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Analyzing homogenous and heterogeneous multi-server queues via neural networks

Eliran Sherzer

TL;DR

The paper tackles predicting the stationary distribution of the number of customers in two multi-server queue models, GI/GI/c and GI/GI_i/2, under non-Markovian inputs. It uses a neural network trained on features derived from the first four moments of inter-arrival and service-time distributions, with Phase-Type sampling to generate diverse inputs and offline simulations to label data. The resulting model achieves state-of-the-art accuracy for GI/GI/c and is the first to predict the full stationary distribution for GI/GI_i/2, with typical errors well below 3% and fast, parallel inference enabling real-time decision support. A key finding is that including moments beyond the fourth provides little to no improvement and can even hurt performance, guiding efficient feature selection. The work also demonstrates practical utility through a numerical optimization example and openly provides code for replication and extension to more complex queueing networks.

Abstract

In this paper, we use a machine learning approach to predict the stationary distributions of the number of customers in a single-staiton multi server system. We consider two systems, the first is $c$ homogeneous servers, namely the $GI/GI/c$ queue. The second is a two-heterogeneous server system, namely the $GI/GI_i/2$ queue. We train a neural network for these queueing models, using the first four inter-arrival and service time moments. We demonstrate empirically that using the fifth moment and beyond does not increase accuracy. Compared to existing methods, we show that in terms of the stationary distribution and the mean value of the number of customers in a $GI/GI/c$ queue, we are state-of-the-art. Further, we are the only ones to predict the stationary distribution of the number of customers in the system in a $GI/GI_i/2$ queue. We conduct a thorough performance evaluation to assert that our model is accurate. In most cases, we demonstrate that our error is less than 5\%. Finally, we show that making inferences is very fast, where 5000 inferences can be made in parallel within a fraction of a second.

Analyzing homogenous and heterogeneous multi-server queues via neural networks

TL;DR

The paper tackles predicting the stationary distribution of the number of customers in two multi-server queue models, GI/GI/c and GI/GI_i/2, under non-Markovian inputs. It uses a neural network trained on features derived from the first four moments of inter-arrival and service-time distributions, with Phase-Type sampling to generate diverse inputs and offline simulations to label data. The resulting model achieves state-of-the-art accuracy for GI/GI/c and is the first to predict the full stationary distribution for GI/GI_i/2, with typical errors well below 3% and fast, parallel inference enabling real-time decision support. A key finding is that including moments beyond the fourth provides little to no improvement and can even hurt performance, guiding efficient feature selection. The work also demonstrates practical utility through a numerical optimization example and openly provides code for replication and extension to more complex queueing networks.

Abstract

In this paper, we use a machine learning approach to predict the stationary distributions of the number of customers in a single-staiton multi server system. We consider two systems, the first is homogeneous servers, namely the queue. The second is a two-heterogeneous server system, namely the queue. We train a neural network for these queueing models, using the first four inter-arrival and service time moments. We demonstrate empirically that using the fifth moment and beyond does not increase accuracy. Compared to existing methods, we show that in terms of the stationary distribution and the mean value of the number of customers in a queue, we are state-of-the-art. Further, we are the only ones to predict the stationary distribution of the number of customers in the system in a queue. We conduct a thorough performance evaluation to assert that our model is accurate. In most cases, we demonstrate that our error is less than 5\%. Finally, we show that making inferences is very fast, where 5000 inferences can be made in parallel within a fraction of a second.

Paper Structure

This paper contains 31 sections, 8 equations, 8 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Literature review
  3. Problem formulation
  4. Solution overview
  5. Training process
  6. Data generating
  7. Input generation
  8. Pre-processing Input
  9. Output generation
  10. Segmentation of the test set by SCV, $\rho$ and number of servers
  11. Datasets: training, validation and test
  12. Network Architecture
  13. Loss function
  14. Experiments
  15. Accuracy metrics
  16. ...and 16 more sections

Figures (8)

  • Figure 1: Work-flow diagram of our learning procedure.
  • Figure 2: Work-flow diagram for inference.
  • Figure 3: SAE values as a function of the number of moments used as input.
  • Figure 4: Total cost as a function of $rate$ and $c$.
  • Figure 5: An example of a feed-forward queueing network.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2