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

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.

Queues With the Dropping Function and Non-Poisson Arrivals

“…The infinite-buffer queue with the dropping function has been studied so far in [21] and [11], with Poisson arrivals only. In both of these papers, a stability analysis was performed first, then followed by derivations of the most important performance characteristics and numerical examples.…”

“…In both of these papers, a stability analysis was performed first, then followed by derivations of the most important performance characteristics and numerical examples. In particular, in [21] the M/M/1 model with the dropping function was studied, while in [11] -the M/G/1 model. The presented herein study on the G/M/1 model is complementary to those papers.…”

“…[25]). Therefore, for queue sizes above N , the new system operates in exactly the same way as the classic G/M/1 system without the dropping function, but with the interarrival time distribution given by (21) and the average value:…”

Strengthening Packet Loss Measurement from the Network Intermediate Point

“…By definition, we can easily identity the unfilled holes in S by sorting the data packets in ascending order according to the sequence number. However, we also note that the byte range identified by the TCP sequence number is 0 to 2 32 . Moreover, in a TCP connection, the sequence number of the first data segment is random.…”

“…In recent years, intermediate-point packet loss estimation has been introduced into Software Defined Network (SDN) applications, i.e., they combined with SDN to achieve packet loss monitoring [28] [29] [30], moreover, for good performance monitoring, Hark et al [31] also evaluated the performance of these techniques to take the appropriate packet loss estimation technique in different network condition. In addition, Andrzej et al [32] realized the packet loss estimation at the intermediate router by leveraging the stationary distribution of the queue size and the dropping function. While Sierra et al [33] estimated packet losses in the network intermediate point by proposing a simplest approach of counting as a retransmission a packet whose sequence number is smaller than the previous one.…”