Simulating Queueing Systems with Python

Purpose

This project demonstrates how to simulate an $$(M/M/1)$$ queueing system and a tandem queue with blocking using the SimPy library in Python. It provides an illustrative example for:

1. Using SimPy to model, simulate, and monitor a queueing system.
2. Comparing simulated performance metrics with theoretical values derived from queueing theory (in the case of $$M/M/1$$).

The project serves as an educational tool to learn about discrete-event simulation and queueing system analysis.

Overview

What is an (M/M/1) Queue?

An (M/M/1) queue is a single-server queue where:

• Arrivals follow a Poisson process (exponentially distributed interarrival times).
• Service times are exponentially distributed.
• There is a single server, and the queue operates on a first-come, first-served basis.

Some Theoretical Formulas for M/M/1

For an $$(M/M/1)$$ queue with arrival rate $$\lambda$$ and service rate $$\mu$$, we have that the average waiting time and average queue size are given by:

$$ T_q = \frac{\lambda}{\mu (\mu - \lambda)} $$

$$ L_q = \frac{\lambda^2}{\mu (\mu - \lambda)} $$

The formulas assume the system operates under steady-state conditions $$\lambda < \mu$$, and they are well known in the queuing theory literature. We will use these formulas and compared them to the results obtained via simulation.

What is a Tandem Queue with Blocking?

A tandem queue with blocking is a system of multiple sequential service stations where:

• Arrivals to the first station follow a Poisson process.
• Service times at each station are exponentially distributed.
• Customers move station-to-station after service.
• Blocking occurs if the next station is full, forcing the customer to wait at the current station.

This setup can lead to delays and reduced throughput due to capacity constraints.