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**11 - 17**of**17**### Network Tomography Based on 1-D Projections

- IMS LECTURE NOTES–MONOGRAPH SERIES
, 2007

"... Network tomography has been regarded as one of the most promising methodologies for performance evaluation and diagnosis of the massive and decentralized Internet. This paper proposes a new estimation approach for solving a class of inverse problems in network tomography, based on marginal distribut ..."

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Network tomography has been regarded as one of the most promising methodologies for performance evaluation and diagnosis of the massive and decentralized Internet. This paper proposes a new estimation approach for solving a class of inverse problems in network tomography, based on marginal distributions of a sequence of one-dimensional linear projections of the observed data. We give a general identifiability result for the proposed method and study the design issue of these one dimensional projections in terms of statistical efficiency. We show that for a simple Gaussian tomography model, there is an optimal set of one-dimensional projections such that the estimator obtained from these projections is asymptotically as efficient as the maximum likelihood estimator based on the joint distribution of the observed data. For practical applications, we carry out simulation studies of the proposed method for two instances of network tomography. The first is for traffic demand tomography using a Gaussian Origin-Destination traffic model with a power relation between its mean and variance, and the second is for network delay tomography where the link delays are to be estimated from the end-to-end path delays. We compare estimators obtained from our method and that obtained from using the joint distribution and other lower dimensional projections, and show that in both cases, the proposed method yields satisfactory results.

### 1. ON THE CHAMBERS ET AL. METHODOLOGY

, 708

"... Abstract. Our comments are in two parts. First, we make some observations regarding the methodology in Chambers et al. Second, we briefly describe another interesting network monitoring problem that arises in the context of assessing quality of service, such as loss rates and delay distributions, in ..."

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Abstract. Our comments are in two parts. First, we make some observations regarding the methodology in Chambers et al. Second, we briefly describe another interesting network monitoring problem that arises in the context of assessing quality of service, such as loss rates and delay distributions, in packet-switched networks.

### TRANSACTIONS ON SIGNAL PROCESSING 1 Fast, Moment-Based Estimation Methods for Delay Network Tomography

, 2008

"... Consider the delay network tomography problem where the goal is to estimate distributions of delays at the link-level using data on end-to-end delays. These measurements are obtained using probes that are injected at nodes located on the periphery of the network and sent to other nodes also located ..."

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Consider the delay network tomography problem where the goal is to estimate distributions of delays at the link-level using data on end-to-end delays. These measurements are obtained using probes that are injected at nodes located on the periphery of the network and sent to other nodes also located on the periphery. Much of the previous literature deals with discrete delay distributions by discretizing the data into small bins. This paper considers more general models with a focus on computationally efficient estimation. The moment-based schemes presented here are designed to function well for larger networks and for applications like monitoring that require speedy solutions. EDICS: SSP-DECO, SSP-HIER, SSP-NGAU, SSP-PARE

### DOI 10.1007/s11134-010-9195-9 Statistical estimation of delays in a multicast tree using

"... Abstract Tomography is one of the most promising techniques today to provide spa-tially localized information about internal network performance in a robust and scal-able way. The key idea is to measure performance at the edge of the network, and to correlate these measurements to infer the internal ..."

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Abstract Tomography is one of the most promising techniques today to provide spa-tially localized information about internal network performance in a robust and scal-able way. The key idea is to measure performance at the edge of the network, and to correlate these measurements to infer the internal network performance. This paper focuses on a specific delay tomographic problem on a multicast diffu-sion tree, where end-to-end delays are observed at every leaf of the tree, and mean sojourn times are estimated for every node in the tree. The estimation is performed using the Maximum Likelihood Estimator (MLE) and the Expectation-Maximization (EM) algorithm. Using queuing theory results, we carefully justify the model we use in the case of rare probing. We then give an explicit EM implementation in the case of i.i.d. ex-ponential delays for a general tree. As we work with non-discretized delays and a full MLE, EM is known to be slow. We hence present a very simple but, in our case, very effective speed-up technique using Principal Component Analysis (PCA). MLE estimations are provided for a few different trees to evaluate our technique.

### Loss Rate Inference in Multi-Sources and Multicast-Based General Topologies

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