Palm problems arising in BAR approach and its applications

TL;DR

This paper investigates Palm distributions in Markov processes with multiple point processes, establishing conditions for inherited Markov structure and developing a framework to handle state changes at counting instants, with applications to queueing models.

Contribution

It affirms the inheritance of Markov structure under Palm distributions and introduces a framework for analyzing state changes at counting instants in the BAR approach.

Findings

Confirmed Markov structure inheritance under Palm distributions.

Developed a framework for state change analysis at counting instants.

Derived heavy traffic limit for a single server queue with finite waiting room.

Abstract

We consider Palm distributions arising in a Markov process with time homogeneous transitions which is jointly stationary with multiple point processes. Motivated by a BAR approach studied in the recent paper Braverman, Dai and Miyazawa (2023}, we are interested in two problems; when this Markov process inherits the same Markov structure under the Palm distributions, and how the state changes at counting instants of the point processes can be handled to derive stationary equations when there are simultaneous counts and each of them influences the state changes. We affirmatively answer to the first problem, and propose a framework for resolving the second problem. We also discuss how those results can be applied in deriving BAR's for the diffusion approximation of queueing models in heavy traffic. In particular, as their new application, the heavy traffic limit of the stationary distribution is derived for a single server queue with a finite waiting room. Those examples show how the BAR approach is useful under the proposed framework.