A situation familiar to everyone is waiting in a line. A typical example might be the line of customers that forms in front of the service windows at a post office. The number of customers in the line grows and shrinks with time, and, as anyone who has had the experience knows, the wait can be highly unpredictable. Because the number in line is a random variable that changes with time, the system of customers and servers fits the definition of a stochastic process.

Other familiar situations where lines or queues form include a ticket booth at a theater, a conference registration desk, a red light at a traffic signal, buffer storage on an assembly line, email on a server, and taxis outside an airport. Basically, a queue results whenever existing demand temporarily exceeds the capacity of the service facility; i.e., whenever an arriving customer cannot receive immediate attention because all servers are busy. This situation is almost always guaranteed to occur at some time in any system that has probabilistic arrival and service patterns. Tradeoffs between the cost of increasing service capacity and the cost of waiting customers prevent an easy solution to the design problem. If the cost of expanding a service facility were no object, then theoretically, enough servers could be provided to handle all arriving customers without delay. In reality, though, a reduction in the service capacity results in a concurrent increase in the cost associated with waiting. The basic objective in most queueing models is to achieve a balance between these costs.

In this section, we are primarily concerned with continuous-time systems operating in steady state. A variety of analytical results are available for these systems mostly under the assumption that the arrival and service processes are Markovian. Formulas embodied in the Queueing add-in compute statistical estimates for such measures as the average number in the queue, the average waiting time for a customer, and the probability that the service mechanism is busy. We do not repeat many formulas in this section, but concentrate on describing the models for which the formulas are available.

One of the key insights gained from studying queueing systems is that they may not be very efficient in terms of resource utilization. Queues form and customers wait even though servers may be idle much of the time. The fault is not in the model or underlying assumptions. It is a direct consequence of the variability of the arrival and service processes. If variability could be eliminated, systems could be designed economically so that there would be little or no waiting, and hence no need for queueing models.