Enhanced Markov Chain Model and Throughput Analysis of the Slotted CSMA/CA for IEEE under Unsaturated Traffic Conditions

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1 Enhance Markov Chain Moel an Throughput nalysis of the Slotte CSM/C for EEE uner Unsaturate Traffic Conitions Chang Yong Jung Stuent Member EEE Ho Young Hwang Stuent Member EEE Dan Keun Sung Senior Member EEE an Gang Uk Hwang Member EEE bstract n this paper we propose an analytical Markov Chain moel of the te CSM/C protocol for EEE uner unsaturate traffic conitions. Our propose Markov Chain moel reflects the characteristics of the EEE MC protocol such as a superframe structure nowlegements an retransmissions with an without limit. We evaluate the throughput performance of the te CSM/C an verify the analytical moel using simulation results. nex Terms EEE CSM/C MC Throughput. NTRODUCTON s wireless sensor networks (WSNs have been wiely eploye in our life it becomes an important issue to connect sensor noes in a simple an efficient manner with low power an low cost. The EEE stanar for lowrate wireless personal area networks (LR-WPN has been introuce to achieve these requirements []. The EEE networks can operate in either a beaconenable moe or a non-beacon-enable moe. n the nonbeacon-enable moe noes in a personal area network (PN communicate with each other accoring to an unte CSM/C protocol. On the other han in the beacon-enable moe noes communicate with each other accoring to a te CSM/C protocol base on a superframe structure. Each superframe consists of an active perio an an inactive perio. The active perio consists of a beacon perio a contention access perio (CP an a contention free perio (CFP. During the inactive perio the coorinator an noes shall not interact with each other an may enter a low-power moe. n orer to investigate the te CSM/C protocol of EEE we consier the beacon-enable moe in this paper. During the CP in the beacon-enable moe EEE has aopte a te CSM/C protocol which is ifferent from that of the EEE 802. WLN [2] [3]. n the case of the WLN the boff count value ecreases by one only if the channel is ile an it is frozen otherwise. On the Manuscript receive July ; revise January This stuy has been supporte in part by a grant Next Generation PC Project from the nstitute of nformation Technology ssessment (T. The review of this paper was coorinate by Prof. Jie Li. C. Y. Jung H. Y. Hwang an D. K. Sung are with the School of Electrical Engineering an Computer Science KST Korea. ( cyjung@cnr.kaist.ac.kr hyhwang@cnr.kaist.ac.kr an ksung@ee.kaist.ac.kr G. U. Hwang is with the Department of Mathematical Sciences KST Korea. ( guhwang@kaist.eu other han in the case of EEE the boff count value ecreases by one regarless of whether the channel is ile or busy. Thus we nee to make a new analytical moel for the CSM/C algorithm use in EEE There have been a few papers regaring this technical issue. Park et al. [4] Tao et al. [5] Pollin et al. [6] an Lee et al. [7] propose analytical Markov chain moels for the te CSM/C of the EEE MC protocol. They assume that the uration of a CP is infinite without consiering the superframe structure uner saturate traffic conitions. Since most applications in wireless sensor networks are expecte to be operate uner unsaturate traffic conitions an they use superframes incluing inactive perios in orer to reuce power consumption we nee an analytical moel consiering unsaturate traffic conitions an the superframe structure. From this point of view Ramachanran et al. [8] propose an analytical moel of the te CSM/C uner unsaturate traffic conitions but they i not consier the inactive perio in the superframe structure. Mi sić et al. [9] also propose a Markov chain moel to analyze the te CSM/C algorithm uner unsaturate traffic conitions consiering the superframe structure nowlegements an retransmission schemes of the EEE MC protocol. However they assume that there was no retransmission limit an they i not verify their analytical moel. Park et al. [4] showe that Mi sić s moel i not match with simulation results uner saturate traffic conitions. n this paper we propose an enhance Markov chain moel to observe the throughput performance of the te CSM/C algorithm uner unsaturate traffic conitions an the propose Markov chain moel reflects the characteristics of the EEE MC protocol such as a superframe structure nowlegements an retransmission schemes with an without retry limit. Through simulations we verify that the analytic results from our moel are well matche with the simulation results. n aition we show that our moel provies more accurate results than Mi sić s moel [9]. The rest of this paper is organize as follows. n Section we briefly escribe the EEE MC protocol. n Section we propose an analytical moel of the te CSM/C for the EEE MC protocol an evaluate the throughput performance of the te CSM/C. Finally we present conclusions in Section V.

2 2. OVERVEW OF THE EEE MC PROTOCOL t the start of each superframe the PN coorinator transmits a beacon frame that carries system parameters such as beacon orer (BO that etermines the length of a beacon interval (B = BO symbols an superframe orer (SO that etermines the length of a superframe uration (SD = SO symbols. n the CP perio each noe communicates with the PN coorinator an other noes using the te CSM/C. The uration of one is aunitboffp erio (efault value = 20 symbols. When a noe has a new ata frame to transmit it initializes relevant parameters such as BE (boff exponent an N B (number of boffs or boff stages which are set to macminbe (efault value = 3 an 0 respectively. n aition it selects a boff counter value uniformly from a winow [0 2 BE ]. The boff counter value is ecremente by one for each time regarless of the channel state an whenever the boff counter value is zero the noe performs carrier sensing that requires two clear channel assessments (CCs at the physical layer before transmission. f the channel is assesse to be ile at the two consecutive CCs then it transmits the ata frame. f the channel is assesse to be busy it increases the values of BE an N B by one an elays the transmission for a ranom number of time s uniformly chosen from [0 2 BE ] where BE is no more than amaxbe (efault value = 5. The above proceure is continue until the successful transmission but if the NB value is greater than macmaxcsmboffs (efault value = 4 then the CSM/C algorithm shall be terminate with a channel access failure. f either a channel access failure occurs or a frame transmission failure occurs ue to a collision the noe retries the above mentione proceure for retransmissions up to amaxframeretries (efault value = 3 times. Since the transmission of an ck frame shall commence at a bounary the uration t from the reception of the last symbol of the ata frame to the transmission of the first symbol of its ck frame is between aturnarountime (efault value = 2 symbols an aturnarountime + aunitboffperio (32 symbols that is one time can be inclue in the uration t at most. fter transmitting a ata frame the noe waits for its ck frame uring the uration specifie by the parameter macckwaitduration (efault value = 54 symbols.. NLYSS OF THE SLOTTED CSM/C N THE EEE MC PROTOCOL. Markov Chain Moels n this paper we consier a single hop wireless network consisting of a PN coorinator an n sensor noes. We assume that all noes are within the transmission range of each other an time-synchronize by PN coorinator s beacon. We also assume that there are no transmission errors an no channel sensing errors. We consier a star topology an an uplink ata transmission scenario so that transmitte frames can be lost only ue to collisions. Note that wireless sensor networks might have a ifferent topology other than a star topology but consiering a general topology is beyon the scope of this paper. Fig.. α T0 n n T0 n P c T n P n c T n T m n n T m n α P ---- α α /W0 /W0 /W W0-22 0W0-2 P P ( P D 0 D D m P P P P ( P ( P /W /W /W /W 0 /W 2 2 W-22 W-2 P P ( P P ( P ( P Markov chain moel / Wm / Wm / Wm / Wm m m2 m2 mwm-22 mwm-2 P ( P P ( P ( P We assume that the ata frame arrives at each noe accoring to a Poisson process with rate λ an that each noe can store a single ata frame. Thus when a noe has a ata frame to transmit it can not accept any more new ata frame from upper layers. We further assume that all n noes are homogeneous an accoringly the performance of all noes is ientical. For this reason we tag an arbitrary noe an call it the tagge noe. For the analysis we construct a iscrete-time Markov chain which moels the operation of the CSM/C algorithm in the tagge noe an captures the key characteristics of the EEE MC protocol such as a superframe structure nowlegements an retransmission schemes with an without retry limit. For convenience when we construct the Markov chain we consier only the time epochs where the states of the Markov chain efine below are change. The state transition iagram of the Markov chain is given in Fig.. The states of the Markov chain at time t are classifie into three types. The first type is of the form {s(t b(t w(t r(t}. Here s(t [0 m] represents the