# On Optimal Cell Flashing for Reducing Delay and Saving Energy in Wireless Networks
## 2 System Model
We use Shannon capacity to model the transmission rate to a user at location \(x\), i.e.,  
\[c_{i}(x)=W\ \text{log}_{2}(1+SINR_{i}(x)), \tag{1}\]  
where \(W\) denotes the channel bandwidth, and \(SINR_{i}(x)\) is the receive signal-to-interference-plus-noise ratio at location \(x\) for the signal from BS \(i\). For analytical tractability, we assume that \(c_{i}(x)\) does not change over time, i.e., we do not consider fast fading or dynamic inter-cell interferences. Instead, \(c_{i}(x)\) can be considered as a time-averaged transmission rate. This assumption is reasonable when the time scale for measuring channel gain \(g_{i}(t)\) is assumed to be much larger than the time scale of fast fading [23; 24]. \(SINR_{i}(x)\) is then given by:  
\[SINR_{i}(x)=\frac{P_{i}g_{i}(x)}{\sigma^{2}+I_{i}(x)}, \tag{2}\]  
where \(P_{i}\) denotes the transmission power spectral density (W/Hz) of BS \(i\), \(g_{i}(x)\) denotes the total channel gain from the BS \(i\) to the user equipment (UE) at location \(x\), including path loss, shadowing, and other factors, if any. In addition, \(\sigma^{2}\) denotes noise power spectral density (W/Hz) and \(I_{i}(x)\) denotes the average interference power spectral density seen by the UE at location \(x\). **Assumption 1** (fast flashing). : We assume that the cell flashing is much faster than flow arrival process, which is reasonable because one frame time in LTE-A is 10 ms, which is much shorter than average file download time that spans seconds to minutes. In addition, we consider that cell flashing period is one frame time of LTE-A or less. Figure 1: Example of cell flashing with four partitions. (**a**) Power configuration in each partition; and (**b**) cell coverage based on SINR. The gray cells represent base stations that are active and the white cells represent base stations that are inactive. Then, transition time between active and sleep mode becomes negligible because transition time in DTX is 17 \(\upmu\)s according to [22]. We can operate cells with non-flashing or flashing mode. If \(\beta\) is 0, all base stations are always on, i.e., non-flashing, and this is same to the case of full time reuse. If \(\beta\) is 1, cellular networks always operate with cell flashing, and each base station is on one third of time in a timely orthogonal fashion. Let \(c_{i}^{0}(x)\) denote the instantaneous capacity at location \(x\) when cells are non-flashing and \(c_{i}^{1}(x)\) denote the instantaneous capacity when one of three cells are on. Thus, the time-averaged capacity \(c_{i}(x)\) at location \(x\) with flashing ratio \(\beta\) is given by:  
\[c_{i}(x,\beta)=(1-\beta)c_{i}^{0}(x)+\frac{\beta}{3}\ c_{i}^{1}(x). \tag{3}\]  
It should be noted that \(c_{i}(x,\beta)\) is location-dependent but not necessarily determined by the distance from the BS \(i\). For example, \(c_{i}(x,\beta)\) can be very small in a shadowed area where \(g_{i}(x)\) is very small. Hence, \(c_{i}(x,\beta)\) can capture shadowing as well [25]. According to Assumption 1, we can use the time-averaged capacity Equation (3) to define the system-load density as below:  
\[\rho_{i}(x,\beta):=\frac{\gamma(x)}{c_{i}(x,\beta)}=\frac{\gamma(x)}{(1-\beta )c_{i}^{0}(x)+\frac{\beta}{3}\ c_{i}^{1}(x)}, \tag{4}\]  
which denotes the fraction of time required to deliver traffic load \(\gamma(x)\) from BS \(i\) to location \(x\). We assume that \(\min_{i}\rho_{i}(x,\beta)\) is finite, i.e., at least one BS has physical capacity to location \(x\in\mathcal{L}\) that is not arbitrarily close to zero [25]. Then, the load of BS \(i\) (or utilization) is given by:  
\[\rho_{i}(\beta)=\int_{\mathcal{L}_{i}}\frac{\gamma(x)}{(1-\beta)c_{i}^{0}(x)+ \frac{\beta}{3}\ c_{i}^{1}(x)}\ dx, \tag{5}\]  
where \(\mathcal{L}_{i}\) is the coverage of BS \(i\). Note that the meaning of \(\rho_{i}(\beta)\) is the busy fractional time of BS \(i\). We do not consider the unstable system, so \(\rho_{i}(\beta)\) will be smaller than 1 [25].