Queueing Theory

Introduction to queuing theory: Queuing theory is a field of operations research. Because the results we got can be used to make business decisions. This is a mathematical study of the waiting line. Queuing theory can perform mathematical analysis on several related processes, including reaching the queue, waiting in the queue, and servicing in front of the queue. Use this theory to measure the average wait time in the queue or system, the expected number of waiting or receiving services, and the probability of encountering the system.

Queuing theory is a mathematical study of queues or queues. We build a queuing model to predict queue length and wait time. Queuing theory is often considered part of operational research because the results are often used in making business decisions about the resources needed to provide a service. Queuing theory was born from the study of Agner Krarup Erlang in developing a model to describe the telephone exchange in Copenhagen. Since then, these ideas have been applied to telecommunications, transport engineering, computing, especially industrial engineering, factory, shop, office and hospital design, and project management.

In queuing theory, the process of birth and death is the most basic example of queuing model, ie M / M / C / K / / FIFO (complete Kendall symbol) queue. This is the queue that Poisson arrives from an infinite population and the C server has exponentially distributed service times with K locations in the queue. This model is assumed to have an infinite population, but it is a model suitable for various telecommunication systems. In the short term, only three types of migration are possible: 1 death, 1 birth, or no birth or death. In the case of birth rate (per unit time) and mortality rate, the probability of the above transition is respectively. In the demographic process, "birth" is a transition to increase the population by one, and "death" is a transition to reduce the population by one.

Single queuing nodes are usually written in A / S / C format using Kendall notation. Where A is the time between arrivals, S is the size of the job, and C is the number of servers on the node. Many of the theorems in queuing theory can be proved by reducing the queue to a mathematical system called Markov chain. This was first described by Andre Markov in the 1906 paper. Danish engineer, Agner Krarup Erlang, who worked in Copenhagen Telephone Exchange in 1909, published the first paper on what is now called queuing theory. He simulated the phone number to the switch through the Poisson process, solved the M / D / 1 queue in 1917, and solved the M / D / K queuing model in 1920. In the symbol of Kendall: