Statistical modeling for network traffic and inference of an M/G/infinity queue model

We consider stochastic modeling and its inference problems for a complex system. The specific motivation of this research is network traffic modeling in which a simple on-off model, an alternating renewal process, arises very naturally as an approximation to the transmission behavior over the Internet. Most studies have focused on the global characterization of the traffic based on the superposition of those simple on-off processes. Because of the ubiquity of small time scale data in the current data networks, a fine scale characterization is also desirable, which could be incorporated into a detailed traffic generation. We focus on an on-off process that has special product structure that has been used to explain different scale behavior. To utilize such a model, much of our attention has been paid to the statistical estimation and simulation of the process. We adopt a parametric estimation scheme and apply to several different types of product processes. We also compare each of them to real data and examine properties of the estimators using asymptotic results and simulation.; The second part of this dissertation deals with nonparametric inference about an M/G/infinity queue model. An interesting connection of the on-off process to an M/G/infinity model has been pointed out. Some parametric approaches have been considered in the first part also to cover both an Exponential family and a Weibull family. This part can be viewed as an extension of the study of the product type of an on-off process but can be regarded as an independent inference problem.; The M/G/infinity queue model is a well known queuing model that has been used in various contexts. A simple description appears as Poisson arrival customers whose service time follows a general distribution. There are an infinite number of servers in the system and depending on whether or not a customer is present, the system is alternating between the two states, busy and idle. Our primary interest is the service time distribution. To be more informative we are particularly interested in the shape of the distribution or the density function. Often however we do not have a direct access to the service times and only the information about busy/idle periods is available. Although the model has been used and studied in various contexts, not much work has been done in that direction with such limited information.; The relationship between busy periods and service times is known through the Laplace transform of the busy periods. Thus this can be viewed as an inversion problem, for which the solution usually is obtained by taking a numerical approximation scheme. Here we adopt a statistical nonparametric approach. We use an expansion idea to exploit the Laplace transform. This enables us to find an explicit inversion formula to recover the service time distribution. This idea is extended to find the density function. Based on that we propose nonparametric estimators with the busy period distribution replaced by its empirical distribution. In particular, the density estimation involves a conventional kernel smoothing estimator whose properties are well known. Theoretical properties of both estimators are studied. We also discuss monotonicity of the distribution function estimator and further incorporate a monotonizing scheme if necessary. Performance issues of the estimators are examined through a simulation study in which both unimodal and bimodal densities are considered.