CHAPTER ONE
1.0  INTRODUCTION
       Waiting in lines is a part of our everyday life. Waiting in lines may be due to the demand at any one time may be more than the capacity of service (over crowded), overfilling or due to congestion. Any time there is more customer demand for a service than can be provided, a waiting lines forms. We wait in lines at the movie theatre, at the bank for a teller, at hospital (s) for diagnosis and treatment at a grocery store. Wait time is depends on the number of people waiting before you, the number of servers serving line, and the amount of service time for each individual customer. Customers can be humans or an object such as customer orders to be process, a machine waiting for repair. Matte analytical method of analyzing the relationship between congestion and delay caused by it can be modelled using queuing analysis. Queuing theory provides tools needed for analysis of system of congestion. Mathematically, systems of congestion appear in many diverse and complicated ways and can vary in extent and complexity.
        A waiting line system or queuing system is defined by two important elements: the population source of its customers and the process or service system. The customer population can be considered as finite or infinite. The customer population is finite when the number of customers affects the potential new customers for the service system already in the system. When the number of customers waiting in line does not significantly affect the rate at which the population generates new customers, the customer population is considered infinite. Customer behaviour can change and depends on waiting line characteristics. In addition to waiting, a customer can choose other alternative. When a customer enters the waiting line but leaves before being serviced, process is called Reneging. When customers changes one line to another to reduce wait time, process is calledjockeying. Balking occurs when a customer do not enter waiting line but decides to come back later. Another element of queuing system is service system. The number of waiting lines, the number of services, the arrangement of the services, the arrival and service patterns, and the service priority rules characterized the service system. Queue system can have channels or multiple waiting lines. Examples of single waiting line are bank counter, airline counters, restaurants, amusement parks etc. In these examples multiple servers might serve customer. In the single line multiple servers has better performance in terms of waiting times and eliminates jockeying behaviour than the system with a singe line for each server. System serving capacity is a function of the number of service facilities and severs proficiency. In queuing system, the terms server and channel are used interchangeable. Queuing systems are either single server or multiple servers. Single server examples includes gas station food mart with single checkout counter, a theatre with a single person selling ticket and controlling admission into the show. Multiple server examples include gas station with multiple gas pumps, grocery stores with multiple cashiers, and multiple tellers in a bank. Services require single activity or services of activities called phases. In a Single-phase system, the service is completed all at once, such as a bank transaction or grocery store checkout counter. In a multiple phase system, the service is completed in a series of phases, such as at fast food restaurant with ordering, pay, and pick up windows. Queuing system is characterised by rate at which customers arrive and served by service system. Arrival rate specifies the average number of customers per time period. The service rate specifies the average number of customer that can be served during a time period.
The service rate governs capacity of the service system.