Long range dependence in network traffic and the closed loop behaviour of buffers under adaptive window control

Abstract

We consider an Internet link carrying http-like traffic, i.e., transfers of finite volume files arriving at random time instants. These file transfers are controlled by an adaptive window protocol (AWP); an example of such a protocol is TCP.

We provide analysis for the auto-covariance function of the AWP-controlled traffic into the link’s buffer; this traffic, in general, cannot be represented by an on–off process. The analysis establishes that, for TCP-controlled transfer of Pareto-distributed file sizes with infinite second moment, the traffic into the link buffer is long range-dependent (LRD).

We also develop an analysis for obtaining the stationary distribution of the link buffer occupancy under an AWP-controlled transfer of files sampled from some distribution. For any AWP, the analysis provides us with the Laplace–Stieltjes transform (LST) of the distribution of the link buffer occupancy process in terms of the functions defining the AWP and the file size distribution. The analysis also provides a necessary and a sufficient condition for the finiteness of the mean link buffer content; these conditions again have explicit dependence on the AWP used and the file size distribution. This establishes the sensitivity of the buffer occupancy process to the file size distribution.

Combining the results from the above analyses, we provide various examples in which the closed loop control of an AWP results in finite mean link buffer occupancy even though the file sizes are Pareto-distributed (with infinite second moment), and the traffic into the link buffer is long range-dependent (with Hurst parameters which would suggest an infinite mean queue occupancy under open loop analysis).

We also study the effect of window reductions due to active queue management and find that window reductions lead to further lightening of the tail of buffer occupancy distribution.

The significance of this work is three-fold: (i) by looking at the window evolution as a function of the amount of data served and not as a function of time, this work provides a new framework for analysing various processes related to the link buffer under AWP-controlled transfer of files with a general file size distribution; (ii) it indicates that the buffer behaviour in the Internet may not be as poor as predicted from an open loop analysis of a queue fed with LRD traffic; and (iii) it shows that the buffer behaviour (and hence the throughput performance for finite buffers) is sensitive to the distribution of file sizes.

Introduction

It was observed in [1] that traffic processes in the Internet display long range dependence. In [2], this phenomenon was traced to the fact that the traffic in the Internet results from the transfer of files that have a heavy-tailed distribution. Models have shown that the transfer of Pareto-distributed files $(P\{V>x\}=\min (1\text{,}\frac{1}{x^{\alpha }})\text{,}1<\alpha <2)$ results in a traffic rate process that has an auto-correlation function that decays as $\frac{1}{{\tau }^{\alpha -1}}$. These observations have been taken to indicate that the buffer occupancy distribution in router buffers will have heavy tails [3]. Such observations are, however, based on an “open loop” analysis of an LRD traffic source feeding a buffer. It has also been noted recently [4] that an understanding of traffic and buffer processes in the Internet should take into account the closed loop nature of Internet congestion control, namely TCP which is an adaptive window protocol (AWP). In this paper, we carry out such an analysis for a particular network scenario.

The Internet carries predominantly elastic traffic; the transfer of such traffic is controlled by TCP [5]. Most of the literature on TCP modelling is concerned with the “throughput” obtained by TCP-controlled file transfers over a single bottleneck link, with or without the assumption of random drops/losses. These works can be divided into two streams; the (chronologically) first stream of work assumes a single bottleneck link that is used to transfer a fixed number of files of very large volumes (see [6], [7], [8], and references therein), whereas the second category deals with the performance of TCP-controlled transfer of http-like (finite volume) files where the number of ongoing transfers is randomly varying (see [9], and references therein). An important consideration in the case of http-like traffic is the distribution of file transfer volumes.

Some of the works that fall in the first category attempt to model the behaviour of the link buffer (see [6], [10]) but, to our knowledge, there is no such analytical study available for TCP-controlled transfer of http-like traffic. In this paper, we develop a framework for analysing the behaviour of the link buffer, and related processes, assuming that the file transfers are controlled using a general adaptive window protocol, explicitly taking into account the distribution of file transfer volumes.

We consider the scenario shown in Fig. 1, where an Internet link connects clients on one side to servers on the other side. We assume that there is no restriction on the number of simultaneous ongoing transfers. The clients generate file transfer requests and the servers send the requested files using an AWP. The servers and clients are connected to the link by very high-speed access links. Hence, the Internet link is the bottleneck; also shown in the figure is this link’s buffer containing data packets from the ongoing file transfers. We make the following system and traffic assumptions:

• The end-to-end propagation delay is negligible (in the sense that the propagation delay between the end nodes of the link is much less than one packet service time; for example, this could be a 34 Mbps link interconnecting two locations 15 km apart in a city for a TCP packet size of 1500 bytes, the bandwidth delay product being 0.3 packets).

• The link buffer on the server side is such that there is no packet loss. (It follows that since the file sizes are finite, the window growth is governed solely by the increase phase of the AWP; the window of each transfer remains finite since the volume of the transfer is finite. We wish to study the tail behaviour of the stationary contents of the buffer; such an analysis would provide some insight into the tail drop loss behaviour with finite buffers.)

