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Fast Discrete-Event Simulation of Markovian Queueing Networks through Euler Approximation

L. Jeff Hong, Yingda Song, Tan Wang

TL;DR

The paper proposes fast discrete-event simulation for large-scale Markovian queueing networks using Euler-type time discretization with backward and forward schemes. By aggregating arrivals and departures over fixed time steps and exploiting vectorization, the methods yield stochastic upper and lower bounds on the state that remain uniformly close over time, with errors that scale linearly with the step size $h$ and vanish relative to network size under a practical selection of $h$. Theoretical results show substantial asymptotic speedups over DES, with accuracy validated by extensive numerical experiments on synthetic and real-world networks, including healthcare and data center topologies. The approach enables scalable performance analysis when exact DES is computationally infeasible, while maintaining rigorous error control and providing practical guidelines for step-size selection and vectorized implementation. The framework also furnishes exact sampling for pure departures, facilitating precise error decomposition and enabling per-customer sojourn-time estimation from aggregated paths.

Abstract

The efficient management of large-scale queueing networks is critical for a variety of sectors, including healthcare, logistics, and customer service, where system performance has profound implications for operational effectiveness and cost management. To address this key challenge, our paper introduces simulation techniques tailored for complex, large-scale Markovian queueing networks. We develop two simulation schemes based on Euler approximation, namely the backward and forward schemes. These schemes can accommodate time-varying dynamics and are optimized for efficient implementation using vectorization. Assuming a feedforward queueing network structure, we establish that the two schemes provide stochastic upper and lower bounds for the system state, while the approximation error remains bounded over the simulation horizon. With the recommended choice of time step, we show that our approximation schemes exhibit diminishing asymptotic relative error as the system scales up, while maintaining much lower computational complexity compared to traditional discrete-event simulation and achieving speedups up to tens of thousands times. This study highlights the substantial potential of Euler approximation in simulating large-scale discrete systems.

Fast Discrete-Event Simulation of Markovian Queueing Networks through Euler Approximation

TL;DR

The paper proposes fast discrete-event simulation for large-scale Markovian queueing networks using Euler-type time discretization with backward and forward schemes. By aggregating arrivals and departures over fixed time steps and exploiting vectorization, the methods yield stochastic upper and lower bounds on the state that remain uniformly close over time, with errors that scale linearly with the step size and vanish relative to network size under a practical selection of . Theoretical results show substantial asymptotic speedups over DES, with accuracy validated by extensive numerical experiments on synthetic and real-world networks, including healthcare and data center topologies. The approach enables scalable performance analysis when exact DES is computationally infeasible, while maintaining rigorous error control and providing practical guidelines for step-size selection and vectorized implementation. The framework also furnishes exact sampling for pure departures, facilitating precise error decomposition and enabling per-customer sojourn-time estimation from aggregated paths.

Abstract

The efficient management of large-scale queueing networks is critical for a variety of sectors, including healthcare, logistics, and customer service, where system performance has profound implications for operational effectiveness and cost management. To address this key challenge, our paper introduces simulation techniques tailored for complex, large-scale Markovian queueing networks. We develop two simulation schemes based on Euler approximation, namely the backward and forward schemes. These schemes can accommodate time-varying dynamics and are optimized for efficient implementation using vectorization. Assuming a feedforward queueing network structure, we establish that the two schemes provide stochastic upper and lower bounds for the system state, while the approximation error remains bounded over the simulation horizon. With the recommended choice of time step, we show that our approximation schemes exhibit diminishing asymptotic relative error as the system scales up, while maintaining much lower computational complexity compared to traditional discrete-event simulation and achieving speedups up to tens of thousands times. This study highlights the substantial potential of Euler approximation in simulating large-scale discrete systems.
Paper Structure (43 sections, 7 theorems, 54 equations, 9 figures, 4 tables, 4 algorithms)

This paper contains 43 sections, 7 theorems, 54 equations, 9 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Simulating the State Dynamics of Markovian Queueing Networks with Euler Approximation
  3. Simulation of Single-Station Queue
  4. Sampling of $A_\tau$
  5. Sampling of $D_\tau$
  6. Sampling of $N_\tau$
  7. Simulation of Queueing Networks
  8. Backward and Forward Approximation as the Stochastic Bounds
  9. Comparison of Computational Complexity between Euler Approximation and DES
  10. Vectorized Euler Approximation for Simulating Queueing Networks
  11. Pure Departure Process
  12. Simulation of Pure Departure Process
  13. Naïve Euler Approximation
  14. A Simple Refinement
  15. The Exact Simulation
  16. ...and 28 more sections

Key Result

Lemma 1

Consider the pure departure process with $m$ servers, each having a service rate of $\mu$. Assume initially there are $x$ customers in the system. Let $D$ be the random number of customers completing service by time $h$.

Figures (9)

  • Figure 1: A Sample Path of Arrival Processes for the True System and Two Euler Approximation Schemes.
  • Figure 2: An Example of Multi-Layer Feedforward Queueing Network.
  • Figure 3: Backward Approximation vs. Forward Approximation.
  • Figure 4: Relation between $h$ and approximation error.
  • Figure 5: Convergence of Relative errors with Decaying Time Intervals.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4