## 3 Network Model  
We consider a multi-tier 5G network where some base stations can be put in low power consumption mode as shown in Figure 1. In our proposed model we are implementing low and zero power consumption mode on the base stations within this multi-tier 5G network. The proposed algorithm is managed in the algorithm management unit, which is included in the base band unit of the network architecture. The Mobility Management Entity (MME) unit of the Evolved Packet Core (EPC) also has some contribution in the algorithm, as MME can inform the BS if any new user is approaching the cell and if any handover is about to happen. Thus, the MME can help the BS to update the total number of active users. Hence the centralized management unit is included in the mobility management entity (MME). The home subscriber server (HSS), mobility management entity (MME), serving gateway (S-GW) and packet data network gateway (P-GW) are included in the evolved packet core (EPC). The nomenclature used in rest of the paper is provided in Table 1.  
**Traffic Model:** We adopt a traffic model where the users arrive according to Poisson process in the coverage area of the BS with a certain arrival rate, \(u_{\lambda}\) and death rate, \(u_{\mu}\). Here we include all the users originating calls in the cell as well as the users being handed over from other cells.  
**Propagation Channel Model:** We consider the log-normal shadowing model with a pathloss exponent, \(\alpha\) and a shadowing variance, \(\sigma\) under AWGN channel. The received signal to noise ratio (SNR) per user is determined by link budget calculation from Equation (1), where \(\text{\textmsbm{1}}_{out}\) is the transmit power of the BS; \(PL\) is pathloss; \(G_{t}\) and \(G_{r}\) are transmit and received antenna gain respectively; \(X\) is log-normal shadowing and \(P_{n}\) is the noise power. Equation (2) is used to find the pathloss \(PL\), where \(d_{0}\) is a reference distance, \(f_{c}\) is carrier frequency, \(c\) is the speed of light and \(d\) is the distance between the BS and the user. Please note that for the sake of simplicity, we consider zero level of interference in the network. The values of \(d_{0}\) and \(c\) are constant and are provided in Table 2.  
\[SNR(dB)=\text{\textmsbm{1}}_{out}+G_{t}+G_{r}-PL-X-P_{n} \tag{1}\]  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Acronyms** & **Description** \\ \hline BS & Base Station \\ MDP & Markov decision model \\ QOS & Quality of service \\ \(BW_{T}\) & Total bandwidth offered by BS \\ \(N_{act}\) & Number of active BS \\ \(N_{std}\) & Number of stand-by BS \\ \(N_{slp}\) & Number of BS in sleep mode \\ \(N\) & Total number of BSs in the network \\ \(\text{\textmsbm{1}}_{out}\) & Transmit power \\ \(\text{\textmsbm{1}}_{act}\) & Power consumption in active mode \\ \(\text{\textmsbm{1}}_{std}\) & Power consumption in stand-by mode \\ \(\text{\textmsbm{1}}_{slp}\) & Power consumption in sleep mode \\ \hline \hline \end{tabular}
\end{table}
Table 1: The acronyms and terms.  
\[PL=20 log(\frac{4\pi d_{0}f_{c}}{c})+10\alpha log(\frac{d}{d_{0}}) \tag{2}\]  
Then [PERSON]'s capacity formula Shannon (1944) is used to determine the required bandwidth for all of the active users for a minimum target data rate, as shown in Equation (3); here \(R_{t}\) is the target data rate and \(snr\) is signal to noise ratio in linear value. This \(BW_{req}\) is the measure of the minimum required QoS. Our proposed algorithm ensures that this QoS requirement is fulfilled while reducing energy consumption and delay.  
\[BW_{req}=\frac{R_{t}}{log_{2}(1+snr)} \tag{3}\]  
Figure 1: Multi-tier architecture for energy efficient 5G networks [1].