Abstract
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.
Highlights
The bufferbloat phenomenon, occurring in the contemporary Internet, consists in frequent overfilling of one or more buffers on the packet delivery path
It may cause significant performance deterioration of the network, especially in terms of the packet delivery time. This phenomenon has been diagnosed and described in several papers, (e.g., [1,2,3]), and the known remedy for it is an application of active queue management in Internet routers. This has been stated in the RFC 7567 document, from 2015, which begins with the following statement: “It presents a strong recommendation for testing, standardization, and widespread deployment of active queue management (AQM) in network devices to improve the performance of today’s Internet”
One way to compute the loss ratio in the queue with the dropping function is by using the stationary distribution of the queue size and the dropping function
Summary
The bufferbloat phenomenon, occurring in the contemporary Internet, consists in frequent overfilling of one or more buffers on the packet delivery path. It may cause significant performance deterioration of the network, especially in terms of the packet delivery time This phenomenon has been diagnosed and described in several papers, (e.g., [1,2,3]), and the known remedy for it is an application of active queue management in Internet routers. The analysis based on the queueing theory research on queues with the dropping function was initiated with approximate solutions given in [10, 11] and followed by exact results [12,13,14,15,16,17,18,19], obtained for various traffic and service assumptions. Analytical solutions of queueing models with the dropping function and the finite buffer are rather complex (see, e.g., [17]) and computationally demanding (see, e.g., [20] for a discussion of this aspect).
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