur the lest ltency to v s children. At the end, tðvþ is converted to slot ssignment sðvþ. Phse is greedy pproch to further reduce the ltency of routers. For exmple, Fig. 5() shows the slot ssignment fter phse. Fig. 5() indictes tht B, C, nd D cn find nother slots nd their ltencies re decresed. This phse cn reduce LðGÞ in some cses. The computtionl complexity of this lgorithm is nlyzed elow. In phse, the complexity of constructing BFS tree is Oðj V jþje jþ. In phse, the cost of sorting is t most Oðj V j Þ nd the computtionl cost to compute tðvþ for ech vertex v is ounded y OðkD I Þ, where D I is the degree of G I. So the time complexity of phse is Oðj V j þ kd I j V jþ. Phse performs similr procedure s phse, so its time complexity is lso OðkD I j V jþ. Overll, the time complexity is Oðj V j þ kd I j V jþ. 4.. A distriuted ssignment scheme In this section, we propose distriuted slot ssignment lgorithm. Ech node hs to compute its direct s well s indirect interference neighors in distriuted mnner. To chieve this, we will refer to the heterogeneity pproch

8 6 M.-S. Pn, Y.-C. Tseng / Computer Communictions (8) A B C t D in [], which dopts power control to chieve this gol. Assuming routers defult trnsmission rnge is r, interference neighors must locte within rnge r. From time-totime, ech router will oost its trnsmission power to doule its defult trnsmission rnge nd send HELLO pckets to its neighor routers. Ech HELLO pcket further contins sender s () depth, () the loction of outgoing superfrme (i.e., slot), nd () numer of interference neighors. Note tht ll other pckets re trnsmitted y the defult power level. When ooting up, ech router will rodcst HELLO pckets climing tht its depth nd slot re NULL. After joining the network nd choosing slot, the HELLO pckets will crry the node s depth nd slot informtion. The lgorithm is triggered y the sink t setting sðtþ ¼k nd then rodcsting its econ. A router v 6¼ t tht receives econ will decide its slot s follows.. Node v sends n ssocition request to the econ sender.. If v fils to ssocite with the econ sender, it stops the procedure nd wits for other econs.. If v successfully ssocites with prent node prðvþ, it computes the smllest positive integer l such tht ðsðprðvþþ lþmodk 6¼ sðuþ for ll ðu; vþ E I nd sðuþ 6¼ NULL. Then v chooses sðvþ ¼ðsðprðvÞÞ lþ modk s its slot. 4. Then, v rodcsts HELLOs including its slot ssignment sðvþ for time period t wit. If it finds tht sðvþ ¼sðuÞ for ny ðu; vþ E I, v hs to chnge to new slot if one of the following rules is stisfied nd goes ck to step. () Node u hs more interference neighors thn v. () Node u nd v hve the sme numer of interference neighors ut the depth of u is lower thn v, i.e., u is closer to the sink thn v. (c) Node u nd v hve the sme numer of interference neighors nd they re t the sme depth ut the u s ID is smller thn v s. The depth of node is the length of the tree pth from the root to the node. The root node is t depth zero. E E I A B C D Fig. 5. () Slot ssignment fter phse. () Slot compcting y phse. t 5. After t wit, v cn finlize its slot selection nd rodcst its econs. In this distriuted lgorithm, slots re ssigned to routers, idelly, in top-down mnner. However, due to trnsmission ltency, some routers t lower levels my find slots erlier thn those t higher levels. Also note tht the time t wit is to void possile collision on slot ssignments due to pcket loss. 5. Simultion results This section presents our simultion results. We first ssume tht the size of sensory dt is negligile nd tht ll routers generte s t the sme time, nd compre the performnces of different convergecst lgorithms. Then we simulte more relistic scenrios where the size of sensory dt is not negligile nd routers need to generte s periodiclly or pssively driven y events rndomly ppering in certin regions in the sensing field. More specificlly, sensors generte s ccording to certin ppliction specifictions. Devices ll run ZigBee nd IEEE protocols to communicte with ech other. Routers cn ggregte child sensors s nd to their prents directly. Ech router hs fix-size uffer. When router s uffer overflows, this router will not ccept further incoming frmes. We lso mesure the goodput of the network, which is defined s the rtio of sensors s successfully received y the sink. Some prmeters used in our simultion re listed in Tle. 5.. Comprison of different convergecst lgorithms We compre the proposed slot ssignment lgorithms ginst rndom slot ssignment (denoted y RAN) scheme nd greedy slot ssignment (denoted y GDY) scheme. In RAN, the slot ssignment strts from the sink nd ech router, fter ssociting with prent router, simply chooses ny slot which hs not een used y ny of its interference neighors. In GDY, routers re given sequence numer in top-down mnner. The sink sets its slot to k. Then the slot ssignment continues in sequence. For node i, it will try to find slot sðiþ ¼sðjÞ l modk, where j is the predecessor of i nd l is the smllest integer letting sðiþ is the slot which does not ssign to ny of i s interference neighors. In the simultions, routers re rndomly distriuted in circulr region of rdius r nd sink is plced in the center. Our centrlized tree-sed scheme nd distriuted slot ssignment scheme re denoted s CTB nd DSA, respectively. We compre the ltency LðGÞ (in terms of slots). Fig. 6 shows some slot ssignment results of CTB nd DSA when r = 5 m nd k ¼ 64. Devices re rndomly distriuted. The trnsmission rnge of routers is set to m. In this cse, CTB performs etter thn DSA.

