Real-Time Scheduling for Event-Triggered and Time-Triggered Flows in Industrial Wireless Sensor-Actuator Networks

Transcription

1 Ral-Tm Schdulng or Evn-Trggrd and Tm-Trggrd Flows n Indusral Wrlss Snsor-Acuaor Nworks X Jn, Abusad Saullah, Chnang Lu, and Png Zng Absrac Wrlss snsor-acuaor nworks nabl an cn and cos-cv approach or ndusral snsng and conrol applcaons. To sas h ral-m rqurmn o such applcaons, hs nworks adop cnralzd schdulng algorhms o opmz h ral-m prormanc basd on global normaon. Exsng cnralzd algorhms mosl ocus on schdulng m-rggrd lows. Th canno cvl schdul vn-rggrd lows du o h dnamcs and unprdcabl o vns. In hs papr, w propos hr undamnal cnralzd algorhms ha rsrv as w rsourcs as possbl or vnrggrd lows such ha h ral-m prormanc o mrggrd lows s no acd. W hn analz hr advanags and dsadvanags. Basd on h analss, w combn hr advanags, ncludng hos n rms o hr rsourc rqurmns, no a cnralzd algorhm. Fnall, w conduc xnsv smulaons basd on boh ral opologs and random opologs. Th smulaons ndca ha or mos s cass h schdulabl o our combnd algorhm s clos o opmal soluons. I. INTRODUCTION Wrlss snsor-acuaor nworks (WSANs) nabl an cn and cos-cv approach or ndusral snsng and conrol applcaons. A larg class o ndusral conrol applcaons mpos ral-m rqurmns bwn snsng and acuaon. For xampl, n h cmn producon procss, h mpraur daa o roar klns mus b dlvrd o h conrol room bor hr dadlns. I a daa pack wh a hgh-mpraur warnng msss s dadln, ml conrol o mpraur als whch ma caus kln xploson. Spccall, xcssv or unboundd lanc ma lad o hghl unsabl ssms, makng ral-m prormanc a crcal rqurmn n ndusral WSANs. Snc Tm Dvson Mulpl Accss (TDMA)-basd approachs can provd prdcabl lanc, h ar prrrd ovr CSMA/-basd ons or ransmsson schdulng n ndusral WSANs []. TDMA-basd approachs ar usuall adopd hrough a cnralzd algorhm ha can xplo h global normaon and ma opmz h ral-m prormanc. Currnl, mos WSANs wh srngn ral-m rqurmns adop cnralzd approachs [2]. Indusral WSANs ma mplo boh m-rggrd lows and vn-rggrd lows such as h lows nvolvng prodc snsng daa and sporadc mrgnc alarms. For a mrggrd low, s packs ar gnrad and sn prodcall X. Jn and C. Lu ar wh h Dparmn o Compur Scnc and Engnrng, Washngon Unvrs n S. Lous, USA. X. Jn and P. Zng ar wh h Shnang Insu o Auomaon, Chns Acadm o Scncs, Chna. A. Saullah s wh h Dparmn o Compur Scnc, Wan Sa Unvrs, USA. Ths work was suppord b NSFC ( and 65335), b NSF (3292 and ), b h Fullgra Foundaon, b Youh Innovaon Promoon Assocaon o h Chns Acadm o Scncs, and b Laonng Provncal Naural Scnc Foundaon o Chna (285229). wh a xd prod. Hnc, rom h rlas m o h rs pack, all subsqun packs rlas ms can b known. Basd on hs, low pahs, and nwork opologs, cnralzd algorhms gnra schduls ha ar dssmnad o ach nod bor h ssm sars. On h ohr hand, or vnrggrd lows, h occurrnc o vns s unprdcabl. Cnralzd algorhms hav dcul n gnrang schduls or hs dnamc vn-rggrd packs. In ral ndusral ssms, whn an vn occurs on a nod, h nod rs snds a mssag o h nwork managr whch hn rgnras h schdul consdrng hs nw low and rdssmnas [3]. Ths approach nroducs a long lanc, acng h ral-m prormanc o vn-rggrd packs. Sa-o-har approachs do no assgn ddcad m slos o vnrggrd packs; nsad h allow hm o prmp h m slos assgnd o m-rggrd packs [4], [5]. Whl hs mhod ma sas h rqurmns o vn-rggrd packs, ma caus som mporan m-rggrd packs mss hr dadlns. In hs papr, w propos cnralzd algorhms o schdul all m- and vn-rggrd lows undr ral-m consrans. Ths algorhms rsrv m slos or vn-rggrd lows such ha h rsrvd m slos ar sucn o handl vn-rggrd lows, and h ral-m prormanc o mrggrd packs s no acd. Ths approach hus avods nroducng long lanc as wll as h prmpon o mrggrd packs. Frs, w propos hr undamnal schdulng approachs: () a vrual-prod mhod (VP), whch changs vn-rggrd lows o vrual m-rggrd lows such ha schdulng algorhms or m-rggrd lows can b usd; (2) a slo-mulplxd mhod (SM), whch s an opmal algorhm whn h objcv s o mnmz h numbr o rsrvd m slos or vn-rggrd lows; (3) a rvrsschdulng mhod (), whch no onl dcrass h numbr o rsrvd m slos, bu also rducs h workload o nwork nods. Scond, w analz h advanags and dsadvanags o h hr mhods, and combn hr advanags o propos an algorhm ha rsrvs as w rsourcs as possbl and maks h schduls as o dssmna. Fnall, w valua our algorhms basd on h opologs o a phscal WSAN sbd and random opologs. Th smulaons ndca ha whn h nod workload s lss han 9% o h maxmum workload, rgardlss o wha h nwork workload s, h drnc o schdulabl raos bwn our combnd algorhm and opmal soluons rmans lss han 3%. In h rs o h papr, Scon II rvws rlad work. Scon III prsns our ssm modl and problm. Scon IV nroducs h hr undamnal algorhms. Scon V compars hr prormancs, and Scon VI proposs an

