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A Generalized Markov Chain Model for Effective Analysis of Slotted IEEE 802.15.4 Pangun Park, Piergiuseppe Di Marco, Pablo Soldati, Carlo Fischione, Karl Henrik Johansson Abstract—A generalized analysis of the IEEE 802.15.4 medium has been studied in [7] for the slotted carrier sense multi- access control (MAC) protocol in terms of reliability, delay and ple access with collision avoidance (CSMA/CA) mechanism energy consumption is presented. The IEEE 802.15.4 exponential of IEEE 802.15.4. However, the energy consumption and backoff process is modeled through a Markov chain taking into account retry limits, acknowledgements, and unsaturated trafﬁc. throughput analysis under unsaturated trafﬁc show a weak Simple and effective approximations of the reliability, delay and matching with the simulation results. energy consumption under low trafﬁc regime are proposed. It In this paper we present an accurate model and analysis is demonstrated that the delay distribution of IEEE 802.15.4 of IEEE 802.15.4 MAC protocol in terms of reliability, de- depends mainly on MAC parameters and collision probability. lay and energy consumption. Unlike previous related works, In addition, the impact of MAC parameters on the performance metrics is analyzed. The analysis is more general and gives more we propose a generalized Markov model of the exponential accurate results than existing methods in the literature. Monte backoff process with retry limits and acknowledgements under Carlo simulations conﬁrm that the proposed approximations unsaturated trafﬁc regime. We show that our Markov chain offer a satisfactory accuracy. gives an accurate model of the reliability, delay and energy Keywords: IEEE 802.15.4 standard, Markov chain model, consumption of IEEE 802.15.4. Evaluating these performance Retry limits and acknowledgement, Model approximation. metrics asks in general for heavy computations. As such, these expressions may not be directly applied to optimize the IEEE 802.15.4 MAC parameters by an in-network processing I. I NTRODUCTION of the nodes [19] since complex computations are out of Wireless Sensor Networks (WSNs) have revolutionized the reach for today’s sensing devices. To overcome this problem, world of distributed systems and have enabled several new we devise a simpliﬁed and effective method that drastically applications. The IEEE 802.15.4 standard has received con- reduces the computation complexity while ensuring a satis- siderable attention in academy and industry as a possible low factory accuracy. Furthermore, we use these results to analyze data rate and low power protocol for WSNs [1]. Understanding the performance of IEEE 802.15.4 as functions of the MAC reliability, delay and energy consumption of IEEE 802.15.4 parameters and collision probability. Monte Carlo simulations networks is essential to characterize the fundamental limita- conﬁrm the validity of our analysis. tions of this protocol and optimize its parameters. The remainder of this paper is as follows. In Section II, we Several simulations-based studies e.g., [2]–[4], as well as describe the slotted CSMA/CA mechanism of IEEE 802.15.4 more recent analytical works, e.g., [5]–[8], investigate the standard. We propose a Markov chain model of CSMA/CA delay, throughput, and energy consumption of IEEE 802.15.4. with retry limits and unsaturated trafﬁc in Section III. In Sec- Most of the theoretical studies are based on the Markov tion IV we present an accurate analysis of the reliability, delay model initially proposed by Bianchi [9] for the IEEE 802.11 and energy consumption. Then, in Section V an approximated standard [10]. The model describes the basic functionalities analysis is developed. In Section VI, we validate our analysis of IEEE 802.11 through a Markov chain under saturated by Monte Carlo simulations. Section VII concludes the paper. trafﬁc and ideal channel conditions. Extensions of this model have been used to analyze the packet reception rate [11], II. OVERVIEW OF THE IEEE 802.15.4 the delay [12], [13], the medium access control (MAC) layer service time [14], [15] and throughput [16], [17] of IEEE The IEEE 802.15.4 standard speciﬁes MAC and physical 802.11. A simple and effective analysis of delay distribution (PHY) layers. The CSMA/CA is used along with a slotted Bi- is studied for IEEE 802.11 in [18]. nary Exponential Backoff (BEB) scheme to reduce collisions due to simultaneous node transmissions. The standard deﬁnes Inspired by Bianchi’s work, a Markov model for IEEE two channel access modalities: the Beacon-enabled modality, 802.15.4 and an extension with acknowledgement mechanism which uses a slotted CSMA/CA and exponential backoff, and have been proposed in [5] and [8], respectively. A modiﬁed a simpler unslotted CSMA/CA without beacons. Markov model including retransmissions with ﬁnite retry limit Consider a node trying to transmit. In slotted CSMA/CA The authors are with the ACCESS Linnaeus Center, Electrical En- of IEEE 802.15.4, ﬁrst the MAC sub-layer initializes four gineering, Royal Institute of Technology, Stockholm, Sweden. E-mails: variables, i.e., the number of backoffs (NB=0), contention {pgpark,pidm,soldati,carlofi,kallej}@ee.kth.se. window (CW=2), backoff exponent (BE=macMinBE) and This work was supported by the EU project FeedNetBack, the Swedish Re- search Council, the Swedish Strategic Research Foundation, and the Swedish retransmission times (RT=0). Then the MAC sub-layer de- Governmental Agency for Innovation System. lays for a random number of complete backoff periods in 978-1-4244-5113-5/09/$25.00 c 2009 IEEE q0 Long frame ACK Short frame ACK −2 ,0 , n −2 , Lc − 1, n Q0 q0 t ack LIFS t ack SIFS q0 1 − q0 q0 Q1 1 − q0 1 − q0 (a) Acknowledged transmission −1, Ls − 1,0 QL0 −1 1 − q0 Long frame Short frame −1,0 ,0 1 LIFS SIFS W0 1 − Pc 1− β 1 −α 0 ,−1,0 0 ,0 ,0 0 ,1,0 0 ,W0 − 2 ,0 0 ,W0 − 1,0 (b) Unacknowledged transmission Pc β α 1 1 Fig. 1. IFS data transmission mechanism with and without acknowledgement. 