# Optimal Base Station Density of Dense Network:
## 2 System Model
### Channel Access Mode of the SmBS Tier  
SmBSs operate in unlicensed bands where the bandwidth is \(W_{\mathrm{m}}\). We assume that they work in LBT mode. A slotted non-persistent carrier sense multiple access (CSMA) protocol is adopted in the SmBS tier for channel sharing. Since the fairness of the resource allocation is not the primary question we concern in network deployment and is beyond the scope of this paper, we assume that all user devices in the SmBS tier are the same and the rate is the average ergodic rate in the small cell. All SmBS users have the same priority, contention window size and scheduling tag. Thus, the channel can be allocated fairly by the protocol. For ease of analysis, we include the acknowledgement (ACK) packet in transmission data. \(a\) is the ratio of channel propagation time to packet transmission time. It depicts the channel state of the SmBS tier in Figure 2. The channel propagation time and transmission time are normalized as \(a\) and \(1\).  
In the SmBS tier, users firstly sense the channel and then send packets only when the channel is sensed idle. If a collision occurs, the packets will be retransmitted according to K-Exponent back off algorithm. For this algorithm, a backlogged packet transfers to state \(i\) after its \(i\)th collision and will be scheduled to retransmit with an exponent probability \(q^{i}\), where \(i=1,2,\ldots K\), \(0<q<1\), and \(K\) is the end state. The contention-based channel shared among the SmBS and its users can be regarded as a multi-queue single-server traffic model. The data of all devices (SmBS and users) are Geo/G/1 queues with a Poisson arrival process. Thus, the fresh and retransmission packets from all devices form a Poisson stream in the channel. We assume the average arrival rate of a device is \(\lambda_{\mathrm{data}}\) and denote the total arrival rate as \(G\). Since the SmBSs are far enough from each other given their low transmit power, the received power from devices in other small cells is lower than the channel sensing threshold. Therefore, we do not consider the inter-cell contention.  
The channel has three states, namely 'idle','successful transmission' and 'collision'. We denote the limiting probability of them as \(\pi_{\mathrm{idle}}\), \(\pi_{\mathrm{suc}}\) and \(\pi_{\mathrm{col}}\), respectively. \(\pi_{\mathrm{idle}}\) is \(e^{-aG}\), indicating that there is no channel access attempt in duration \(a\). \(\pi_{\mathrm{suc}}\) is \(aGe^{-aG}\) since the transmission is successful when there is only one device sending data. Therefore, \(\pi_{\mathrm{col}}\) is \(1-e^{-aG}-aGe^{-aG}\). The duration of these three state are, respectively, \(t_{\mathrm{idle}}=a\), \(t_{\mathrm{suc}}=1+a\) and \(t_{\mathrm{col}}=1+a\).  
Among the three states, the channel is utilized only in the successful transmission state. Otherwise, the channel is either idle or backing off. Thus, the channel is productive only in a portion of time. Let \(f_{\mathrm{m}}\) be the portion of the time that there is a successful transmission in the channel and the expected link rate of the channel is \(R\). Therefore, the throughput of an SmBS is  
\[S_{\mathrm{m}}=R\cdot f_{\mathrm{m}}. \tag{3}\]  
Figure 2: Channel state in a non-persistent carrier sense multiple access (CSMA) network.  
To analyze and maximize the throughput of SmBS, we need to solve two problems:  
1. the expected link rate \(R\);
2. the productive fraction of time \(f_{\rm m}\).  
By controlling the parameters that influence \(R\) and \(f_{\rm m}\), the maximization of \(S_{\rm m}\) can be realized.