2 algorhm combnng hr advanags. Scon VII prsns our smulaon rsuls. Scon VIII concluds h papr. II. RELATED WORK Ral-m schdulng or WSANs has bn wdl sudd [6]. Th work n [7] provs ha h schdulng problm o WSANs s NP-hard, and hn drvs a srong ncssar condon or schdulabl. Ar ha, o mprov h ralm prormanc o WSANs, man cnralzd schdulng algorhms ar proposd, such as assgnng xd prors [8], [9], assgnng sgmnd slos [], lmnang bolnck [], and addrssng spaal r-us [2]. Howvr, hs cnralzd algorhms assgn communcaon rsourcs onl o m-rggrd packs. I hs algorhms ar usd o handl vn-rggrd packs ha ar rlasd a vr m slo, a larg numbr o rsourc rsrvaons mus lad o xrml low schdulabl. To handl vn-rggrd packs, som xsng approachs allow o prmp h rsourc o m-rggrd packs [4] or assgn m slos ha ar shard b boh ps o packs [3]. In such approachs, m-rggrd packs ma b droppd or mss dadlns. In h approach proposd n [5], bor vn-rggrd packs ar ransmd, all nods swch o an mrgnc sa, and sop ransmng m-rggrd packs. Schdulng and roung proposd n [4], [5] ams o rduc h prmpon o m-rggrd packs b vn-rggrd packs. In conras, w propos ral-m schdulng algorhms ha do no allow vn-rggrd packs o prmp m-rggrd packs and rsrv as w m slos as possbl or vn-rggrd packs such ha h ral-m rqurmns o all packs can b guarand. III. PROBLEM STATEMENT A WSAN s characrzd b a wo-upl < N, L >. Th nod s N ncluds a gawa n and snsor/acuaor dvcs n. Th gawa conncs wh an accss pon, and ach snsor/acuaor dvc s quppd wh a sngl hal-duplx ranscvr. Thror, a nod canno rcv and snd smulanousl. Th nwork managr sowar, ncludng schdulng, roung and mannanc algorhms, s mplmnd n h gawa. Th lnk s L dnos h nwork opolog. I nods n and n j can drcl communca wh ach ohr, hn h lnk l,j n s L s qual o ; ohrws, l,j =. Th low s s dnod b F = {, 2,...,, 2,...}. Each m-rggrd low F, whr F F, gnras a pack prodcall wh a prod p, and s rlav dadln s mplc,.., qual o s prod p. Th prods o mrggrd lows ar harmonc,.., p = p 2 x, whr x s an ngr, and p s h un prod ha conans a cran numbr o m slos. For a m- and vn-rggrd hbrd nwork, h prod o h schduls s no lss han h las common mulpl o h prods o m-rggrd lows. Harmonc prods ar wdl usd n ndusral WSANs [3] and ral-m mbddd ssms [6] as h hpr-prod (las common mulpl) rmans small ha smpls schdulng. W assum ha a m slo all o h m-rggrd lows rlas hr rs packs. Thn, m-rggrd low rlass s j-h pack a m slo j p, and s absolu dadln s (j + ) p. Evn-rggrd low F, whr F F, s aprodc. An vn-rggrd pack can b rlasd a an m bu mus b dlvrd o s dsnaon whn s absolu dadln + d. Th m nrval [, + d ] s calld s acv nrval. W assum ha whn low rlass a pack, dos no rlas anohr bor s dadln. I an vn-rggrd low has o rlas mulpl packs, can b rgardd as mulpl lows, and ach o hm rlass on pack. Th roung pah π ( s a wld-card characr, rprsnng and ) o low s rom a snsor sn va h gawa n o an acuaor dn, and conans c hops. Snc roung s alrad wll-sudd [7], w do no propos an roung algorhm and consdr ha h rous ar alrad gnrad basd on som xsng algorhms [8]. WSANs suppor 6 non-ovrlappng channls dnd n h IEEE sandard som o whch ma rman unavalabl du o xrnal nrrnc. W us m ( m 6) o dno h numbr o avalabl channls. Th schdulng algorhm s basd on h m slod channl hoppng (TSCH) MAC proocol [9]. TSCH has wo dmnsons: m slos (T S) and channls (CH) as shown n Fg. (b). All nods ar m snchronzd, and can accss all channls. Schdulng algorhms assgn a m slo and a channl o ach ransmsson. A ransmsson τ,j dnos h j-h hop o a pack o low. Nods chang hr workng mods, Transm (T x) or Rcv (Rx), basd on assgnmn normaon. Two ransmssons ha nvolv a common nod canno b schduld a h sam m slo. Ths suaon s calld nod conlc. CH CH2 CH CH (a) Thr lows suprram TS 3:7-2:3-4 3:-4 2:4-2:-5 :2-4 :4- TS TS2 TS3 TS4 TS5 2:-3 :-6 TS TS TS2 TS3 TS4 TS5 :Tx 2:Rx 3:Rx 2:Tx :Rx Workng mods o Nod 4, w 4=5 (b) Suprram rpa TS6 TS7 3:7- :4- :2-4 2:-3... TS TS TS2 TS3 TS4 TS5 3:Rx :Rx 3:Tx 2:Rx 2:Tx :Tx Workng mods o Nod, w =6 (c) Dssmnad workng-mod abls Fg.. An xampl o a radonal m-rggrd schdul Assgnmn normaon s organzd no suprrams. I all lows ar m-rggrd, h lngh o a suprram s qual o h las common mulpl o hr prods. Th nwork managr dssmnas h workng mods o on suprram o ach nod. Thus, nods sor hr own workngmod normaon n local mmor and chang hr mods accordngl, and hn, h nwork runs connuousl. Fg. shows an xampl. Whn h hr m-rggrd lows hav h sam prod 6, h suprram lngh s 6. A m slo T S and on channl CH, h scond hop o s sn rom n 4 o n ; hnc, n 4 and n ar n T x mod and Rx mod, rspcvl. In Fg. (c), h workng-mod abls o n 4 and n ar shown. W us w o dno h numbr o workngmod nrs o nod n n on suprram, and w 4 = 5, w = 6. Snc h nods ar mmor-consrand, h numbr o workng-mod nrs s rsrcd. Th uppr bound o w s W. Snc vn-rggrd lows ar no prodc, h radonal

