# Buffer with N Policy and Active Management
## 2 Related Work  
To the author's best understanding, the findings of this study are new. Specifically, there are several published articles on the N policy, but without the active queue management component. Conversely, there are many papers on active queue management, but without the N policy. The combination of the N policy with active management considered here makes the N policy more usable, enabling better control of performance. As indicated previously, the combined model requires a different mathematical approach due to the absence of vertical homogeneity, and constitutes a non-trivial generalization of the original model.  
The general idea of a system with alternating idle and work phases, switched by a critical queue size and zero queue size, dates back to [1]. Much later, it was noticed that such an organization of a buffer suits very well the purpose of saving energy in wireless sensor networks [2]. Since then, several papers on the N policy have been published with this application in mind, e.g., [2; 3; 4; 5; 6; 7; 8; 9]. These papers differ in various details of the model and studied performance characteristics. Specifically, in [3; 4], a fully Markovian model is studied, with exponentially distributed service and Poisson arrivals. In [5], a queuing model of two, rather than one, sensor nodes is considered, again under fully Markovian assumptions. A non-Markovian model, with general service times, is explored in [2; 6; 7]. In particular, the stationary characteristics are shown in [2], whereas [6; 7] are devoted to transient analysis. Moreover, pure Poisson arrivals are assumed in [2; 6], while compound Poisson arrivals are considered in [7]. Finally, in [8; 9], the whole network performance, rather than a single node, is investigated. However, due to the complexity of the model, the results are obtained via simulations rather than mathematical analysis, as in [2; 3; 4; 5; 6; 7].  
Herein, we study the model of the N policy with generally distributed service times and Poisson arrivals, but it is crucial that the system is extended by the active queue management component, which is not present in [1; 2; 3; 4; 5; 6; 7; 8; 9].  
The literature on active management of a buffer, but without the N policy, is broad. Firstly, there are many simulation studies in which various forms of the dropping function are tested, including linear [10], polynomial [11; 12; 13], trigonometric [14], exponential [15], Gaussian [16], beta [17], and composed of a few different elementary functions [18; 19; 20; 21]. Secondly, there are plenty of mathematical modeling papers focusing on various performance characteristics, which are obtained for various arrival processes, service distributions, and dropping functions, see, e.g., [22; 23; 24; 25; 26; 27; 28; 29; 30]. However, all the models of [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20] lack the N policy, which is the starting point here. It needs to be noted that active queue management has been postulated to be applied in sensor networks in many papers, see e.g., [31; 32; 33; 34; 35].  
Finally, the dropping function is not the sole viable method of active buffer management. Many other active management algorithms have been considered in the literature, e.g., [36; 37; 38; 39; 40; 41; 42]. In these algorithms, the dropping probability may not be based solely on the buffer occupancy, but it may depend on other factors as well, e.g., on the packet waiting times. The great advantage of active management basing dropping functions is its simplicity, which translates into low computational complexity compared with other advanced algorithms [43]. Again, in none of the papers [36; 37; 38; 39; 40; 41; 42] the N policy is embedded.