On the Number of Losses in an MMPP Queue

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.

“…From (27), it follows that all integrals in (31) and 32are finite for every k and l. Therefore 28is finite, as a finite sum of finite summands. This completes the proof of Eθ < ∞.…”

Section: Stabilitymentioning

confidence: 99%

Queues With the Dropping Function and Non-Poisson Arrivals

2020

IEEE Access

Self Cite

“…It is one of the most important characteristics of queues occurring in packet network nodes. It has been measured experimentally in TCP/IP networks (see, e.g., [24][25][26][27]) and studied analytically in classic queueing models (see, e.g., [28]). 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.…”

Section: Loss Ratio and Output Ratementioning

confidence: 99%

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.

“…Therefore, several papers have been published on the loss ratio measurements in the Internet, see e.g., [15][16][17][18][19]. Analytical formulas for the loss ratio in queueing models with finite buffers and different arrival processes can be found in [20][21][22]. In particular, in [20] the formula for the case of Poisson arrivals is shown, in [21]the formula for the case of Markov-modulated arrivals, while in [22]-for the case of batch Markovian arrivals.…”

Section: Related Workmentioning

confidence: 99%

Burst ratio in a single-server queue

Samociuk

2018

Telecommun Syst

Self Cite

In contemporary packet networks, the possibility of packet loss is a negative, but inevitable aspect of the network design. One of the most important characteristics of the packet loss process is the burst ratio-a characteristic describing the tendency of losses to occur in long series, one after another. In this paper, we study the burst ratio in the queueing system with the finite buffer for packets. This is motivated by the fact, that most packet losses in wired networks occur due to queueing of packets in routers, and overflowing routers' buffers. We firstly derive the exact formula for the burst ratio. Then we study its behaviour as the buffer size grows, and obtain a simplified formula for large buffers. Thirdly, we present numerical results for different system parameterizations as well as the comparison with simulation results. Then we show results of measurements of the burst ratio in the networking laboratory. Finally, we draw conclusions on the influence (or lack of it) of several factors on the burst ratio.