3 suprram s no suabl or a m- and vn-rggrd hbrd nwork. In Scons IV and V, w wll dscuss how o rsrv m slos or vn-rggrd lows. Our objcv s o schdul h ransmssons so as o m h ollowng hr consrans. Ral-m consran: All o h m-rggrd packs and vn-rggrd packs hav o b dlvrd o dsnaons bor hr dadlns. Nod-conlc consran: A nod can srv (.., ransm or rcv) a mos on ransmsson a a m slo. Rsourc consran: For ach nod, h numbr o workng-mod nrs s no grar han W. A s o lows s calld schdulabl, has a asbl schdul ha ms all h abov consrans. A schdulng algorhm s opmal, can nd a asbl schdul whnvr hr xss on. No ha h schdulng problm sudd n [7] has F = and hnc s a spcal cas o our problm. Snc h problm n [7] s NP-hard, our problm s also NP-hard. Thror, w wll dsgn hghl cn hursc algorhms. Our algorhms am o guaran h ral-m prormanc and rlabl o all lows. Alhough h rsrvd m slos or vn-rggrd lows ar no usd whn no vnrggrd low occurs, h m-rggrd lows can sll b schduld, and h rsrvaons do no consum xra nrg. IV. FUNDAMENTAL METHODS FOR SCHEDULING Compard o radonal m-rggrd nworks, h mrggrd and vn-rggrd hbrd nworks hav hr nw acors ha ac h schdulabl: prodc, m slo rsrvaon and nod rsrvaon or vn-rggrd lows. Frs, schduls ar no prodc, or h prod s no shor nough o sas h rsourc consran, h nwork canno b schduld vn hough a asbl soluon has bn gnrad. Scond, h mor m slos and nods ha ar rsrvd or vn-rggrd lows, h hardr h schdulng o m-rggrd lows bcoms. In hs scon, w propos hr mhods and analz hm n rms o schdul lngh and rsourc rqurmns. A. Vrual-Prod Mhod Alhough h ral-m schdulng or vn-rggrd lows s sll an opn problm, hr ar man suds on mrggrd lows. An nuv schdulng approach or vnrggrd lows s o convr hm o vrual m-rggrd lows and hn schdul hm usng h sam algorhms wh ohr m-rggrd lows,.g. RM [6] and C-LLF [7]. Th onl drnc bwn vn-rggrd lows and mrggrd lows s ha vn-rggrd lows ar no prodc. Thror, w assgn vrual prods p o vn-rggrd lows. Th calculaon o vrual prods s shown n Thorm. Whn an vn-rggrd pack s rlasd a m slo, was o b schduld unl h subsqun vrual prod sars. Thorm provs ha hs vrual prod wll hav an nsanc ha s rlasd and nshs n h nrval [, +d ]. Thror, h vn-rggrd pack wll no mss s dadln. Thn, w can oban h suprram lngh H vp = max F {p }, as all prods ar harmonc. A ach m slo n h suprram, an dl channl xss, h ransmsson ha has h hghs pror and dos no nvolv a nod ha s common wh h schduld lows n hs slo s schduld on ha channl. Thorm. I an vn-rggrd pack s rlasd a m slo, and s absolu dadln s + d, hn hr mus xs a harmonc vrual prod p = p 2 log2( d + 2p ) wh an nsanc ha s ull conand n nrval [, + d ]. Proo. W s h vrual prod o low o p 2 x (x Z). x should b as larg as possbl, as n ach vrual prod, c m slos hav o b occupd. A pack s rlasd a m slo ( N), and s dadln s + d. Thus, our objc s o nd h maxmum x such ha j (p 2 x ) < (j + ) (p 2 x ) + d, p 2 x p 2, +d x + p 2 x whr j can b an ngr. W know +d + j and j p 2. So, n h rang o [ ], hr mus x b a las on ngr. W dscuss wo cass as ollows. I ( +d + p 2 ) ( x p 2 ) <, hn log x 2 ( d + 2p ) < x log 2 ( d + p ). Th maxmum x s log 2 ( d + p ). In hs cas, hr ar man valus o, d and p ha mak no ngr xs n h rang,.g., = and d = p = 4. Thror, hs cas dos no hold. I ( +d + p 2 ) ( x p 2 ), hr mus b an ngr x n h rang rgardlss o hs valus. Thn, w oban x log 2 ( d + 2p ). Thror, p = p 2 log2( d + 2p ). For h gnrad schduls, h rsourc consran mus b sasd. Th numbr o workng-mod nrs ha low nroducs o n j s w,j(h vp ) = H vp / p n j = sn or n j = dn 2 Hvp / p n j π and n j sn and n j dn ohrws. () I n j s h sourc or dsnaon, s onl usd onc n on prod. Ohrws, h nod rcvs a pack and hn snds ; hus, h nod s usd wc. Thn, h numbr o workngmod nrs o nod n j s w vp j = F w,j (Hvp ). I n j, w vp j W, h gnrad schduls ar asbl. Thn, or hs vrual-prod mhod (VP), w calcula h rsrvaon o m slos and nods. W consdr a cran m nrval [, ), and h packs wh dadlns bor m slo mus b dlvrd. Thus, h numbr o m slos rsrvd or low s p c. In ac, or vn-rggrd low, h smalls rlas nrval bwn wo conscuv packs s d +. Thror, n h wors cas, onl d + c m slos ar usd. Comparng h m slos ha ar acuall usd and h rsrvd m slos, w can oban ha d + c = p c d + c p 2 log 2 ( d + 2p ) c d + d Thus, h ulzaon o rsrvd m slos s no grar han 5%, and a las hal o h m slo rsrvaons ar wasd. Th calculaon o nod rsrvaons s smlar o ha o m slo rsrvaons. In ach rsrvd slo, wo nods ar rsrvd. Thror, h numbr o nod rsrvaons or low s c 2, and lss han 5% o rsrvaons ar usd. p

4 B. Slo-Mulplxd Mhod Th VP mhod wass som rsrvd m slos. To mnmz such was, w propos a slo-mulplxd mhod (SM). For low, s m slo rsrvaon s rpad ar vr d + m slos. W dn h d + m slos as a rsrvaon nrval. In drn rsrvaon nrvals, h rsrvd m slos mus b a h sam placs. For xampl, Fg. 2 shows smpl SM schdulng, whr h dadln o h vnrggrd low s 4 and h prod o h wo m-rggrd lows s. Thr roung pahs ar h sam as hos n Fg. (a). Th gra blocks ar rsrvd or h vn-rggrd low. Th ar placd a h rs and scond m slos o vr rsrvaon nrval. In ach rsrvd m slo, all nods n s roung pah ar rsrvd or hs low. I a nod s passd hrough b an vn-rggrd low, a on rsrvd m slo, s assgnd o work n wo mods Rx and T x,.g. nod n. Is workng mhod s as ollows: h nod has no rcvd h pack, wll lsn o h channl; h nod has rcvd h pack a a m slo, wll snd a h nx rsrvd m slo. For xampl, nod n rcvs h pack a m slo, and hn wll snd a m slo 5. In hs wa, whnvr an vn-rggrd low rlass a pack, n s acv nrval, hr mus b c m slos ha ar rsrvd or ransmng h pack. W prov hs n Thorm 2. Thn, w can know ha whn h objcv s o mnmz h numbr o rsrvd m slos, h asbl soluon ound b SM s