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The article discusses the three-dimensional representation of traffic flow. The traffic flow theory, particularly for single-lane traffic, contains many methods for displaying graphically the evolution of various traffic characteristics, such as flow, q, density, k, and velocity, v. Which graphical representation one uses depends upon which characteristics one wishes to analyze. Each method tends to display certain features in a convenient way, and others in a less convenient way. This three-dimensional surface is like a staircase, because cars are discrete, but people usually are not interested in such fine details. In the three-dimensional representation, the relation between these two families of curves is very simple. The x-t curves represent a contour map of the surface n (x, t), i.e., lines of constant height n (x, t) = j. The n-t curves also describe a contour map of the same surface but with the axes rotated so that the "base variables" are n and t. In the three-dimensional representation, these waves are paths of constant direction of the vector A and could be traced quite easily if one locates all points of the surface n (x, t) at which a plane of given orientation can be made tangent to the surface.

The methods of analysis used in part I are extended to the analysis of a vehicle-actuated signal at the intersection of two two-way streets (four-way intersection) at which there is no turning traffic. The main conclusion is that the high efficiency of a vehicle-actuated signal as compared with a fixed-cycle signal for one-way streets as discussed in part I, does not necessarily extend to the case of two-way streets. In particular, if the flows in opposite direction on the two-way street are nearly equal and the intersection is nearly saturated, then it is very inefficient (even worse than a fixed-cycle) for a vehicle-actuated signal to follow a policy of holding a signal green until the last of the two discharging queues has vanished (it is even worse to switch when only one queue has vanished). In all cases, the competition between incompatible desirable policies for the two opposing traffic streams leads to compromise strategies that give considerably higher delays on all approaches than occurs on one-way streets. [ABSTRACT FROM AUTHOR]

A vehicle-actuated traffic light controls two intersecting traffic streams keeping the light green for lane i, i=1 or 2 until any existing queue has been discharged and a headway of duration at least βi is detected in the subsequent arrivals. The object of this paper is to investigate how one should choose the βi so as to minimize the average delay per vehicle at the intersection. A model is considered in which the arrival headways are exponentially distributed random variables, departure headways have any specified distribution, and there is a random lost time for each switch of the traffic light. Formulas are derived for the moments of the cycle times, the average wait per vehicle, and the optimal βi. [ABSTRACT FROM AUTHOR]

A crude mathematical model is constructed to represent the flow of traffic in tunnels or other finite sections of highway on which passing is impossible The main ingredients of the model are (1) each car has a `desired speed which is considered to be a random variable, (2) the actual speed of any car is the smaller of its own desired speed and the actual speed of the car ahead of it, (3) if the latter speed is the smaller, the trailing car maintains some fixed spacing Δ behind its predecessor The consequences of the model are (1) the capacity of a tunnel is a monotone decreasing function of its length, (2) the velocity of any car is an increasing function of its distance from the entrance and is equal to the smallest of its own desired speed and those of all other cars in the tunnel ahead of it, (3) the spacing between cars is equal to Δ for most cars but when the spacing is larger than a, it is usually much larger and increases with the distance from the entrance [ABSTRACT FROM AUTHOR]