1 W1 1 − Pc 1− β 1 −α 1,−1,0 1,0 ,0 1,1,0 1,W1 − 2 ,0 1,W1 − 1,0 β α Pc the range [0, 2BE − 1] units. When the backoff period is 1 zero, the node performs the ﬁrst clear channel assessment Wm (CCA). If two consecutive CCAs are idle, then the node 1 − Pc 1− β 1 −α m ,−1,0 m ,0 ,0 m ,1,0 m ,Wm − 2 ,0 m ,Wm − 1,0 β α commences the packet transmission. If either of the CCA Pc fails due to busy channel, MAC sublayer will increase the −1, Ls − 1, n −2 ,0 ,0 −2 , Lc ,0 value of both NB and BE by one up to a maximum value 1 W0 macMaxCSMABackoffs and macMaxBE, respectively. Hence, −1,0 , n 1 − Pc 1− β 0 ,−1, n 1 −α 0 ,0 , n 0 ,1, n 0 ,W0 − 2 , n 0 ,W0 − 1, n the value of NB and BE depend on the number of CCA Pc β α 1 failures of a packet. Once the BE reaches macMaxBE, it W1 remains at the value of macMaxBE until it is reset. If NB 1 − Pc 1− β 1 −α 1,−1, n 1,0 , n 1,1, n 1,W1 − 2 , n 1,W1 − 1, n β α exceeds macMaxCSMABackoffs, then the packet is discarded Pc due to the channel access failure. Otherwise the CSMA/CA 1 algorithm generates a random number of complete backoff 1 − Pc 1− β 1 −α Wm periods and repeat the process. Here, the variable macMaxC- m ,−1, n m ,0 , n m ,1, n m ,Wm − 2 , n m ,Wm − 1, n β α Pc SMABackoffs represents the maximum number of times the Fig. 2. Markov chain model for CSMA/CA algorithm for IEEE 802.15.4 CSMA/CA algorithm is required to backoff. If channel access is successful, the node starts transmitting packets and waits for acknowledgement (ACK). The reception of the corresponding ACK is interpreted as successful packet transmission. If the scenario for many WSN applications. We study the behavior node fails to receive ACK due to collision or ACK timeout, the of a single node by using a Markov model. variable RT is increased by one up to macMaxFrameRetries. Let s(t), c(t) and r(t) be the stochastic processes represent- If RT is less than macMaxFrameRetries, the MAC sublayer ing the backoff stage, the state of the backoff counter and the initializes two variables CW=0, BE=macMinBE and follows state of retransmission counter at time t, respectively, experi- the CSMA/CA mechanism to re-access the channel. Otherwise enced by a node to transmit a packet as indicated in Fig. 2. the packet is discarded due to the retry limit. Note that the By assuming independent probability that nodes start sensing, default MAC parameters are macMinBE = 3, macMaxBE = the stationary probability τ that a node attempts a ﬁrst carrier 5, macMaxCSMABackoffs = 4, macMaxFrameRetries = 3. sensing in a randomly chosen time slot is constant and inde- To account for the data processing time required at the MAC pendent of other nodes, and the tuple (s(t), c(t), r(t)) is a three sublayer, two successive frames transmitted from a device are dimensional Markov chain. We denote the MAC parameters by separated by at least an Inter-Frame Spacing (IFS) period; if W0 2macMinBE , m0 macMinBE, mb macMaxBE, m the ﬁrst transmission requires an acknowledgment, the sepa- macMaxCSMABackoffs, n macMaxFrameRetries. The states ration between the ACK frame and the second transmission is from (i, Wm − 1, j) to (i, W0 − 1, j) represent the backoff at least an IFS period. Fig. 1 illustrates the IFS period of data states. States (Q0 , . . . , QL0 −1 ) consider the idle state when the frame with and without ACK. Note that the waiting time to packet queue is empty and the node is waiting for new packet receive ACK is in the range aTurnaroundTime (12 symbols) arrivals. Note that the idle states (Q0 , . . . , QL0 −1 ) take into to aTurnaroundTime + aUnitBackoffPeriod (12 + 20 symbols). account the unsaturated trafﬁc condition. States (i, 0, j) and The IFS period depends on the length of the transmitted data (i, −1, j) represent CCA1 and CCA2, respectively. Let α be frames. See [1] for further details. the probability that CCA1 is busy, β the probability that CCA2 is busy. Let q0 the probability of going back to the idle state III. M ARKOV C HAIN M ODEL Q0 , and let Pc be the probability that a transmitted packet en- counters a collision. The states (−1, k, j) and (−2, k, j) model In this section, we propose a generalized analytical model the successful transmission and packet collision, respectively. of the slotted CSMA/CA mechanism of beacon enabled IEEE The state transition probabilities associated with the Markov 802.15.4 with retry limits for each packet transmission. chain of Fig. 2 are We consider a star network with a personal area network (PAN) coordinator, and N nodes with beacon-enabled slotted P (i, k, j|i, k + 1, j) = 1, for k ≥ 0 , (1) CSMA/CA and ACK. All nodes contend to send data to the PAN coordinator, which is the data sink. Assume that the α + (1 − α)β P (i, k, j|i − 1, 0, j) = , for i ≤ m , (2) network generates an unsaturated trafﬁc, which is a natural Wi (1 − α)(1 − β)Pc From Eqs. (8), (9), (10), we have P (0, k, j|i, 0, j − 1) = , for j ≤ n , (3) W0 m Wi −1 n P (Q0 |m, 0, j) = q0 (α + (1 − α)β), for j < n , (4) bi,k,j (12) P (Q0 |i, 0, n) = q0 (1 − α)(1 − β), for i < m , (5) i=0 k=0 j=0 m n W +1 i P (Q0 |m, 0, n) = q0 , (6) = (α + (1 − α)β)i b0,0,j 1 − q0 i=0j=0 2 P (0, k, 0|Q0 ) = , for k ≤ W0 − 1 . (7) b0,0,0 1−(2x)m+1 1−xm+1 1−y n+1 W0 2 1−2x W0 + 1−x 1−y if m ≤ mb − m0 Eq. (1) is the decrement of backoff counter, which happens with probability 1. Eq. (2) represents the probability of ﬁnding = mb −m0 +1 mb −m0 +1 b 1−(2x) busy channel either in CCA1 or CCA2 and of selecting a 0,0,0 2 1−2x W0 + 1−x 1−x + state uniformly the in the next backoff stage. Eq. (3) gives (2mb + 1)xmb −m0 +1 1−xm−mb +m0 1−y n+1 1−x 1−y the unsuccessful transmission probability after ﬁnding an idle channel in both CCA1 and CCA2, and a node picks uniformly otherwise, a state in the next retransmission stage. Eqs. (4) and (5) where x = α + (1 − α)β and y = Pc (1 − xm+1 ). Similarly, represent the probability of going back to the idle stage due to m n m n the channel access failure and retry limits, respectively. Eq. (6) bi,−1,j = (1 − α)(α + (1 − α)β)i b0,0,j is the probability of going back to the idle stage at backoff