# Background

As a traffic engineer faced with the optimal design of a highway segment which is uncongested, safe, and a pleasant experience, one needs to understand the relationships between the fundamental design and the performance variables in order to make an informed choice.

That the relationship between these variables is quite complex is well-known. In order to proceed with our task, we should try and answer the following questions:

• What are the fundamental design and performance variables involved in designing an uncongested highway segment?
• How are these fundamental design (d), performance (p), and contextual (c) variables related?

p= f(c,d)

• What type of mathematical model might be developed so that we can predict and control these variables so that the highway segment
can be optimized?

Congestion occurs mainly as a result of increased number of vehicles competing for the limited space of a roadway segment. Drivers tend to pick the shortest routes from any given origin point to a destination and this behavior results in an under utilization on certain road links while other links are subject to more traffic than they can handle, thereby resulting in congestion on these latter links. In essence, an appropriate mathematical model needs to be developed so as to study the dynamic, stochastic effects of congestion on traffic flow.

The essential components of our mathematical model are:

• The service rate (speed) decays within the roadway as a direct result of increased vehicular traffic demand. This relationship between speed and vehicular density is a monotonically decreasing function of density .
• Since there is a finite amount of available space within each roadway segment, the density of vehicles in a road segment has an upper limit, often called the jam density.

There are essentially four fundamental performance variables which are the most critical to a dynamic traffic network model:

• Flow (q):= Output volume or throughput as it is sometimes called is the equivalent hourly rate at which vehicles pass a point on a highway segment.
• Density (k):= the number of vehicles travelling over a unit length of highway at an instant of time.
• Speed ( µ ) := is the distance travelled by the unit time.
• Time (t):= is the average time a vehicle spends on a given road segment of length L

The most critical design variables we will be concerned with here in the dynamic model are:

• Length of the highway segment L
• Number of lanes along the segment N

Other design variables such as pavement material, geometric curve design, changes in elevation, etc. are considered to be less critical to the dynamic traffic model than the two above, but could be related to the empirical speed/density curve which will be talked about shortly and which can be incorporated into the mathematical model through the speed/density curve.

One final contextual or given variable is the incoming flow or Volume of traffic entering the highway segment under study:

• Input Volume ():= the total number of vehicles that enter highway segment during a given time interval.

There are certainly other contextual variables to consider such as weather conditions, climate, wind conditions, etc. but these are felt of be not as critical as Volume to the dynamic traffic model. Again, as stated above, these contextual factors could be accommodated through the empirical speed/density curves. As a graphical way of relating the key performance variables, we need to examine a fundamental graph of the speed along a highway as a function of the density of vehicular traffic. Since the speed/density function is not the same for every roadway segment, there are a family of curves possible for this relationship as indicated in the figure.

While some researchers argue that one must make a choice between a specific linear or nonlinear curve, we will develop a mathematical model that allows us to use either a linear or nonlinear relationship between speed and density. This will become one of the key flexible features of our mathematical model. The calibration of the model can be fine tuned for the specific context in which the highway segment is located, although you will not be asked to do this in the laboratory experiments.

For a linear model we have the following mathematical relationship between the speed of the vehicles and the capacity of the roadway segment:

µ = A/C (C + 1 - n)

where A is the average vehicle speed of a lone vehicle on the roadway and C is the road link capacity which is a function of the size of the vehicles and the length of the roadway segment L and the number of traffic lanes: C= α (L,N). Finally, n represents the number of vehicles using the roadway.

For a nonlinear (exponential) model we have the relationship between vehicle speed and capacity of the roadway:

µ = A exp [- (n-1 β) γ]

here β and γ are the scale and shape parameters of the exponential distribution and the other variables are as defined previously.

What the linear and nonlinear models will do for us is to give each vehicle travelling along the road segment a dynamic speed based on the density of cars along the segment. This type of functional relationship for dynamically updating the vehicle speed along the roadway as a function of the density of the roadway is called a "state dependent queueing model." Once we have this state dependent function, we can incorporate it into a queueing representation of the traffic along the roadway so as to predict the flow volume q, the travel time t and the number of vehicles occupying the highway segment at steady-state. These performance measures will then allow us to optimize the design variables L, N and also control the input volume V of cars to the roadway segment. In order to demonstrate these powerful tools, let's carry out the following experimental design.