Models
Model Classes
 General Markov Non-Markov Networks
 Model Classes General Model
 The figure shows the basic components of a queuing system. Potential and actual customers are represented by the small circles and may be persons, parts, machines or almost any other quantity. The servers are represented by the numbered rectangles and may be any resource, such as a person, machine or repair shop, that performs a function. Customers arrive to the system from some input source and enter service immediately if any of the servers are idle. If all servers are busy, the customer waits in a queue until one is available. After some finite amount of time the customer departs. The details of the process depend on the various assumptions and parameter values adopted for the components of the system. The input source, also known as the calling population, is the collection of potential customers that might have need for the services offered by the system. It is characterized by its size, N, which is often assumed to be infinite for modeling purposes, and the probability distribution governing the interarrival times. The queue is the number of customers waiting for service, and may be concentrated at a fixed location such as a bank foyer or may be distributed in time and space such as airplanes approaching a runway. The queue discipline defines the rules by which customers are selected for service. A common discipline is first-come-first-served (FCFS), otherwise known as first-in-first-out (FIFO), but other possibilities are priority schemes or random selection. The service mechanism is the process by which customers are served. The usual assumption is that service is provided by one or more identical servers (channels) operating in parallel. When dealing with a network of queues, however, various configurations will be considered. The characteristics of service are the number of channels, s, and the service time probability distribution. The queuing system is the combination of the queue and the service channels. The measures used to analyze the queuing system primarily involve the number of customers in the system. This number is represented by the state of the system. In the figure below we show a state as a circle with the number in the circle indicating the number of customers associated with the state. State 0 is the empty state when there are no customers and all servers are idle. In states 1 through s, the customers are all being served and there are none in the queue. For states greater than s, all servers are busy and some customers must wait in the queue. The arcs represent events. An arrival, denoted by a, causes the number in the system to increase by 1 while a departure, denoted by d, causes the number in the system to decrease by 1. We use subscripts on these designations to indicate that the associated process may depend on the state of the system. Since both the interarrival and service times are random variables the state of the system is a stochastic process. For stable systems, there exist steady-state probabilities for the number of customers in the system. We call the steady-state probability of n customers, . The steady-state probabilities have two meanings: probability of finding the system in state n at some randomly selected time, or proportion of time the system is in state n. Queuing theory provides formulas for computing the steady-state probabilities for several different configurations of the queuing system. Most of these require that the interarrival times and service times are governed by the exponential probability distribution. Approximate results are available when the distributions are not exponential. These formulas are implemented in the Excel Queuing add-in described elsewhere on this site. Given the steady-state probabilities, one can compute a variety of measures of interest to the designer or operator of a queuing system. These include the average number of customers in the system, average time in the system and efficiency of the servers. We use the term average as synonomous with the statistical terms mean or expected value. Average numbers and times can also be computed for the customers in service or in the queue. The terminology page of this section provides a list of the steady-state results that can be obtained.

Operations Research Models and Methods
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by Paul A. Jensen