Optimised Delay-Energy Aware Duty Cycle Control for IEEE with Cumulative Acknowledgement

Transcription

1 2014 IEEE 25th Internatonal Symposum on Personal Indoor and Moble Rado Communcatons Optmsed Delay-Energy Aware Duty Cycle Control for IEEE wth Cumulatve Acknowledgement Yun L Kok Keong Cha Yue Chen Jonathan Loo School of Electronc Engneerng and Computer Scence Queen Mary Unversty of London Emal: yun.l mchael.cha yue.chen@qmul.ac.uk Department of Computer and Communcatons Engneerng Mddlesex Unversty Emal: J.Loo@mdx.ac.uk Abstract IEEE beacon-enabled mode adopts duty cycle to acheve energy effcency and provdes an optonal acknowledgement (ACK mechansm to ensure the transmsson relablty. However frequently sendng ACK ntroduces addtonal ACK transmsson energy consumpton and ncreases endto-end delay. In ths paper we focus on a duty cycle optmsaton problem wth jont consderaton on energy effcency end-toend delay and relablty for IEEE networks. We frst formulate a cumulatve ACK enabled duty cycle optmsaton problem as an nventory control problem. Then the optmal soluton to the problem s derved by applyng dynamc programmng (DP. Furthermore a low complexty delay-energy aware duty cycle control (DE-DutyCon s proposed to reduce the computatonal complexty of mplementng the control on computaton lmted sensor devces. The jont-cost upper bound of DE-DutyCon s also provded. DE-DutyCon acheves an exponental reducton of computatonal complexty compare wth DP optmal control. Smulaton results show that the proposed DE-DutyCon acheves close performance n terms of energy effcency end-to-end delay and packet drop rato compare wth DP optmal control under varous network traffc. I. INTRODUCTION The emergng wreless sensor networks (WSNs are featured as ntegratng varous applcatons and provdng relable tmely data delvery whle achevng energy effcency [1] [2]. IEEE [3] whch supports cluster-tree network has been wdely adopted for WSNs n varous applcatons such as home automaton ndustral management and health care [4]. The low duty cycle n IEEE mproves the energy effcency. However t ncreases the endto-end delay as devces may swtch nto nactve perods to save energy. In addton the unformed duty cycle control n current standard may not meet varous requrements of dfferent applcatons. Some work have been done to mprove energy effcency and reduce delay by desgnng adaptve duty cycled controls. The delay reducton of UMAC [5] s acheved by controllng the length of actve perods based on a utlsaton functon whch s the rato of the actual transmsson and receptons performed by the devce. However the unformed duty cycle control for all devces s not flexble when each devce generates dfferent amount of traffc wth dfferent qualty of servce (QoS requrements. A duty cycle control algorthm called Traffc-adaptve Dstance-based Duty Cycle Assgnment (TDDCA s proposed n [6] wth the am of mprove latency. The duty cycle s ncreased when contenton s reported. Otherwse the duty cycle s decreased every tme perod down to a mnmum preset value. However to enable the control contenton reports pggyback flags and modfcatons of the packet header are needed. The aforementoned work focus on delay reducton for duty cycle based MAC protocols wthout consderaton on relablty. To jontly consder energy effcency delay and relablty the duty cycle learnng algorthm (DCLA n [7] adapts the duty cycle usng renforcement learnng process to mnmse power consumpton whle balancng the probablty of successful data delvery and delay constrants of the applcaton. The problem of ths approach s the learnng processes present slow convergence due to the selecton of a large state-acton space. More recently an adaptve optmal dutycycle algorthm (AODC runnng on top of IEEE MAC protocol s proposed n [4]. AODC ams at mnmsng energy consumpton whle meetng the relablty and delay requrements. However ths work focus on the non-beaconenabled mode of IEEE In our prevous work [8] a low complexty duty cycle control RADutyCon was proposed wth jont consderaton of energy consumpton and end-to-end delay for IEEE networks wth stop-and-wat automatc repeat request (ARQ transmsson scheme. In ths paper we further mprove energy