# Priority-Based Machine-To-Machine Overlay Network over LTE for a Smart City
## 4 Modelling PRACH for H2H and PM2M Traffic  
We now present the analytical model of random access, beginning with H2H terminals. Although the population of H2H terminals in a single LTE/LTE-A cell is large, they attempt random access occasionally and infrequently, which means that we may assume that H2H requests arrive according to a Poisson distribution. The mean arrival rate \(\lambda_{w}\) of H2H traffic can be calculated for a single PRACH resource as, \(\lambda_{WH}=(\lambda_{w}T_{f})/c_{f}\), where \(T_{f}\) is the LTE superframe time and \(c_{f}\) determines the number of PRACH resource blocks per LTE frame.  
LTE PRACH is overloaded by preamble transmissions with potential collisions and external interference caused by the random access in surrounding cells. To detect a signal successfully at the eNodeB, the Signal to Interference Noise Ratio (SINR) should exceed a certain threshold value as indicated in Figure 17.12 in [2]. For example, a threshold value of \(18 dB\) guarantees that the probability of preamble missed detection is smaller than \(10^{-2}\) and the probability of false alarm is less than \(10^{-3}\) for eNodeB. For H2H terminals, SINR is the ratio of preamble sequence energy over noise power density, for \(n\) H2H terminals concurrently performing access including initial and handoff requests. Assuming all transmissions use the same power level, the aforementioned ratio is:  
\[\begin{split} E_{seq}/N_{0}&=\frac{\frac{P_{seq}I_{ 1}}{R}}{\frac{(n-1)I_{1}+N_{M}I_{1}+\eta_{p,1}+\eta_{0}W}{W}}\\ &=P_{seq}\frac{W/R}{n-1+N_{M}+\eta_{p,1}/I_{1}+\eta_{0}W/I_{1}} \end{split} \tag{1}\]  
where  
\(P_{seq}\) is the preamble sequence length in bits;  
\(W\) is the LTE PRACH bandwidth;  
\(R\) is the LTE PRACH preamble data rate;  
\(I_{1}\) is the received signal power;  
\(\eta_{p,1}\) is the power of external (Gaussian) interference;  
\(\eta_{0}\) is the spectral density of white noise; and  
\(N_{M}\) is the number of preambles that are always active for the PM2M overlay.  
The ratio of the outer cell interference and received signal power \(\eta_{p,1}/I_{1}\) of PRACH follows a Gaussian distribution with mean and variance denoted by \(k_{m,1}\) and \(k_{p,1}\)[10]. In the case of the threshold  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**UP** & **Traffic** & \(CW_{min}\) & \(CW_{max}\) \\ \hline
0 & Background (BK) & 16 & 64 \\
1 & Best Effort (BE) & 16 & 32 \\
2 & Excellent Effort (EF) & 8 & 32 \\
3 & Controlled Load (CL) & 8 & 16 \\
4 & Video (VI) & 4 & 16 \\
5 & Voice & 4 & 8 \\
6 & Network control & 2 & 8 \\
7 & Emergency or medical event data & 1 & 4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: IEEE 802.15.6 Access Categories (AC) with User Priorities (UP) and Contention Window (CW) values.  
being exceeded, the overload condition occurs, and the preamble cannot be detected successfully. The probability of PRACH overload in the first handshake step is:  
\[O\_PRACH_{n}=P_{r}(n+\frac{\eta_{p,1}}{I_{1}}>\frac{W}{RT_{1}}-N_{M}-\frac{W\eta _{0}}{I_{1}}) \tag{2}\]  
The PRACH overload for third handshake step for \(j\) concurrent H2H terminals is:  
\[Q\_PRACH_{j}=P_{r}(j+\frac{\eta_{p,3}}{I_{3}}>\frac{W_{3}}{R_{3}T_{3}}-\frac{W_ {3}\eta_{0}}{I_{3}}) \tag{3}\]  
We assume that spectral efficiency in the third handshake step is the same as the first step, \(\frac{W_{3}}{R_{3}}=\frac{W}{R}\). The SINR threshold in this step is \(-5\) dB because it is observed over a single bit. The ratio of the outer cell interference and received signal power \(\frac{\eta_{p,3}}{I_{3}}\) of PRACH follows a Gaussian distribution with mean and variance denoted by \(k_{m,3}\) and \(k_{v,3}\)[10].