## 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?

### 7. Conclusions

Upon observing the contributions of the researchers in the field of queueing theory, it can be noticed that plenty of works have been done pointing out many extendable areas. Change of one parameter in any of the proposed models might cause a huge change in the result of performance measures. Small change in the arrival rate may create large queue or no queue, and small change in service rate may make the customers very happy for the quick service or may have to wait for a long time. For any of the queue, time plays a vital role. It is very important how long a customer waits in the queue to get the service and how fast a server provides the service. To make the service more effective, sometimes we need to add the servers and increase the efficiency. We have seen the proposed queueing models for finite and infinite capacities. Some of the queueing models have finite capacity and some are ready to serve for any number of customers. Some are time dependent studied under transient fashion and some are used for the optimization model. We have chosen Markovian queuing model with finite number of customers for a single or multiple servers. Besides the usual and standard mathematical modelling in queueing theory, consideration of customers' behaviours, servers' breakdown or vacation along with the limitation in arrivals can be introduced to make the model more realistic and challenging. On the other hand, suggesting some mathematical models considering those limitations may not always be reliable, so verifying those models in the real life situations with the use of computer simulation would be a remarkable contribution in the study of queueing theory.

All the studies carried out by the researchers have several applications in the real life. These applications are specially focused for making the life easier specially by saving time and money. In this process, number of models are proposed with applications in the different areas. The other motivation is to get the maximum profit with the minimum utilization of the limited resources, called the constraints. We intend to study the conditions for the optimal solution in order to maximize the production or the benefits using those limited constraints. These basic phenomena can be applied in telecommunication, traffic control, employee allocation, computer scheduling, supermarket, hospitals and many other fields. Variations of arrival and service disciplines in a queueing problem is the challenging work to tackle in the days to come. In addition, the simultaneous study of queueing operations with manufacturing and logistics may yield some interesting interlinks. Some of the literatures are described in the former Section. Such comparative study can be applied in the real problems of the industries to optimize production and distribution operations. The detail study of the application of queueing theory in supply chain networks will be our due course.