# Buffer with N Policy and Active Management
## 5 Numeric Examples
### Impact of Active Management during the Transmission Phase  
The intention of this part is to assess the influence of active management solely during the transmission phase. Therefore, passive queue management is assumed during the energy-saving phase, i.e., \(d_{1}(n)=d^{*}(n)\), where \(d^{*}(n)\) is given in (1).  
Conversely, in the transmission phase, active management is implemented in the following form:  
\[d_{2}(n)=d_{2}(n,b)=\left\{\begin{array}{ll}0,&\text{if}\quad n<20,\\ \\ ((n-20)/80)^{b},&\text{if}\quad 20{\leq}n{<}100,\\ \\ 1,&\text{if}\quad n\geq 100.\end{array}\right. \tag{49}\]  
Specifically, \(d_{2}(n)\) depends on a parameter \(b\in[0,\infty)\). A higher value of \(b\) makes \(d_{2}\) less aggressive, meaning the dropping probability is lower across the entire operating range from 20 to 80, and vice versa. For \(b=1\), the function \(d_{2}\) is linear, for \(b>1\), it is convex; and for \(b<1\), it is concave. Several selected shapes of \(d_{2}\) are illustrated in Figure 1.  
In Figure 2, the average queue length versus \(b\) is depicted. As observed, the function is mostly increasing, which is expected. Higher values of \(b\) (corresponding to smaller dropping probabilities) generally lead to longer average queues. However, particularly intriguing is the graph for \(b\in(0,1)\). It clearly shows a non-monotonic trend, reaching a minimum around \(b=0.39\). This indicates that for low values of \(b\), decreasing dropping probabilities across all queue sizes actually decreases the average queue size, rather than increasing it, which is very surprising.  
Figure 1: Dropping function \(d_{2}(n,b)\) for several values of parameter \(b\).  
This phenomenon can be understood by considering the two components that influence the average queue length in the system with the N policy, i.e., the duration of the transmission phase, \(\overline{R}\), and the queue sizes in the transmission phase, embedded in \(B(k)\) (see Formula (17)). For \(b\in(0,0.39)\) in Figure 2, the following happens: Increasing \(b\) decreases dropping probabilities in the transmission phase, leading to larger queue sizes in this phase. Simultaneously, the duration of the transmission phase, \(\overline{R}\), also increases. Therefore, we have growing \(B(k)\) in the numerator of (17) and growing \(\overline{R}\) in the denominator. The resulting \(\overline{X}\) depends on which grows faster with \(b\), \(A(k)\), or \(\overline{R}\). Apparently, for small values of \(b\), the duration of the transmission phase may grow faster with \(b\) than queue sizes in this phase. Such a phenomenon does not occur in classic queueing systems without the N policy.  
In Figure 3, the energy efficiency versus \(b\) is depicted. As observed, higher values of \(b\) result in smaller energy efficiency of the system. This is expected because a higher \(b\) prolongs the transmission phase, thereby decreasing \(E\). Clearly, \(b\) can be adjusted to achieve a desired energy efficiency. Notably, not all values of \(E\) can be attained: \(E\) ranges from \(0.50\) for \(b=\infty\) to \(0.63\) for \(b=0\). Therefore, every value of \(E\) within the interval \((0.50,0.63)\) can be achieved. For example, to achieve \(E=0.55\), \(b\) should be set to \(1.695\). The range of achievable values of \(E\) can be expanded by incorporating active management in the energy-saving phase as well, which will be explored later.