value of NB at time t where m = macmaxcsmboff. b(t represents the value of the boff counter at time t. When s(t = i b(t is in [0 W i ] where W 0 = 2 macminbe an W i = W 0 2 min(iamaxbe macminbe i m. w(t { 2} represents the remaining number of CCs to be one for transmission at time t. Thus if b(t = 0 an w(t = 2 then the tagge noe performs the first CC at time t. Similarly if b(t = 0 an w(t = the tagge noe performs the secon CC at time t. P an P in Fig. enote the probabilities that the channel is ile when performing the first CC an the secon CC respectively. Finally r(t [0 R] represents the value of the retransmission plane at time t #0 # # R

3 3 which is shown by a rectangular box in Fig. where R = amaxf rameretries. Note that our moel can simply be applie to the retransmission scheme without limit if R tens to. The secon type is of the form {Ts(t n r(t} an {Ts(t r(t}. These states represent the non-eferre an eferre transmissions in the s(t-th boff stage on the r(t-th retransmission plane at time t. Since we consier the superframe structure with finite CP an inactive perio note that the tagge noe nees to efer a transmission until the start of the next CP when the transmission can not be complete within the current CP. Pc n an Pc in Fig. enote the collision probabilities for non-eferre transmissions an eferre transmissions respectively. The thir type is of the form {D s(t r(t}. These states represent the waiting state in the s(t-th boff stage on the r(t-th retransmission plane at time t in orer for the tagge noe to efer the transmission until the next CP ue to l of the remaining s in the current CP. P is the probability that a eferre transmission occurs. n Fig. we have in fact one more state { } which represents the state where the tagge noe has no ata frame to transmit. The transition probability α is the probability that a new ata frame occurs between state transition times at the tagge noe. Now to complete the construction of the Markov chain in Fig. we nee to etermine the probabilities P P Pc n Pc P an α. s shown below we can irectly etermine the probabilities P an α but it seems to be ifficult to etermine P P Pc n an Pc irectly. To solve this problem assuming that P P Pc n an Pc are inepenent of the boff stages an retransmission planes we obtain expressions of P P Pc n an Pc in terms of the steay state probabilities of the Markov chain an solve them numerically. The etails are summarize in the following subsections. B. Probabilities P an α To compute the probability P note that the total number of time s neee for a single transmission is 2 + Nata + Nt + N. Here 2 s are inclue ue to the number of time s for performing two CCs Nata is the number of time s for the transmission of a frame Nt is the number of time s for the perio t an N is the number of time s for the transmission of an ck frame. Since the transmission is eferre ue to the l of the remaining times in a CP P is approximate by P N ef NCP = 2 + N ata + N N CP t + N where NCP is the total number of time s in a CP. Next α is the probability that a new ata frame occurs between state transition times at the tagge noe. Since the ata frame arrives accoring to a Poisson process with rate λ α is expresse as α = N CP T NCP λe λτ τ + To 0 NCP λe λτ τ 0 ( where T o = T + T CF P + T nactive + T BEP. Here T T CF P T nactive an T BEP are the urations of one time a contention free perio (CFP an inactive perio an a beacon perio respectively. Note that N CP is the N CP probability that an arbitrary time is not the last time of a CP. C. Probabilities Pc n an Pc Let b ijkl = lim t P {s(t = i b(t = j w(t = k r(t = l} b T n i l = lim t P {Ts(t n = Ti n r(t = l} b T i l = lim t P {Ts(t = Ti r(t = l} b D i l = lim t P {D s(t = D i r(t = l} i [0 m] j [0 W i ] k [ 2] l [0 R] be the stationary probabilities of the Markov chain. We assume that the probabilities P P P n c an Pc are inepenent of the boff stages an retransmission planes. Then some transition probabilities in Fig. can be expresse as b T i l = P b i2l i [0 m] l [0 R] b T n i l = C b i2l = P P ( P b i2l i [0 m] l [0 R] b i+2l = C 2 b i2l = ( P P ( P b i2l i [0 m ] l [0 R] b 02l+ = C 3 b 02l = [ m ] (Pc n C + Pc P C2 i + C2 m+ b 02l l [0 R ] where C represents the transition probability from the state where the tagge noe performs the first CC to the state where the tagge noe performs a non-eferre transmission C 2 an C 3 represent the transition probabilities between boff stages an retransmission planes respectively. Let τ n an τ be the stationary probabilities that the tagge noe carries out non-eferre an eferre transmissions respectively. Then they satisfy τ n = τ = R m l=0 i=0 R m l=0 i=0 i=0 b T n i l b T i l. Let τ be the stationary probability that the tagge noe transmits a ata frame. Then it follows that τ = τ n + τ. The collision probabilities P n c an P c for non-eferre an eferre transmissions of the tagge noe respectively are expresse as Pc n = ( τ n (n C (2 Pc = ( PCP tx (n (3

4 4 where τ n C in Eq. (2 is the conitional probability that the tagge noe performs the first CC given that a non-eferre transmission occurs. PCP tx in Eq. (3 is the probability that the tagge noe efers the transmission in a CP. To compute PCP tx note that τ N CP represents the average number of eferre transmissions in a CP at the tagge noe where is the average number of time s staying in an arbitrary state of the Markov chain in Fig.. is given by = ( τ + τ( (Nata + Nt + N +τ (Nata + Ntimeout where Ntimeout is the number of time s until the ck timer expires ue to no ck. Since there is at most one eferre transmission in a CP for the tagge noe we have P tx CP CP = τ N. (4 From Eqs. (2 an (3 the overall collision probability for the tagge noe is expresse as = Pc n τ n + P τ c. τ + τ n τ + τ n D. Probabilites P an P To obtain the probabilities P an P that the channel is ile at the first an secon CCs respectively we nee to compute the average number of busy an ile time s in a CP from the viewpoint of the network or channel. The average number of busy s in a CP is N b CP = [N s tx net (N ata + N + N c tx net N ata] (5 where Nnet s tx an Nnet c tx are the average numbers of successful transmissions an collie transmissions in the network except the tagge noe in a CP respectively an can be expresse as N s tx net = (n [N N c tx net = (n [ n 2 n tx N noe n tx noe P n i= c n 2 + N tx Pc n n 2 n 2i + Nnoe tx i + i= noep c n 2 ] (6 Pc ] n 2i i + n tx where Nnoe is the average number of non-eferre transmissions for a non-tagge noe in a CP an given by n tx Nnoe an N tx noe is the average number of eferre transmissions for a non-tagge noe in a CP an given by Nnoe tx = τ N CP which is the same as P tx CP in Eq. (4. n aition Pc n n 2i is the conitional collision probability that uner the given conition that one noe among n non-tagge noes carries out a non-eferre transmission i noes among n 2 other non-tagge noes carry out noneferre transmissions an i + noes accoringly collie an is expresse as = τ nn CP P n c n 2i = ( n 2 i (7 ( τ n C i ( τ n C n 2 i. n the same manner Pc n 2i for eferre transmissions is expresse as ( n 2 Pc n 2i = P tx i CP ( P tx CP n 2 i. i Since Pc n n 2 in Eq. (6 is the probability that a noneferre transmission of one noe among n non-tagge n tx noes is successful (n Nnoe n n 2 in Eq. (6 is the average number of successful non-eferre transmissions for the network except the tagge noe in a CP. Regaring Eq. (7 we consier the case that i+ non-tagge noes collie. From the viewpoint of the network or channel it is consiere as a single collie transmission. Therefore we can say that each of i+ non-tagge noes involve in the collision contributes the amount of i+ portion to the single n tx n 2 P collie transmission. (n N n c n 2i noe i= i+ in Eq. (7 is the average number of non-eferre collie transmissions for the network except the tagge noe in a CP. n the same way (n N tx in Eq. (6 an n 2 noe i= noe P c n 2 (n N tx P c n 2i i+ in Eq. (7 are the average numbers of successful eferre transmissions an collie eferre transmissions for the network except the tagge noe in a CP. Note that the first an secon CCs for non-eferre transmissions can not be performe at any time s of a CP but can be performe in a limite range of a CP. The length of the limite range is NCP Nef. N CP b in Eq. (5 can inclue the number of busy s in the range where the tagge noe oes not perform the first CC ue to eferre transmissions. Since