In most of the previous work, it has been commonly assumed that the probability of assessing the channel is same for all nodes within a network, independent of the length of the back-off windows and is distributed uniformly over the whole service
M. Baz, PhD Thesis, University of York 2014
time e.g., [143-147]. Although this assumption reduces the complexity of modelling, it does not fully reflect the stochastic behaviour of the protocol. This chapter proposes a more realistic approach by assuming that the probability of assessing the channel varies between stages as well as differing from one node to another. Justification for the first part of this assumption is drawn from the fact that a node assesses the channel only after its choice of back-off duration has elapsed, and while this duration is selected randomly it is still dependent on the probability distribution of the back-off, which differs between stages. The second part of the assumption is justified by considering that a node assesses the channel only when it has a packet ready to transmit, and this depends on the traffic intensity that typically differs between nodes in multihop WSNs.
Let 𝜙𝑛(𝑚)(𝑘) be the probability that a node 𝑛 assesses the channel in the 𝑘th slots during the 𝑚th stage (i.e., probability that node 𝑛 is in the mth stage and 𝑘 slot and it assesses the channel), where 𝑘 represents the number of slots that are elapsed since a node 𝑛 commences servicing a packet, i.e., 1 ≤ 𝑘 ≤ ∑𝑀𝑖=0𝑊𝑖 and 𝑚 is an arbitrary stage, i.e., 0 ≤ 𝑚 ≤ 𝑀. Let us assume further that 𝜙𝑛 is the average probability that node 𝑛 assesses the channel per the complete service time, hence the value of 𝜙𝑛 can be derived as a weighted sum of the moments of 𝜙𝑛(𝑚)(𝑘) as:
𝜙𝑛 = ∑ (1 − 𝛼𝑛)𝑚 ( ∑𝑊(𝑚)𝜙𝑛(𝑚)(𝑘) 𝑘=1 𝑊(𝑚) 𝑀 𝑚=0 + ∑ ( ∑ (𝜙𝑛(𝑚)(𝑘) −∑𝑊 𝜙𝑛(𝑚)(𝑘) (𝑚) 𝑘=1 𝑊(𝑚) ) 𝑓 𝑊(𝑚) 𝑘=1 𝑊(𝑚) ) 1 𝑓 𝐹 𝑓=2 ) 𝑊(𝑚) = (∑ 𝑊𝑖 𝑚 𝑖=0 ) ; 0 ≤ 𝑚 ≤ 𝑀 (5.20)
where (1 − 𝛼𝑛)𝑚 is a weighting factor that qualifies the stages of service time (as a node goes to the next stage only when it has assessed the channel as busy in the previous stage). The term ∑ 𝜙𝑛
(𝑚)
(𝑘)
𝑊(𝑚) 𝑘=1
𝑊(𝑚) is the average of all probabilities by which
node 𝑛 assesses the channel during the 𝑚th stage. The term ( (𝑊(𝑚))−1∑ (𝜙𝑛(𝑚)(𝑘) − (𝑊(𝑚))−1∑ 𝜙 𝑛(𝑚)(𝑘) 𝑊(𝑚) 𝑘=1 ) 𝑓 𝑊(𝑚) 𝑘=1 ) represents the 𝑓th
M. Baz, PhD Thesis, University of York 2014
moment of 𝜙𝑛(𝑚)(𝑘)) while the sum from 2 to 𝐹 accounts for the first 𝐹th higher moments of probabilities of assessment of the channel in a stage. Equation (5.20) expresses the average probability of assessing the channel based on the behaviour of the node itself which is the first equation required to obtain the value of 𝛼𝑛.
It is of interest to note that equation (5.20) is able to account for stochastic characteristics of the probabilities of channel assessment using the higher moments rather than the average value as is the case in most of the analytical models reported in the literature. Moreover, this equation is simple as it only requires computing the values of 𝜙𝑛(𝑚)(𝑘) in each stage which is determined in the rest of this section for the IEEE 802.15.4 protocol.
