Stationary IPA Estimates for Non-Smooth G/G/1/$\infty$ Functionals via Palm Inversion and Level-Crossing Analysis

TL;DR

This paper develops stationary derivative estimates for non-smooth workload functionals in G/G/1/∞ queues, using Palm inversion and level-crossing analysis to handle discontinuities and parameter variations.

Contribution

It introduces a novel global level-crossing analysis method that avoids limiting arguments, enabling derivative estimation for non-smooth functionals with discontinuities.

Findings

Provides stationary derivative estimates for workload functionals with discontinuities.

Extends analysis to parameters like service speed and input time scale.

Employs Palm inversion and level-crossing analysis without limiting arguments.

Abstract

We give stationary estimates for the derivative of the expectation of a non-smooth function of bounded variation f of the workload in a G/G/1/$\infty$ queue, with respect to a parameter influencing the distribu- tion of the input process. For this, we use an idea of Konstantopoulos and Zazanis based on the Palm inversion formula, however avoiding a limiting argument by performing the level-crossing analysis thereof globally, via Fubini's theorem. This method of proof allows to treat the case where the workload distribution has a mass at discontinuities of f and where the formula has to be modified. The case where the parameter is the speed of service or/and the time scale factor of the input process is also treated using the same approach.