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If the time required for a vehicle (particularly buses or elevators) to load passengers is an increasing function of the number of passengers loaded, then, on a route served by more than one vehicle, the vehicles tend to form pairs. If a vehicle runs behind (ahead of) schedule, it typically will pick up more (less) passengers and get even further behind (ahead of) schedule. The following is concerned with a shuttle type of route having just two vehicles and serving passengers who arrive at a constant rate. There is one control point at which vehicles can be intentionally delayed. The object is to devise a strategy of control that will minimize the average waiting time of the passengers. The strategy must correct for random fluctuations in trip time so that the headways will not become sufficiently unequal as to initiate effects of pairing. An approximate solution of the optimization problem suggests that the optimal control involves sufficiently tight control that the pairing effect has little influence on the waiting times. [ABSTRACT FROM AUTHOR]

Vehicles load passengers at a single service point and, after traversing some route, return for another trip. The travel times of successive trips are independent identically distributed random variables with a known distribution function. After a vehicle returns to the service point, one has the option of holding it, or dispatching it immediately. Passengers arrive at a uniform rate and the objective is to minimize the average wait per passenger. The problem of determining the optimal strategy (dispatch or hold) for a system of m vehicles is formulated as a dynamic programming problem. It is analyzed in detail for m = 1 and m = 2. For m = 1, the optimal strategy will hold a vehicle if it returns within less than about half the mean trip time. For m = 2, and for a small coefficient of variation of trip time C( T), the optimal strategy will control the vehicles so as to retain nearly equally spaced dispatch times, within a range of time proportional to C4/3(T). [ABSTRACT FROM AUTHOR]

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.

Consider a circular central business district of a city with n evenly spaced radial streets and m circular streets. We assume that during the evening rush hour driver origins are all inside the city and exits at the circumference with a joint distribution that depends only upon the relative angle between original and final radii, and the radial coordinate of the origin. Travel time on any directed link is assumed to be a function of the flow along that directed link and the radial coordinate of the link. For a flow of Q vehicles per hour leaving the city, our problem is to find an assignment of drivers to paths such that every driver follows a path of minimum travel time between his origin and exit. This extends some work of Smeed who considers a similar model with no flow dependence of the travel time and a uniform distribution of origins. It is first shown that this circularly symmetric system gives rise to a circularly symmetric optimal flow pattern. This flow pattern can be generated by assigning each driver to a path that uses at most one of the circular streets. Eventually the assignment is expressed explicitly as a function of 2(m - 1) unknown parameters that satisfy a system of equations and/or inequalities. The general solution of these equations were not found, but solutions were obtained for a number of special cases that showed the effect of congestion at the center and the effect of a belt expressway. Differential equations were also obtained for n → ∞ and m → ∞. [ABSTRACT FROM AUTHOR]

We consider a one-way street that intersects ii one-way (side) streets. Traffic approaching the first street is steady as are the flows on all side streets. Traffic signals are idealized as perfect on-off switches and traffic is treated as a fluid moving with a constant velocity on the main street. Total delay and total number of stops are evaluated for several types of signal coordination schemes. Some conclusions are: 1. For any given common cycle time and given splits at each inter- section, there is a choice of offsets (phases) that simultaneously minimizes both the total delay and stops, but it is not necessarily the one that produces a `through band' of maximum bandwidth. 2. There are signal settings for which some signals operate on half or a third the cycle time of other lights, which give a main Street delay equal to that for the optimal common cycle time setting but give less delay to the side street. 3. Similar models are commonly used to find maximum through bands for two-way traffic, but it is not obvious that this is a suitable objective even for one-way streets. [ABSTRACT FROM AUTHOR]