effcency and reduce end-to-end delay of IEEE by applyng Go-Back-N ARQ transmsson scheme. Go-Back-N ARQ uses cumulatve ACK scheme where the recever only sends one ACK sgnfyng that t has receved all packets n a certan tme perod [9]. We propose a delay-energy aware duty cycle control (DE-DutyCon for IEEE wth cumulatve ACK scheme. Wth less ACK transmsson both energy consumpton and averaged end-to-end delay are reduced. The contrbutons of the paper are summarsed as follows: the delay-energy aware duty cycle optmsaton problem s formulated nto an nventory control problem whch jontly takng energy consumpton end-to-end delay and packet /14/$31.00 c 2014 IEEE /14/$ IEEE 1051

2 drop nto consderaton. And then the optmal soluton to the problem s derved by applyng dynamc programmng (DP. Furthermore a low complexty suboptmal control DE- DutyCon s proposed to reduce the computatonal complexty of runnng DP on sensor devces. Last but not the least the upper bound of DE-DutyCon s derved. The remander of the paper s organsed as follows. In secton II we gve the system model and the background of IEEE MAC protocol. Problem formulaton s gven n Secton III. Secton IV presents the dervaton of the optmal soluton and the proposed DE-DutyCon. Smulaton results and concluson are gven n secton V and secton VI respectvely. II. SYSTEM MODEL As the mult-hop case can be vewed as the combnaton of several two-hop scenaros ths paper s dedcated to the analyss of two-hop cluster-tree network as shown n Fg 1. The level of the devce s denoted as l n. The coordnator n 0 s n level-1 thus l n0 = 1; there are N full-functon devces (FFDs n (1 N n level-2. Reduced-functon devces (RFDs are n level-3. We denote the chld RFDs set of FFD n as ch n and the number of RFDs n ch n s M n. Cumulatve ACK s used n Go-Back-N ARQ whch means that only one ACK s requred for each transmsson par. The recever wll send one ACK to nform the sender that all packets between ths transmsson par have been receved correctly at the end of each transmsson. A. IEEE (2011 In IEEE (2011 beacon-enabled mode the duraton between two consecutve beacons s called Beacon Interval (BI whle the duraton of an actve perod s called Superframe Duraton (SD. Specfcally BI = abasesuperf rameduraton 2 BO (1 SD = abasesuperf rameduraton 2 SO (2 where Beacon Order (BO and Superframe Order (SO are two ntegers rangng from 0 to 14 (0 SO BO 14 and abasesuperf rameduraton = 15.36ms at 2.4 GHz wth 250 kbps data rate. The duty cycle = SD/BI = 2 SO BO s the rato of the actve porton over each tme perod. In mult-hop transmsson each FFD dvdes ts BI nto two superframes named ncomng superframe and outgong superframe [3]. The FFD n receves the beacon from the coordnator n the ncomng superframe and transmts ts beacon to ts chld RFDs n the outgong superframe. Whle the ncomng superframe duty cycle s decded by the parent FFD of the devce and enclosed n the receved beacon each FFD can control ts outgong superframe duty cycle (refer as duty cycle n ths paper. To smplfy the synchronsaton we assume all FFDs have same BO. Thus the duty cycle control of FFD n s acheved by settng ts outgong SO n. RFD FFD Coordnator B. Queue and Traffc Models q k+1 n n n 0 level-1 Fg. 1. Network Model. n j level-2 level-3 We assume all generated packets are avalable at the begnnng of each tme perod. All the packets are forwarded to the coordnator n 0 for uplnk transmsson and qn max s the maxmum queue length of devce n. The new arrved packets wll be dropped f the queue length reaches ts maxmum. Smlar to [5] the queue length of tme perod k+1 of devce n s gven as ( = mn [qn k + rn k fn k + gn k qn max (3 where 0 k K 1 [ = max(0 gn k s the number of packets beng generated by devce n n tme perod k; fn k s the number of packet transmted by devce n n tme perod k; and rn k s the number of packets receved by devce n n tme perod k. We assume each RFD generates and sends ndependent Posson dstrbuted number of packets wthn the actve perod of each BI whch means fn k and gn k are ndependent random varables. III. PROBLEM FORMULATION In ths secton we formulate the duty cycle control problem as a dynamc programmng nventory control problem [10] to mnmse the