9 M.-S. Pn, Y.-C. Tseng / Computer Communictions (8) Tle Simultion prmeters Prmeter Vlue Length of frme s heder nd til 8 Bytes Length of sensor s 6 Bytes Becon length 8 Bytes Mximum length of frme 7 Bytes Bit rte 5 kps Symol rte 6.5 k symols/s BseSuperfrmeDurtion 96 symols UnitBckoffPeriod symols CCATime 8 symols McMinBE MxBE 5 McMxCSMABckoffs 4 Mximum numer of retrnsmissions Next, we oserve the impct of different r, C R (numer of routers), nd T R (trnsmission distnce). Fig. 7() shows the impct of r when k ¼ 64, T R ¼ 5 m, nd C R ¼ ðr=þ. CTB performs the est. DSA performs slightly worse thn CTB, ut still significntly outperforms RAN nd GDY. It cn e seen tht RAN nd GRY could result in very long convergecst ltency. Both CTB nd DSA re quite insensitive to the network size. But this is not the cse for RAN nd GDY. Fig. 7() shows the impct of T R when C R ¼, r ¼ m, nd k ¼ 64. Since lrger trnsmission rnge implies higher interference mong routers, the ltencies of CTB nd DSA will increse linerly s T R increses. The ltency of RAN lso increses when T R ¼ 7 m ecuse of the incresed interference. After T R P m, the ltency of RAN decreses ecuse tht the network dimeter is reduced. Bsiclly, GDY ehves the sme s CTB nd DSA. But when the trnsmission rnge is lrger, the ltency slightly ecomes smll. Fig. 7(c) shows the impct of C R when r ¼ m, T R ¼ m, nd k ¼ 8. As lrger C R mens higher network density nd thus more interference, the ltencies of CTB nd DSA increse s C R increses. Since the network dimeter is ounded, the ltency of RAN is lso ounded. GDY is sensitive to the numer of routers when there re less routers. This is ecuse tht ech router cn own slot nd the ltency increses proportionlly to the numer of routers. With r ¼ m, C R ¼, nd T R ¼ m, Fig. 7(d) shows the impct of routers duty cycle. Note tht lower duty cycle mens lrger numer of ville slots. Interestingly, we see tht the ltencies of CTB, DSA, nd GDY re independent of the numer of slots. Contrrily, with rndom ssignment, RAN even incurs higher ltency s there re more freedom in slot selection. Next, we ssume tht sensors re instructed to their dt in periodiclly mnner. We set r ¼ m, T R ¼ m, nd C R ¼ with 6 rndomly plced sensors ssocited to these routers, nd we further restrict router cn ccept t most sensors. BO SO is fixed to six, so k ¼ BO SO ¼ 64. Since the erlier simultions show tht CTB nd DSA perform quite close, we will use only CTB to ssign routers slots. Sensors re required to generte every 5.66 s (the length of one econ intervl when BO ¼ 4). We set the uffer size of ech router is KB. We llocte two mini-slots for ech child router of the sink s the GTS slot. Since ðbo SOÞ is fixed, smll BO implies smller slot size (nd thus smller unit size of LðGÞ). So, smller slot size seemingly implies higher contention mong sensors if they ll intend to to their prents simultneously. In fct, smller BO does not hurt the overll ing times of sensors if we cn properly divide sensors into groups. For exmple, in Fig. 8, when BO ¼ 4, ll sensors of router cn in every superfrme. When BO ¼, if we divide sensors into two groups, then they cn lterntely in odd nd even superfrmes. Similrly, when BO ¼, four groups of sensors cn lterntely. Since the length of superfrmes re reduced proportionlly, the intervls of sensors ctully remin the sme in these cses. In the following experiments, we groups sensors ccording to their prents IDs. A sensor elongs to group m if the modulus of its prent s ID is m. Fig. 9 shows the theoreticl nd ctul ltencies under different BOs. Note tht my e delyed due to uffer