Some reflected autoregressive processes with dependencies

TL;DR

This paper investigates complex reflected autoregressive processes with dependencies, providing exact Laplace transform expressions for waiting times and analyzing a novel retrial queueing model, broadening understanding of dependent queueing systems.

Contribution

It introduces a broad class of autoregressive processes with dependencies, offering exact analytical expressions and modeling a new retrial queueing system with dependent orbit searching times.

Findings

Explicit Laplace transform expressions for waiting times.

Probability generating function for orbit queue length as an infinite product.

Analysis of a multidimensional autoregressive process.

Abstract

Motivated by queueing applications, we study various reflected autoregressive processes with dependencies. Amongst others, we study cases where the interarrival and service times are proportionally dependent with additive and/or subtracting delay, as well as cases cases where interarrival times depends on whether the service duration of the previous arrival exceeds or not a random threshold. Moreover, we study cases where the autoregressive parameter is constant as well as a discrete or a continuous random variable, as well as cases where . More general dependence structures are also discussed. Our primary aim is to investigate a broad class of recursions of autoregressive type for which several independence assumptions are lifted, and for which a detailed exact analysis is provided. We provide expressions for the Laplace transform of the waiting time of a customer in the system in terms of an infinite product of known Laplace transforms. An integer-valued reflected autoregressive process that can be used to model a novel retrial queueing system with orbital searching time to depend on whether the last busy period starts with an empty or a non empty orbit queue, is also discussed. For such a model the probability generating function of the stationary orbit queue length is given as an infinite product of known generating functions. A first attempt towards multidimensional setting is also analyzed. Some additional generalizations with more general dependence structure are also discussed.