The objective of this research is to study a class of closed queueing networks that are prevalent in telecommunication systems. In particular, we will consider systems with subexponential processing times since there is strong evidence that in many telecommunications applications, processing times have subexponential distributions. This means that the tail of the probability distributions decay much slower than exponential implying that the probability of extremely large observations is non-negligible. The intellectual merit of this project involves analyzing certain characteristics of two key performance measures: cycle times (time spent in the system) and waiting times (from arrival until joining the server at each node). In telecommunication systems, one is interested in the probability that any one of these performance measures is greater than a large value which is referred as tail asymptotics. The objective is to understand the tail asymptotics of transient and stationary cycle times and waiting times. An approach that involves developing upper and lower bounds on these two performance measures and obtaining expressions on the tail probabilities using these bounds will be employed.
Even though there has been a growing interest in queueing systems with subexponential processing times, most of the existing research has focused on single stage queues. Thus, there is a gap between the types of telecommunication systems that can be modeled using the existing analytical models and those that arise in telecommunication settings. If successful, the results of this research will bridge this gap. Moreover, this research project is long term. The analytical models developed during this phase will be (eventually) used to design control strategies for telecommunication systems that will ultimately lead to improvements in the operations of these systems.
The objective of this research is to study a class of closed queueing networks that are prevalent in telecommunication systems. In particular, we consider systems with subexponential processing times since there is strong evidence that in many telecommunications applications, processing times have subexponential distributions. For example, it has been shown that the FTP (File Transfer Protocol) transfers or the file sizes on world wide web have subexponential distributions. This means that the tail of the probability distributions decays much slower than exponential implying that the probability of extremely large observations is non-negligible. The intellectual merit of this project involves analyzing the tail asymptotics of transient and stationary cycle times (time spent in the system) and waiting times (from arrival until joining the server at each node). Our main intellectual contribution was developing analytical expressions for the tail asymptotics of transient and stationary cycle times and waiting times in a class of networks (including tandem queues and fork and join queues). We have also conducted numerous numerical experiments that illustrate that these tail asymptotics provide good approximations for the actual tail probabilities even for moderate values in the tail. Furthermore, we have developed optimal admission policies for loss networks that are prevalent in telecommunications settings. One of the broader impacts of this project is that the analytical models developed during this phase can be used to improve the operations of these systems and ultimately advance the state of the art in the analysis and control of telecommunication networks. Finally, our greatest impact on the infrastructure of science and engineering is as educators. This grant has provided partial support to a former PhD student and a current PhD student. The direct involvement of these PhD students in this project educated new researchers in the area of stochastic processes.
