An extensive body of research deals with estimating the correlation and the Hurst parameter of Internet traffic traces. The significance of these statistics is due to their fundamental impact on network performance. The coverage of Internet traffic traces is, however, limited since acquiring such traces is challenging with respect to, e.g., confidentiality, logging speed, and storage capacity. In this work, we investigate how the correlation of Internet traffic can be reliably estimated from random traffic samples. These samples are observed either by passive monitoring within the network, or otherwise by active packet probes at end systems. We analyze random sampling processes with different inter-sample distributions and show how to obtain asymptotically unbiased estimates from these samples. We quantify the inherent limitations that are due to limited observations and explore the influence of various parameters, such as sampling intensity, network utilization, or Hurst parameter on the estimation accuracy. We design an active probing method which enables simple and lightweight traffic sampling without support from the network. We verify our approach in a controlled network environment and present comprehensive Internet measurements. We find that the correlation exhibits properties such as long range dependence as well as periodicities and that it differs significantly across Internet paths and observation times.
The injection of test packets into a network for inferring network performance, i.e., active probing, has attracted considerable attention in recent years. End-to-end packet delays or inter packet times are metrics commonly used to estimate network characteristics such as the average available bandwidth or even to reconstruct statistics of the cross-traffic @cite_3 @cite_23 @cite_37 @cite_1 . Under the assumption of FIFO scheduling, cross traffic intensity can be estimated from the dispersion of back-to-back probing packets @cite_26 @cite_23 @cite_12 @cite_32 .
@cite_37 @cite_26 @cite_1 @cite_32 @cite_3 @cite_23 @cite_12
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Most existing available-bandwidth measurement techniques are justified using a constant-rate fluid cross-traffic model. To achieve a better understanding of the performance of current bandwidth measurement techniques in general traffic conditions, this paper presents a queueing-theoretic foundation of single-hop packet-train bandwidth estimation under bursty arrivals of discrete cross-traffic packets. We analyze the statistical mean of the packet-train output dispersion and its mathematical relationship to the input dispersion, which we call the probing-response curve. This analysis allows us to prove that the single-hop response curve in bursty cross-traffic deviates from that obtained under fluid cross traffic of the same average intensity and to demonstrate that this may lead to significant measurement bias in certain estimation techniques based on fluid models. We conclude the paper by showing, both analytically and experimentally, that the response-curve deviation vanishes as the packet-train length or probing packet size increases, where the vanishing rate is decided by the burstiness of cross-traffic. We develop a new metric for quantifying end-to-end throughput in multihop wireless networks, which we term random access transport capacity, since the interference model presumes uncoordinated transmissions. The metric quantifies the average maximum rate of successful end-to-end transmissions, multiplied by the communication distance, and normalized by the network area. We show that a simple upper bound on this quantity is computable in closed-form in terms of key network parameters when the number of retransmissions is not restricted and the hops are assumed to be equally spaced on a line between the source and destination. We also derive the optimum number of hops and optimal per hop success probability and show that our result follows the well-known square root scaling law while providing exact expressions for the preconstants, which contain most of the design-relevant network parameters. Numerical results demonstrate that the upper bound is accurate for the purpose of determining the optimal hop count and success (or outage) probability. Although packet-pair probing has been used as one of the primary mechanisms to measure bottleneck capacity, cross-traffic intensity, and available bandwidth of end-to-end Internet paths, there is still no conclusive answer as to what information about the path is contained in the output packet-pair dispersions and how it is encoded. In this paper, we address this issue by deriving closed-form expression of packet-pair dispersion in the context of a single-hop path and general bursty cross-traffic arrival. Under the assumptions of cross-traffic stationarity and ASTA sampling, we examine the statistical properties of the information encoded in inter-packet spacings and derive the asymptotic average of the output packet-pair dispersions as a closed-form function of the input dispersion. We show that this result is different from what was obtained in prior work using fluid cross-traffic models and that this discrepancy has a significant impact on the accuracy of packet-pair bandwidth estimation. Packet delay and loss are two fundamental measures of performance. Using active probing to measure delay and loss typically involves sending Poisson probes, on the basis of the PASTA property (Poisson Arrivals See Time Averages), which ensures that Poisson probing yields unbiased estimates. Recent work, however, has questioned the utility of PASTA for probing and shown that, for delay measurements, i) a wide variety of processes other than Poisson can be used to probe with zero bias and ii) Poisson probing does not necessarily minimize the variance of delay estimates. In this paper, we determine optimal probing processes that minimize the mean-square error of measurement estimates for both delay and loss. Our contributions are twofold. First, we show that a family of probing processes, specifically Gamma renewal probing processes, has optimal properties in terms of bias and variance. The optimality result is general, and only assumes that the target process we seek to optimally measure via probing, such as a loss or delay process, has a convex auto-covariance function. Second, we use empirical datasets to demonstrate the applicability of our results in practice, specifically to show that the convexity condition holds true and that Gamma probing is indeed superior to Poisson probing. Together, these results lead to explicit guidelines on designing the best probe streams for both delay and loss estimation.