In The Queue, Please!

In Brief, About Queuing Theory:

Queuing theory (or queueing theory) refers to the mathematical study of the formation, function, and congestion of waiting lines, or queues. The study of queues comes under a discipline of Operations Research called Queueing Theory and is a primary methodological framework for evaluating resource performance besides simulation. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.

At its core, a queuing situation involves two parts:

• Someone or something that requests a service—usually referred to as the customer, job, or request.
• Someone or something that completes or delivers the services—usually referred to as the server.

Queueing theory is used in software development for purposes such as project management kanban boards, inter-process communication message queues, and devops continuous deployment pipelines. A queueing model is constructed so that queue lengths and waiting time can be predicted.

Queueing theory notations:

Queueing theory uses notation with Greek letters. We use some of the popular notation; we also add some custom notion that help us with software projects.

Kendall Notation:

David G. Kendall devised a shorthand notation to describe a queueing system containing a single waiting queue: (A / B / X / Y / Z).

• A : Customer arriving pattern (Inter-arrival-time distribution).
• B : Service pattern (Service-time distribution).
• X : Number of parallel servers.
• Y : System capacity.
• Z : Queueing discipline.

Examples:

• M / M / 1 / ∞ / FCFS
• M / M / 1 / K

Inter-arrival rate, service rate, dropout rate:

The most important notation:

• λ: inter-arrival rate. This measures how fast new customers are coming into the queue.
• μ: service rate. This measures how fast customers in the queue are being handled.
• σ: dropout rate. This measures how fast customers are skipping out the queue unhandled.

Examples:

• λ = μ means the arrival rate equals the service rate; the queue is staying the same size, other than dropouts.
• λ > μ means the arrival rate is greater than the service rate; the queue is getting larger, other than dropouts.
• λ < μ means the arrival rate is less than the service rate; the queue is getting smaller, other than dropouts.

Utilization ratio:

The most important notation that summarizes a queue:

• ρ: utilization ratio = λ / μ

Examples:

• ρ = 1 means the arrival rate is equal to the service rate; the queue is staying the same size.
• ρ > 1 means the arrival rate is greater than the service rate; the queue is getting larger.
• ρ < 1 means the arrival rate is less than the service rate; the queue is getting smaller.

Count:

We count customers often, and we use this notation:

• κ: count (capacity of the system).

Example:

• κ = 100 means there are 100 customers.
• κ > 100 means there are more than 100 customers.
• κ ≫ 100 means there are many more than 100 customers.

Standard notations:

Standard notations for queueing theory also uses these symbols:

• n: the customer number in the system.
• C: the number of parallel servers.
• M: the initial number of customers.

Our Project:

Overview:

The understanding and prediction of the stochastic behavior of these queues will provide a theoretical insight into the dynamics of these shared resources and how they can be designed to provide better utilization. And the modeling and analysis of waiting queues/networks is the main implemented subject in this project.

Goals and Specifications:

We implemented, using Java and JavaFX GUI, the Deterministic and Stochastic Models:

• Model D/D/1/K-1
• Model M/M/1
• Model M/M/1/K
• Model M/M/C
• Model M/M/C/K

Including graph result (if exists for a model), ending up with a cool CSS style!

Based on the model you choose and the inputs you have, the project can asnwer you the following questions:

• Sketching number of the customers at time (t), (Only for Model D/D/1/K-1).
• Expected number of the customers in the system (L).
• Expected number of the customers in the queue (Lq).
• Expected waiting time spent in the system (W).
• Expected waiting time spent in the queue (Wq).

How to work?

After downloading the project and running it, it would be very easy steps:

1. You will get the home application page, as shown in the screenshots below.
2. Choose the model you want to make analysis with.
3. Once you've chosen a model, only the needed parameters will be enabled.
4. Only Model D/D/1/K-1 can be sketched. Also, you can make a query in.
5. Enjoy!

Screenshots:

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Contributors:

This project exists thanks to all wonderful people who contribute: