Queueing Systems: Simulation & Numerical Methods

A Python package for simulating and analyzing queueing systems (QS) and networks.

About

This repository focuses on solving steady-state problems in queueing theory.

Key Features:

• Simulate various types of queueing systems and networks.
• Numerical methods for solving queueing theory problems.
• Analyze system performance metrics such as waiting times, soujourn times, load factor and etc.

Use Cases

• Modeling cloud computing infrastructure.
• Designing efficient call centers.
• Optimizing transportation systems.
• Network traffic analysis.

Contributing

Contributions are welcome! If you find any issues or have suggestions, please open an issue. Your pull requests are also appreciated. You can write me at xabarov1985@gmail.com

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Installation

Install most-queue with pip

  pip install most-queue

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Description of the Project

Most_queue consists of two main parts:

• most_queue.theory contains programs that implement methods for calculating queueing theory models.
• most_queue.sim contains simulation programs.

See examples in the tests folder:

FIFO QS

#	Kendall Notations	Description	Example	Tutorial
1.	Ek/D/c	Numerical calculation of a multi-channel system Ek/D/n	link
2.	GI/M/1	Solving for QS GI/M/1	link
3.	GI/M/c	Solving for QS GI/M/c	link
4.	M/D/c	Solving for QS M/D/c	link	link
5.	M/G/1	Solving for QS M/G/1	link
6.	M/H2/c	Numerical calculation of QS M/H2/c by the Takahashi-Takami method with complex parameters when approximating the serving time by the H2-distribution	link	link
7.	M/M/c/r	Solving for QS M/M/c/r	link	link

QS with priorities

#	Kendall Notations	Description	Example	Tutorial
1.	M/Ph/c/PR	Numerical calculation of QS M/Ph/c with 2 classes and PR - priority. Based on the approximation of busy periods	link
2.	M/M/c/PR	Numerical calculation of QS M/M/c with 2 classes, PR - priority by the Takahashi-Takami numerical method based on the approximation of the busy period by the Cox distribution	link
3.	M/M/c/PR	Numerical calculation of QS M/M/c with 3 classes, PR - priority by the Takahashi-Takami numerical method based on the approximation of busy period by the Cox distribution	link
4.	M/G/1/PR	Calculating QS with preemtive priorities (single-channel).	link	link
5.	M/G/1/NP	Calculating QS with non-preemtive priorities (single-channel).	link	link
6.	M/G/c/Priority	Calculating QS with NP and PR (multi-channel) by method of relation	link	link

Fork-Join QS

#	Kendall Notations	Description	Example	Tutorial
1.	M/M/c/Fork-Join	Solving for Fork-Join queueing system	link
1.	M/G/c/Split-Join	Solving for Split-Join queueing system	link

QS with Batch Arrival

#	Kendall Notations	Description	Example	Tutorial
1.	Mx/M/1	Solving for the of Mx/M/1 QS with batch arrival	link

QS with Vacations

#	Kendall Notations	Description	Example	Tutorial
1.	M/H2/c	Numerical calculation of the M/H2/c system with H2-warming using the Takahasi-Takagi method.	link	link
2.	M/G/1	Solving for QS M/G/1 with warm-up
3.	M/Ph/c	Multichannel queuing system with H2-serving time, H2-warm-up, H2-cold delay and H2-cold (vacations). The system uses complex parameters, which allows you to calculate systems with arbitrary serving, warm-up, cold-delay and cold variation coefficients	link
4.	M/M/c	Multichannel queuing system with exp serving time, H2-warm-up and H2-cold (vacations). The system uses complex parameters, which allows to calculate systems with arbitrary warm-up and cold variation coefficients	link

QS with Impatience

#	Kendall Notations	Description	Example	Tutorial
1.	M/M/1/D	Solving for M/M/1 with exponential impatience	link

Closed QS (with finite number of sources)

#	Kendall Notations	Description	Example	Tutorial
1.	M/M/1/N	Solving for the Engset model for M/M/1 with a finite number of sources.	link

Queuing Networks

#	Kendall Notations	Description	Example	Tutorial
1.	General Network	Numerical calculation of queuing network with priorities in nodes	link

Usage

• Look here for examples
• Look here for jupyter tutorials