Multitype branching processes with immigration in random environment and polling systems

TL;DR

This paper analyzes multitype branching processes with immigration in random environments to determine the tail distribution of accumulated final products and applies these results to study busy periods in polling systems with random service and switch-over times.

Contribution

It introduces a method to find the tail distribution of the final product in multitype branching processes within random environments and applies it to polling systems.

Findings

Derived the tail distribution of the final product in the process.

Analyzed the tail distribution of busy periods in polling systems.

Provided insights into the impact of random service disciplines and switch-over times.

Abstract

For multitype branching processes with immigration evolving in a random environment and producing a final product we find the tail distribution of the size of the final product accumulated in the system for a life period. Using this result we investigate the tail distribution of the busy periods of the branching type polling systems with random service disciplines and random positive switch-over times