![]() ![]() That sounds abstract, but everyday people encounter queue lines all the time. Simply, a queue line is an ordered group of people waiting for their turn to do something. We don’t mind if you prefer to spell it differently, but we want you to thoroughly understand what queue lines are and how they affect your business operations. “Cue line” might be something a director says to an assistant when an actor needs to start rehearsing a scene-maybe “cue que” comes from asking “what cue?” Or perhaps a cue line is the abstract path in which one moves a pool cue in billiards. In Spanish and Portuguese, “que” means “what,” so saying “que line” is sort of saying “what line”-which might come off as ironic in a bad way if the line is long. Īfter the 1940s, queueing theory became an area of research interest to mathematicians.When you are patiently waiting at a coffee shop for your morning latte, are you in a que line, a cue line, a Q-line, a cue que, or a queue line?Īfter seeing this spelled so many ways over the years, we feel we need to set the record straight. The M/G/1 queue was solved by Felix Pollaczek in 1930, a solution later recast in probabilistic terms by Aleksandr Khinchin and now known as the Pollaczek–Khinchine formula. If the node has more jobs than servers, then jobs will queue and wait for service. k describes the number of servers at the queueing node ( k = 1, 2, 3.D stands for "deterministic", and means jobs arriving at the queue require a fixed amount of service.M stands for "Markov" or "memoryless", and means arrivals occur according to a Poisson process.He modeled the number of telephone calls arriving at an exchange by a Poisson process and solved the M/D/1 queue in 1917 and M/D/ k queueing model in 1920. In 1909, Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on what would now be called queueing theory. The two-stage one-box model is common in epidemiology. The system transitions between values of k by "births" and "deaths", which occur at the arrival rates λ i If k denotes the number of jobs in the system (either being serviced or waiting if the queue has a buffer of waiting jobs), then an arrival increases k by 1 and a departure decreases k by 1. The behaviour of a single queue (also called a queueing node) can be described by a birth–death process, which describes the arrivals and departures from the queue, along with the number of jobs currently in the system. A setting with a waiting zone for up to n customers is called a queue with a buffer of size n. A setting where a customer will leave immediately if the cashier is busy when the customer arrives, is referred to as a queue with no buffer (or no waiting area). Each cashier processes one customer at a time, and hence this is a queueing node with only one server. (There are other models, but this one is commonly encountered in the literature.) Customers arrive, are processed by the cashier, and depart. Server c has just completed service of a job and thus will be next to receive an arriving job.Īn analogy often used is that of the cashier at a supermarket. Server b is currently busy and will take some time before it can complete service of its job. Server a is idle, and thus an arrival is given to it to process. Jobs (also called customers or requests, depending on the field) arrive to the queue, possibly wait some time, take some time being processed, and then depart from the queue.Ī queueing node with 3 servers. In fact, one of the flagship journals of the field is Queueing Systems.Ī queue or queueing node can be thought of as nearly a black box. The spelling "queueing" over "queuing" is typically encountered in the academic research field. These ideas have since seen applications in telecommunication, traffic engineering, computing, project management, and particularly industrial engineering, where they are applied in the design of factories, shops, offices, and hospitals. Queueing theory has its origins in research by Agner Krarup Erlang, who created models to describe the system of incoming calls at the Copenhagen Telephone Exchange Company. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is the mathematical study of waiting lines, or queues. In the study of queue networks one typically tries to obtain the equilibrium distribution of the network, although in many applications the study of the transient state is fundamental. In this image, servers are represented by circles, queues by a series of rectangles and the routing network by arrows. Queue networks are systems in which single queues are connected by a routing network. ![]()
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