# Power-Efficient Random Access Design for Machine Type Communication
## 3 LTE-nMTC Random Access Design
### Effective Bandwidth and Optimal Resource Configuration
That is,  
\[w^{\dagger\dagger}(\alpha,N)=\underset{w\leq w^{\dagger}(\alpha)}{\mathrm{ argmin}}\Delta\tau_{\mathrm{c}}(w,\alpha,N)\;, \tag{17}\]  
where  
\[\Delta\tau_{\mathrm{c}}(w,\alpha,N)=\frac{\tau_{\mathrm{c}}(w,\alpha,N,|M|)}{ \tau(w^{\dagger},\alpha,|M|)}-1\;. \tag{18}\]  
The optimal random access channel bandwidth in (17) is hence a function of the number of contending devices \(N\) per random access opportunity. Figure plots the time penalty, \(\underset{w\leq w^{\dagger}(\alpha)}{\mathrm{min}}\Delta\tau_{\mathrm{c}}(w, \alpha,N)\) in (17), as a function of \(N\), which is a monotonically increasing function of \(N\), for the four coverage classes. In Figure 5, \(L_{i}\) denotes the \(L\) for CC \(i\). The transmission power of the device is \(p=20\) dBm, receiver noise figure \(\zeta=5\) dB, and \(\beta=-6\) dB. At an, e.g., 10% penalty, the devices that can be supported by \(W=180\) kHz is \(N=7\) for CC1 (\(\alpha=\)155 dB), and 3 for CC2 (\(\alpha=150\) dB), whereas for CC3 (\(\alpha=145\) dB) and CC4 (\(\alpha=140\) dB) \(N\) dwindles down to less than two. On the other hand, in order for all high coverage classes to support the same number of devices as CC 1, i.e., \(N=7\), they have to live with much higher penalties, i.e., \(\underset{w\leq w^{\dagger}(\alpha)}{\mathrm{min}}\Delta\tau_{\mathrm{c}}(w^{ \dagger\dagger}(\alpha,N),\alpha,N)\gg 10\%\) (for \(\alpha=150\), 145, and 140 dB). This phenomenon can be better explained using Figure 6, where we observe that for CC 1 (\(\alpha=\)155 dB) and at \(N=7\), the optimal bandwidth equals the effective bandwidth, i.e., \(\frac{w^{\mathsf{tt}}(a)}{w^{\mathsf{tt}}(a)}=1\). The 10% performance penalty is thus purely attributable to collisions. Whereas in order for higher coverage classes to support \(N=7\), each random access channel of these classes has to settle with an optimal bandwidth \(w^{\mathsf{tt}}(\alpha)\) that is much smaller than the effective bandwidth \(w^{\mathsf{tt}}(\alpha)\), i.e., \(\frac{w^{\mathsf{tt}}(a)}{w^{\mathsf{tt}}(a)}\ll 1\) (\(\alpha=150\), 145, and 140 dB) as shown in Figure 6, in order to make room for a sufficient number of FDM channels, \(\frac{W}{w^{\mathsf{tt}}}\). The resultant bandwidth deficit (i.e., \(\frac{w^{\mathsf{tt}}(a)}{w^{\mathsf{tt}}(a)}<1\)) gives rise to high performance penalty, i.e., \(\Delta\tau_{\mathrm{c}}(w^{\mathsf{tt}},\alpha,N)\gg 10\%\). A solution to maintaining the collision at a reasonable low probability (corresponding to, e.g., 10% penalty) for high coverage classes without incurring high bandwidth deficit penalty is to add more random access channels per random access occasion in a time division multiplexing (TDM) fashion so that the total number of random access channels is increased by \(L\) times, i.e., \(N_{\mathrm{RACH}}=LW/w\), where \(L\) is the multiplicity of the TDM random access resources in time domain (cf. Figure 7). The collision probability becomes  
\[p_{\mathrm{c}}(w,N,L)=1-\Big{(}1-\frac{w}{LW}\Big{)}^{N-1}. \tag{19}\]  
Clearly, (19) is less than (15) for \(L>1\). Nevertheless, more TDM resources are needed to compensate for the shortage in bandwidth. Referring back to Figure 5 (dotted lines), we show the values of \(L\) needed for high coverage classes, i.e., \(L_{2}=3\), \(L_{3}=12\), and \(L_{4}=36\), such that \(N=7\) can be supported without incurring the bandwidth deficit penalty, i.e., \(\frac{w^{\mathsf{tt}}(\alpha)}{w^{\mathsf{tt}}(\alpha)}\approx 1\) for \(\alpha=150\), 145, and 140 dB. That is, the penalty due to contention is below 10% as long as the number of concurrent contending devices is less than 7 (cf. Figure 6). In the following discussion, \(N\) is hence assumed to be 7 unless otherwise specified. The implication of \(N\) on the overall random access capacity will be discussed in Section 4.4. The optimal channel configurations are summarized in Table 2, whereas Table 3 is for \(W=1.08\) MHz to show the bandwidth scalability of the design.