i=0j=0 i=0j=0 counter m and retransmission stage n, as function of the trafﬁc 1 − xm+1 1 − y n+1 conditions q0 . Eq. (7) models the probability of going back to = (1 − α) b0,0,0 , (13) 1−x 1−y the ﬁrst backoff stage from the idle stage. In the following, we use Eqs. (1)–(7) to compute the stationary distribution of and the Markov chain. n Ls −1 Lc −1 Let bi,k,j = limt→∞ Pr(s(t) = i, c(t) = k, r(t) = j), i ∈ b−1,k,j + b−2,k,j (14) j=0 k=0 k=0 (−2, m), k ∈ (−1, max(Wi − 1, Ls − 1, Lc − 1)), j ∈ (0, n) 1 − y n+1 be the stationary distribution of the Markov chain where = (Ls (1 − Pc ) + Lc Pc )(1 − xm+1 ) b0,0,0 . Ls , Lc are the time period for successful transmission and 1−y packet collision, respectively. Next, we derive the closed form By considering that the successful transmission and the failure expression for such distribution chain. Owing to the chain events are due to the limited number of backoff stages m and regularities and Eqs. (1)– (7), we have the retry limits n, the idle state probability is Wi − k n bi,k,j = bi,0,j , (8) Q0 =q0 QL0 −1 + q0 (α + (1 − α)β) bm,0,j Wi j=0 where m m n + Pc (1 − β) bi,−1,n + (1 − Pc ) (1 − β) bi,−1,j i=0 i=0j=0 2i W0 i ≤ mb − m0 Wi = q0 xm+1 (1 − y n+1 ) 2mb −m0 W0 i > mb − m0 . = + Pc (1 − xm+1 )y n 1 − q0 1−y From Eq. (2), for i ≤ m we obtain (1 − xm+1 )(1 − y n+1 ) +(1 − Pc ) b0,0,0 , (15) i 1−y bi,0,j = (α + (1 − α)β) b0,0,j . (9) where L0 is the idle state length without generating packets L0 −1 From Eq. (3), b0,0,j is rewritten as follows and l=0 Ql = L0 Q0 . Note that Eqs. (12)–(15) give the state values bi,k,j as a function of b0,0,0 . By replac- m ing Eqs. (12)–(15) in the normalization condition given by b0,0,j = (1 − α)(1 − β)Pc bi,0,j−1 (10) i=0 Eq. (11), we obtain the expression for b0,0,0 . m j = (1 − α)(1 − β)Pc (α + (1 − α)β)i b0,0,0 . i=0 IV. ACCURATE A NALYSIS In this section, we derive the accurate expressions of the By the normalization condition, we know that reliability, delay and energy consumption offered by IEEE m Wi −1 n m n 802.15.4 by using the Markov chain developed in the previous bi,k,j + bi,−1,j section. i=0 k=0 j=0 i=0j=0 n Ls −1 Lc −1 L0 −1 + b−1,k,j + b−2,k,j + Ql = 1 . (11) A. Reliability j=0 k=0 k=0 l=0 To derive the probability of successful packet reception, or We next derive the expressions of each term in Eq. (11). reliability, we derive ﬁrst the probability τ that a node attempts a ﬁrst carrier sensing (CCA1) in a randomly chosen time slot B. Delay is The average delay for a successfully received packet is m n 1−xm+1 1−y n+1 deﬁned as the time interval from the instant the packet is at τ= bi,0,j = 1−x 1−y b0,0,0 . (16) the head of its MAC queue and ready to be transmitted, until i=0 j=0 an ACK for such a packet is received. If a packet is dropped The probability τ depends on the probability Pc that a trans- due to either the limited backoffs m or the ﬁnite retry limit mitted packet encounters a collision, the probability α that n, its delay is not included into the average delay. CCA1 is busy, and the probability β that CCA2 is busy. We Let Dj be the event that a node sends a packet successfully study these three probabilities next. at the jth time. Then, from the Marov model, the random The term Pc is the probability that at least one of the N − 1 variables (Dj − Dj−1 ) and (Dj+1 − Dj ) are independent. remaining nodes transmits in the same time slot. If all nodes Let Th,i be the random time needed to obtain two successful transmit with probability τ , Pc is CCAs from the selected backoff counter value in backoff level i. Recalling from Section II, a node transmits the packet when Pc = 1 − (1 − τ )N −1 , the backoff counter is 0 and two successful CCAs occur. The transmission may be successful with probability 1 − Pc , where N is the number of nodes. Similarly to [8], we derive or collide with probability Pc . The total delay D to have a the busy channel probabilities α and β as follows. Since successful transmission within n unsuccessful attempts is n α = α1 + α2 , (17) D= 1(Aj |At ) Dj , j=0 where α1 is the probability of ﬁnding channel busy during j CCA1 due to data transmission, namely where Dj = Ls + j Lc + h=0 Th , Th is the backoff stage delay, Ls and Lc are the time periods for successful packet α1 = L(1 − (1 − τ )N −1 )(1 − α)(1 − β) , transmission and collided packet transmission, respectively. The event Aj denotes the occurrence of a successful packet and α2 is the probability of ﬁnding the channel busy during transmission at time j + 1 given j previous unsuccessful CCA1 due to ACK transmission, which is transmissions, whereas the event At denotes the occurrence of a successful packet transmission within n attempts. By N τ (1 − τ )N −1 α2 = Lack (1 − (1 − τ )N −1 )(1 − α)(1 − β) , knowing the time duration of ACK frame, ACK timeout, IFS, 1 − (1 − τ )N data packet length and header duration, we compute Ls , Lc as where Lack is the length of the ACK. Finally, Ls = L + tack + Lack + IF S , 1 − (1 − τ )N −1 + N τ (1 − τ )N −1 Lc = L + tm,ack , β= . (18) 2 − (1 − τ )N + N τ (1 − τ )N −1 where L is the total length of packet including overhead and The expressions of the carrier sensing probability τ and the payload, tack is ACK waiting time, Lack is the length of ACK busy channel probabilities α and β form a system of non-linear frame, IF S is Inter-Frame Spacing and tm,ack is the timeout equations that can be solved through a numerical method. of ACK, see the details in Section II and [1]. We then have In slotted CSMA/CA, packets are discarded due to two j Pc (1 − xm+1 )j Pr(Aj |At ) = , reasons: (i) channel access failure (ii) retry limits. Channel n m+1 ))k k=0 (Pc (1 − x access failure happens when a packet fails to obtain idle j 1 − Pc (1 − xm+1 ) Pc (1 − xm+1 )j channel in two consecutive CCAs within m + 1 backoffs. = n+1 (22) Furthermore, a packet is discarded if the transmission fails 1 − (Pc (1 − xm+1 )) due to repeated collisions after n + 1 attempts. Following the where Pc is the collision probability per sending attempt and Markov model presented in Fig. 2, the