total expected jont-cost of energy consumpton and end-to-end delay. To ensure the costs of energy consumpton and end-to-end delay are addtve we defne the transmttng energy cost E t (fn k recevng energy cost E r (rn k dle lstenng energy cost E l (rn k and end-to-end delay cost D(rn k of the devce n n terms of number of packets. As the energy consumpton of ACK packets transmsson exsts only when the FFD receves packets. Thus a fxed ACK transmsson energy cost A s ntroduced along wth the recevng energy consumpton. The specfc defnton of each cost s gven as: r E r (rn k A + c r k n = qn max l n f rn k > 0 (4 0 f rn k = 0. E f (f k n = c f f k n q max n l n (5 E l (q k n = c l [f k n g k n q k n r k n q max n l n (6 D(qn k = c d [qk n + rn k + gn k fn k qn max (7 l n 1052

3 where A = c f M n ; c f c r c l and c d are the cost coeffcents of transmttng recevng dle lstenng and delay IV. DUTY CYCLE CONTROL We decompose the problem P n nto a sequence of subproblems S(rn k c l t would never be optmal to receve new packet at the respectvely. Note that c r < c l as f c r were greater than by applyng the DP algorthm. The objectve of each S(rn k last perod and possbly n earler perods. Dfferent devce s to mnmse the cost-to-go functon U(qn k back from tme perod K to tme perod k. Thus the optmal buffer sze qn max and level of the devce l n have also been soluton to S(rn 0 ntegrated nto the problem formulaton to dfferentate FFDs s the soluton to P n. Then the cost-to-go functons U(qn k and RFDs. of S(rn k s defned as ( We further ntroduce α and β to assgn the weghtngs U(qn k of energy effcency and end-to-end delay requrements of = mn E α E r (rn k π n D + E f (fn k + E l (qn k dfferent applcatons. The expected weghted-sum jont-cost functon for devce n at tme perod k s + βd(qn k + EU(qn k+1. (12 ( J(rn k = E α E f (fn k + E r (rn k + E l (qn k + βd(qn k To reduce the number of notatons n the equatons we. denote m k n = qn k + rn k and n k n = fn k gn k. If δ(0 = 0 (8 δ(rn k = 1 for rn k > 0 based on (4 - (8 we have where 3α + β = 1 whch follows [8]. IEEE U(q (2011 beacon-enabled mode apples slotted carrer-sense n k = mn E Aδ(rn k π n D + W (m k n αc r qn k q max n l n multple access wth slotted collson avodance (CSMA/CA (13 for packet transmsson. We assume devces need to perform two clear channel assessments (CCAs before packet transmsson. The beacon transmsson duraton s D bcn. Thus for where each FFD the total packet transmsson duraton s gven as W (m k n = αe f (fn k + αe r (rn k + αc l [nk n m k n M n P D = SD D bcn = Dmj + Mn δ + D ACK j=1 where D mj s the packet transmsson duraton of RFD m j ch n δ and D ACK are watng tme and transmsson duraton of the ACK packet respectvely. Then the number of packets that can be receved by devce n at k th tme perod s rn k = M n m j=1 η Dmj /D p where Dp s transmsson duraton per packet and η s the throughput lmtaton coeffcent [11] whch shows mpact of the backoff and contenton durng CSMA/CA transmsson. Accordng to (2 the relatonshp between the mnmum SO and rn k s gven as SO(rn k = (10 r k log 2 ( n D p + M n δ + D ACK + D bcn. η Our objectve s to fnd the control of the optmal duty cycles πn (2 SO BO for each devce n whch mnmse the overall expected jont-cost over K tme perods. Hence the jont optmsaton problem s: P n : mn π n D E K 1 k=0 s.t. q K n = 0 r k n J(r k n r max n where D s vald duty cycle sets of devce n and rn max the maxmum number of packets devce n could receve. (9 (11 s A. Optmal Soluton q max n l n + βc d [mk n n k n q max n l n + U([m k n n k n. (14 It s not trval to solve S(rn k as functon U(qn k s not a convex functon due to the lmted buffer sze of devces. However t has been proved by Scarf that an optmal mult perod(s S soluton exsts f U(qn k s A convex functon [12]. Defnton 1: The real-valued functon f s an A convex functon f A 0 for all z 0 b > 0 f satsfes the A convexty property ( f(y f(y b A + f(z + y f(y + z. (15 b Theorem 1: If the cost-to-go functon U(qn k s A-convex the optmal duty cycle control s a mult-perod polcy: when the queue length qn k s smaller than the Tn k SOn k s set based on (10 otherwse SO equals to zero: SOn k SO(r k = n f q k n < Tn k 0 f qn k Tn k (16 where rn k s the optmal number of packets the devce should receved at each tme perod. Proof: For k = K functon U([m K n n K n s the zero functon and αc l [nk n m K n qn max l n + βc d [m K n n K n q max n l n s a convex functon. Thus W (m K 1 n s the sum of two convex functons so W (m K 1 n s convex. Therefore A convex. Gven the A convexty of W (m K 1 n f W (Tn K 1 gves the mnmum value of the functon W (m K 1 n Tn K 1 = m K 1 n = arg mnm K 1 and t K 1 n R W (mk 1 n n 1053

