Transient analysis of a stationary L\'evy-driven queue

TL;DR

This paper derives simple transform formulas for the transient workload distribution in a stationary Lévy-driven queue, without assuming jumps are one-sided, using ladder process exponents.

Contribution

It provides novel explicit expressions for transient workload transforms in Lévy queues without restrictions on jump directions.

Findings

Derived the transform of the minimum workload over an exponential interval.

Provided explicit formulas involving bivariate Laplace exponents.

Extended analysis to general Lévy input without one-sided jump assumptions.

Abstract

In this paper we study a queue with L\'evy input, without imposing any a priori assumption on the jumps being one-sided. The focus is on computing the transforms of all sorts of quantities related to the transient workload, assuming the workload is in stationarity at time 0. The results are simple expressions that are in terms of the bivariate Laplace exponents of ladder processes. In particular, we derive the transform of the minimum workload attained over an exponentially distributed interval.