Granular model of packet pair separation in Poissonian traffic

Abstract

We introduce a theoretical model of packet pair separation based on a transient solution of the Takács integro-differential equation. We show that in addition to the parameters of the fluid approximation (physical bandwidth and the average cross traffic rate) a new parameter characterizing the granularity of the cross traffic is necessary. These three parameters determine the dynamics of the queue in the diffusive approximation and all important distributions and averages of the packet separation even in multi-hop scenarios (assuming independent cross traffic on different hops). The model describes correctly the data collected in simulations, laboratory and Internet experiments. The adjustable model parameters are the physical bandwidth, the available bandwidth and the weighted average of the packet size of the cross traffic. We show that an implementation of the theoretical results can be used to estimate such parameters in packet chirp type measurements and can be a good candidate for improved available bandwidth estimation.

Introduction

In recent years the characterization of network traffic has been the interest of measurements in the Internet [1], [2], [3], [4]. The importance of measuring the main parameters of network traffic is evident. For many applications, such as audio and video streaming, it is essential to know the available bandwidth of the network path [5], [6]. Usually, end-to-end measurements are used to obtain information about the traffic conditions on a path and various probing strategies emerged.

Analysis of the dispersion process of packet pairs plays an essential role in determining the properties of cross traffic and other network conditions from experiments. The statistical analysis should rely on certain theoretical assumptions on the details of this dispersion process. The simplest approach is the fluid model where the cross traffic is regarded as a continuum of infinitely small packets. Dovrolis et al. [5] extended the work of Jacobson [7] on packet pair spacing without any cross traffic and showed the effect of fluid cross traffic in the dispersion of packet trains. The finite packet size, the granular nature of the traffic has an important impact on the observable packet dispersion, which is disregarded in the fluid model.

The packet-pair spacing is the variable that plays a key role in the analysis of the end-to-end packet pair measurements. The available bandwidth and other parameters of the route can be estimated by studying its statistics. In a recent theoretical work Liu et al. showed that the sample path average of the output packet-pair spacing can be expressed with the δ-interval available bandwidth sample-path frequency distribution [8], [9]. The results are non-parametric in nature and valid at a very broad set of conditions, which are very important from the point of view of a complete theoretical description. It is however, a prohibitive task to calculate the sample-path frequency distribution at cases of practical importance.

In order to take the next step, in this paper we derive a parametric expression and develop a framework based on the transient solution of the Takács integro-differential equation of queuing theory [10], [11], [12]. We show that this approach takes into account the granular nature of the cross traffic and can provide an exact formula for practical traffic and network parameter estimation. Our conclusion is that the physical bandwidth, the average cross traffic rate and the new granularity parameter P_g—introduced in this paper—determine the important parameters of queuing systems in the diffusion approximation and are sufficient to characterize their behaviour and all the important properties of the packet separation. The concept can easily be generalized for multi-hop systems, if the cross traffic is independent on different hops, and a useful approximation for the average output packet spacing can be developed. Our model also describes correctly the data collected in simulations and laboratory experiments. We implement the theoretical results in a measurement tool which can estimate various parameters like the physical bandwidth, the available bandwidth and the weighted average of the cross traffic packet size in the bottleneck link with remarkable accuracy.

In Section 2 we introduce our queuing theory based framework for the distribution of the output spacing. In Section 3 we derive a closed expression for the average output spacing in terms of zero workload probability. In Section 4 we give an explicit solution for the M/D/1 case. In Section 5 we validate the results with packet level simulations at various cross traffic distributions. In Section 6 we introduce granularity (P_g) that can characterize the deviation from the fluid model. This weighted average is an effective packet size of the cross traffic. We show that hops with the same physical bandwidth, average cross traffic rate and granularity parameter are identical from the point of view of the diffusive approximation of queuing theory. This parameter triple determines all their important distributions In Section 8 we show that the results can be generalized from single to multi-hop systems. In Section 9 we develop a traffic parameter estimation tool based on the theoretical results. We study the shape of the χ² surface and the confidence of the parameter fitting. The applicability and accuracy of the method is tested in Section 10 in various real laboratory measurements.