Queues with Dropping Functions and Autocorrelated Arrivals

2020

IEEE Access

Self Cite

We deal with the single-server queueing system, in which an arriving job (packet, customer) is not allowed to the queue with the probability depending on the queue size. Such a rejected job is lost and never returns to the queue. The study is motivated, but not limited to, active queue management in Internet routers. The exponential service times and general interarrival times are assumed, what makes the model to be a generalization of classic G/M/1 and G/M/1/N queueing models. Firstly, a replacement for the ρ < 1 stability condition, which is too excessive in the considered system, is proven. Then, several popular performance characteristics are derived, including the distribution of the queue size, waiting time, workload and the time to reach a given level, as well as the loss ratio. Finally, numerical examples are presented, demonstrating the impact of the standard deviation of the interarrival time on the system performance, as well as the performance of the system for different parameterizations of the dropping function.

Section: Related Workmentioning

confidence: 99%

Queues With the Dropping Function and Non-Poisson Arrivals

2020

IEEE Access

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“…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. In particular, in [10], an approximate analysis 2 Mathematical Problems in Engineering of the model with batch Poisson arrivals and linear dropping function was presented.…”

Section: Introductionmentioning

confidence: 99%

“…In [15], the model with multiple service stations was studied. In [16,17], systems with autocorrelated arrivals were considered, with and without batch arrivals, respectively. Finally, [18,19] deal with a generalization of the model to the case with continuous packet volumes, continuous buffer capacity, and an acceptance function (rather than dropping function), which enqueues an arriving packet with probability that depends on the remaining buffer capacity.…”

Section: Introductionmentioning

confidence: 99%

“…The finite-buffer assumption seems reasonable; indeed, buffers in real switches and routers are finite. Unfortunately, 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). The reason of this is that the finite-buffer model does not enable simplification of some large, but finite sums are appearing in its solution.…”

Section: Introductionmentioning

confidence: 99%

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The Single-Server Queue with the Dropping Function and Infinite Buffer

Mathematical Problems in Engineering

Barczyk

Samociuk

2018

Self Cite

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.

“…To the best of the author’s knowledge, the results presented herein are new. For studies on other characteristics (the queue size, loss probability, response time) of systems with the dropping function, or carried out under different assumptions on the arrival process and service times, we refer the reader to [ 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 ]. On the other hand, there are several papers on the time to reach a given level in classic queueing models, i.e., without the dropping function—see, for example, [ 29 , 30 , 31 , 32 , 33 ] and the references given there.…”

Section: Introductionmentioning

confidence: 99%

On the Transient Queue with the Dropping Function

2020

Entropy

Self Cite

We deal with a queueing system, in which arriving packets are being dropped with the probability depending on the queue size. Such a scheme is used in several active queue management schemes proposed for Internet routers. In this paper, we derive and analyze a selected transient characteristic of the model, i.e., the probability that in a given time interval the queue size is kept under a predefined level. As the main purpose of the discussed queueing scheme is to maintain the queue size low, this is a natural characteristic to study. In addition to that, the average time to reach a given level is derived. Theoretical results for both characteristics are accompanied by numerical examples. Among other things, they demonstrate that the transient behavior of the queue may vary significantly with the shape of the dropping function, even if the steady-state performance remains unaltered.