## 3 Proposed Dynamical DOSS Model
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Description** \\ \hline K & Number of tiers in the network \\ \hline \(\mathrm{P(N=1)}\) & The probability of having exactly one user \\ \hline \(\mathrm{P(N=0)}\) & The probability of having no users \\ \hline \(\mathrm{P_{a}}\) & The probability of a BS being active \\ \hline \(\mathrm{italicSP_{s}}\) & Probability of a BS transitioning to sleep mode \\ \hline \(\mathrm{q_{ON}}\) & Power consumption in the ON state \\ \hline \(\mathrm{q_{standby}}\) & Power consumption in the standby state \\ \hline \(\mathrm{q_{sleep}}\) & Power consumption in the sleep state \\ \hline \(\mathrm{q_{swoff}}\) & Power consumption in the switch-off state \\ \hline \(\mathrm{P_{f}}\) & Transmit power of the Femto-tier base station \\ \hline \(\mathrm{P_{T}}\) & Total power consumption \\ \hline \(\mathrm{P_{i}}\) & Transmit power of the \(i\)-th tier base station \\ \hline \(\mathrm{\alpha}\) & Exponent of path loss \\ \hline \(\mathrm{A(\alpha,\beta i,\beta min)}\) & A function involving SINR and path loss exponent \\ \hline \(\mathrm{\beta_{i}}\) & SINR threshold for the \(i\)-th tier \\ \hline \(\mathrm{\beta_{min}}\) & Minimum threshold of SINR \\ \hline \(\mathrm{\beta f}\) & SINR threshold for the Femto-tier \\ \hline \(\mathrm{P_{CON}}\) & Probability of coverage when the BSs are in the ON state \\ \hline \(\mathrm{\beta_{Th}}\) & SINR threshold for QoS maintenance \\ \hline \(\mathrm{d_{min}}\) & Delay constraint \\ \hline \(\mathrm{P_{s}}\) & Probability to go into sleep mode \\ \hline \(\mathrm{\lambda_{f}}\) & Small Base Station density or arrival rate of small BS \\ \hline \(\mathrm{F_{X(A)}}\) & Probability density function of X \\ \hline EE & Energy efficiency Function \\ \hline \(\mathrm{\beta_{f}}\) & SINR threshold for the small-cell tier \\ \hline \(\mathrm{\beta_{macro}}\) & SINR threshold for the macro BS tier \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._\[(N\leq n)=\sum_{i=1}^{K}\sum_{j=1}^{N}\frac{\lambda_{u_{i}}^{K}\big{(}\rho\lambda_{BS _{j}}\big{)}^{p}\Gamma(i+\rho)}{\Gamma(\rho)K!\Big{(}\lambda_{u_{j}}+\rho\lambda _{BS_{j}}\big{)}^{i+\rho}} \tag{2}\]  
Equation (2) represents the cumulative probability of the number of users being less than or equal to n in a multi-tier network, incorporating user densities and base station densities, along with their respective system parameters. \((N\leq n)\) is the probability that the number of users N is less than or equal to n, \(K\) is the number of tiers in the network, \(N\) is the total number of users in the network, \(\lambda_{\{u_{j}\}}\) is user density or arrival rate for the Jth user, \(\lambda_{\{BS_{j}\}}\) is the density or arrival rate of BSs for the ith BS. In a special case (ii), when N = 1 or N = 0, i.e., when BS has at least a single user or it has no user, it can be represented as follows:  
\[(N=1)=\sum_{i=1}^{K}\left(1+\frac{\lambda_{u}/\lambda_{BS_{i}}}{\rho}\right)^ {-(\rho+1)}\frac{\lambda_{u}}{\lambda_{BS_{i}}} \tag{3}\]  
\[(N=0)=\sum_{i=1}^{K}\!\left(1+\frac{\lambda_{u}/\lambda_{BS_{i}}}{\rho}\right) ^{-\rho} \tag{4}\]  
The probability of an active BS with at least one user can be written as follows:  
\[P_{a}=1-[P(N=0)]=1-\sum_{i=1}^{K}\!\left(1+\frac{\lambda_{u}/\lambda_{BS_{i}}} {\rho}\right)^{-\rho} \tag{5}\]  
The above equations indicate that the probability of a BS staying in active mode depends upon the user-BS density ratio denoted as \(\vartheta=\lambda_{u}/\lambda_{BS_{i}}\), when \(\lambda_{u}\gg\lambda_{BS_{i}}\), then \(P(N=0)\approx 0\) and \(p_{a}\approx 1\) representing a situation when the BS density is sufficiently high to ensure at least a single user per BS. The probability of BS to turn into the sleep mode will be \(p_{s}\), which can be written as follows:  
\[P_{s}=1-P_{a} \tag{6}\]  
\(p_{s}\) represents the combined probability of BS to turn into the sleep mode in different modes, as per its activity level, i.e., each cluster has a fraction on \(1-p_{s}\) off total BS, which is in an active state.