Notation for Queues

Kendall’s notation and A/S/n notation are two popular notations for describing queues.

• Kendall’s notation: This describes a queue by using a set of symbols to represent the queue’s various characteristics. It is represented by a three-letter notation, with each letter representing a different aspect of the queue. The first letter denotes the arrival process, the second the service process, and the third the number of servers. For example, an M/M/1 queue has a Poisson arrival process (represented by the letter M), an exponential service time distribution (also represented by the letter M), and one server (indicated by the number 1).
• A/S/n notation: In this notation A represents the probability distribution of the interarrival time, S represents the service time distribution, and n represents the number of servers.

M/M/1, for example, denotes a queue with a Poisson arrival process, an exponential service time distribution, and one server. M/M/c denotes a queue with Poisson Arrival and Exponential service with c servers, implying that the service is provided by more than one server while the service time remains exponential.
These notations are widely used in queueing theory and analysis because they allow for a quick understanding of the queue’s characteristics and the selection of appropriate mathematical models to represent the queueing system and thus evaluate its performance.

In general, there is no fixed set of processes that run on systems; thus, measuring the exact processing requirements of processes is impossible. We can, however, measure the distributions of CPU bursts and I/O bursts over the course of a process and derive a mathematical formula that identifies the probability of a specific CPU burst. The arrival rate of processes in the system can be approximated in the same way. The development of queuing theory, a branch of mathematics, resulted from the use of mathematical models for evaluating the performance of various systems. 

The fundamental model of queuing theories is the same as the model of a computer system. Each computer system is represented as a collection of servers such as CPUs and I/O devices, each with its own queue. The article focuses on discussing the Queuing System in Operating System.

The following topics will be discussed here:

1. Components of Queuing System
2. Number of Servers
3. Measures of Performance for Queuing Systems
4. Notation for Queues
5. Queue Discipline
6. Queuing Models

Let’s start discussing each of these topics in detail.