Approximating G(t)/GI/1 queues with deep learning

TL;DR

The paper addresses transient analysis of time-varying single-server queues by proposing a Moment-Based Recurrent Neural Network (MBRNN) that predicts the full transient distribution $P(t)$ using only the first four moments of the inter-arrival and service-time distributions and the initial state. It combines moment-based input preprocessing, an LSTM architecture, and a loss function that emphasizes both overall accuracy and tail behavior, achieving substantial speedups over simulation while delivering high fidelity across diverse test cases. The authors show that MBRNN outperforms fluid and diffusion approximations, remains robust under varying cycle utilization and SCV, and scales to longer horizons with controllable accuracy loss; they also provide an open-source implementation. The approach enables real-time or data-driven decision making in optimization and inference scenarios, and the framework can extend to more complex queueing systems such as $G(t)/GI/c$ and queueing networks, making transient analyses broadly accessible in practice.

Abstract

In this paper, we apply a supervised machine-learning approach to solve a fundamental problem in queueing theory: estimating the transient distribution of the number in the system for a G(t)/GI/1. We develop a neural network mechanism that provides a fast and accurate predictor of these distributions for moderate horizon lengths and practical settings. It is based on using a Recurrent Neural Network (RNN) architecture based on the first several moments of the time-dependant inter-arrival and the stationary service time distributions; we call it the Moment-Based Recurrent Neural Network (RNN) method (MBRNN ). Our empirical study suggests MBRNN requires only the first four inter-arrival and service time moments. We use simulation to generate a substantial training dataset and present a thorough performance evaluation to examine the accuracy of our method using two different test sets. We show that even under the configuration with the worst performance errors, the mean number of customers over the entire timeline has an error of less than 3%. While simulation modeling can achieve high accuracy, the advantage of the MBRNN over simulation is runtime, while the MBRNN analyzes hundreds of systems within a fraction of a second. This paper focuses on a G(t)/GI/1; however, the MBRNN approach demonstrated here can be extended to other queueing systems, as the training data labeling is based on simulations (which can be applied to more complex systems) and the training is based on deep learning, which can capture very complex time sequence tasks. In summary, the MBRNN can potentially revolutionize our ability to perform transient analyses of queueing systems.

Figures (23)

• Figure 1: An arrival process example.
• Figure 2: Work-flow diagram of our learning procedure.
• Figure 3: Work-flow diagram for inference.
• Figure 4: Input of arrival process.
• Figure 5: Arrival pattern illustrations.

Theorems & Definitions (2)

• Remark 1
• Remark 2