opmal. Ths s bcaus h rsrvd m slos n a rsrvaon nrval ar lss han c, h pack rlasd a h bgnnng o h nrval s unschdulabl. CH2 CH -4 :3-4 : : : : TS TS5 Rsrvaon nrval 2:4-2: :-3 TS, d +=5, p = 2, p 2= Fg. 2. An llusraon o h slo-mulplxd mhod Thorm 2. For h SM mhod, whnvr an vn-rggrd low rlass a pack, n s acv nrval hr mus b m slos ha ar rsrvd o ransm h pack. c Proo. Suppos ha low rlass a pack a m slo. Is acv nrval s [, + d ]. Whn h acv nrval s a rsrvaon nrval, hr mus b c m slos. Ohrws, w s ha h rsrvaon nrval ncluds o, and h ndng s 2. R(, ) dnos h numbr o rsrvd m slos or bwn and. W know R(, )+R(, 2 ) = c. Snc h rsrvd m slos ar a h sam placs n drn nrvals, R(, ) = R( 2 +, +d ). Hnc, R(, )+ R(, 2 ) = R( 2 +, +d )+R(, 2) = R(, +d ) = c. In SM, h suprram lngh s H sm = LCM(d +, d 2 +,..., p, p 2,...). I d + s rlavl prm wh p j, hn h suprram lngh ma b vr long, makng dcul o sas h rsourc consran W. Hnc, w rs chck h consran. For nod n j, h numbr o workng-mod nrs nroducd b m-rggrd low can b calculad basd on Eq. (). I n j s n h pah o vn-rggrd low, wll b usd a ach rsrvd slo. Thus, h numbr o workngmod nrs nroducd b s calculad as ollows. { w,j(h sm c ) = Hsm / (d +) n j π ohrws. Thror, h numbr o workng-mod nrs o n j s wj sm = F w,j (Hsm ) + F w,j (Hsm ). I n j, wj sm W, w nvok Algorhm o schdul lows. Algorhm Tm slo and channl assgnmn n SM Inpu: V, F Oupu: Schdul S, asbl : v k V, c k = ; 2: or = o (H sm ) do 3: V = V ; 4: whl dlch and V do 5: nd h pack v k wh h arls absolu dadln n V ; 6: V = V {v k }; 7: > h absolu dadln o v k hn 8: rurn Unschdulabl; 9: v k blongs o an vn-rggrd low hn : A = { + q (d Hsm + ) q [, d )}; + : a A, dlch a and n j π, n j usdnod a hn 2: S = S + {<, a, b > a A, b dlcha}; 3: upda dlch a and usdnod a; 4: (+ + c k ) == c hn V = V {v k}; 5: v k blongs o a m-rggrd low hn 6: n j τ k,c, n j usdnod hn 7: S = S + {< τ k,c,, b > b dlch }; 8: upda dlch and usdnod ; 9: (+ + c k ) == c hn V = V {v k}; 2: V == hn rurn S; Algorhm assgns m slos and channls o h packs n s V. In h bgnnng, V ncluds all m-rggrd packs ha ar rlasd n h suprram and h vn-rggrd packs ha ar rlasd a slo. Th ohr vn-rggrd packs ar xcludd bcaus h subsqun assgnmns ar rpad. V s h s o packs ha hav no bn ampd o b schduld a h currn slo. For ach pack v k V, c k dnos h numbr o m slos ha hav bn assgnd o v k. dlch a and usdnod a ar h s o dl channls and h s o usd nods a m slo a, rspcvl. Each lmn < (or τk,g ), T S, CH > S dnos vn-rggrd low or h g-h hop o h m-rggrd pack v k ha occups m slo T S and channl CH. Ths algorhm s basd on h EDF (Earls Dadln Frs) polc [6], whch assgns a hghr pror o h pack wh an arlr absolu dadln. A ach m slo, hr xs dl channls n dlch and packs n V (ln 4), hn h pack wh h hghs pror s ampd o b schduld (ln 5). Whn h slcd pack v k s vn-rggrd (ln 9), w mus chck whhr h -h m slo n all rsrvaon nrvals can b assgnd o h pack or no (lns ). I hr s no nod conlc bwn h currn vn-rggrd low and h occupd nods, hn h vn-rggrd low can b schduld a h -h m slo and on a channl n dlch a (ln 2). Th occupd channl s rmovd rom dlch a, and h usd nods ar addd no usdnod a (ln 3). I c m slos ar assgnd o, s schdul nshs (ln 4). For mrggrd pack v k (lns 5 9), s schdul s smlar o vn-rggrd packs. Th drnc s ha w onl chck whhr h currn ransmsson τ k,c can us m slo or no, and do no consdr h -h m slo n h nx prod. I a pack msss s dadln (ln 7), h nwork

5 s unschduld (ln 8). Ohrws, h rsul S s rurnd whn all schduls ar compld (ln 2). Th m complx o hs algorhm s O( V H sm ). In a cran nrval [, ), h numbr o m slos rsrvd or low s d + c, and h numbr o nod rsrvaons s d + c (c + ) bcaus n ach m slo all nods n s roung pah ar rsrvd. C. Rvrs-Schdulng Mhod Th SM mhod mnmzs h numbr o rsrvd m slos, bu all nods n roung pahs hav o b occupd n hs rsrvd m slos. W propos a rvrs-schdulng mhod () o rduc boh m slo and nod rsrvaons. In hs mhod, a rsrvd m slo s allowd o b usd b onl on ransmsson unlss no ohr m slo s avalabl or anohr pack o h sam low. Th m slos rsrvd or ransmng on pack ar calld an arrangmn o m slos. W assum ha a pack s rlasd a a m slo, s rad o b ransmd n ha slo. An arrangmn o m slos can provd srvc or h packs rlasd bor or a h rs m slo o h arrangmn, and hr dadlns ar no lar han h las slo o h arrangmn. Fg. 3 shows h schduls gnrad b or h sam low s consdrd n Fg. 2. Th packs rlasd bwn T S and T S3 can us h rs arrangmn. Howvr, h pack rlasd a T S4 msss h rs hop a T S3. I has o wa or h nx arrangmn o b schduld, and h las m slo o h nx arrangmn canno b ar s dadln. In hs wa, alhough vn-rggrd packs can b rlasd a an m slo, onl par o hm nd o b schduld,.g., h packs rlasd a T S and T S4, and w call hm crcal packs. Th ohrs can rus h rsrvd m slos. Howvr, hs rsrvd slos ar sll ddcad snc wo conscuv packs o h sam low canno xs smulanousl. CH2 CH dadln dadln dadln TS TS5 TS s arrangmn 2 nd arrangmn 3 rd arrangmn Fg. 3. An llusraon o h mhod For ach vn-rggrd low, h rlas m o h rs crcal pack s r, =, and s dadln s d, = d. W dn h nd-o-nd lanc l,j o h j-h crcal pack as h