A L\'evy input fluid queue with input and workload regulation

TL;DR

This paper analyzes a complex queuing model with Le9vy process inputs and workload regulation, using advanced fluctuation theory to derive steady-state workload distributions.

Contribution

It introduces a novel queuing model with Le9vy inputs and dynamic workload modifications, applying fluctuation theory for steady-state analysis.

Findings

Derived steady-state workload distribution

Applied fluctuation theory to queue analysis

Enhanced understanding of workload regulation mechanisms

Abstract

We consider a queuing model with the workload evolving between consecutive i.i.d.\ exponential timers $\{e_q^{(i)}\}_{i=1,2,...}$ according to a spectrally positive L\'evy process $Y_i(t)$ that is reflected at zero, and where the environment $i$ equals 0 or 1. When the exponential clock $e_q^{(i)}$ ends, the workload, as well as the L\'evy input process, are modified; this modification may depend on the current value of the workload, the maximum and the minimum workload observed during the previous cycle, and the environment $i$ of the L\'evy input process itself during the previous cycle. We analyse the steady-state workload distribution for this model. The main theme of the analysis is the systematic application of non-trivial functionals, derived within the framework of fluctuation theory of L\'evy processes, to workload and queuing models.