4 s the smallest value of m K 1 n that W (t K 1 n = W (Tn K 1 + A. we have approprate scalars t K 1 n Tn K 1 and U(q K 1 n = A + W (Tn K 1 α cr qk 1 n q max W (q K 1 α cr qk 1 n n n l n qn K 1 qn max l n qn K 1 < t K 1 n t K 1 n. (17 For k = K 2 1 as U(qn K 1 and W (m K 2 n are all A convex by repeatng the above process recursvely we have the A convexty of U(qn k. The proof of the theorem s fnshed. B. Suboptmal DE-DutyCon Based on the above analyss the optmal duty cycle of devce n can be found by runnng DP. However DP needs to conduct exhaustve search over all possble solutons at each tme perod whch s very energy neffcent and tme consumng. Thus t s dffcult or mpractcal for computatonally-lmted sensor devces to run DP. Rollout algorthms have demonstrate excellent performance on a varety of dynamc optmsaton problems. Interpreted as an approxmate DP algorthm a rollout algorthm estmates the cost-to-go at each tme perod by estmatng future costs whle followng a heurstc control referred to as the base polcy. The heurstc n ths paper s nspred by the threshold structure of the optmal control. To ensure the stable of the queue length the most straght forward approach s to set T n equals to the mean value of fn k for each devce n. Based on (17 the heurstc base control of P n s gven as SOn k SO(rn k = f qn k < fn k 0 f qn k fn k (18 The proposed DE-DutyCon s the one that attans the mnmum of the cost-to-go functon [ U(rn k = mn π n D E α(e r (rn k + E f (fn k + E l (rn k (19 ] +βd(rn k + EŨ(rk+1 n where Ũ(rk+1 n s the approxmaton of U(rn k+1 based on the heurstc. Gven the approxmatons Ũ(rk n whch s calculated based on the heurstc the computatonal savng of DE-DutyCon s evdent as only a sngle mnmsaton problem has to be solved at each tme perod. Notced that even wth readly avalable approxmatons Ũ(rk+1 n the calculaton of the mnmsaton over π n D may nvolve substantal computaton. To further save the computaton a subset D of the promsng controls s dentfed n the proposed DE-DutyCon. Thus the mnmsaton over D n (20 s replaced by a mnmsaton over a subset D D. Theorem 2: Let s denote Û(rk n as the estmate cost-togo of DE-DutyCon whose control range s D D. U(rn k as the expected actual cost-to-go ncurred by DE-DutyCon. Then we have U(rn k Ũ(rk n whch means Ũ(rk n s the cost-to-go upper bound of DE-DutyCon. Proof: For k = 0 1 K 1 denote [ Û(rn k = mn E α(e r (r k π n D n + E f (fn k + E l (rn k ] + βd(rn k + EŨ(rk+1 n. Thus for all q k n we have Û(rk n Ũ(rk n let (20 Û(r K n = G(r K n (21 = α(e r (r K n + E f (f K n + E l (r K n + βd(r K n. Applyng backward nducton on k we have U(rn k = Û(rn K = G(rn K for all qn K. Assumng that Ū(rn k+1 Û(rn k+1 for all qn k+1 we have U(rn k = E G(rn k + Ū(rk+1 n E G(rn k + Û(rk+1 n E G(rn k + Ũ(rk+1 n = Û(rk n. (22 The frst equalty above follows from the defnton of the cost-to-go U(r k n of DE-DutyCon whle the frst nequalty follows from the nducton hypothess and the second nequalty follow from the assumpton Û(rk n Ũ(rk n. Then we have U(r k n Û(rk n Ũ(rk n for all q k n. Thus the Ũ(r k n s a readly obtanable performance upper bound for the cost-to-go functon U(r k n. In addton two remarks are gven as follows. Remark 1: The proposed DE-DutyCon has lower computatonal complexty as compared to DP optmal control. If D s the average search range of the devces the computatonal complexty of DP algorthm s O(KD N+D whle that of the suboptmal control s only O(KN D Remark 2: The proposed suboptmal controls has lower synchronsaton overhead as compared to controls n [2]. The proposed control does not need addtonal SYNC packet to ensure the devces are actve at the same tme as t employs the same BO as defned n IEEE (2011 and all