## A Survey on Queueing Systems with Mathematical Models and Applications

Read this paper on waiting line analysis and queues. It provides a good survey of the theory and uses of this type of analysis. Pay particular attention to sections 1 through 3. How might these models be used to balance firm costs with different levels of customer satisfaction?

### 1. Introduction

Queueing theory is one of the branches of applied mathematics which studies and models the waiting lines. Danish mathematician A. K. Erlang (1878–1929), who published his first paper entitled "The Theory of Probability and Conversations" in 1909, is considered as the father of queueing theory. Further going back to the history, it can be observed that a viable queueing theory was developed by French Mathematician S. D. Poisson (1781–1840), who created a probability distribution function for the total outcomes of independent trials. He used statistical approach for these distributions which can be applied to the situations where excessive demands are to be fulfilled on a limited resource. During the late 1800s, all telephone calls used to be switched manually to the recipient by an operator. Each customer used to call the operator first and the operator used to fix the call for the customer. In this process, telephone companies were facing problem to appoint more operators. Callers who were unable to reach to an operator may simply hung up for several minutes with frustration and might think that it was a busy time for the operators. On the other hand, some would be waiting their turn to talk to the operator. And some others would call repeatedly thinking that the operator would be sufficiently annoyed by repeated calls to serve them next. These type of behaviour of the customers caused problems for traffic engineers because they affected the level of demand for service from an operator. A call which was not reached to the operator could be lost and could be effectively out of the system. To overcome this situation and to reduce the number of switchboards in an area, the most important application of queuing theory was developed. Those callers who repeatedly try for the operator increased demands on the system by appearing several requests. Poisson's formula was meant only for the repeated callers. Kendell presented a paper that opened a general review of some points in congestion theory to enhance the study for a single server queue where input is Poisson and service time is generally distributed. In  Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain, The Annals of Mathematical Statistics, he further extended the study of the stochastic processes for the theory of queues and their analysis by the method of the imbedded Markov chain. The study was carried out first reviewing on single-server queues and using the similar technique to the analysis of many server queuing system.

Stochastic process is a key factor to specify in queueing systems because it describes the arrival pattern as well as the structure and the discipline of the service facility. Queueing system deals with queue length and waiting times. The concept of queue is applied not only in the waiting system by the human beings but also in modern technology of computer and other service providers by the devices. In general, it is not necessary that service will be immediately available to address the demand of all the customers, so that they are forced to line up. In the queueing system, the one who demands the service is referred as customer, which may be a person, a task, or a commodity. The other element of the queueing system is the one who provides the service with some defined discipline, called the server. It may be people, machine or objects. Some of the service disciplines are first come first served (FCFS), last come first served (LCFS), service in random order (SIRO), priority, processor sharing (PS), round-robin (RR). We study performance measures of a queueing system where only the limited number of customers are served and arrival or service or both occur in a batch. If any of the customers come after the prescribed quota has already been served, the server does not provide the service to the new comer.

The main goal of this study is to present and analyse measures of performance effectiveness for some specific queuing models. The models we investigate have important applications in the study of machine repair problem, tele-traffic, computer and flexible manufacturing systems, production processes, transportation, monitoring, controlling, and managing complex engineering systems that have finite buffer system. Transient study makes the problem more realistic. The situations considered are also applicable to various day-to-day activities. It has been attracting the attention of mathematicians and operations research scientists in the field of social and liberal globalization of economy. The study provides a prospect to the development of research and design in related fields.

Rest of the paper is organized as follows: Section 2 presents the state of arts of the queueing system and associated theories developed by various researchers. Section 3 describes the different components of queueing system along with the standard notations used in queueing models. Some of the mathematical formulae for different queueing models and the derivation of simple Markovian queue are explained in Section 4. Section 5 includes Little's law with its use for the calculation of performance measures in queueing systems. Some applications of queueing models in supply chain management are observed in Section 6. Finally, Section 7 concludes the paper.