# Priority-Based Machine-To-Machine Overlay Network over LTE for a Smart City
## 5 Modelling the PM2M Backoff Procedure with Backoff Error
The incorrect idle medium probability in CSMA slots where all \(n_{i}\) nodes with priority \(i\) perform access is:  
\[m\_idle_{Aerr}=\sum_{i=0}^{7}PB_{err}(1-\tau_{i})^{n_{i}} \tag{17}\]  
and the probability that the medium is busy is:  
\[m\_busy=1-m\_idle_{Aerr} \tag{18}\]  
Then, we evaluate the probability of the medium being idle as observed by UE of class \(k\in(0,1)\) during the backoff countdown (which means that other nodes do not access the medium during the current RAP) as:  
\[m\_err_{k}=\frac{m\_idle_{Aerr}}{1-\tau_{k}} \tag{19}\]  
During a given backoff countdown, there is a possibility that there is not enough time in the superframe (RAP) to complete the countdown and the transmission. The probability of this event is:  
\[S\_time_{k}=\frac{1}{X_{RAP}-T_{succ}-C_{k}} \tag{20}\]  
where  
\(T_{succ}=(data_{s}+ack_{s}+sifs)\) is the successful transmission time in slots;  
Figure 6: Markov chain for \(UP_{k}\), modified from [13]. \(C_{k}=\frac{CW_{k,min}+CW_{k,max}}{4}\) is the approximate mean backoff counter value; and \(T_{coll}=(data_{s}+ack_{s}+sifs)\) is the unsuccessful transmission time. In all cases, SIFS refers to short interframe space [11]. Now, we have to calculate the probability that the backoff counter value of a node will be unfrozen (i.e., decremented) for \(j=1\ldots W_{k,m_{k}}\):  
\[\sigma=(1-BER_{PM2M})^{dt_{s}+ak_{s}} \tag{15}\]0  
The indices of state probabilities \(b_{k,i,j}\) of the Markov chain as shown in Figure 6 are \(k=0\ldots 1\), \(i=0\ldots R\) and \(j=0\ldots W_{k,i}\). The medium access probability is calculated as \(\tau_{k}=\sum_{0}^{R}b_{k,i,o}\). We have to calculate the zeroth backoff phase, which depends on the probability of the CSMA slot being in the idle state, the probability that the queue is empty due to successful transmission or dropped because of the exceeded retry limit. In our model, we adopt the expression for the probability of the queue being empty \(Q_{ent}\) from [13]. Then, the probability of being in the idle state may also refer to the wrong medium status in the presence of backoff error. Assuming that the probability of the data frame arrival during the interval between two successive Markov points is denoted as \(\alpha\), the probability of the idle state is calculated as:  
\[\sigma=(1-BER_{PM2M})^{dt_{s}+ak_{s}} \tag{15}\]1  
Thus, the sum of all DTMC states belonging of the zeroth backoff phase for traffic class \(k\) is:  
\[\sigma=(1-BER_{PM2M})^{dt_{s}+ak_{s}} \tag{15}\]2  
\[\sigma=(1-BER_{PM2M})^{dt_{s}+ak_{s}} \tag{15}\]3 Extended Markov chain for \(UP_{k}\) modified from [13]. The normalization condition for DTMCs of each class requires that the sum of all the state probabilities is one. By solving the system of equations belonging to the DTMC of each traffic class and coupling the equation for the behaviour of medium, we calculate the medium access probability for each traffic class with backoff error, \(\tau_{k}\), \(k=0,1\), during RAP as:  
\[\sigma=(1-BER_{PM2M})^{dt_{s}+ak_{s}} \tag{15}\]4  
During the CSMA-CA mechanism when a node tries to get access to the medium, it has to backoff and lock the backoff counter if the channel is busy due to a transmission by another node, be it ultimately successful or not; or there is not enough time in the current superframe to complete the frame transmission. The node unlocks the backoff counter again when the channel is idle for the SIFS period within the superframe time, and the current superframe has enough time to complete the frame transaction. All of the above conditions work fine in the absence of backoff error. However, in the presence of backoff error, the node could decide that the medium is idle and unlock the backoff counter when the medium is actually busy, which could lead to excessive collisions in the overlay network. Alternatively, the node may freeze the backoff counter when it should not; this leads to longer backoffs and possibly to relegating a transmission to the next superframe.