# TARS: A Novel Mechanism for Truly Autonomous Resource Selection in LTE-V2V Mode 4
## 4 System Model Assumptions and Proposed Mechanism
### System Model  
To observe the behavior of RB selection and collision, it is important to model _Process 1_, _Process 2_, and _Process 3_. The probability of RB collision in the context of _Process 1_ in ARS is elementary. For \(n\) users and a number of RBs \(r\), RB selection is an independent event of RB selection from a single pool, performed by every vehicle, and it is given by \(n/n(n-r)!\). Therefore, the system model only considers _Process 2_ and _Process 3_. Symbols/acronyms and their abbreviations are listed in Table 1.  
Considering _Process 2_ and _Process 3_ as the DTMC, the following assumptions were made:  
* If a user has data to transmit, then the user has already selected an RB for the transmission of the data; hence, _Process 1_ is omitted;
* The number of transmitting users is directly proportional to the RB collisions;
* A user is considered as a transmitting user only after reselecting the same resource for further transmission. The transmitting users are the ones that have more data, and they contend for more resources, hence resulting in more collisions. Therefore, to observe the behavior of _Process 2_ and _Process 3_ in terms of the number of transmitting users, let \(X_{n}\) be the number of transmitting users in ARS at any instant of time,  
\[X_{n}\ \in\ S\ :=\{0,1,2,\ldots,N\} \tag{1}\]  
where \(X\) represents the ARS process, \(S\) is the state space, \(n\) is the time slot, and \(N\) is the total number of transmitting users. A user that has data to transmit always enters _Process 2_ to select a counter. The probability of _Process 2_ is defined by the probability that a user has data to transmit. Therefore, the probability that a user has data to transmit and that it is in _Process 2_ in time slot \(n\) is given by \(\gamma\). It was assumed that after _Process 2_, a simultaneous event also occurs, i.e., \(X\) enters in _Process 3_ at time \(n\). Then, the probability of reselecting the same RB is given by \(\delta\) \begin{table}
\begin{tabular}{c l} \hline \hline
**Acronyms** & **Abbreviations** \\ \hline \(X\) & represents the ARS process \\ \(n\) & time slot \\ \(X_{n}\) & number of transmitting users at \(n\) \\ \(S\) & state space \\ \(N\) & total number of transmitting users \\ \(\gamma\) & the probability that a user has data to transmit at \(n\) \\ \(\delta\) & the probability of reselecting the same RB \\ \(P\) & transition probability matrix \\ \(x\) & current state \\ \(t\) & next state \\ \(P_{x,t}\) & transition probability matrix from \(x\) to \(t\) \\ \(p_{1}\) & the probability of moving from the current state to the next state \\ \(p_{2}\) & the probability of moving back to the previous state \\ \(p_{3}\) & the probability of staying in the same state \\ \(\pi_{i}\) & the frequency of being in a certain state \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of acronyms/symbols. and the probability of not selecting the same RB is given by \((1-\delta)\). Once a counter and reselection counter are selected, the user is considered as a transmitting user at time slot \(n+1\). The state transitions in \(X\) happen when \(X\) enters in _Process 3_. When a user has data to transmit at time \(n\), the number of transmitting nodes \(X_{n}\) either increases or decreases at \(n+1\) depending on the probabilities of _Process 2_ and _Process 3_. The transition probability matrix \(P\) is given by:  
\[P_{x,t}=\left\{ \begin{array}{cl}p_{1}=\gamma(\delta),&\{t=x+1|x=1,2,3, \ldots\}\\ p_{2}=\gamma(1-\delta),&\{t=x-1|x=1,2,3,\ldots\}\\ p_{3}=(1-\gamma)(1-\delta),&\{t=x|x=1,2,3,\ldots\}\\ \gamma,&\{x=0,t=1\}\\ (1-\gamma),&\{x=0,t=0\}\end{array} \right\} \tag{2}\]  
where \(P_{x,t}\) is the transition probability matrix from the current state \(x\) to the next state \(t\). The transitions happen when the user enters _Process 2_ and _Process 3_. Observe that, at \(n\), if \(\gamma\) and \(\delta\) are selected, then \(X_{n}\) increases by one, i.e., \(X_{n+1}=X_{n}+1\). \(X_{n}\) decreases if, after counter selection, the same RB is not selected.