the number of those busy s is very small compare with that of s that the tagge noe can perform the first CC that is (NCP N ef we assume that this effect can be negligible. Therefore the probability P is approximate as (NCP P N ef N CP b NCP N ef. (8 Next we observe that the secon CC can occur when the b i channel is ile at the first CC. Note that NCP is the average number of s where the channel is ile but it is busy at the next. Since those s can just occur before b i ata or ck frame transmissions N is expresse as CP b i NCP = [Nnet s tx ( + Nt + Nnet c tx ]. Thus the probability P is approximate as P (NCP N ef N b b i CP NCP P NCP N ef. (9 Finally each parameter of P P P n c an P c can be numerically solve from Eqs. (2 (3 (8 (9 an the normalization conition of the Markov chain as mentione before. E. Throughput nalysis S = Let S be the system throughput. Then S is compute as nd R τ( E[T P ayloa ] ( τe[t ] + τ( E[T suc ] + τ E[T col ]

5 5 where D R is ata rate (bps E[T P ayloa ] is the average uration of a ata frame payloa an E[T suc ] an E[T col ] are the average urations of a successful transmission an a collie transmission respectively an are expresse as E[T suc ] = T Nsuc E[T col ] = T Ncol = T (N ata + N t + N = T (N ata + N timeout. E[T ] is the average uration of a non-transmission. Note that the transmission s inclue s of ata an ck frame transmissions an t for successful transmissions an s of ata frame transmissions an waiting until the ck timer expires for collie transmissions. To compute E[T ] we first consier the average number of transmission s of tx noe the tagge noe in a CP enote by N. Since the average number of transmissions of the tagge noe in a CP is given by N tx it follows that noe = N n tx noe noe = τn CP + N tx tx Nnoe = Nnoe[( tx Nsuc + Ncol ]. Observing that among the non-transmission s the last in a CP is followe by a CFP perio an inactive perio an a beacon perio E[T ] is given by E[T ] = (N CP N tx noe N CP where T o is given in Eq. (. F. Performance Evaluation T + T o N tx noe n orer to evaluate the throughput performance of the EEE MC protocol we consier 2.4 GHz physical layer an D R = 250 (kbps. ata frame consists of PHY an MC heaers an MC payloa. Let the size of ata frames (L ata be 32 (bytes the size of PHY an MC heaers of ata frames be 5 (bytes an the size of ck frames (L be (bytes. The number of noes in a network is 0. Each perio is set as T BEP = 60 2 SO (symbols T CP = SO (symbols T CF P = SO (symbols an T nactive = BO SO (symbols respectively. Fig. 2 shows the normalize system throughput S/D R when we vary the normalize offere loa per noe λ n (= λl ata /D R an the values of SO an BO. n Fig. 2 we consier the case of retransmissions without limit to compare our results with Mi sić s results. Note that we can obtain the results for retransmissions without limit by letting the maximum number of retransmissions R go to. s shown in Fig. 2 Mi sić s moel fails to match with the simulation results as the normalize offere loa λ n increases. There are some reasons for the mismatch of Mi sić s moel. First they i not obtain the collision probabilities for non-eferre an eferre transmissions separately. Secon when calculating the average number of busy s in a CP to obtain the probability that the channel is ile at the first CC they i not consier the fact that there are no nowlegements for collie transmissions an they i not properly obtain the Normalize System Throughput (S/D R Misic s moel SO=BO= (simulation SO=BO=2 (simulation SO=BO=3 (simulation SO=BO= (Our moel SO=BO=2 (Our moel SO=BO=3 (Our moel SO=BO= (Misic s moel SO=BO=2 (Misic s moel SO=BO=3 (Misic s moel Normalize Offere Loa per Noe (λ n Fig. 2. Normalize system throughput for varying SO an BO values an retransmissions without limit average number of transmissions in a CP in the viewpoint of the network. Thir when obtaining the probability that the channel is ile at the secon CC the conition that the channel is ile at the first CC was not consiere in Mi sić s moel. These factors affect inaccurate results of Mi sić s moel. On the other han our moel is well-matche with simulation results. For the normalize offere loa λ n smaller than 0 2 since most of