We present an analysis of queues with the dropping function and infinite buffer. In such queues, the arriving packet (job, customer, etc.) can be dropped with the probability which is a function of the queue size. Currently, the main application area of the dropping function is active queue management in routers, but it is applicable also in many other queueing systems. So far, queues with the dropping function have been analyzed with finite buffers only, which led to complicated, computationally demanding formulas. Assuming infinite buffers enabled us herein to obtain formulas in compact, easy to use forms. Moreover, a model with the infinite buffer can often be used as a good approximation of the real queue, in which the buffer is large. We start with noticing that the classic stability condition, ρ<1, cannot be used for queues with the dropping function and infinite buffer. For this reason, we prove a few new, easy to use conditions, which guarantee system stability or instability. Then we prove several theorems on popular performance characteristics, including the queue size, busy period, loss ratio, output rate, and system response time. Additionally, we derive a special, very important characteristic called the burst ratio, which may influence severely the quality of real-time multimedia transmissions. All the theorems are illustrated with numerical examples, demonstrating in particular how the system stability may be tested and how the shape of the dropping function may affect different performance characteristics.

Active Queue Management (AQM) is recommended by Internet Engineering Task Force to mitigate the bufferbloat phenomenon in the Internet. In this paper, we show the results of comprehensive measurements carried out in our university network, in which a device with an AQM algorithm, designed and programmed for this purpose, was running. The implemented AQM algorithm was based on the dropping function, i.e. arriving packets were dropped randomly, with the probability being a function of the queue length. Several different dropping function forms, proposed in the networking literature, were used, in addition to the classic FIFO queue (no AQM). The experiment lasted over a month, during which the state of the network was measured and recorded several thousand times. This made the results independent of the natural fluctuations of the users’ behavior and the network load. Conclusions on the general performance improvement offered by the implemented AQM, as well as the differences in the performance between particular forms of the dropping function, were reached. Some of these conclusions differ from those drawn previously from simulations. This underlines the need for carrying measurements of new AQMs in real, operating networks, with complex, natural traffic.