Large deviation analysis of the single server queue
The author develops the large deviation principle (LDP) for a sequence of probability measures which is based on the view that it is an analogue of weak convergence. The known queueing processes in the \(GI/GI/1\) model (queue-size and virtual waiting time) are used as a touchstone to show the advantages and possibilities of the approach in the asymptotic analysis. As the author asserts, all known previous results on the asymptotics of large deviations for the \(GI/GI/1\) queue deal with one-dimensional distributions, while his goal is to obtain functional large deviation limit theorems that treat queueing processes as elements of certain functional spaces. At first it is studied the LDP of the mentioned processes (scaled and normalized by a large factor) when they have certain exponential moments, and the corresponding rate functions are found in explicit form. The analysis makes no assumptions about the value of the traffic (by contrast with the heavy traffic hypothesis in weak convergence). Then there are obtained explicit results concerning the logarithmic asymptotics of the probabilities that the process exceeds high levels at large times in normal traffic. There are also derived LDP and some conditional limit theorems for renewal processes.
scientific article; zbMATH DE number 851504
scientific article