Queuing Systems Theory

This study aims at introducing a multi-server queues with vacations according to two system performance metrics (average system waiting time, average vacation time). We examine how such a dynamic impacts common system measures such as stability, expected number of customers in the system, probability of anexpected time. We have designed a method to calculate performance metrics for the previous queue system defined as average system uptime and average vacation time. We have changed the different features of access rates, service rates and number of servers to achieve the best results and try to improve and develop the system which makes it easier for servers to perform their service to obtain the appropriate vacations for them without being

Introduction

1.1 Queuing Systems

Queuing Systems theory is a mathematical study for the queuing rows or queuing lines. The queuing system consists of customers or units that require a type of service where the lines that customers enter when the service is not available immediately and there are cases where customers leave the system without considering the line and the service center that provides service to the customers.

The year 1910 was the real birth of the Queuing Systems theory by the Danish scientist Erlang, where he began that year to conduct experiments related to the problem of congestion in the center of telephone exchange through the workers in the telephone call center as it was found that callers are often subjected to some delays during the periods in which telephone calls are frequent because of the inability of the staff to meet orders in a synchronous manner with the speed with which they occur.

Erling was determined to calculate the delay duration for one worker in the division and then he circulated his study and the results to a number of workers. The development of the telephone traffic continued on the bases set by Erling and by the end of the Second World War, the use of this method was expanded and included a number of general cases, which characterized by Queuing lines (Saaty,

Service Discipline

The service discipline is the way in which the service is provided to customers. The most popular customer service disciplines are: "Service in order of access" means that it is the first who served or the first who arrives is the first to leave (First -In- First-Out) or (FIFO). This discipline is the common discipline in most of the queuing systems. "Service by Reverse Access Order" or who finally arrives is served first (Last-In-First-Out) or (LIFO). This disciplineis used in most storage systems where the products that are finally stored are consumed because they are within reach. "Random service" where items are randomly selected to serve them regardless of their arrival time (Selection- In –Random-Order) or (SIRO) as in some of the data entry operations. "service by priority" or (PRI), where the preference is given to some customers such as patients arriving at hospitals or clinics in serious condition and cases of important military messages. Or a preference according to a particular order or