constrint. As cn e seen, the ctul ltency does not lwys fvor smller BO. Our results show tht BO ¼ performs etter. Fig. 9() shows the goodput of sensory s, chnnel utiliztion t the sink, nd the numer of dropped frmes t the sink. When BO ¼ 4, lthough there is no frmes eing dropped t the sink, the goodput is still low. This is ecuse lot of collisions hppen inside the network, cusing mny sensory s eing dropped t intermedite levels ( frme is dropped fter exceeding its retrnsmission limit). Fig. shows log of the numers of frmes received y sink s child router when BO ¼ 4. We cn see tht more thn hlf of the ctive portion is wsted. Overll, BO ¼ produces the est goodput nd shorter ltency. Some previous works cn e lso integrted in this periodicl ing scenrio, such s the dptive GTS lloction mechnism in [] nd the ggregtion lgorithms for WSNs in [7,]. Fig. shows n experiment tht routers cn compress s from sensors with rte cr when BO ¼. If router receives n s nd ech s size is 6 Bytes (s in Tle ), it cn compress the size to 6 n ð crþ. The ltencies decrese when the cr ecomes lrger. By compressing the dt, 5.. Periodicl ing scenrios Currently, there re some pltforms which re equipped with lrger RAMs. For exmple, Jennic JN5 [5] hs 96 KB RAM nd CC4DBK [] hs KB RAM. There re sixteen mini-slots per ctive portion (slot).

10 8 M.-S. Pn, Y.-C. Tseng / Computer Communictions (8) 999 k = 64 k = 64 L(G)=9 L(G)= Fig. 6. Slot ssignment exmples y CTB nd DSA. Averge L(G) 5 5 CTB DSA RAN GDY Averge L(G) 5 5 CTB DSA RAN GDY Network rdius (m) Trnsmission rnge (m) c 7 d 5 Averge L(G) CTB DSA RAN GDY Averge L(G) CTB DSA RAN GDY Numer of ZigBee routers Network duty cycle (%).98 Fig. 7. Comprison of ltencies under different configurtions. the goodput cn up to 98% nd the cn rrive to the sink more quickly. 5.. Event-driven ing scenrios In the following, we ssume tht sensors ing ctivities re triggered y events occurred t rndom loctions in the network with rte k. The sensing rnge of ech sensors is m nd ech event is disk of rdius of 5 m. A sensor cn detect n event if its sensing rnge overlps with the disk of tht event. Ech router hs n KB uffer. When sensor detects n event, it only tries to tht event once. All other settings re the sme s those in Section 5.. Fig. shows the simultion results when k = /5s, / 5s, nd /s. From Fig. (), we cn oserve tht when

11 M.-S. Pn, Y.-C. Tseng / Computer Communictions (8) BO=4 # of groups = econ n th superfrme ll sensors econ (n+)th superfrme ll sensors econ (n+)th superfrme... BO= # of groups = econ n th superfrme group econ (n+)th superfrme econ (n+)th superfrme econ (n+)th superfrme group group group econ (n+4)th superfrme n th (n+)th (n+)th (n+)th (n+4)th (n+5)th (n+6)th (n+7)th (n+8)th superfrme superfrme superfrme superfrme superfrme superfrme superfrme superfrme superfrme BO= # of groups = 4 group group group group group group group group econ econ econ econ econ econ econ econ econ Fig. 8. An exmple of scheduling under different vlues of BO L(G) x slot-size (in seconds) Theoreticl Actul Goodput or chnnel utiliztion (%) Goodput Chnnel utiliztion The numer of dropped frmes Numer of dropped frmes 8 9 BO BO 4 Fig. 9. Simultions considering uffer limittion nd contention effects: () Theoreticl vs. ctul ltencies nd () goodput, chnnel utiliztion, nd numer of dropped frmes. Report (Becon) intervl: 5.66 s Report (Becon) intervl: 5.66 s Numer of frmes received Active portion:.9 s Active portion:.9 s Active portion:.9 s time (s) Fig.. A log of the numer of frmes received y sink s child router when BO ¼ 4. BO is smll, the ltency cn not chieve to the theoreticl vlue. This is ecuse tht n ctive portion is too smll to ccommodte ll s from sensors, thus lengthening the ltency. When BO ecomes lrger, the theoreticl nd ctul curves would meet. However, the good put will degrde, s shown in Fig. (). This is ecuse s re likely to e dropped due to uffer overflow. How to determine proper BO, which cn contin most of the s nd gurntee low ltency, is n importnt design issue for such scenrios. 