Queueing theory is the mathematical analysis of queues/waiting lines. The idea is that a queue has some capacity for some number of customers or tasks waiting for service from some number of servers who/that provide service at some efficiency and in some order. The origin of the discipline is the work done by Danish telephone engineer Agner Krarup Erlang (1878-1929).
Over time, the application of queueing theory expanded to industries beyond just telecommunication, such as customer service for any enterprise; retail and grocery stores; bars and restaurants; manufacturing, shipping, and delivering; transportation and traffic engineering; and hospitals and other healthcare-related organizations.
The goals for any queue are to operate as efficiently as possible such that the quality of the service is maximized and the wait time is minimized. In a business setting, failing to meet these goals can directly negatively impact customer satisfaction and retention as well as profit. In a hospital setting, such as an emergency room, failing to meet these goals can quickly become a matter of life or death.
The complexity of a queueing model is dependent on many factors and assumptions, such as the behavior of the arrival rate of the customers or tasks to be serviced, the quantity of servers in the queue, the deterministic or stochastic variability of the service time, and the efficiency of the servers to serve the queue.
In an effort to design a notation that can be used to describe a queueing system, Kendall (1953) proposed the three characteristics A/S/c, where A is the time between arrivals, S is the distribution of the service time, and c is the number of service channels or servers. Three additional characteristics were developed to further describe the behavior of a queue: K/N/D, where K is the capacity of the queue, N is how many customers or tasks (total population) can arrive for service, and D is the discipline of the queue. Thus, the six characteristics of a queue are written as A/S/c/K/N/D. Although there are six characteristics, the fundamental, obligatory characteristics to describe a queue are still A/S/c, as K/N/D, if not otherwise defined, are defaulted to ∞/∞/FIFO and can be omitted from the notation.
References:
Kendall, D. G. (1953, September). Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain. The Annals of Mathematical Statistics, 24(3), 338-354. https://www.jstor.org/stable/2236285
Winston, W. L. (2004). Queuing theory. In Operations research: Applications and algorithms (4th ed.) (pp. 1051-1061). Brooks/Cole—Thomson Learning.
Daniel M. Packer 