Nonparametric inference for stochastic feedforward networks based on cross-spectral analysis of point processes

Abstract

In this paper we study a nonparametric estimation problem for stochastic feedforward systems with nodes of type G/G/$\infty.$ We assume that we have observations of the external arrival and external departure processes of customers of the system, but no information about the movements of the indistinguishable customers in the network. Our aim is the construction of estimators for the characteristic functions and the densities of the service time distributions at the nodes as well as for the routing probabilities. Since the system is only partly observed we have to clarify first if the parameters are identifiable from the given data. The crucial point in our approach is the observation that in our stochastic networks under study the influence of the arrival processes on the departure processes can be described in a linear and time-invariant model. This makes it possible to apply cross-spectral techniques for multivariate point processes. The construction of the estimators is then based on smoothed periodograms. We prove asymptotic normality for the estimators. We present the statistical analysis for a tandem system of two nodes in full details and show afterwards how the results can be generalized to feedforward systems of three or more nodes and to systems with positive feedback probabilities at the nodes.