# Power-Efficient Random Access Design for Machine Type Communication
## 3 LTE-nMTC Random Access Design
### Simplified Random Access Signaling
The access ID in our design is similar to the temporary ID in LTE random access procedure to distinguish the devices within the cell. The device randomly selects an access ID from a pool of \(2^{10}\) IDs. The 10-bit CRC is using to check the decoding status at the receiver. The false alarm of decoding access message is caused by the CRC check error which is very small and can be ignored. Figure 3 shows the simplified random access procedure for LTE-nMTC. Although the access message decoding does not require the timing information due to the protection of the long CP in the current design, the round-trip propagation delay of the access signal, \(\Delta t\), still remains unknown and needs to be estimated from the detected access request signal by the base station. The estimated delay is then attached to the uplink resource assignment message (cf. Figure 3) so that the device can adjust its transmission time (according to \(\Delta t\)) to ensure the time alignment of the subsequent uplink transmissions (including data and control channels) from different devices. A smaller CP can then be used only to accommodate the multipath spread on the traffic channel. The propagation delay estimation using the access request signal is described in Section 3.4. Figure 2: Illustration of the proposed random access signal and the corresponding arrive time. The probability of an access channel which is selected by more than one device is  
\[p_{\text{ch}}=1-\left(\frac{N_{\text{RACH}}-1}{N_{\text{RACH}}}\right)^{N-1}=1- \left(1-\frac{1}{N_{\text{RACH}}}\right)^{N-1}\!\!. \tag{4}\]  
This is the random access channel collision probability. Another collision situation is that more than one device select the same access ID. It can be expressed as  
\[p_{\text{ID}}=1-\left(1-\frac{1}{N_{\text{ID}}}\right)^{N-1}\!\!, \tag{5}\]  
where \(N_{\text{ID}}=2^{10}\) is the size of the access ID pool. Therefore, the total collision probability can be written as  
\[p_{\text{c}}=1-(1-p_{\text{ch}})(1-p_{\text{ID}}). \tag{6}\]  
In general, the number of access channels \(N_{\text{RACH}}=S/s\) is smaller than the number of total access IDs. Therefore, \(p_{\text{c}}\approx p_{\text{ch}}\). When an access channel is selected by more than one device, there will be a decoding failure at the base station. The device will prepare another random access attempt when there is no uplink resource assignment to itself in a given duration. Apparently, the collision will increase the air time. Reducing the collision probability requires an increase in the number of access channels, \(N_{\text{RACH}}\). However, for a given total random access bandwidth of \(W\), a larger \(N_{\text{RACH}}\) means a smaller bandwidth (\(w=W/N_{\text{RACH}}\)) per random access channel, a lower data rate that each channel supports, and a longer transmission time for a device. The following question then arises: Given a bandwidth of \(W\), what is the resource configuration scheme for the random access channels (i.e., the bandwidth of a random access channel) that minimizes the overall access request message transmission time? We use the \"effective bandwidth\" to solve this question [22].