For the IEEE 802.15.4 protocol, in the 0th stage, 𝜙𝑛(0)(𝑘) refers to the probability that a node assesses the channel in the 0th stage and slot 𝑘, where 1 ≤ 𝑘 ≤ 𝑊0. As a node in this stage selects a slot to assess the channel randomly based on a uniform distribution between 1 and 𝑊0, thereby 𝜙𝑛(0)(𝑘) is:
𝜙𝑛(0)(𝑘) = { 1
𝑊0; 1 ≤ 𝑘 ≤ 𝑊0 0; otherwise
(5.21)
In the 1st stage, the probability of assessing the channel in the 𝑘𝑡ℎ slot, denoted by 𝜙𝑛(1)(𝑘), is the probability that a node assesses the channel in slot 𝑗 where 0 ≤ 𝑗 ≤ 𝑘 − 1 in the 0th stage and then backs-off for 𝑘 − 𝑗 − 1 slots in the 1st stage. For example, the probability that a node assesses the channel in the 3rd slot in the 1st stage is the sum of the probabilities that a node assesses the channel in slot 0, 1, or 2 in the 0th stage, finds the channel busy, and then backs-off for 2, 1 or 0 slots respectively in the 1st stage. An illustration of this concept is given in figure 5.3.
Figure 5.3 Relation between sensing and backing-off in the 0th and 1st stages
Accordingly, 𝜙𝑛(1)(𝑘) can be given as:
……
……
A node assesses the channel in the
0thstage
Slot Slot Slot Slot Slot Slot
……
A node assesses the channel in the
1ststage
M. Baz, PhD Thesis, University of York 2014
𝜙𝑛(1)(𝑘) = ∑ 𝜙𝑛(0)(𝑗)Φ(1, 𝑘 − 𝑗 − 1); 1 ≤ 𝑘 ≤ 𝑊0+ 𝑊1 𝑗=𝑘−1
𝑗=1
(5.22)
where Φ(𝑚, 𝑗) is defined as the probability that a node selects to back-off for 𝑗 slots at the beginning of stage 𝑚 which is given as:
Φ(𝑚, 𝑗) = 1
𝑊𝑚; 1 ≤ 𝑗 ≤ 𝑊𝑚; 0 ≤ 𝑚 ≤ 𝑀 (5.23) The methodology used to derive 𝜙𝑛(1)(𝑘) can be generalised for the other stages, as the relationship between the channel assessment and back-off of any two consecutive stages follows the same procedure. This gives:
𝜙𝑛(𝑚)(𝑘) = ∑ 𝜙𝑛(𝑚−1)(𝑗)Φ(𝑚, 𝑘 − 𝑗 − 1) 𝑗=𝑘−1 𝑗=1 ; 2 ≤ 𝑚 ≤ 𝑀; 1 ≤ 𝑘 ≤ ∑ 𝑊𝑖 𝑖=𝑚 𝑖=0 (5.24)
Figure 5.4 depicts the probability of assessing the channel in the IEEE 802.15.4 protocol. It is shown that in the 0th stage, a node assesses the channel with a constant probability equal to 1 𝑊⁄ 0. This results from the fact that the probability of assessing the channel in this stage depends on the selection of a back-off period with a uniform distribution from one to 𝑊0. In the remaining stages, the probability of assessing the channel differs significantly between slots, and this variation increases with an increasing number of stages. For example, 𝜙𝑛(1)(𝑘) is an isosceles trapezoid function that results from the convolution between two independent uniform distributions: 𝜙𝑛(0)(𝑘) and Φ(1, 𝑗) each having different ranges, while 𝜙𝑛(4)(𝑘) is closer to a normal distribution which results from the central limit theorem.
M. Baz, PhD Thesis, University of York 2014 Figure 5.4 Probability of channel assessment in IEEE 802.15.4
Using the values of 𝜙𝑛(𝑚)(𝑘) shown in figure 5.4, the average probability of assessment the channel per service time 𝜙𝑛 can be derived using equation (5.20) up to predefined moments (by assigning the value for 𝐹 in equation 5.20).
Figure 5.4 demonstrates one of the key limitations of the IEEE 802.15.4 CSMA-CA protocol, that is the probability of channel assessment in a back-off stage decreases considerably compared to the previous stages. This characteristic makes a node that has failed to access the channel in a specific back-off stage is less sensitive for the channel conditions in the subsequent stages. Hence a node can defer its transmission for longer periods even if the channel is idle which in turn leads to magnify the end- to-end delay of packets and to reduce the throughput of the network. Evaluations for the effect of this characteristic over multihop networks are presented in the results of this chapter.