# Buffer with N Policy and Active Management
## 6 Conclusions  
We studied a buffer with the N policy, enhanced with active management during both energy-saving and transmission phases. This extension allowed for much greater control over queue performance, compared to the N policy with passive management. We derived formulae on the queue length distribution, throughput, and energy efficiency of this extended policy. These formulae were proven for a general queue model, accommodating arbitrary service distribution and arbitrary parameterizations of active management in both phases.  
Numerical results demonstrated how the queue's performance can be tuned using active management in the transmission phase, energy-saving phase, or both simultaneously. Additionally, we presented solutions to optimization problems where the cost function balanced trade-offs between queue size and throughput.  
We highlighted that without active management, adjusting the system's energy efficiency through the threshold value manipulation is severely limited. Conversely, when active management is implemented, not only can we adjust energy efficiency using active management parameters, but the threshold value can also effectively serve this purpose. Furthermore, we observed that the system occasionally exhibits paradoxical behavior owing to its complexity. For example, increasing dropping probabilities in the transmission phase, energy-saving phase, or both, can sometimes increase the average queue size. Such phenomenon is impossible in a classic queue without the N policy.  
Future research directions could explore models with general distributions of interarrival times. Exact solutions for such models may be exceedingly difficult, akin to the challenges posed by the G/G/1 queue, but various approximation techniques could be investigated to obtain approximate solutions. Alternatively, we may seek a solution to the model using the Markovian Arrival Process (MAP) [46]. It allows for precise modeling of the marginal distribution of interarrival times as well as the internal dependence structure of consecutive interarrival times. The main difficulty would be obtaining the distribution of the state of the modulating Markov chain at switching points, i.e., when the queue reaches the critical threshold \(K\) and when the buffer becomes empty.  
Finally, we used active management based on the dropping function. This is a popular approach, but not the only possible one, to queue management. Other methods, e.g., [36; 37; 38; 39; 40; 41; 42], often utilize packet waiting time as an alternative or additional factor in the calculation of the dropping probability. Unfortunately, their queueing models are analytically intractable, even when applied to buffers without the N policy. The N policy, which introduces additional, non-trivial complexity, would make analytical solutions of such models completely out of reach, given the current state of queueing theory. However, such models can be studied using discrete-event network simulators. Similarly, models of whole networks of nodes exploiting the N policy with active management can be studied via simulations.  
This research received no external funding.