THE theory of Fourier integrals arises out of the elegant pair of reciprocal formulae The Laplace Transform By David Vernon Widder. (Princeton Mathematical Series.) Pp. x + 406. (Princeton: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

The Laplace Transform

Today's Internet is a massive, distributed network which contin- ues to explode in size as e-commerce and related activities grow. The hetero- geneous and largely unregulated structure of the Internet renders tasks such as dynamic routing, optimized service provision, service level verification and detection of anomalous/malicious behavior extremely challenging. The problem is compounded by the fact that one cannot rely on the cooperation of individual servers and routers to aid in the collection of network traffic measurements vital for these tasks. In many ways, network monitoring and inference problems bear a strong resemblance to other "inverse problems" in which key aspects of a system are not directly observable. Familiar sig- nal processing or statistical problems such as tomographic image reconstruc- tion and phylogenetic tree identification have interesting connections to those arising in networking. This article introduces network tomography, a new field which we believe will benefit greatly from the wealth of statistical the- ory and algorithms. It focuses especially on recent developments in the field including the application of pseudo-likelihood methods and tree estimation formulations.

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Network Tomography: Recent Developments

In this paper we consider the problem of inferring link-level loss rates from end-to-end multicast measurements taken from a collection of trees. We give conditions under which loss rates are identifiable on a specified set of links. Two algorithms are presented to perform the link-level inferences for those links on which losses can be identified. One, the minimum variance weighted average (MVWA) algorithm treats the trees separately and then averages the results. The second, based on expectation-maximization (EM) merges all of the measurements into one computation. Simulations show that EM is slightly more accurate than MVWA, most likely due to its more efficient use of the measurements. We also describe extensions to the inference of link-level delay, inference from end-to-end unicast measurements, and inference when some measurements are missing.

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Network tomography on general topologies

In network performance tomography, characteristics of the network interior, such as link loss and packet latency, are inferred from correlated end-to-end measurements. Most work to date is based on exploiting packet level correlations, e.g., of multicast packets or unicast emulations of them. However, these methods are often limited in scope-multicast is not widely deployed-or require deployment of additional hardware or software infrastructure. Some recent work has been successful in reaching a less detailed goal: identifying the lossiest network links using only uncorrelated end-to-end measurements. In this paper, we abstract the properties of network performance that allow this to be done and exploit them with a quick and simple inference algorithm that, with high likelihood, identifies the worst performing links. We give several examples of real network performance measures that exhibit the required properties. Moreover, the algorithm is sufficiently simple that we can analyze its performance explicitly

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Network Tomography of Binary Network Performance Characteristics

In this paper we explore the use of end-to-end unicast traffic as measurement probes to infer link-level loss rates. We leverage on of earlier work that produced efficient estimates for link-level loss rates based on end-to-end multicast traffic measurements. We design experiments based on the notion of transmitting stripes of packets (with no delay between transmission of successive packets within a stripe) to two or more receivers. The purpose of these stripes is to ensure that the correlation in receiver observations matches as closely as possible what would have been observed if the stripe had been replaced by a notional multicast probe that followed the same paths to the receivers. Measurements provide good evidence that a packet pair to distinct receivers introduces considerable correlation which can be further increased by simply considering longer stripes. We then use simulation to explore how well these stripes translate into accurate link-level loss estimates. We observe good accuracy with packet pairs, with a typical error of about 1%, which significantly decreases as stripe length is increased to 4 packets.

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Inferring link loss using striped unicast probes

Packet delay greatly influences the overall performance of network applications. It is therefore important to identify causes and locations of delay performance degradation within a network. Existing techniques, largely based on end-to-end delay measurements of unicast traffic, are well suited to monitor and characterize the behavior of particular end-to-end paths. Within these approaches, however, it is not clear how to apportion the variable component of end-to-end delay as queueing delay at each link along a path. Moreover, there are issues of scalability for large networks. In this paper, we show how end-to-end measurements of multicast traffic can be used to infer the packet delay distribution and utilization on each link of a logical multicast tree. The idea, recently introduced in Caceres et al. (1999), is to exploit the inherent correlation between multicast observations to infer performance of paths between branch points in a tree spanning a multicast source and its receivers. The method does not depend on cooperation from intervening network elements; because of the bandwidth efficiency of multicast traffic, it is suitable for large-scale measurements of both end-to-end and internal network dynamics. We establish desirable statistical properties of the estimator, namely consistency and asymptotic normality. We evaluate the estimator through simulation and observe that it is robust with respect to moderate violations of the underlying model.

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Joe Horowitz

Papers

Multicast-based inference of network-internal delay distributions