As you wait impatiently for the traffic to move, do you ever wonder why you have to wait so long in traffic? There are professionals, called traffic engineers, who actually study such questions and use their knowledge of math and science to resolve the problem of long lines or queues. If you would like to learn more about the math behind queues, continue to read, or skip to a queuing activity.

## Queuing Theory: Math to the rescue!

A queue occurs when vehicles or people wait in line. A queue in transportation refers to a line of vehicles waiting to be served, like at a toll booth. Queuing theory is the mathematical study of those lines. Engineers apply queuing theory principles to develop and analyze models to measure roadway performance.

## Models and assumptions

A model (like, a miniature car or a mathematical model) is a representation of reality — phenomena and/or systems. Models help explain, predict, or control the phenomenon being represented. Modeling, the art of designing models, is an essential part of most sciences. However, models—as a representation of the reality—are imperfect. A “good” model should be realistic and include all the key elements of the phenomenon represented, but in order to be easily designed, manipulated, and used, it needs to be simple. In order to simplify, we have to make assumptions.

For understanding queues, we have to make assumptions about the arrival/departure patterns and queue characteristics. Taking the toll booth example mentioned earlier, assume that the number of arriving and departing vehicles increases linearly with time, following a movement pattern of first-in, first-out (FIFO). If there is only one channel (line)/one exit at the toll booth, the pattern is known as “deterministic arrival and deterministic departure with one channel (D/D/1)”, which is also the simplest queuing model.

The term deterministic in this context refers to the assumption that there is a fixed arrival rate (which often is a rough assumption) and a fixed departure rate, which corresponds to a fixed service rate, i.e., each vehicle needs the same amount of service. In other words, the vehicles’ arrival and departure rates are not random. Such a traffic flow is represented as a straight line in Figure 1. From this information, we can calculate the **queue length** (number of vehicles waiting at a given time) and **vehicle delay** (number of delayed vehicles in a period of time multiplied by this period, e.g., one vehicle-minutes delay corresponds to one vehicle delayed for one minute).

As shown in Figure 1, a queue would be generated, when the number of arrival vehicles is higher than the departure vehicles. The mathematical expression of a queue is the difference between the number of arriving and departing vehicles. Therefore, if the difference equals to zero, the queue would disappear. This phenomenon is also called “queue dissipation”. Since the queue length is the number of vehicles in the queue at a given point in time, the longest vehicle queue occurs when the difference between the number of arriving and departing vehicles is the greatest. Similarly, another significant indicator of roadway performance is the total traffic delay, which is measured by the area of the lower triangle in the graph.