m bwn s rs rsrvd m slo and s dadln. Thus, h rlas m o h scond crcal pack s r,2 = d, l, +, and h dadln s d,2 = r,2 + d. Thror, h rlas m and dadln o h k-h crcal pack ar r,k = (k ) d + (k ) l,j (2) d,k = k d + (k ) j [,k) j [,k) l,j. (3) In a cran nrval [, ), w nd o rsrv k arrangmns o m slos or low,.., d,k d,(k+). From Eq. (2) and (3), w know ha w wan o dcras h numbr o rsrvd arrangmns, h lanc j [,k) l,j should b rducd. Thror, rsrvs m slos or crcal vnrggrd packs rom hr dadln o hr rlas m. Th basc da o, calld Basc, s as ollows: Sp (): Th m-rggrd packs rlasd a m slo and h rs crcal packs o all vn-rggrd lows ar ncludd n pack s V. Sp (2): Th pack wh h hghs EDF pror n s V s slcd, brakng s b prorzng h low wh h smallr ID. Thn, h pack s m-rggrd, h mhod nds c avalabl m slos rom s rlas m o s dadln; h pack s vn-rggrd, h mhod nds c avalabl m slos rom s dadln o s rlas m. Sp (3): Ar a pack s schduld, h nx pack o h sam low s addd o V. Thn, h xcuon jumps o Sp(2). Th mhod should run unl on suprram H rs s nshd. Howvr, w canno calcula h suprram lngh H rs usng h sam approachs as H vp and H sm. Crcal packs ar aprodc. Th rlas m o a crcal pack s drmnd b h rsrvaons o s prvous crcal pack, and h rsrvaons ar unknown bor h prvous crcal pack s schduld succssull. Thror, or, onl whn h schduls n on suprram ar gnrad, can w know h suprram lngh. In VP and SM, h schduls ar xcud rpadl bwn h rs and las m slos o a suprram. Smlarl, n, hr also xs wo spcal m slos α and β (α < β). Th m h ollowng condons: () h schduls a m slos α + g and β + g ( < g) ar h sam; (2) h schduls a m slos α g and β g ( g) ar drn. Thus, h nwork rs xcus h schduls rom T S o T Sβ, hn jumps o T Sα and rpas h schduls bwn T Sα and T Sβ. Thror, β corrsponds o h suprram lngh. In Thorm 3, w analz s uppr bound. Thorm 3. Whn h Basc mhod s usd o schdul a nwork, h suprram lngh s no grar han Π Π η [,d +] mn{η +, c + }, whr = max F {p }. Proo. W us ξ, =< ɛ, η > and ξ, =< ɛ, η, ɛ 2, η 2,... > o dscrb h saus o low and low a m slo, rspcvl. For a m-rggrd low, ξ, mans ha a h bgnnng o m slo, s pack sll has ɛ hops o b ransmd n h ollowng η m slos. For an vn-rggrd low a on m slo, hr ar probabl mulpl crcal packs o b schduld. Th j-h pack o hs corrsponds o (ɛ j, η j ). W us ξ =< ξ,, ξ2,,..., ξ,, ξ2,,... > o dno h nwork saus a m slo. A wo m slos α and β, ξ α = ξ β, hn h schduls ar m slo β ar h sam as hos ar m slo α. Addonall, η rlcs h dadln o h rs pack and h sarng os o h ollowng packs, and ɛ and c rprsn h xcuon m o h rs pack and ollowng packs, rspcvl. Whn h dadlns and xcuon ms o all packs ar drmna, basd on h EDF polc h schduls mus b unqu. Hnc, h uppr bound o h suprram lngh s a mos β. W know ha, ξ, = ξ, = ξ,2 =... =< c, p >. W assum ha hr ar a mos x drn sauss or vn-rggrd lows. Hnc, h nwork runs or x m slos, hn hr mus b wo m slos ha hav h sam saus. Thror, β x +, and h uppr bound o h suprram lngh s no grar han x. Thn, w calcula x. W know ha or low, η j [, d + ].

6 Whn η j aks a valu n hs rang, ɛ j probabl aks ε(η j ) valus, whr ε(η j ) = mn{η j +, c + }. Tha s bcaus whn η j [, c ] and h nwork s schdulabl, ɛ j canno b grar han η j. I ɛ j > η j, hn hr ar no nough m slos o ransm h rmanng ɛ j hops. Hnc, ɛ j can b an valu n {,,..., η j },.., η j + valus. Whn η j s grar han c, ɛ j can b a valu n {,,..., c },.., c + valus. In h xrm cas, a on m slo, d + crcal packs ha blong o h sam low nd o b schduld, and hr η j dr rom ach ohr b on m slo. Thus, ξ, =< ɛ,, ɛ 2, 2,..., ɛ d +, d + >. Thror, hr ar a mos Π η [,d +] mn{η +, c + } sauss a on m slo or an vn-rggrd low. All vn-rggrd lows ar ndpndn o ach ohr. Thus, whn w consdr all o hm, hr ar a mos Π Π η [,d +] mn{η +, c + } sauss. Thror, h uppr bound o h suprram lngh s Π Π η [,d +] mn{η +, c + }. Algorhm 2 Th rvrs-schdulng mhod Inpu: F Oupu: S, α, β : or (β = ( ); ; β+ = ) do 2: S = S + Basc(β ( ), β); 3: n j, wj rs > W hn 4: rurn Unschdulabl; 5: or (α = ; α < β; α+ = ) do 6: V = Chck(S, α, β); 7: Schdulabl(S, α, V ) hn 8: rurn S, α and β; Mhod Chck(S, α, β) Inpu: S, α, β Oupu: Unschdulabl pack s V : V = h vn-rggrd packs ha hav no nshd a m slo β; 2: or v k V (v k blongs o ) do 3: (ɛ a, η a) ξ,α, ɛ k ɛ a, η a η k hn 4: V = V {v k }; 5: rurn V ; In h wors cas, nods nd a hug local mmor o sor schdul normaon o such a long suprram. Thror, w propos Algorhm 2 o consruc a shor suprram such ha h rsourc consran can b sasd. In Algorhm 2, w rs us Funcon Basc(sarng TS, ndng TS) o gnra schduls n ach (ln 2). Whn a on- schdul s gnrad, w chck all nods can sas h rsourc consran (ln 3). Th numbr o workng-mod nrs wj rs can b cound basd on h gnrad schdul S. I an nod dos no sas h rsourc consran, hn h nwork s unschdulabl (ln 4). Ohrws, w r o nd h sarng m slo α o prodc schduls (ln 5). Funcon Chck() chcks hr ar unschdulabl vn-rggrd packs whn h schduls ar rpad bwn α and β (ln 6). In Chck(), s V ncluds h vn-rggrd packs ha hav no nshd b T Sβ (ln o Chck()). For ach pack n V, hr ar nough rsrvd m slos ar T Sα (lns 2 3 o Chck()), hn h