Modelling structured multivariate point process data has wide ranging applications like understanding neural activity, developing faster file access systems and learning dependencies among servers in large networks. In this paper, we develop the Poisson network model for representing multivariate structured Poisson processes. In our model each node of the network represents a Poisson process. The novelty of our work is that waiting times of a process are modelled by an exponential distribution with a piecewise constant rate function that depends on the event counts of its parents in the network in a generalised linear way. Our choice of model allows to perform exact sampling from arbitrary structures. We adopt a Bayesian approach for learning the network structure. Further, we discuss fixed point and sampling based approximations for performing inference of rate functions in Poisson networks.