Computing the steady-state probabilities of a tandem queueing system, a Machine Learning approach

TL;DR

This paper introduces a neural network-based method to accurately approximate steady-state probabilities in tandem queueing systems with general renewal processes, outperforming existing models and analyzing the influence of moments and auto-correlation.

Contribution

It presents a novel neural network approach that leverages moments and auto-correlation to estimate steady-state distributions in complex queueing networks, advancing beyond traditional models.

Findings

Neural network approach outperforms existing models.

First five moments nearly suffice to determine probabilities.

First two auto-correlation lags almost fully determine steady-state probabilities.

Abstract

Tandem queueing networks are widely used to model systems where services are provided in sequential stages. In this study, we assume that each station in the tandem system operates under a general renewal process. Additionally, we assume that the arrival process for the first station is governed by a general renewal process, which implies that arrivals at subsequent stations will likely deviate from a renewal pattern. This study leverages neural networks (NNs) to approximate the steady-state distribution of the marginal number of customers at each station in the tandem queueing system, based on the external inter-arrival and service time distributions. Our approach involves decomposing each station and estimating the departure process by characterizing its first five moments and auto-correlation values, without limiting the analysis to linear or first-lag auto-correlation. We demonstrate that this method outperforms existing models, establishing it as state-of-the-art. Furthermore, we present a detailed analysis of the impact of the i^th moments of inter-arrival and service times on steady-state probabilities, showing that the first five moments are nearly sufficient to determine these probabilities. Similarly, we analyze the influence of inter-arrival auto-correlation, revealing that the first two lags of the first- and second-degree polynomial auto-correlation values almost completely determine the steady-state probabilities of a G/GI/1 queue.