A mathematical method of analyzing is the congestions and delays of waiting in line. Queuing theory examines every component of waiting in line to be served, including the arrival process, service process, number of servers, number of system places and the number of customers (which might be people, data packets, cars, etc.). Queuing theory deals with one of the most unpleasant experiences of life, that is, waiting. Queuing is quite common in many fields, for example, in telephone exchange, in traffic flow, in shipping orders, in a supermarket, at a petrol station, at computer systems, in telecommunications such as call centers, etc. The author has mentioned the telephone exchange first because the first problem of Queuing theory was raised by calls and Erlang was the first who treated congestion problems in the beginning of 20th century.

Queuing theory is the mathematics of waiting lines. It is extremely useful in predicting and evaluating system performance. Queuing theory has been used for operations research, manufacturing and systems analysis. Traditional queuing theory problems refer to customers visiting a store, analogous to requests arriving at a device (Singh, 2009).

Queuing theory is the mathematical study of waiting lines that enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue, and being served by the Service Channels at the front of the queue (Anand, 2010). In Queuing theory, the author studies situations where units of some kind arrive at a service facility for receiving service of some description, some of the units have to wait for service, and depart after service.

In Queuing theory, a model is constructed so that queue lengths and waiting times can be predicted. Queuing theory is generally considered as a branch of operations research, because the results are often used when making business decisions about the resources needed to provide a service (Sundarapandian, 2009).

• Science of waiting
• Predict customer waiting time
• Evaluate the impact of variability on the key performance measures of service
• A queuing system is comprised of the following components: Arrival process, Service process and a Queue
• It helps in capacity planning decisions

1.1 Purpose of the study

The purpose of the present study is to give the fundamental ideas about the queuing theory. It deals with the day-to-day problems which every individual has experienced in his life i.e., waiting in line. Sometimes, it is a pleasant experience, but many times it can be extremely frustrating for both the customer and the store manager. A customer waiting too long automatically leads to losing a customer. To understanding this nature and how to manage them is one of the most important areas. In this paper, the author has discussed some concepts and applications related to queuing theory.

1.2 Applications of Queuing Theory

• Telecommunications
• Traffic control
• Determining the sequence of computer operations
• Predicting computer performance
• Health services (e.g.. control of hospital bed assignments)
• Airport traffic, airline ticket sales
• Layout of manufacturing systems.

• Capacity Planning in services poses challenges because of:
  • Lack of control over customer demand
  • Presence of customers in the service process
  • Trade-off between customer waiting and idle service capacity
• It is difficult to control customer demands because of the variability in customer demand
• It is very important to manage variability and predict customer waiting time
• Faster service delivery has become one of the important components of competitive advantage for service organizations
• It is important to address these questions in the context of different service sectors using queuing theory.

1.3 Queues abound in everyday life

• Supermarket checkout
• Traffic lights
• Waiting for the elevator
• Waiting at a gas station
• Waiting at passport control
• Waiting at a doctor's office
• Paperwork waiting at somebody's office to be processed

1.4 Queuing Theory for Telecommunications

Discrete Time Modeling of a Single Node System focuses on discrete time modeling and illustrates that most queuing systems encountered in real life can be set up as a Markov chain. This feature is very unique because the models are set in such a way that matrix-analytic methods are used to analyze them (Attahiru SuleAlfa, 2010).

1.5 Single Queuing nodes

A single node queue is a system in which a customer comes for service only at one node. The service may involve feedback into the same queue. When it finally completes service, it leaves the system. If there is more than one server, then they must all be in parallel. It is also possible to have single node queues in which we have several parallel queues and a server or several servers moving around from queue to queue to process the items waiting; an example is a polling system (Attahiru SuleAlfa, 2010)

Queuing Network is a system model set of service centers representing the system resources that provide service to a collection of customers that represent the users. Queuing Network is a model in which jobs departing from one queue arrive at another queue or possibly the same queue (Bose, 2002). A queuing network describes the system as a set of interacting resources.

2.1 Classification of Queuing Networks

The queuing networks consist of several connected systems. In an open queuing network, customers enter the network from outside, receive service at systems and leave the network. In a closed network, the number of customers is constant (Filipowicz & Kwiecien, 2008). If a new customer enters the network exactly when one customer departs, we can model this situation as closed queuing network. Queuing Networks are classified as follows (Raj Jain, 2008).

• Open queuing network: External arrivals and departures
• Closed queuing network: No external arrivals or departures
• Mixed queuing networks: Open for some workloads and closed for others
• Series queuing networks: Each individual queue can be analyzed independently of other queues

A queuing system describes the system as a unique resource. The queuing system is defined as the arrival process, the service process, the number of servers and their service rate, the queuing discipline process, the population constraints and the system or queue capacity (Balsamo & Marin, 2007). We study the phenomena of standing, waiting, and serving, and we call this study Queuing Theory. Any system in which arrivals place demands upon a finite capacity resource may be termed a Queuing system.

3.1 Key elements of queuing systems

• Customer: refers to anything that arrives at a facility and requires service, e.g., people, machines, trucks, emails.
• Server: refers to any resource that provides the requested service, eg., repair-persons, retrieval machines, runways at airport.

