One queuing system with correlated input flow

A. K. Bergovin^a, A. M. Ryazanov^a, V. G. Ushakov<sup>ab

a</sup> Faculty of Computational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University, 1-52 Lenin- skie Gory, GSP-1, Moscow 119991, Russian Federation
^b Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation

Abstract: A single-line queuing system with an infinite number of waiting places, an arbitrary distribution of service time, and a Poisson incoming flow with random intensity is considered. The intensities are subordinated to the autoregressive dependence of the first order. The joint distribution of the number of total customers in the system is obtained as well as the sojourn time of a customer in the system in a nonstationary regime. Expressions for stationary distributions and their probabilistic characteristics are also presented. The average sojourn time in the system in the stationary regime is numerically studied and illustrated under different assumptions on the distribution of service time and on the characteristics of the incoming flow. Comparison is made with the classical $M|G|1$ queue.

Keywords: random intensity, queue length, waiting time, passive traffic analysis, quality of service.

Received: 12.12.2024
Accepted: 15.01.2025

DOI: 10.14357/19922264250107