6. Conclusions In this pper, we hve defined new minimum dely econ scheduling (MDBS) prolem for convergecst with the restrictions tht the econ scheduling must e complint to the ZigBee stndrd. We prove the MDBS prolem is NP-complete nd propose optiml solutions for specil cses nd two heuristic lgorithms for generl cses. Simultion results indicte the performnce of our heuristic lgorithms decrese only when the numer of interference

12 M.-S. Pn, Y.-C. Tseng / Computer Communictions (8) 999 L(G) x slot-size (in seconds) Theoreticl Actul Goodput or chnnel utiliztion (%) Goodput Chnnel utiliztion Numer of dropped frmes.5.5 Numer of dropped frmes 5 7 Compression rte (%) Compression rte (%) 9 Fig.. Simultions considering dt compression: () Theoreticl vs. ctul ltencies nd () goodput, chnnel utiliztion, nd numer of dropped frmes. L(G) x slot-size (in second) Theoreticl Actul(λ=/5s) Actul(λ=/5s) Actul(λ=/s) BO Goodput (%) λ=/5s λ=/5s λ=/s BO Fig.. Simultion results of event-driven scenrios: () theoreticl vs. ctul ltencies nd () goodput. neighors is incresed. Compred to the rndom slot ssignment nd greedy slot ssignment scheme, our heuristic lgorithms cn effectively schedule the ZigBee routers econ times to chieve quick convergecst. In the future, it deserves to consider extending this work to n synchronous sleep scheduling to support energy-efficient convergecst in ZigBee mesh networks. Acknowledgements Y.-C. Tseng s reserch is co-sponsored y Tiwn MoE ATU Progrm, y NSC Grnts 9-75-E-7--PAE, ET, 95--E-9-58-MY, 95-- E-9-6-MY, 95-9-E-9-7, 95-8-E-9-9, nd 94-9-E-7-9, y Reltek Semiconductor Corp., y MOEA under Grnt No. 94-EC-7-A-4-S-44, y ITRI, Tiwn, y Microsoft Corp., nd y Intel Corp. References [] Chipcon CC4DBK. Aville from: < [] Dust network Inc. Aville from: < [] Design nd construction of wildfire instrumenttion system using networked sensors. Aville from: < [4] Hitt monitoring on gret duck islnd. Aville from: < [5] Jennic JN5. Aville from: < [6] Motes, smrt dust sensors, wireless sensor networks. Aville from: < [7] S.-J. Bek, G. de Vecin, X. Su, Minimizing energy consumption in lrge-scle sensor networks through distriuted dt compression nd hierrchicl ggregtion, IEEE Journl on Selected Ares in Communictions (6) (4) 4. [8] H. Choi, J. Wng, E.A. Hughes, Scheduling on sensor hyrid network, in: Proceedings of IEEE Interntionl Conference on Computer Communictions nd Networks (ICCCN), Sn Diego, USA, 5. [9] S. Gndhm, Y. Zhng, Q. Hung, Distriuted miniml time convergecst scheduling in wireless sensor networks, in: Proceedings of IEEE Interntionl Conference on Distriuted Computing Systems (ICDCS), Liso, Portugl, 6. [] D. Gnesn, B. Greenstein, D. Perelyuskiy, D. Estrin, J. Heidemnn. An evlution of multiresolution storge for sensor networks, in: Proceedings of ACM Interntionl Conference on Emedded Networked Sensor Systems (SenSys), Los Angeles, USA,. [] B. Hohlt, L. Doherty, E. Brewer, Flexile power scheduling for sensor networks, in: Proceedings of ACM/IEEE Interntionl Conference on Informtion Processing in Sensor Networks (IPSN), Berkeley, USA, 4.

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