probability that the (1 − xm+1 ) is the probability of successful channel accessing packet is discarded due to channel access failure is within the maximum number of m backoff stages. Note that the probability of the event Aj is normalized by considering n xm+1 (1 − y n+1 ) all the possible events of successful attempts At . Hence, the Pcf = x bm,0,j = . (19) j=0 1−y expected value of D is n The probability of a packet being discarded due to retry limits E[D] = Pr(Aj |At ) E[Dj ] , (23) is j=0 m j Pcr = Pc (1 − β)bi,−1,n = y n+1 . (20) where E[Dj ] = Ts + j Tc + h=0 E[Th ] . i=0 By following a similar approach as the one for the char- acterization of D, we see that the total backoff delay Th is Therefore, by using Eq. (19) and (20), the reliability is given modelled by by m Th = 1(Bi |Bt ) Th,i , R = 1 − Pcf − Pcr . (21) i=0 where where Pi , Psc , Pt , Pr and Psp are the average energy con- i sc i b sumption in idle-listen, channel sensing, transmit, receiving, Th,i = 2 Tsc + Th,k + Th,k , (24) and sleep states, respectively. We assume that the radio is k=1 k=0 i set in idle-listen state during the backoff stages and the sc and where 2Tsc is the successful sensing time, k=1 Th,k timeout of ACK, tm,ack = Lack + 1, in time units Sb . In is the unsuccessful sensing time due to busy channel during Eq. (27), the ﬁrst and second terms take into account the i b CCA, and k=0 Th,k is the backoff time. The event Bi energy consumption during idle backoff state and channel denotes the occurrence of a busy channel for i-th times, and sensing state, respectively. The third, fourth and ﬁfth terms then of idle channel at the i + 1th time. By considering all the consider the energy consumption of packet transmission stage. possibilities of busy channel during two CCAs, the probability The last term is the energy consumption of idle stage without of Bi is conditioned on the successful sensing event within m packet generation. By substituting Eqs. (12)–(15) to Eq. (27), attempts Bt , given that the node senses an idle channel in we obtain the average energy consumption in closed form. CCA. It follows that 2i k V. A PPROXIMATED A NALYSIS k=1 Cαβ (i) Pr(Bi |Bt ) = m , (25) In previous sections we presented a generalized Markov k=0 Cαβ (k) chain model of the CSMA/CA mechanism, and we gave where Cαβ (i) gives all possibilities of choosing i elements the expressions of the reliability, delay for successful packet from a set of busy channel probabilities {α, (1 − α)β} and delivery, and energy consumption. These expressions are based k Cαβ (i) is one of the elements in the set Cαβ (i). Hence, the on the nonlinear Eqs. (16)-(18), which must be solved through total number of combinations for i elements is equal to 2i a numerical method. However, these expressions may be and Cαβ (i) returns one combination out of 2i . The expected k computationally demanding and inadequate for usage in sensor backoff delay is devices. For instance, a node may need to solve locally m an optimization problem where the cost function is given E[Th ] = Pr(Bi |Bt ) E[Th,i ] . i=0 by the energy (27), and the constraints are imposed by the Note that E[Th,i ] follows from Eq. (24). The unsuccessful reliability (21) and delay (23) expressions. We argue that i sc sensing time k=1 Th,k in Eq. (24) is related to the picking simpler expressions for such an optimization problem are of i elements in the set Cαβ (i). For instance, the combination needed for an in-network solution [19]. (α, α) returns the unsuccessful sensing delay Tsc + Tsc and In this section, we approximate the accurate model and the combination (α, (1 − α)β) gives the unsuccessful sensing analysis developed in Section III by simpler expressions. The b delay Tsc + 2Tsc . Furthermore, the backoff time Th,k of k un- key idea is that sensor nodes can easily estimate the busy successful sensing tries is uniformly distributed in [0, Wk − 1]. channel probabilities α, β and the probability τ . Therefore, Hence, we can rewrite the expected backoff delay E[Th ] as we propose some approximated expressions where nodes exploit local measurements to evaluate reliability, delay, and m i Wk − 1 E[Th ] =2 Tsc + Pr(Bi |Bt ) Sb energy consumption, rather than solving nonlinear equations. k=0 k=0 2 In the following, we give these approximations. Recall that we Tsc m 2 i deﬁned x = α + (1 − α)β and y = Pc (1 − xm+1 ). + m k k C k (i)(Nα (i) + 2 Nβ (i)) , ˆ k=0 Cαβ (k) i=0 k=1 αβ A. Reliability k where Sb is the time unit aUnitBackoffPeriod, and Nα (i), To approximate the reliability expression of Eq. (21), we k Nβ (i) return the number of α and (1−α)β of the combination ˆ ﬁrst consider the carrier sensing probability τ of Eq. (16), k Cαβ (i), respectively. where the state b0,0,0 follows from the normalization condition By a similar approach, the variance of the total delay is in Eq. (11). Given z ≥ 0, note that n σ 2 [D] = Pr(Aj |At ) E[Dj ] − (E[D])2 . 2 (26) 1 − z m+1 ≈1+z if z 1 (28) j=0 1−z C. Energy Consumption By using this approximation, Eq. (12) is approximated as m Wi −1 n b0,0,0 By considering the Markov chain model given in Fig. 2, the bi,k,j ≈ [(1 + 2x)W0 + 1 + x] (1 + y) average energy consumption is given as follow i=0 k=0 j=0 2 m Wi −1 n m n (29) Etot =Pi bi,k,j + Psc (bi,0,j + bi,−1,j ) i=0 k=1 j=0 i=0 j=0 Similarly, Eq. (13) is approximated by n L−1 n m n + Pt (b−1,k,j + b−2,k,j ) + Pi (b−1,L,j bi,−1,j ≈ b0,0,0 (1 − α)(1 + x)(1 + y) (30) j=0 k=0 j=0 i=0 j=0 n L+Lack +1 and Eq. (14) is approximated by + b−2,L,j ) + (Pr b−1,k,j + Pi b−2,k,j ) n Ls −1 j=0 k=L+1 Lc −1 b−1,k,j + k=0 b−2,k,j L0 −1 j=0 k=0 + Psp Ql , (27) l=0 ≈ b0,0,0 Ls (1 − xm+1 )(1 + y), (31) where we assume that the successful packet service time is results. The approximated distribution is obtained by using a equal to the packet collision time, namely Ls = Lc . Finally, moment matching approach. Namely, the discrete probability let K0 = L0 q0 /(1 − q0 ), then the approximate idle stage of distribution function of the delay is approximated by known Eq. (15) is distributions whose average and