Delay Network Tomography Using a Partially Observable Bivariate Markov Chain

Estimation of link delay densities in a computer network, from source-destination delay measurements, is of great importance in analyzing and improving the operation of the network. In this paper, we develop a general approach for estimating the density of the delay in any link of the network, based on continuous-time bivariate Markov chain modeling. The proposed approach also provides the estimates of the packet routing probability at each node, and the probability of each source-destination path in the network. In this approach, the states of one process of the bivariate Markov chain are associated with nodes of the network, while the other process serves as an underlying process that affects statistical properties of the node process. The node process is not Markov, and the sojourn time in each of its states is phase-type. Phase-type densities are dense in the set of densities with non-negative support. Hence, they can be used to approximate arbitrarily well any sojourn time distribution. Furthermore, the class of phase-type densities is closed under convolution and mixture operations. We adopt the expectation-maximization (EM) algorithm of Asmussen, Nerman, and Olsson for estimating the parameter of the bivariate Markov chain. We demonstrate the performance of the approach in a numerical study.

I. Introduction

Network tomography aims at estimating internal parameters of a computer network from some measurements taken from accessible nodes or links of the network. Network tomography was pioneered by Vanderbei and Iannone [34] and Vardi [35]. In [34], the rate of traffic over source-destination pairs of the network was estimated from aggregated traffic counts at input and output nodes. In [35], the rate of traffic over source-destination pairs was estimated from traffic counts on some links of the network. Both formulations led to similar sets of under-determined linear equations of the form where is a column vector of say traffic count measurements, is a column vector of say traffic variables of interest, , and is a zero-one routing matrix with if contributes to , and otherwise. In [35], represents traffic over links, and represents traffic over source-destination pairs, which is modeled as a vector of independent Poisson random variables. It was shown in [35] that the unknown source-destination rates are identifiable provided that does not contain duplicate columns or zero columns. A Bayesian solution to the rate estimation problem of [35] was developed in [31]. Similar tomography problems arise in other networks such as road and rail networks [31], as well as in image deblurring [30], [36].