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Related Experiment Video

Updated: Jun 14, 2026

Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures
10:56

Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures

Published on: May 20, 2014

Queueing process with excluded-volume effect.

Chikashi Arita1

  • 1Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan. airta@math.kyushu-u.ac.jp

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 7, 2010
PubMed
Summary
This summary is machine-generated.

This study extends the M/M/1 queueing model with spatial structure and excluded volume, analyzing particle hopping dynamics. Researchers identified a critical line for the stationary state and particle behavior in this exclusion process.

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Area of Science:

  • Statistical Mechanics
  • Queueing Theory
  • Stochastic Processes

Background:

  • The M/M/1 queue is a fundamental model in queueing theory.
  • The totally asymmetric simple exclusion process (TASEP) models particle movement on a line with exclusion.
  • Integrating spatial structure and excluded-volume effects into queueing models presents unique challenges.

Purpose of the Study:

  • To introduce and analyze an extended M/M/1 queueing process incorporating spatial structure and excluded-volume effects.
  • To investigate the particle hopping dynamics analogous to the totally asymmetric simple exclusion process (TASEP).
  • To determine the conditions for a stationary state and analyze system properties.

Main Methods:

  • Constructing a stationary-state solution using a matrix product form derived from the open TASEP.
  • Deriving the critical line that delineates parameter regions for the existence of a stationary state.
  • Calculating the average system length and particle number.

Main Results:

  • A stationary-state solution was successfully constructed for the extended model.
  • The critical line separating parameter space based on the existence of a stationary state was identified.
  • Monotonicity of the probability distribution of system length in the stationary state was demonstrated.

Conclusions:

  • The extended M/M/1 queue with spatial structure and excluded volume exhibits a well-defined stationary state under specific conditions.
  • The model provides insights into the interplay between queueing dynamics and exclusion processes.
  • Further generalizations, including backward hopping and coupled systems, were considered, expanding the scope of analysis.