variance is matched to the L0 −1 actual average and variance of the delay. More speciﬁcally, Ql ≈ b0,0,0 K0 1 + y + Pc (1 − xm+1 )(y n − y − 1) . let Da be an approximating delay distribution having average l=0 2 2 µDa and variance σDa . Then, we impose that µDa and σDa are (32) given by Eqs. (23) and (26), respectively. Typical distribution By summing together Eqs. (29)–(32), the approximated state for Da should be one-sided, as the Exponential, Log-normal, probability b0,0,0 is Poisson, and Chi-square ones, since the delay is positively W0 distributed. In Section VI, we evaluate the accuracy of the b0,0,0 ≈ (1 + 2x) (1 + y) + Ls (1 − x2 )(1 + y) approximated probability distribution function of the delay as 2 −1 given by these one-sided distributions. + K0 (Pc (1 − x2 ))2 (Pc (1 − x2 ))n−1 + 1 +1) , where we neglect the term in Eq (30) and use 1 − xm+1 ≈ C. Energy Consumption 1 − x2 . Finally, we propose an approximation of the average energy In a similar way, the carrier sensing probability given by consumption. From Eq. (12), the average energy consumption Eq. (16) is approximated as τ = (1 + x)(1 + y)b0,0,0 . Hence, of the backoff stage is the approximated reliability is m Wi −1 n m+1 n+1 Pi bi,k,j R=1−x (1 + y) − y , (33) i=0 k=1 j=0 Pi τ (1 − x)(1 − (2x)m+1 ) where y = (1 − (1 − τ )N −1 )(1 − x2 ). R is a function of the = W0 − 1 , (37) busy channel probability α, β, the collision probability Pc and 2 (1 − 2x)(1 − xm+1 ) the MAC parameters m0 , mb , m, n. where we assume that the carrier sensing probability τ is measured by the node, i.e., it is not computed analytically. B. Delay By putting together Eqs. (12), (13) and (16), the average energy consumption of the sensing state is The average delay given by Eq. (23) is approximated as m n E[D] = PT D (34) Psc (bi,0,j + bi,−1,j ) = Psc (2 − α)τ . (38) i=0 j=0 where P = [Pr(A0 |At ) · · · Pr(An |At )]T ∈ R(n+1)×1 , D = Similarly, by substituting Eq. (14) and Eq. (16), the average [d0 · · · dn ]T ∈ R(n+1)×1 , dj = Ts + j Tc + (j + 1)E[T ], energy consumption for packet transmission including both and where Pr(Aj |At ) is given by Eq. (22). E[T ] is the successful transmission and packet collision is approximation of the average backoff period: n L−1 n m i W0 2k − 1 Pt (b−1,k,j + b−2,k,j ) + Pi (b−1,L,j + b−2,L,j ) E[T ] = 2Tsc + P (Bi |Bt ) Sb + 2Tsc k j=0 k=0 j=0 i=0 k=0 2 n L+Lack +1 T + (Pr b−1,k,j + Pi b−2,k,j ) (39) = 2Sb 1 + P T (35) j=0 k=L+1 = (1 − α)(1 − β)τ (Pt L + Pi + Lack (Pr (1 − Pc ) + Pi Pc )) . where P = [P (B0 |Bt ) · · · P (Bm |Bt )]T ∈ R(m+1)×1 , T = [t0 · · · tm ]T ∈ R(m+1)×1 , P (Bi |Bt ) is given by Eq. (36) We assume that the energy consumption at sleeping state is and ti = (2i+1 − 1)W0 + 3i − 1 /4. The approximation negligible, namely Psp ≈ 0. By summing up Eqs. (37), (38) considers the worst case, i.e., a failure of the second sensing and (39), the approximated average energy consumption is (CCA2), which implies that Tsc = Sb and that each sensing Pi τ (1 − x)(1 − (2x)m+1 ) failure takes 2Tsc . Under these assumptions, the probability of Etot = W0 − 1 + Psc (2 − α)τ 2 (1 − 2x)(1 − xm+1 ) the event Bi in Eq. (25) is approximated by + (1 − α)(1 − β)τ (Pt L + Pi + Lack (Pr (1 − Pc ) max(α, (1 − α)β)i +Pi Pc )) . (40) P (Bi |Bt ) = m , (36) k=0max(α, (1 − α)β)k where we did a further approximation by not considering all VI. M ODEL VALIDATION AND P ERFORMANCE A NALYSIS the possibilities of busy channel during two CCAs. Here we present extensive Monte Carlo simulations of slot- Now, we are in the position to give an approximation ted IEEE 802.15.4 to validate our accurate and approximated of the discrete probability distribution function of the delay. expressions of the reliability, delay and energy consumption. A probability generation function approach can be used to The simulations are based on the speciﬁcations of the IEEE compute the discrete probability distribution of the delay. 802.15.4 [1] with several values of the trafﬁc condition and However, such an approach is computationally quite expen- MAC parameters. A performance analysis is also conducted. sive. For analysis and optimization, some continuous well- We investigate the effects of MAC parameters m0 , mb , m, n known distributions are used to approximate the simulation on the performance metrics. Details follow in the sequel. 1 1 1 0.95 0.9 0.9 0.9 0.8 0.8 sim, q 0 = 0 0.85 acc, q 0 = 0 0.7 0.7 reliability reliability reliability 0.8 app, q 0 = 0 0.6 0.6 sim, q 0 = 0 sim, q 0 = 0 0.75 sim, q 0 = 0.3 0.5 0.5 acc, q =0 acc, q 0 = 0 0 acc, q 0 = 0.3 app, q 0 = 0 app, q 0 = 0 0.7 0.4 0.4 app, q 0 = 0.3 sim, q 0 = 0.3 sim, q 0 = 0.3 0.65 0.3 acc, q 0 = 0.3 0.3 acc, q 0 = 0.3 sim, q 0 = 0.9 app, q 0 = 0.3 app, q = 0.3 0 0.6 acc, q 0 = 0.9 0.2 0.2 sim, q 0 = 0.9 sim, q 0 = 0.9 0.55 app, q 0 = 0.9 0.1 acc, q = 0.9 0.1 acc, q = 0.9 0 0 app, q 0 = 0.9 app, q = 0.9 0 0.5 0 0 3 4 5 6 7 8 2 3 4 5 0 1 2 3 4 5 6 7 MAC parameter, m MAC parameter, n MAC parameter, m0 (a) m0 = 3, . . . , 8, mb = 8, m = 4, n = 3, (b) m = 2, . . . , 5, m0 = 3, mb = 8, n = 3, (c) n = 0, . . . , 7, m0 = 3, mb = 8, m = 4 Fig. 3. Reliability as a function of the trafﬁc conditions q0 = 0, 0.3, 0.9, and MAC parameters m0 = 3, . . . , 8, mb = 8, m = 2, . . . , 5, n = 0, . . . , 7. The length of the packet is L = 7 and the number of nodes is N = 10. 