The class of distributions associated with the generalized Pollaczek-Khinchine formula

TL;DR

This paper characterizes the distribution class of the maximum of certain Lévy processes, providing a new explicit identity that generalizes classical queueing and Brownian motion results.

Contribution

It introduces a novel distributional identity for Lévy processes combining Brownian motion and subordinator components, extending known formulas.

Findings

Derived a new distributional identity for the maximum of Lévy processes.

Generalized the Pollaczek-Khinchine formula to broader Lévy process classes.

Connected stationary distributions of reflected processes with classical queueing results.

Abstract

The goal is to identify the class of distributions to which the distribution of the maximum of a L\'evy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the L\'evy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing L\'evy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczeck-Khinchine formula for stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.