This paper studies the feasibility and algorithms for inferring the delay at each link in a communication network based on a large number of end-to-end measurements. The restriction is that we are not allowed to measure directly on each link and can only observe the route delays. It is assumed that we have considerable flexibility in choosing which routes to measure. We investigate two different cases: 1) each link delay is a constant and 2) each link delay is modeled as a random variable from a family of distributions with unknown parameters. We will answer whether such indirect inference is possible at all, and when possible, how it can be carried out. The emphasis is on developing the maximum-likelihood estimators for scenario 2) when the link delays are modeled by exponential random variables or mixtures of exponentials. We have derived solutions based on the EM algorithm and demonstrated that, even though they do not necessarily reflect the true model parameters, they do seem to maximize the likelihood in most cases and that the resulting probability density functions match the true functions on regions where the probability mass concentrates