1 40 sim, q = 0.3 0 0.9 acc, q = 0.3 0 35 app, q 0 = 0.3 0.8 sim, q = 0.6 0 30 acc, q = 0.6 average delay (ms) 0.7 0 app, q 0 = 0.6 25 reliability 0.6 sim, q = 0.3 0 sim, q = 0.9 0 acc, q = 0.3 acc, q = 0.9 0 0 0.5 20 app, q 0 = 0.3 app, q 0 = 0.9 0.4 sim, q = 0.6 0 15 acc, q = 0.6 0 0.3 app, q 0 = 0.6 10 0.2 sim, q = 0.9 0 acc, q = 0.9 5 0.1 0 app, q 0 = 0.9 0 0 10 20 30 40 50 60 10 15 20 25 30 35 40 45 50 55 60 number of nodes number of nodes Fig. 4. Reliability as a function of trafﬁc condition q0 = 0.3, 0.6, 0.9 with Fig. 5. Average delay as a function of trafﬁc condition q0 = 0.3, 0.6, 0.9 a given MAC parameters (m0 = 3, mb = 8, m = 4, n = 3) and packet with a given MAC parameters (m0 = 3, mb = 8, m = 4, n = 3) and packet length L = 7. length L = 7. A. Reliability Validation are necessary but not sufﬁcient for high reliability under high trafﬁc conditions. Fig. 4 illustrates the reliability as obtained by Monte Carlo simulations and the accurate expression Eq. (21) and approxi- mated one Eq. (33) as a function of the trafﬁc q0 = 0.3, 0.6, 0.9 B. Delay Validation with a length of the packets L = 7, and MAC parameters In Fig. 5 we report the average delay as obtained by Monte m0 = 3, mb = 8, m = 4, n = 3. The accurate analytical Carlo simulations, the accurate expression given by Eq. (23), model and approximated model match the simulation results and approximated one given by Eq. (34). The average delay quite well under low trafﬁc condition q0 = 0.6, 0.9. However, is reported as a function of the trafﬁc q0 = 0.3, 0.6, 0.9, the approximated expression shows a weak matching for the with a length of the packet L = 7 and MAC parameters high trafﬁc q0 = 0.3 and large number of nodes N ≥ 30. The m0 = 3, mb = 8, m = 4, n = 3. Similarly to the reliability, reason is that the approximation given by Eq. (28) holds if both the accurate and approximated expressions predict well x 1, but x increases as the trafﬁc and the number of nodes the simulation results under low trafﬁc condition q0 = 0.6, 0.9, increases. whereas the approximation becomes less accurate for high Fig. 3 shows the reliability as obtained by Monte Carlo trafﬁc q0 = 0.3 and large number of nodes N ≥ 30. simulations, the accurate and approximated expressions as Fig. 6 shows the average delay as a function of different a function of the trafﬁc conditions q0 = 0, 0.3, 0.9 with a trafﬁc conditions q0 = 0.3, 0.6, 0.9 with a given number of given number of nodes N = 10 and different MAC param- nodes N = 10 and different MAC parameters m0 , m, n. eters m0 , m, n. The accurate and approximated expressions Both the accurate and approximated expressions match well match quite well the simulation results. The expressions are the simulation results, but the approximated model does not closer to simulation results under unsaturated trafﬁc condition predict well the simulation results under high trafﬁc condition q0 = 0.3, 0.9 than saturated trafﬁc condition q0 = 0. The q0 = 0.3 due to the approximation given by Eq. (28). Observe reliability approaches 1 under very low trafﬁc regime q0 = 0.9. that the average delay increases as trafﬁc condition increases In Figs. 3(a), 3(b), the reliability increases as MAC parameters due to high busy channel probability and collision probability. m0 , m increase, respectively. In Fig. 3(c), it is interesting to Fig. 6(a) shows that the average delay increases exponentially observe that the reliability does not improve as the retry limits as m0 increases. Hence, we conclude that m0 is the key n increases for high trafﬁc conditions q0 = 0. Notice that the parameter in terms of average delay with respect to m and reliability saturates to 0.6 if n ≥ 2. Hence, the retransmissions n. 70 15 15 sim, q 0 = 0.3 sim, q = 0.3 sim, q 0 = 0.3 0 14 acc, q = 0.3 14 acc, q 0 = 0.3 60 acc, q 0 = 0.3 0 app, q = 0.3 13 app, q = 0.3 13 0 app, q 0 = 0.3 0 average delay (ms) sim, q 0 = 0.6 sim, q 0 = 0.6 average delay (ms) sim, q = 0.6 12 12 acc, q = 0.6 average delay (ms) 50 0 0 acc, q = 0.6 app, q 0 = 0.6 0 acc, q = 0.6 11 11 0 app, q 0 = 0.6 sim, q = 0.9 0 40 app, q = 0.6 0 10 sim, q 0 = 0.9 10 acc, q 0 = 0.9 sim, q = 0.9 acc, q = 0.9 app, q 0 = 0.9 0 0 9 9 30 acc, q 0 = 0.9 app, q = 0.9 0 app, q 0 = 0.9 8 8 20 7 7 6 6 10 5 5 0 4 4 3 4 5 6 7 8 2 3 4 5 0 1 2 3 4 5 6 7 MAC parameters, m0 MAC parameters, m MAC parameters, n (a) m0 = 3, . . . , 8, mb = 8, m = 4, n = 3, (b) m = 2, . . . , 5, m0 = 3, mb = 8, n = 3, (c) n = 0, . . . , 7, m0 = 3, mb = 8, m = 4 Fig. 6. Average delay as a function of the trafﬁc conditions q0 = 0.3, 0.6, 0.9 and MAC parameters m0 = 3, . . . , 8, mb = 8, m = 2, . . . , 5, n = 0, . . . , 7. The length of the packet is L = 7 and the number of nodes is N = 10. 1 1 1 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.6 0.6 0.6 CDF CDF CDF 0.5 0.5 0.5 0.4 0.4 0.4 0.3 sim 0.3 sim 0.3 sim exp exp exp 0.2 logn 0.2 logn 0.2 logn poiss poiss poiss 0.1 0.1 0.1 chi2 chi2 chi2 0 0 0 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 5 10 15 20 25 30 35 40 delay (ms) delay (ms) delay (ms) (a) m0 = 3, mb = 5, q0 = 0, Pc = 0.775 (b) m0 = 3, mb = 5, q0 = 0.3, Pc = 0.22 (c) m0 = 3, mb = 5, q0 = 0.9, Pc = 0.0044 1 1 1 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.6 0.6 0.6 CDF CDF 0.5 0.5 CDF 0.5 0.4 0.4 0.4 sim sim sim 0.3 0.3 0.3 exp exp exp 0.2 logn 0.2 logn 0.2 logn poiss poiss poiss 0.1 0.1 chi2 0.1 chi2 chi2 0 0 0 50 100 150 200 250 300 350 50 100 150 200 250 10 20 30 40 50 60 delay (ms) delay (ms) delay (ms) (d) m0 = 5, mb = 8, q0 = 0, Pc = 0.2766 (e) m0 = 5, mb = 8, q0 = 0.3, Pc = 0.1347 (f) m0 = 5, mb = 8, q0 = 0.9, Pc = 0.0044 Fig. 7. Cumulative distribution function of the delay for successfully received packets as a function of the trafﬁc conditions q0 = 0, 0.3, 0.9 and different MAC parameters m0 = 3, 5, mb = 5, 8, m = 4, n = 3. The length of the packet is L = 7 and the number of nodes is N = 20. The Exponential, Log-normal, Poisson, and Chi-square distribution are used for moment matching. We check the validity of the delay distribution by using Figs. 7(c), 7(f)). By comparing Figs. 7(a), 7(b) to 7(d), 7(e), we the approximation given by the moment matching approach observe that a larger MAC parameter gives longer tails. From described in Section V-B. Fig. 7 shows the cumulative dis- Figs. 7(c) and 7(f), we conclude that a good approximation tribution function (CDF) of packet delay as obtained by of the distribution depends on both MAC parameters and Monte Carlo simulations and the approximated distribution collision probability. as a function of different parameters m0 = 3, 5, mb = 5, 8, To validate the accuracy of the approximated distributions m = 4, n = 3, the packet length L = 7, the number of by using a moment matching with the Exponential, Log- nodes N = 20 and different trafﬁc conditions q0 = 0, 0.3, 0.9. normal, Poisson, Chi-square distributions, the correlation co- The moment matching has been obtained by using the Expo- efﬁcients ρ2 between the simulation results and approximated nential, Log-normal, Poisson, and Chi-square distributions. In distribution has been evaluated. Recall that the closer ρ2 to Figs. 7(a), 7(b), 7(d), 7(e), we see that the Exponential distri- 1, the better the approximation. In the following, we validate bution predicts well the CDF for high collision probabilities. the dependence between collision probability and correlation By contrast, in Figs. 7(c), 7(f), we observe that the Log-normal coefﬁcient of the approximated distributions. or Poisson distributions provide a good approximation for low collision probability. In addition, the Exponential distribution Figs. 8 show the relation of the correlation coefﬁcient ρ2 provides us with a fair approximation except for cases of very between the simulation results and the approximated distri- low collision probabilities. For these probabilities, the delay bution over different collision probabilities as a function of distribution is more similar to a deterministic distribution (see the different trafﬁc condition q0 = 0.6, 0.9 and parameters m0 = 3, 5, mb = 5, 8, m = 4, n = 3, the length of packet 1 1 0.95 0.95 correlation coe cient correlation coe cient 0.9 0.9 exp, q =0.6 exp, q 0 =0.6 0 0.85 logn, q 0 =0.6 0.85 logn, q =0.6 0 poiss, q 0 =0.6 poiss, q =0.6 0 0.8 chi2, q 0 =0.6 0.8 chi2, q 0 =0.6 exp, q 0 =0.9 exp, q 0 =0.9 logn, q 0 =0.9 logn, q 0 =0.9 0.75 0.75 poiss, q 0 =0.9 poiss, q 0 =0.9 chi2, q 0 =0.9 chi2, q 0 =0.9 0.7 0.7 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 0.25 collision probability collision probability (a) m0 = 3, mb = 5, L = 7, q0 = 0.6, 0.9 (b) m0 = 3, mb = 5, L = 14, q0 = 0.6, 0.9 1 1 0.95 0.95 correlation coe cient correlation coe cient 0.9 0.9 exp, q =0.6 exp, q 0 =0.6 0 0.85 logn, q 0 =0.6 0.85 logn, q 0 =0.6 poiss, q =0.6 poiss, q 0 =0.6 0 0.8 chi2, q 0 =0.6 0.8 chi2, q 0 =0.6 exp, q =0.9 exp, q 0 =0.9 0 logn, q 0 =0.9 logn, q 0 =0.9 0.75 0.75 poiss, q 0 =0.9 poiss, q 0 =0.9 chi2, q 0 =0.9 chi2, q 0 =0.9 0.7 0.7 0 0.04 0.08 0.12 0.16 0.1782 0 0.05 0.1 0.15 0.2 collision probability collision probability (c) m0 = 5, mb = 8, L = 7, q0 = 0.6, 0.9 (d) m0 = 5, mb = 8, L = 14, q0 = 0.6, 0.9 Fig. 8. Correlation coefﬁcients of approximated delay cumulative distribution function including Exponential, Log-normal, Poisson, Chi-square distributions as a function of different parameters m0 = 3, 5, mb = 5, 8, m = 4, n = 3, the length of packet L = 7, 14 and the number of node N = 10, 20, 30, 40, 50, 60. L = 7, 14 and the different number of nodes N . Hence, six −3 x 10 2 sim, q = 0.3 different correlation coefﬁcients and collision probabilities are 0 1.8 acc, q 0 = 0.3 app, q 0 = 0.3 displayed for a given trafﬁc condition. From Fig. 8, we observe 1.6 sim, q 0 = 0.6 that the correlation coefﬁcient depends mainly on collision power consumption (W) acc, q 0 = 0.6 1.4 app, q 0 = 0.6 probability Pc . In Fig. 8(a) and 8(b), the correlation coefﬁcient 1.2 sim, q acc, q 0 = 0.9 = 0.9 is reported for a given set of MAC parameter (m0 = 3, 0 1 app, q = 0.9 0 mb = 5) and similar collision probabilities as those of Fig. 8. 0.8 Observe that the correlation coefﬁcient varies smoothly over 0.6 different collision probabilities (see Figs. 8(c) and 8(d)). 0.4 Therefore, we conclude that if the BEB mechanism does not 0.2 have a strict limitation on the maximum value of the backoff 0 10 15 20 25 30 35 40 number of nodes 45 50 55 60 exponent mb (as in IEEE 802.11), then the delay distribution Fig. 9. Average energy consumption as a function of trafﬁc condition q0 = is mainly dependent on collision probability (see Figs. 8(c) 0.3, 0.6, 0.9 with MAC parameters (m0 = 3, mb = 8, m = 4, n = 3) and and 8(d)). Otherwise, if MAC parameters (m0 , mb , m, n) a given length of packet L = 7. have strict limitations as currently done in IEEE 802.15.4, then the delay distribution depends on both MAC parameters and collision probability. choose the best approximated distribution out of Exponential, Log-normal, Poisson, Chi-square distributions by measuring Fig. 8 shows a good matching between the CDF of sim- collision probability. ulation results and the approximated distributions. Notice that the best correlation coefﬁcient for the MAC parameters C. Energy Consumption Validation m0 = 5 and mb = 8 is very close to 1. The Exponential In Fig. 9 we reported the average energy consumption as distribution gives the better match with the simulation results achieved by Monte Carlo simulations, for the accurate expres- for Pc > 0.1. In [14], the delay distribution of IEEE 802.11 sion given by Eq. (27) and the approximated expression given matches well with a Log-normal distribution for almost all by Eq. (40). The curves depend on the trafﬁc q0 = 0.3, 0.6, 0.9 cases. However, the Log-normal distribution does not match with a length of the packet L = 7, and MAC parameters well the simulation results for IEEE 802.15.4 as the collision m0 = 3, mb = 8, m = 4, n = 3. We observe that both probability increases. The reason is that the delay distribution the accurate and approximated expressions predict well the for IEEE 802.15.4 does not have long tails compared to IEEE simulation results under different trafﬁc conditions. 802.11, because the MAC parameters have a strict limitation. Fig. 10 shows the energy consumption as a function of dif- Hence, for given MAC parameters, from our results we can ferent trafﬁc conditions q0 = 0, 0.3, 0.9 with a given number of −3 −3 −3 x 10 x 10 x 10 6 8 7 sim, q 0 = 0 sim, q 0 = 0 sim, q 0 = 0 acc, q =0 acc, q 0 =0 acc, q 0 =0 0 7 6 app, q 0 = 0 5 app, q 0 = 0 app, q 0 = 0 power consumption (W) sim, q = 0.3 power consumption (W) 0 sim, q 0 = 0.3 sim, q 0 = 0.3 power consumption (W) 6 acc, q 0 = 0.3 acc, q = 0.3 5 acc, q = 0.3 0 app, q 0 = 0.3 0 4 app, q 0 = 0.3 app, q 0 = 0.3 sim, q 0 = 0.9 5 sim, q 0 = 0.9 4 acc, q = 0.9 sim, q 0 = 0.9 0 acc, q = 0.9 app, q = 0.9 0 0 3 acc, q = 0.9 4 0 app, q 0 = 0.9 app, q 0 = 0.9 3 3 2 2 2 1 1 1 0 0 0 3 4 5 6 7 8 2 3 4 5 0 1 2 3 4 5 6 7 MAC parameter, m0 MAC parameter, m MAC parameter, n (a) m0 = 3, . . . , 8, mb = 8, m = 4, n = 3, (b) m = 2, . . . , 5, m0 = 3, mb = 8, n = 3, (c) n = 0, . . . , 7, m0 = 3, mb = 8, m = 4 Fig. 10. Average energy consumption as a function of the trafﬁc conditions q0 = 0.3, 0.6, 0.9 and MAC parameters m0 = 3, . . . , 8, mb = 8, m = 2, . . . , 5, n = 0, . . . , 7. The length of the packet is L = 7 and the number of nodes is N = 10. nodes N = 10 and different MAC parameters m0 , m, n. The [2] G. Lu, B. Krishnamachari, and C. Raghavendra, “Performance eval- accurate analytical model and approximated model match well uation of the IEEE 802.15.4 MAC for low-rate low-power wireless networks,” in Proc. of IEEE IPCCC, 2004. the simulation results for all trafﬁc conditions. It is interesting [3] J. Zheng and M. L. Lee, “A comprehensive performance study of IEEE to observe that the energy consumption decreases as the MAC 802.15.4,” in IEEE Press Book, 2004. parameters m0 , m, n increase under a high trafﬁc condition [4] A. Koubaa, M. Alves, and E. Tovar, “A comprehensive simulation study of slotted CSMA/CA for IEEE 802.15.4 wireless sensor networks,” in q0 = 0. In essence, as the MAC parameters increase, the node IEEE International Workshop on Factory Communication Systems, Jun may stay more time in idle backoff stage than transmit or 2006, pp. 183–192. receiving mode i.e., Pr > Pi > Psp and Pt > Pi > Psp . [5] S. Pollin, M. Ergen, S. C. Ergen, B. Bougard, L. V. D. Perre, F. Catthoor, I. Moerman, A. Bahai, and P. Varaiya, “Performance analysis of slotted We observe that the energy consumption increases as MAC carrier sense IEEE 802.15.4 medium access layer,” in Proc. of IEEE parameters (m0 , m, n) increase under low trafﬁc condition GLOBECOM, 2006, pp. 1–6. q0 = 0.3, 0.9. Since the node needs to stay more time in so so [6] J. Mi˘i´ , S. Shaf, and V. Mi˘i´ , “Performance of a beacon enabled IEEE 802.15.4 cluster with downlink and uplink trafﬁc,” in IEEE Trans. idle sleep stage without packet generation under low trafﬁc Parallel and Distributed Systems, 2006, pp. 361–376. condition q0 = 0.3, 0.9, the main component of average energy [7] P. K. Sahoo and J. P. Sheu, “Modeling IEEE 802.15.4 based wireless consumption is the idle backoff time rather than transmit or sensor network with packet retry limits,” in PE-WASUN, 2008, pp. 63– 70. receiving energy consumption. It is interesting to observe that [8] S. Pollin, M. Ergen, S. C. Ergen, B. Bougard, F. Catthoor, A. Bahai, and the energy consumption has a weaker dependence on the retry P. Varaiya, “Performance analysis of slotted carrier sense IEEE 802.15.4 limits n than the other MAC parameters m0 , m. acknowledged uplink transmissions,” in Proc. of IEEE WCNC, 2008, pp. 1559–1564. [9] G. Bianchi, “Performance analysis of the IEEE 802.11 distributed cordi- nation function,” in IEEE Journal on Selected Areas in Communications, VII. C ONCLUSIONS vol. 18, March 2000. [10] IEEE Std 802.11 Wireless LAN Medium Access Control (MAC) and In this paper, we presented a generalized approach to ana- Physical Layer (PHY) Speciﬁcations, IEEE, 1999. [Online]. Available: lyze performance of the slotted CSMA/CA mechanism in the http://www.ieee802.org/11 IEEE 802.15.4 standard. The approach is based on a Markov [11] P. Chatzimisios, A. C. Boucouvalas, and V. Vitsas, “IEEE 802.11 packet delay a ﬁnite retry limit analysis,” in Proc. of IEEE GLOBECOM, 2003, chain that considers retry limits, the acknowledgement mecha- pp. 950–954. nism, and unsaturated trafﬁc, which are important components [12] P. Chatzimisios, V. Vitsas, and A. C. Boucouvalas, “Throughput and of most wireless sensor network applications. We derived the delay analysis of IEEE 802.11 protocol,” in Proc. of IEEE IWNA, 2002, pp. 168–174. reliability, delay and energy consumption expressions offered [13] Z. Hadzi-Velkov and B. Spasenovski, “Saturation throughput-delay by the slotted IEEE 802.15.4 standard by both an accurate and analysis of IEEE 802.11 in fading channel,” in Proc. of IEEE ICC, computationally demanding approach, and an approximate and 2003. [14] H. Zhai, Y. Kwon, and Y. Fang, “Performance analysis of IEEE 802.11 simple approach. We showed that the approximated analysis MAC protocols in wireless LANs: Research articles,” in Wirel. Commun. is effective for low trafﬁc. Furthermore, unlike 802.11, we Mob. Comput., 2004, pp. 917–931. observed that the delay distribution of IEEE 802.15.4 depends [15] O. Tickioo and B. Sikdar, “Queueing analysis and delay mitigation in IEEE 802.11 random access MAC based wireless networks,” in Proc. mainly on the MAC parameters and the collision probability. of IEEE INFOCOM, 2004. In addition, we analyzed the impact these parameters on the [16] H. Wu, Y. Peng, K. Long, S. Cheng, and J. Ma, “Performance of reliability, delay and energy consumption. reliable transport protocol over IEEE 802.11 wireless LAN: Analysis and enhancement,” in Proc. of IEEE INFOCOM, 2002, pp. 599–607. Future investigations include the use of the aforementioned [17] F. Cali, M. Conti, and E. Gregori, “Dynamic tuning of the IEEE 802.11 achievements to the systematic design of optimized IEEE protocol to achieve a theoretical throughput limit,” in IEEE Trans. 802.15.4 MAC based on speciﬁc application requirements. Networking, Dec 2000, pp. 785–790. [18] P. Raptis, A. Banchs, and K. Paparrizos, “A simple and effective delay distribution analysis for IEEE 802.11,” in Proc. of IEEE PIMRC, 2006, pp. 1–5. R EFERENCES [19] A. Giridhar and P. R. Kumar, “Toward a Theory of In-network Compu- tation in Wireless Sensor Networks,” IEEE Communication Magazine, [1] IEEE Std 802.15.4-2996, September, Part 15.4: Wireless Medium Access pp. 97–107, April 2006. Control (MAC) and Physical Layer (PHY) Speciﬁcations for Low-Rate Wireless Personal Area Networks (WPANs), IEEE, 2006. [Online]. Available: http://www.ieee802.org/15

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