3.2 Characteristics of a Queuing system

• Distribution of time between arrivals
• Distribution of service times
• Number of parallel servers
• Maximum number of customers a system can accommodate

3.3 Notations used for Queuing models

Different characteristics are used for queuing models. The performance measures for all queuing models are same. The author can define the notations used for various performance measures of the queuing systems as given below (Daniel, 1995)

• τ - is job inter-arrival time
• λ - Mean number of arrivals per time period
• µ - Mean number of customers served per time period
• ρ - Utilization factor for the service system
• L_s - Average number of customers in the system
• L_q - Average number of customers in the queue
• W_s - Average time a customer spends in the system
• W_q - Average time a customer spends waiting in the queue
• ρ_n - Probability of exactly ' n' customers in the system
• n - Number of customers in the system

3.4 Queuing Delay

Queuing delay is the time a job waits in queue until it can be executed.

• Delay between the point of entry of a packet in the transmit queue to the actual point of transmission of the message.
• Delay depends on the load on the communication link.

3.5 Queue Structure

• First-come-first-served (FCFS)
• Last-come-first-serve (LCFS)
• Service-in-random-order (SIRO)

Queuing models can help in balancing the cost related to capacity planning of service system and the cost incurred due to waiting time of customers (Dowdy, Almeida, & Menasce, 2004). Various performance measures which can help in managing the above mentioned trade-offs are mentioned below.

• Average number of customers in the service system
• Probability of having specific number of customers in the service system
• Average number of customers waiting in the queue or queue length
• The average time each customer spends in the service system: It is represented by the total time a customer spends in a queue and is being served or service time
• The average time each customer spends in the queue
• The utilization factor for the service system

To solve practical problems, the first step is to identify the appropriate queuing system and then to calculate the performance measures. Of course, the level of modeling heavily depends on the assumptions. It is recommended to start with a simple system and then if the results do not fit to the problem, continue with a more complicated one. Various software packages help the interested readers in different level. Smart-Soft is a comprehensive suite of software programs that run under the Windows operating system and can be installed on one or more PCs to provide a complete, feature rich, easy to use, Queue Management System. Applications for Queue Management come in all sizes: A simple system with a single Queue and complex system with multiple Queues. Smart-Soft is highly configurable and can handle a wide variety of Queuing applications (Alex and Simmens, 2013).

5.1 Infinite-Source Queuing Systems

In finite-source models, the arrival intensity of the request depends on the state of the system which makes the calculations more complicated. In the case of infinite-source models, the arrivals are independent of the number of customers in the system resulting in a mathematically tractable model. In queuing networks, each node is a queuing system which can be connected to each other in various ways. The main aim of this chapter is to know how these nodes operate (Sztrik, 2012).

5.2 Finite-Source Queuing Systems

Depending on the assumptions on source, service times of the requests and the service disciplines applied at the service facility, there are a great number of queuing models at different levels to get the main steady-state performance measures of the system. It is also easy to see that, depending on the application, we can use the terms request, customer, machine, message, job equivalently. The above mentioned models are referred to as (Sztrik, 2001). A finite model is generally more complicated analytically, because the number of customers already in the queuing system at any point of time affects the number of potential customers remaining in the input source. In some cases, however, we are forced to assume a finite queuing system if the rate at which the input source generates arrivals is significantly affected by the number of customers in the queuing system. Care must be exercised in the selection of a queuing system model (Kobayashi, & Konheim, 1977).

• The waiting space for the customer is usually limited
• The arrival rate may be state dependent
• The arrival process may not be stationary
• The population of customers may not be infinite and the queuing discipline may not be first come first serve
• Services may not be rendered continuously
• The Queuing system may not have reached the steady state. It may be, instead, in transient state

The studies of these systems can benefit from the integration of knowledge acquired in existing disciplines, which specialize in modeling systems displaying such aspects. Queuing theory is an example of such a discipline. We have demonstrated here the application of modeling and analysis techniques, borrowed from queuing theory, to the description of an arbitrary genetic network. This allowed for the derivation of the probability distribution function of the network. Queuing theory is extensive and diverse, and further delving into it, while still keeping in mind the biological problems at hand, can probably yield several other useful results. Also we have suggested that queuing theory may benefit as well from this interdisciplinary dialogue. To satisfy the customer, we will follow some motto incorporating planning, designing, operating, evaluating, organizing and analyzing.

Queuing theory is not just some esoteric branch of operations research used by mathematicians. It is a practical operations management technique that is commonly used to determine staffing, scheduling and inventory levels, and to improve customer satisfaction, by understanding queues and learning how to manage them through simple models and equations. Designing and operating can help to improve customer-facing and internal processes to give organizations a competitive advantage.

This article describes queuing systems and queuing networks which are successfully used for the performance analysis of different systems such as computer, communications, transportation networks and manufacturing. Also it has described characteristic, queuing software, queuing structure and so on. In addition, examples of queuing theory applications are given. Thus the present article gives some fundamental concepts of queuing theory and their applications.