Some reflected autoregressive processes with dependencies

Ioannis Dimitriou () and Dieter Fiems ()
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Ioannis Dimitriou: University of Ioannina
Dieter Fiems: Ghent University

Queueing Systems: Theory and Applications, 2024, vol. 106, issue 1, No 2, 67-127

Abstract: Abstract Motivated by queueing applications, we study various reflected autoregressive processes with dependencies. Among others, we study cases where the interarrival and service times are proportionally dependent on additive and/or subtracting delay, as well as cases where interarrival times depend 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. 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 distribution of a customer in the system in terms of an infinite sum of products of known Laplace transforms. An integer-valued reflected autoregressive process that can be used to model a novel retrial queueing system with impatient customers and a general dependence structure is also considered. For such a model, we provide expressions for the probability generating function of the stationary orbit queue length distribution in terms of an infinite sum of products of known generating functions. A first attempt towards a multidimensional setting is also considered.

Keywords: Workload; Queueing systems; Laplace–Stieltjes transform; Generating function; Recursion; Wiener–Hopf boundary value problem; 60K25; 62M10; 60K10; 60J05; 90B22 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s11134-023-09899-3

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