pack can b schduld. Ohrws, s r-schduld rom T Sα (ln 7), and h occupd m slos b h packs n V can b r-usd. I all packs n V ar schduld succssull, h schduls bwn h currn α and β can b xcud rpadl (ln 8), and β s h suprram lngh H rs. Th m complx o Chck() s O( V 2 ). For Algorhm 2, h numbr o raon o or loop n ln s O( Hrs ), whr H rs s lmd b h rsourc consran W. Th m complxs o lns 2, 5 and 7 ar O( V ), O( Hrs ) and O( V ), rspcvl. Thror, h m complx o s O(( Hrs ) 2 H rs V ). In a cran nrval [, ), h numbr o m slos rsrvd or low s no lss han (d +) (c ) c bcaus n h bs cas, h lanc s c, and hn, h drnc bwn h rlas ms o wo squnc crcal packs s (d +) (c ). Smlarl, h numbr o nod rsrvaons s no lss han (d +) (c ) c 2. V. COMPARISON A rlav comparson o h hr mhods n rms o schdul lngh and rsourc rqurmns s summarzd n Tabl I. As h abl shows, h smpls mhod VP has h shors suprram, whl mor han hal o rsrvd communcaon rsourcs ar wasd. Thror, communcaon rsourcs ar ampl, VP s h bs choc. Th SM mhod rsrvs h mnmum numbr o m slos. Howvr, s nod rsrvaon rqurmn s vr hgh and h suprram lngh ma b vr long whn h paramrs n LCM() ar rlavl prm. Hnc, onl a small numbr o nods ar occupd and d + s no rlavl prm wh h ohrs, hn SM bcoms an cv approach. For h mhod, w canno calcula s suprram lngh. Howvr, s nod rsrvaon ma b h las, and, unlk VP, dos no was rsourcs. Thror, bcoms an cv approach whn s schduls can b consrucd. Comparng h hr algorhms, w know ha VP s h smpls; SM has h bs m-snsv prormanc; s h mos lxbl. Thus h hr mhods hav drn advanags. Thror, vnrggrd lows should b schduld b drn algorhms basd on hr aurs. In h nx scon, w propos an algorhm ha combns h advanags o h hr mhods o solv h schdulng problm. VI. A COMBINED ALGORITHM Consdrng F vn-rggrd lows and 3 mhods, hr ar 3 F possbl combnaons or schdulng. Whn h numbr o lows s vr small, h xcuon m o xplor all combnaons ma b accpabl. Howvr, or mos cass, w nd a as algorhm o nd a asbl soluon. Our proposd combnd algorhm () s shown n Algorhm 3. In h bgnnng o, all vn-rggrd lows ar schduld b VP bcaus hr ar nough communcaon rsourcs, VP s h ass and smpls. Thn, h combnaon s chckd o vr h schdulabl. I s no schdulabl, w adjus lows o b schduld b SM or unl a asbl soluon s ound. Th assgnmn procss s as ollows. S F vp (F sm or F rs ) ncluds h lows ha ar schduld b VP (SM or ). Thn, F vp F sm F rs = and F vp F sm F rs = F. Frs, all vn-rggrd lows ar n F vp (ln ). Thn, w rmov a low rom F vp o on o h ohr wo ss. To rduc h rsourc was, h rmovd low has h largs rsourc c ulzaon d + (ln 8). I h prod o s no rlavl

7 TABLE I COMPARISON AMONG THREE METHODS Mhod Prodc (suprram lngh) Nod rsrvaon Tm slo rsrvaon VP Mhod = max F {p } = p c 2 = p c SM Mhod = LCM(d +, d 2 +,..., p, p 2,...) = d + c (c + ) = d + c Mhod Π Π η [,d +] mn{η +, c + } H rs d c +2 c 2 d c +2 c prm wh hos o ohr m-rggrd lows and h rmoval dos no add nod rsrvaon (ln 9), s addd o F sm (ln ). Ohrws, s addd o F rs (ln 2). Funcon AddRs() chcks h addon o nod rsrvaon. I c (c + ) > d + p c vp 2, hn AddRs() rurns ru. I all lows n F hav bn movd ou and h nwork s sll unschdulabl, hn h low havng h maxmum nod rsrvaons n F sm s rmovd o F rs unl F sm = (lns 3 5). Fnall, all vn-rggrd lows ar n F rs. Consdrng boh m slo rsrvaons and nod rsrvaons, ma hav h ws rsrvaons. I s oppos o VP. Thror, h adjusmn s o rmov lows rom F vp o F rs. Algorhm 3 Th combnd algorhm Inpu: F Oupu: Schdul S, asbl : F vp = F ; F sm = F rs = ; 2: whl ru do 3: F = F + V P (F vp ); 4: all ncssar condons ar sasd hn 5: (F, F sm ) rurns S hn 6: rurn S; 7: F vp hn 8: nd a low wh h largs c d + n F vp ; 9: (j, (d + ) / p j or (d + ) = p j ) and!addrs( ) hn : F sm = F sm + { }; F vp = F vp { }; : ls 2: F rs = F rs + { }; F vp = F vp { }; 3: ls F sm hn 4: nd a low wh h mos nod rsrvaons n F sm ; 5: F rs = F rs + { }; F sm = F sm { }; 6: ls brak; 7: rurn Unschdulabl; For ach combnaon o F vp, F sm and F rs, w us h ollowng algorhm o schdul hm. Frs, h lows n F vp ar changd o vrual-prod lows (ln 3). Thn, w propos hr ncssar condons as ollows. I s F dos no sas hs condons, mus b unschdulabl. Thus, h xcuon m can b rducd cvl. Condon : h ulzaon o ach nod s no grar han,.., n j N, δ,j + ε,j c p F d F sm + + δ F rs,j d + 2 c, whr δ,j = n j s h sourc or dsnaon o low and δ,j = 2, passs hrough n j; ohrws, δ,j =. I uss n j, ε,j = ; ohrws, ε,j =. Condon 2: h nwork ulzaon s no grar han h numbr o channls m,.., c p F + c F sm d + + c F rs d + 2 c m. Condon 3: h lowr bound o h numbr o workngmod nrs s no grar han W. For h mhod, h shors suprram s. F vp s addd no F. F sm dos no ncras h suprram lngh. Thus, h lowr bound o h suprram lngh s. Thror, rs )+ w,j(h rs )+ ŵ,j(h rs ) W, F sm F rs F w,j(h whr h lowr bound o s as ollow. H rs / d +2 c n j = sn or n j = dn ŵ,j(h rs ) = 2 H rs / d +2 c nj π and n j sn and n j dn ohrws. In Funcon (), h lows n F sm nd o rsrv all nods on hr pahs. Ths s h onl drnc rom Algorhm 2. Thus, h m complx o () s O( N ( Hrs ) 2 H rs V ). Th numbr o raons o whl loop n ln 2 s O( F ). Thror, h m complx o Algorhm 3 s O( F N ( Hrs ) 2 H rs V ). VII. EVALUATION In hs scon, w wll valua our algorhm basd on h opologs o a phscal WSAN sbd and random opologs. Two mrcs ar usd or prormanc valuaon: () schdulabl rao s h prcnag o s cass or whch an algorhm s abl o nd a asbl schdul, and (2) xcuon m s h m rqurd o gnra a asbl schdul. W compar our algorhm and h hr undamnal mhods VP, SM and wh hr mhods SS [3], EDF and UP. UP shows h prcnag o cass ha sas h hr ncssar condons, hrb showng a consrvav uppr bound o schdulabl rao ha an opmal algorhm can achv. Th orgnal SS mhod allows m-rggrd packs o b dscardd whn vn-rggrd packs ar usng shard m slos. To mak suabl or our problm, SS dos no assgn shard m slos. Th pur EDF polc s an cv schdulng algorhm or m-rggrd lows. In our valuaon, h EDF mhod schduls no onl m-rggrd packs bu also all possbl vn-rggrd packs. I wo vn-rggrd packs blong o h sam low, EDF can assgn h sam m slos o hm. All algorhms ar wrn n C and run on a Wndows machn wh 3.4GHz CPU and 6GB mmor. Th paramrs usd n hs scon ar summarzd n Tabl II. Th numbr o lows s 2 n, and hr ar 2 n vn-rggrd lows. Sourc nods and dsnaon nods ar randoml slcd, and ar no rpadl usd. Random opologs ar gnrad basd on h nod dns ρ. Th harmonc prods ar randoml slcd n { 2 [, ]}, and h dadlns o vn-rggrd lows ar n { [2, 2 ]}. W us Condons and 2 o calcula h ulzaon o h accss pon and nwork. Th accss pon s h hospo, and s ulzaon s largr han nwork ulzaon and s a k acor ha acs h schdulabl. Th schdul normaon ncluds sourc, dsnaon, Rx/Tx, TS and CH. Rx/Tx and

8 schdulabl rao schdulabl rao schdulabl rao schdulabl rao schdulabl rao schdulabl rao TABLE II PARAMETE n Numbr o nods m Numbr o channls ρ Nod dns o a nwork Fracon o sourcs and dsnaons Fracon o vn-rggrd lows u Ulzaon o h accss pon W Uppr bound o h numbr o workng-mod nrs CH can b sord n an unsgnd char. Hnc, a workngmod nr nds 5 bs. W s h mmor sz o 5kB and kb. Thr corrspondng W ar 24 and 248, rspcvl. A. Ral Topologs W us h opologs o an ndoor sbd [2] dplod n Joll Hall o Washngon Unvrs n S. Lous. Th sbd consss o 7 TlosB mos, ach quppd wh CC242 rados and complan wh h IEEE sandard. All nods ak urns broadcasng packs. I h pack rcpon rao bwn wo nods s hghr han 8%, a rlabl lnk s consdrd bwn hm. Thus, w oban wo ral opologs a ransmsson powr lvls dbm and 5dBm..8.6 VP.4 SM.2 SS EDF (a) u [6%, 7%) (b) u [7%, 8%) (c) u [8%, 9%) (d) u [9%, %) Fg. 4. Schdulabl rao wh a opolog o dbm Fg. 4 shows h comparson o schdulabl raos whn TX powr s dbm. Th paramrs ar [.2,.9], n = 7, m = 6, =.2 and W = 24. For ach, s cass ar randoml gnrad. Accordng o h ulzaon u, hs s cass ar prsnd n drn sub-gurs. Th rsuls ar normalzd wh UP as h basln. As u and ncras, h schdulabl raos dcras bcaus s dcul o nd a asbl soluon whn h soluon spac bcoms mor and mor complx. Among all mhods, has h hghs schdulabl rao. Whn u > 9%, can sll solv mor han hal o s cass. VP and SS ar smlar. In h SS mhod, vn-rggrd packs ar consdrd mor mporan han m-rggrd packs. Ar schdulng all vn-rggrd packs, m-rggrd packs sar o b schduld. SS pas mor anon o h crcal, bu no h mporal. VP consdrs onl mporal. Thror, has br schdulabl han SS. For h SM mhod, s suprram lngh s vr long, and ach rsrvd m slo nds mulpl workng-mod nrs o dscrb. Th local mmor o a TlosB mo s no sucn o sor all hs nrs. Onl whn s vr small can a small par o s cass b schduld. Alhough SM rsrvs h mnmum numbr o m slos, usng SM alon ma no b praccal. EDF has o handl all possbl vn-rggrd packs. Hgh workload lads o low schdulabl rao. In h ollowng, h smulaon rsuls wll no nclud SM and EDF. (a) u [8%, 9%), (b) u [9%, %), (c) u [8%, 9%), (d) u [9%, %), Fg. 5. Excuon m.2.2 dbm,w=5k,=.2,m=6-5dbm,w=5k,=.2,m=6 dbm,w=k,=.2,m=6.8.8 dbm,w=5k,=.3,m=6 dbm,w=5k,=.2,m= (a) u [8%, 9%) (b) u [9%, %) Fg. 6. Schdulabl rao undr varng paramrs Fg. 5 shows h xcuon ms o and. Th corrspondng s cass ar h sam as Fg. 4. VP and SS can nd a asbl soluon whn 2 ms. As h numbr o lows and h accss pon ulzaon ncras, h xcuon m o ncrass bcaus hr ar mor ransmssons o b dal wh. rpadl xcus o nd a asbl soluon. Thror, h mpac o u and on s no obvous. For all s cass, h xcuon m o s lss han 7s. Ths s accpabl o handl dnamc r-dplomns o ndusral applcaons. Fg. 6 shows h comparson o schdulabl raos undr varng paramrs. A bar corrsponds o h avrag o a curv rom =.3 o.9. From hs gurs, w nd ha: Th dcras o powr lvl lads o a slghl dcras o schdulabl raos bcaus whn h ransmsson powr rducs, h numbr o hops ncrass. Whn h local mmor doubls, h schdulabl rao ncrass b 4% and % n h wo sub-gurs. Fg. 7 shows h mmor sz rqurd b and VP undr varng u. Whn u > 7%, VP can hardl nd asbl soluons. Hnc, hr s no corrspondng subgur. Mmor ndd b s abou our ms ha b VP. rads mmor or schdulabl. 5kB mmor s sucn or mos s cass. I h numbr o channls ncrass, hn mor rsourcs can b usd, and h prormanc should b mprovd. Howvr, h ncras o m has almos no mpac on h schdulabl rao. Th rason s ha rgardlss o how man channls ar usd, h ulzaon u s unchangd. Whn h racon o vn-rggrd lows ncrass rom.2 o.3, h schdulabl rao obvousl dcrass. Each combnaon o hs paramrs corrsponds o wo subgurs ha ar smlar o Fg. 4(c) and (d). W do no show.4.2