devces are actvated at the begnnng of each BI V. SIMULATION RESULTS AND ANALYSIS In ths secton the performance of DE-DutyCon s evaluated n Matlab. We consder a two-hop cluster-tree network as explan n secton II. We appled ON/OFF traffc model n the smulaton. When the devce s actve (ON the dstrbuton of the traffc rate follows a Posson dstrbuton. When the devce s nactve (OFF t s dle and does not generates any packets. To focus on the performance of duty cycle control we assume there s no packet loss durng transmsson. Packets are dropped only when the queue length of the devce reaches ts buffer sze qn k fn k + rn k > qn max. The maxmum queue length of FFDs s 50 packets and that of RFDs s 20 packets. Energy consumpton parameters are based on CC2420 data sheet 1054

5 TABLE I SIMULATION PARAMETERS Parameter Value Parameter Value frequency 2.4 GHz α 0.2 data rate 250kbps β 0.4 transmt power 36.5 mw packet sze 100 bytes receve power 41.4 mw CCA sze 8 symbols dle lsten power 41.4 mw ACK packet sze 10 symbols sleep power mw unt backoff perod 20 symbols [13] and MAC layer parameters are based on IEEE (2011 standard [3]. The BI s 0.49s (BO = 5 and fn k follows posson dstrbuton wth the mean value equals to 30 packets per actve perod and the number of observaton tme perods K s 100. The results are the averaged values of 1000 runs of the devce n. More parameters are gven n TABLE I. The performance of DE-DutyCon s compared wth a control DP optmal control and a heurstc base control. Benchmark control: to reduce the end-to-end delay the control ams at maxmsng the number of receved packets rn k. The SO s determned based on (10. DP optmal control: exhausted search of the optmal rn k s processed at each tme perod based on (13 and m k n = qn k + rn k then the value of the optmal SO s determned based on (16. Heurstc : the heurstc has a threshold equals to fn k. Thus rn k = fn k qn k and the value of the SO s determned baed on (18. DE-DutyCon: DE-DutyCon searches at each tme perod to fnd the mnmum value of (19 whle the future cost s estmated by applyng the heurstc based control. The value of the optmal SO s determned based on (10 and the maxmum SO s bounded the search range D at each tme perod. Accordng to Remark 1 to reduce computatonal complexty the search range D s set to be 15 packets. Fg.2 shows the jont-costs of the evaluated control mechansms. The cost functon reducton of the the proposed DE- DutyCon between 60 kbps - 80 kbps are approxmately 47% energy effcency (kb/j DP optmal 1000 packet arrval rate (kbps Fg. 3. Energy effcency versus packet arrval rate. - 41% compared to the control. The jont-cost functon value of DE-DutyCon s close to that of DP optmal control. As stated n Theorem 2 the jont-cost upper bound of DE-DutyCon s that of the heurstc. The mprovement of DE-DutyCon to the heurstc s acheved by searchng the mnmum of (24 at each tme perod. Accordng to Remark 1 DE-DutyCon wll be more benefcal when devce n has large number of chld RFDs wth an exponental reducton n computatonal complexty. The energy effcency curves n Fg. 3 have an ncrease trend along wth the ncrease of traffc arrval rate. The convex change of energy effcency curve of DE-DutyCon between kbps s because the hgher dle lstenng energy consumpton due to radcal ncrease of SO value. The proposed DE-DutyCon acheves hgher energy effcency compared to the control and the heurstc from 25 kbps. After 60 kbps the number of transmtted packets the network can support s relevant stable whch means the network s saturated. Thus the energy effcency curves keep flat for all examned controls. The end-to-end delay curves n Fg. 4 have same trend wth the results n [2]. End-to-end delay of DE-DutyCon s cost functon value DP optmal 0 packet arrval rate (kbps Fg. 2. DE-DutyCon performance bound versus packet arrval rate. average end to end delay per packet (s DP optmal 0 packet arrval rate (kbps Fg. 4. End-to-end delay versus packet arrval rate. 1055

6 packet drop rato DP optmal 0 packet arrval rate (kbps Fg. 5. Packet drop rato versus packet arrval rate. lower than that of DP optmal. The end-to-end delay curve of DE-DutyCon begns to decrease after the traffc arrval rate s hgher than 30 kbps. Ths s due to the fact that a packet can only be sent out once the exstng buffered packets are cleared. Wth more generated packets by RFDs the ncreased number of dropped packets reduces the number of buffered packets whch are generated n earler tme perods. Thus the buffer tme s shortened for the packets generated n later tme perods thereby the averaged end-to-end delay s decreased. Combned wth Fg.3 t s clear that the ncrease of energy effcency s at the cost of ncreasng end-to-end delay. Fg. 5 shows the packet drop rato of the evaluated control mechansms. The packet drop rato of DE-DutyCon has close performance compared wth that of the DP optmal control. The packet drop rato of DE-DutyCon s hgher than that of the control and the. Ths s because wth hgher requrement on energy effcency DE-DutyCon reduced the length of actve perods whch leads to hgher possblty of packet been buffered n the queue. The packet drop rato s ncreased due to lmted maxmum queue length. VI. CONCLUSION In ths paper we proposed DE-DutyCon a low complexty duty cycle control for acknowledgement enabled IEEE based networks wth jont consderaton of energy effcency and end-to-end delay. The cumulatve ACK from Go-Back-N ARQ transmsson scheme s appled so as to reduce the number of ACK transmssons. The duty cycle control problem s formulated based on DP nventory control problem where the jont-cost functon demonstrated the trade-off between energy effcency and end-to-end delay. Notced that the derved DP optmal duty cycle control has a heurstc structure the low complexty DE-DutyCon s proposed based on a heurstc control. The proposed DE- DutyCon reduces the computaton complexty of DP optmal control at an exponental level. In addtonal we analytcally proved that the heurstc s the jont-cost upper bound of DE-DutyCon. Smulaton results shown that the reduced jont-cost of energy consumpton and end-to-end delay has leaded to slghtly ncrease of packet drop rato when devces have lmted buffer sze. Moreover whle DE-DutyCon acheved an exponental reducton of computaton complexty t has slghtly hgher jont-cost compared to that of the DP optmal control. REFERENCES [1] M. R. Palattella N. Accettura M. Dohler L. A. Greco and G. Bogga Traffc Aware Schedulng Algorthm for Relable Low-Power Mult- Hop IEEE e Networks n Proc. IEEE Internatonal Symposum on Personal Indoor and Moble Rado Communcatons PIMRC September [2] H. Byun and J. Yu Adaptve Duty Cycle Control wth Queue Management n Wreless Sensor Networks IEEE Transactons on Moble Computng vol. 12 no. 6 pp June [3] IEEE std Part. 15.4: Wreless Medum Access Control (MAC and Physcal Layer (PHY Specfcatons for Low-Rate Wreless Personal Area Networks (LR-WPANs IEEE Std [4] P. Park S. C. Ergen C. Fschone and A. S. Vncentell Duty- Cycle Oprmzaton for IEEE Wreless Sensor Networks ACM Transactons on Sensor Networks 10(1 12:1 12:32 November [5] S. H. Yang H. W. Tseng E. Wu and G.-H. Chen Utlzaton Based Duty Cycle Tunng MAC Protocol for Wreless Sensor Networks n Proc. IEEE Global Telecommuncatons GLOBECOM. Dallas Texas November pp [6] Z. Yuqun F. Chen-Hsang I. Demrkol and W.B. Henzelman Energy- Effcent Duty Cycle Assgnment for Recever-Based Convergecast n Wreless Sensor Networks n Proc. IEEE Global Telecommuncatons Conference GLOBECOM pp. 1-5 December [7] R. Alberola and D. Pesch. Duty cycle learnng algorthm (DCLA for IEEE beacon-enabled wreless sensor networks Ad Hoc Networks 10( June [8] Y. L K. K. Cha Y. Chen and J. Loo Low Complexty Duty Cycle Control wth Jont-optmazaton of Delay and Energy Effcency for Beacon-enabled IEEE Networks n Proc. 20th European Wreless EW 14 May [9] A. S. Tanenbaum and D. J. Wetherall Computer Networks 5th Edton Prentce Hall [10] D. P. Bertsekas Dynamc Programmng and Optmal Control 3rd Edton Athena Scentfc [11] T. R. Park T. H. Km J. Y. Cho S. Cho and W.H. Kwon Throughput and energy consumpton analyss of IEEE slotted CSMA/CA Electroncs Letters vol 41 no [12] H. Scarf The Optmalty of (ss Polces for the Dynamc Inventory Problem n Proc. of the 1st Stanford Symposum on Mathematcal Methods n the Socal Scences Stanford CA [13] CC GHz IEEE Complant and ZgBee Ready RF Transcever onlne: