A Generalized Markov Chain Model for Effective Analysis of Slotted

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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-
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