9 schdulabl rao schdulabl rao schdulabl rao schdulabl rao schdulabl rao schdulabl rao hos smlar rsuls. Onl h paramr combnaon wh =.3 has a drnc ha s shown n Fg. 8. Whn =.3 and u [8%, 9%), h schdulabl rao o s vr low. Whn u s xd, h wr lows, h mor ulzaon ach low has. Hnc, hr prods and dadlns ar shor. Ths lad o mor conlcs and long lancs. In hs cas, s dcul o consruc h prodc. changs som vnrggrd lows o m-rggrd lows. Alhough h chang maks h lows consum mor rsourcs, h consumpon s accpabl whn u [8%, 9%). In Fg. 8(b), h schdul rao o sharpl rducs bcaus whn u > 9%, hr s almos no rdundan rsourcs. (a) u [6%, 7%), V P (b) u [6%, 7%), (c) u [8%, 9%), (d) u [9%, %), Fg. 7. Comparson o mmor sz VP.2 SS (a) u [8%, 9%) (b) u [9%, %) Fg. 8. Schdulabl rao whn =.3 B. Random Topologs To valua h mpac o n and ρ on schdul raos, w randoml gnra opologs basd on h quaon A = nd2 27 2πρ [2], whr A s a squar ara, and d dnos h ransmng rang o 4 m. A gawa s dplod a h cnr o h squar A, and n nods ar randoml dplod n A. I h dsanc o wo nods s lss han d, hr s a rlabl lnk bwn hm. Fg. 9 shows h schdulabl rao undr varng n. Th ohr paramrs ar m = 6, ρ =, =.8, =.2 and W = 24. Our combnd algorhm ouprorms h ohrs. Whn u s xd, h hghr h numbr o nods, h lssr h nod conlcs. Thus, as n ncrass, h schdulabl rao o and ncrass. Fg. shows h schdulabl rao undr varng ρ. Whn u [8%, 9%), as h ncras o ρ, h raos slghl ncras bcaus h numbr o avalabl rla nods ncrass, and h numbr o hops dcrass. Whn u [9%, %), h s cass ar hard o b schduld rgardlss o h valu o ρ. VIII. CONCLUSION In hs papr, w ocus on h ral-m schdulng problm or m-rggrd and vn-rggrd hbrd nworks. In xsng approachs, m-rggrd packs ar droppd whn vn-rggrd packs ar ransmd. Our algorhms do no n VP SS n (a) u [8%, 9%) (b) u [9%, %) Fg. 9. Schdulabl rao undr varng n (a) u [8%, 9%) (b) u [9%, %) Fg.. Schdulabl rao undr varng ρ ρ= ρ=2 ρ=3 drop m-rggrd packs and schdul all m- and vnrggrd packs undr ral-m consrans. Our proposd combnd algorhm rsrvs as w m slos as possbl and consrucs asbl suprrams. Th smulaons ndca ha our combnd algorhm sgncanl ouprorms h ohrs. In h uur, w wll sud dsrbud schdulng algorhms whr ach nod dnamcall gnras schduls basd on h bhavors o vn-rggrd lows. W shall also consdr prmpv schdulng whr vn-rggrd lows ar allowd o prmp low-crcal m-rggrd lows undr ssm sabl consrans. REFERENCES [] D. Chn, M. Nxon, and M. Alosus, WrlssHART T M Ral-m msh nwork or ndusral auomaon. Sprngr, 2. [2] P. Park, S. C. Ergn, C. Fschon, C. Lu, and K. H. Johansson, Wrlss nwork dsgn or conrol ssms: A surv, IEEE Commun. Surv. & Tuor., 27. [3] IEC, Ic 6259: Indusral communcaon nworks wrlss communcaon nwork and communcaon prols wrlsshar, 29. [4] X. Zhu, X. Tao, T. Gu, and J. Lu, Targ-awar, ransmsson powradapv, and collson-r daa dssmnaon n wrlss snsor nworks, IEEE Trans. Wrl. Commun., 25. [5] D. Yang, J. Ma, Y. Xu, and M. Gdlund, Sa-wrlsshar: A novl ramwork nablng sa-crcal applcaons ovr ndusral wsns, IEEE Trans. Ind. Inorm., 28. [6] R. T. Hrmo, A. Gallas, and F. Tholr, Schdulng or sch and slow channl hoppng mac n low powr ndusral wrlss nworks: A surv, Compu. Commun., 27. [7] A. Saullah, Y. Xu, C. Lu, and Y. Chn, Ral-m schdulng or wrlsshar nworks, n RTSS. IEEE, 2. [8], Pror assgnmn or ral-m lows n wrlsshar nworks, n ECRTS. IEEE, 2. [9], End-o-nd dla analss or xd pror schdulng n wrlsshar nworks, n RTAS. IEEE, 2. [] D. Yang, Y. Xu, H. Wang, T. Zhng, H. Zhang, H. Zhang, and M. Gdlund, Assgnmn o sgmnd slos nablng rlabl ral-m ransmsson n ndusral wrlss snsor nworks, IEEE Trans. Ind. Elcron., 25. [] X. Jn, F. Kong, L. Kong, W. Lu, and P. Zng, Rlabl and mporal opmzaon or mulpl coxsng wrlsshar nworks n ndusral nvronmns, IEEE Trans. Ind. Elcron., 27. [2] W.-B. Pönr, H. Sdl, J. Brown, U. Rodg, and L. Wol, Consrucng schduls or m-crcal daa dlvr n wrlss snsor nworks, ACM Trans. Sn. Nw., 24. [3] B. L, L. N, C. Wu, H. Gonzalz, and C. Lu, Incorporang mrgnc alarms n rlabl wrlss procss conrol, n ICCPS. ACM, 25. [4] X. Jn, F. Kong, L. Kong, H. Wang, C. Xa, P. Zng, and Q. Dng, A hrarchcal daa ransmsson ramwork or ndusral wrlss snsor and acuaor nworks, IEEE Trans. Ind. Inorm., 27. [5] C. Xa, X. Jn, L. Kong, and P. Zng, Schdulng or mrgnc asks n ndusral wrlss snsor nworks, Snsors, 27. [6] J. W. Lu, Ral-m ssms. Prnc hall, 2. [7] W. Rhan, S. Fschr, M. Rhan, and M. H. Rhman, A comprhnsv surv on mulchannl roung n wrlss snsor nworks, J. Nw. Compu. Appl., 27. [8] M. Nobr, I. Slva, and L. A. Guds, Roung and schdulng algorhms or wrlssharnworks: a surv, Snsors, 25. [9] D. D Gugllmo, S. Brnza, and G. Anasas, I : A surv, Compu. Commun., 26. [2] (28) Wusl wrlss snsor nwork sbd. [Onln]. Avalabl: hp://cps.cs.wusl.du/ndx.php/tsbd [2] T. Camlo, J. S. Slva, A. Rodrgus, and F. Boavda, Gnsn: A opolog gnraor or ral wrlss snsor nworks dplomn, n SEUS. Sprngr, 27.