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The following is mostly a discussion of some issues relating to the design of minimum cost bus routes serving a multiple origin-multiple destination trip distribution. The main difficulty in determining any "optimal" routing origina tes from the fact that the objective function (total cost) is a nonc onvex function of the assignment; the higher the demand for tripson a route, the better is the ice that one can provide . One consequence of this is that a square grid of straight line bus routes is not likely to be an optimal geometry even under highly idealized conditions. "Good" geometries are more likely to focus routes onto a single street and past a common junction. [ABSTRACT FROM AUTHOR]

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]

We consider here an idealized infinite rectangular grid of roads with a translationally symmetric O-D distribution. The total cost of travel on all links approaching each junction is approximated by a quadratic function of the four flows N, 5, E, and W at that junction. If any of the four eigen valves of this quadratic form is negative, the system optimal assignment problem is non-convex. If there are economies of scale (due possibly to construction costs) then all eigenvalues could be negative and the optimal assignment will lead to a hierarchical type of flow distribution (city streets, arterials, freeways, etc.). If costs arise only from congestion, however, it is possible that one or more of the eigenvalues is negative particularly if the cost of travel N, for example, is more sensitive to the flows E and/or W than to the flow N, or is more sensitive to the flow S than N. If it is more sensitive to the flow E-W an efficient assignment would seem to be one in which the space is divided into subregions such that in certain subregions traffic will be predominantly N or S and in other subregions it is predominately E or W The optimal assignment is expected to be highly unstable to changes in the O-D distribution. If it is more sensitive to the flow S. a user optimal assignment may be stable and translationally symmetric but not the system optimal. The conclusion is that a non-convex assignment problem is not only a computational nightmare, but may be inconsistent with social objective or impractical to implement. [ABSTRACT FROM AUTHOR]

We imagine that drivers in the morning commute must traverse some homogeneous section of highway of length 1>0 and finite capacity qm to reach their workplaces. Each driver has a specified latest time at which he can exit the road and still be at work on time, but, because of the capacity and other traffic flow restrictions, it may be necessary that some drivers be at work earlier than their scheduled times if no one is to be late. Traffic is treated as a continuous fluid in the manner proposed by Lighthill and Whit/tam. This represents a generalization to 1> 0 of a theory described by Hendrickson and Kocur for point bottlenecks (1 = 0) and a revision of a theory presented by Mahmassani and Herman. That waves can travel with only finite speed causes the schedule delay to be larger for 1> 0 than for I = 0 and possibly be positive even if the demand rate is always less than qm. The conclusions, however, are quite different from those given by Mahmassani and Herman. If there is a cost per unit of trip time and another (lower) cost per unit of schedule delay, there is a well-defined system optimal assignment. There is some question, however, regarding the stability of the user optimal assignment. [ABSTRACT FROM AUTHOR]

The following is a survey and critique of the literature on airport capacity. It was originally written as a set of lecture notes for a university course on airport planning. The purpose is to describe, in as simple a way as possible, how the capacity of a runway configuration depends upon the strategy for sequencing various types of operations (heavy or light arriving or departing aircraft), the runway geometry, the instrument flight rules, etc. [ABSTRACT FROM AUTHOR]

S. L. Albin has described extensive simulations of queue behavior for a system with a single server and an arrival process that is a superposition of n renewal processes. The simulations show, among other things, that as n increases for a fixed traffic intensity p, the queue behavior approaches that of the M/M/1 system. The rate of convergence, however, becomes much slower as the traffic intensity p comes closer to 1. Several qualitative effects shown in the simulations are explained here. In particular it is shown that the approach to the M/M/1 system requires that n(1 - p)2>>1. [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.

Suppose that a given number n of vehicles (buses, trains, etc.) may be dispatched at any times from some source to serve passengers along a route. The arrival rate of passengers is some given (nonconstant) smooth function of time, typically having one or more peaks. One wishes to choose the dispatch times tj , j = 1, . . , ii in such a way as to minimize the total waiting time of all passengers. It is shown that if the capacity of the vehicles is sufficiently large to serve all waiting pas- sengers and n is large, then the optimal flow rate of vehicles (reciprocal of the headways) and the number of passengers served per vehicle, both vary with time approximately as the square root of the arrival rate of passengers. If the vehicles have limited capacity, their dispatch schedule will be distorted so that certain vehicles are dispatched as soon as they are full. [ABSTRACT FROM AUTHOR]

Presence detectors on highways record whether or not there are any cars on some nonzero length section of highway. The absence of ears is used as a criterion for .switching a vehicle-actuated signal. The problem considered here is the following. Cars pass the detector according to a Poisson process, at velocities that are independent, identically distributed random variables. Starting at some arbitrary time origin, how long will it be before the detector of length L is empty for the first time. It is shown that, in most cases, this time is nearly the same as if all cars had the same velocity q//c, q =flow, k = spacial density, and is rather insensitive to the distribution of the velocities. [ABSTRACT FROM AUTHOR]

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]

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]

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]

We consider here the delays suffered by a car in passing through a long sequence of synchronized traffic signals, assuming that the density of traffic is low enough that one can neglect the interactions between cars, and that each car has a desired speed which it maintains at all times except when stopped by a traffic light By considering the special case of equally spaced lights, we conclude that for short distances between lights the relative phase of the traffic lights that gives a minimum average delay per light depends upon the fraction of green time of the traffic cycle For long distances between lights, it is found that the usual progressive timing of lights may cause greater delays than a random synchronization but in such cases there are other schemes of synchronization that are still better Finally, some qualitative effects of velocity fluctuations are discussed [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]