• The link buffer on the server side implements a per-flow round-robin scheduling discipline with a service quantum of one packet. Examples of such scheme are Deficit Round Robin (DRR, see [11]) and weighted fair queueing (WFQ).

• Each request is for the transfer of a single file, and the files have independent and identically distributed sizes.

• The starting instants of file transfers constitute a Poisson process (of rate $\lambda$). (The instants at which new user sessions start is now accepted to be well modelled by a Poisson process (see [4]); our model thus assumes that each session transfers just one file.)

• We first assume that the link buffer does not drop or mark the packets owing to any active queue management mechanisms. This assumption is later relaxed and random marking of packets are also considered.

The first assumption above implies that the link buffer contains all unacknowledged data from the ongoing file transfers (sessions). This also implies that the link is busy whenever at least one session is active.

It has been shown [12], [13] that for Pareto-distributed file sizes (with tail $\frac{1}{x^{\alpha }}$), the data departure rate process ($d(t)$ in Fig. 1) is long range-dependent (LRD) with Hurst parameter $\frac{3-\alpha }{2}$. This result follows from the observation that, owing to zero propagation delay, the $d(t)$ process corresponds to the busy idle process of a work conserving queue. Further, $d(t)$ is not affected by the feedback control used. Clearly, however, the input process to the link buffer depends on the feedback control used and hence it is interesting to study the correlation structure of the data arrival rate process into the link buffer (denoted by $a(t)$ in Fig. 1); this is one contribution of the work presented here.

Extensive analysis of Internet data has confirmed that Internet traffic is LRD (see [1]). It has been argued that the LRD behaviour of Internet traffic is related to heavy-tailed file transfer volumes [2]. Recent studies (see [12], [14], [15]) show that the stationary distribution of a queue fed with LRD traffic will have a non-exponential tail; for example, it has been shown that an arrival rate process auto-covariance that is $\text{O}(\frac{1}{{\tau }^{\alpha -1}})\text{,}1<\alpha <2$, leads to a stationary distribution of buffer occupancy that has a tail that is $\text{O}(\frac{1}{x^{\alpha -1}})$. The above observations are usually combined to conclude that the link buffer occupancies in the Internet will be heavy-tailed. Such observations are, however, based on an “open loop” analysis of an LRD traffic source feeding a buffer. Recent numerical studies [4], [16], [17] suggest that an understanding of traffic and buffer processes in the Internet should take into account the closed loop nature of Internet congestion control, namely TCP which is an adaptive window protocol. The second contribution of this paper is to carry out such an analysis for the network scenario of Fig. 1 and for a general AWP.

It is easy to see that the behaviour of a buffer for a given input process can be strikingly different in a feedback loop as compared to when the same process is applied to the buffer (i.e., “open loop”). In Fig. 2, we provide a simple example. Fig. 2(a) depicts a closed queueing system where a single customer is fed back to the queue (with a new service requirement distributed as exponential($\mu$)) as soon as it gets served; the system is clearly stable as there is always a single customer in the system. Note that the customer arrival instants to the queue form a Poisson process of rate $\mu$. Fig. 2(b) depicts an M/M/1 queue with a Poisson arrival process of rate $\mu$, and exponentially distributed service requirement with mean $\frac{1}{\mu }$; this queue is clearly unstable (the queue length process being a null recurrent Markov chain).

It is intuitive that introduction of window reductions due to presence of active queue management scheme at the link buffer would result in a well-behaved buffer occupancy distribution. We study this phenomenon for two specific AWPs and find that the results are in accordance with the intuition.

Assuming an AWP and a general file size distribution, we study the auto-covariance function of the data arrival rate process into the link buffer (the $a(t)$ process, see Fig. 1). We then analyse the link buffer occupancy process for a general AWP and file size distribution and provide a necessary and a sufficient condition for the existence of the mean buffer occupancy. Combining the results from above two analysis, it is shown that it is possible to have a finite mean link buffer occupancy even when the file size requirements are heavy-tailed and the $a(t)$ process is LRD. This does not contradict the result of [12], [14] as the model analysed there does not include any feedback control from the queue. Next, we consider specific AWPs to study the effect of window reductions owing to random packet markings/drops and find that, as expected, window reductions result in further lightening in the tail of the buffer occupancy distribution while the traffic into the buffer remains LRD.

The paper is organised as follows. In Section 2, we develop a queueing equivalent model of the scenario of Fig. 1, introduce some notation we use in the paper and give some queueing results required later in the work. In Section 3, we introduce some characterising functions associated with an AWP. Section 4 presents a study of the auto-covariance function of the $a(t)$ process. In Section 5, we give the analysis of the link buffer occupancy process. In Section 6, we consider two specific AWPs and study the effect of introducing random marking of packets on the link buffer occupancy process and the $a(t)$ process. Section 7 concludes the paper.