Let us suppose we have a system that we can measure a fixed number of times, but at any chosen interval. Motivated by an example of probing data networks, we model this as a black box system: we can either chose to open the box or not at any time period, our aim to find out how the system evolves over time.
We use the statistical principles of design of experiments to model numerical experiments that can be designed optimally. We demonstrate how to analyse the evolution of a system as a Markov Chain, and deduce its likelihood function, and hence the Fisher information matrix. From this, numerical results provide a guide to the best design for the experiment for different values of input parameters, and we show that we can find estimators whose variance is close to the minimum variance possible. We further develop our ideas to show what happens when we take into account the effect of the observations interfering with the experiment, as would always be the case with packet probing. We present examples, and demonstrate how this could be useful to many fields, with particular reference to experiments on data networks.