transmissions are successful ue to low traffic Mi sić s moel seems to be well-matche with simulation results. s the value of BO increases with a fixe SO value the normalize system throughput ecreases for the value of λ n larger than 0 2. This is because the proportion of a CP is getting smaller ue to longer inactive perio uner these traffic conitions. On the other han for the value of λ n smaller than 0 3 there is almost no ifference in the system throughput regarless of the uration of a beacon interval (B ue to low traffic. Fig. 3 shows the normalize system throughput S/D R when we vary the normalize offere loa per noe λ n an the ata size. n this case we consier retransmissions with limit (R = 3. Let the values of SO an BO be set to. s the ata size increases the normalize system throughput increases. Since most of transmissions are successful in low traffic loa such as λ n smaller than 0 2 large ata frame sizes yiel high throughput performance. n high traffic loa such as λ n larger than 0 2 if a noe transmits a large ata frame it occupies the channel for a long time. This fact causes low transmission probability an low collision probability which results in high throughput performance. Fig. 4 shows the normalize system throughput S/D R when we vary the number of noes (n an the value of the retransmission limit (R. n this case the normalize offere loa per noe λ n is set to 0.03 the values of SO an BO are set to an the maximum value of boff stages m is set to 0. s the value of R increases the normalize system throughput slightly increases for low traffic loa such as n smaller than 6. Retransmissions ecrease the probability of frame rops. However for high traffic loa such as n larger than 2 the system throughput ecreases as the value of R

6 6 Normalize System Throughput(S/D R Data Size = 22 bytes (sim 0 3 Data Size = 22 bytes (ana Data Size = 32 bytes (sim Data Size = 32 bytes (ana Data Size = 52 bytes (sim Data Size = 52 bytes (ana Normalize Offere Loa per Noe (λ n Fig. 3. Normalize system throughput for varying ata sizes an retransmissions with limit [3] G. Bianchi EEE 802. Saturation Throughput nalysis EEE Commun. Lett. Vol. 2 No. 2 pp Dec [4] T. R. Park T. H. Kim J. Y. Choi S. Choi an W. H. Kwon Throughput an Energy Consumption nalysis of EEE Slotte CSM/C EE Electr. Lett. Vol. 4 No. 8 pp Sep [5] Z. Tao S. Panwar D. Gu an J. Zhang Performance nalysis an a Propose mprovement for the EEE Contention ccess Perio Proc. EEE Wireless Communications an Networking Conference (WCNC 2006 Las Vegas Nevaa U.S. pril [6] S. Pollin M. Ergen S. C. Ergen B. Bougar L. V. Perre F. Catthoor. Moerman. Bahai an P. Varaiya Performance nalysis of Slotte Carrier Sense EEE Meium ccess Layer Proc. EEE GLOBECOM 2006 San Francisco California U.S. Nov Dec [7] T. Lee H. R. Lee an M. Y. Chung MC Throughput Limit nalysis of Slotte CSM/C in EEE WPN EEE Commun. Lett. Vol. 0 No. 7 pp July [8]. Ramachanran. K. Das an S. Roy nalysis of Contention ccess Perio of EEE UWEE Technical Report UWEETR Feb [9] J. Mi sić an V. B. Mi sić ccess Delay for Noes with Finite Buffers in EEE Beacon Enable PN with Uplink Transmissions Compuer Communications Vol. 28 No. 0 pp June Normalize System Throughput (S/D R R=3 (ana R=3 (sim R=5 (ana 0.02 R=5 (sim R=7 (ana R=7 (sim Number of Noes (n Fig. 4. Normalize system throughput for varying the value of the retransmission limit (R increases. smaller value of R causes earlier frame rops. Since these frame rops reuce the number of contening noes in the network the collision probability ecreases. This yiels higher system throughput. V. CONCLUSONS n this paper we propose an analytical Markov chain moel of the te CSM/C in the EEE MC protocol consiering a superframe structure nowlegements an retransmissions with an without limit uner unsaturate traffic conitions. With the propose moel we evaluate the throughput performance of the te CSM/C. We valiate our propose analytical moel by simulation. REFERENCES [] EEE Specification Wireless Meium ccess Control (MC an Physical Layer (PHY Specifications for Low Rate Wireless Personal rea Networks (LR-WPNs Oct [2] EEE 802. Specification Wireless LN Meium ccess Control (MC an Physical Layer (PHY Specificaions June