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In this paper the authors study the queues that build up in a minor stream of traffic waiting to cross or merge with a major stream The criterion for joining or crossing is that a gap between veincles in the major stream be greater than or equal to some constant value Several authors have studied the case where bulk service is provided for the minor stream queue with the appearance of each gap, in this paper the authors emphasize the higher flow rate situations where one car in the minor stream queue is released in a first-come first-serve fashinon with the appearance of a gap It is assumed that the minor stream is Poisson-fed while the intervehicle headways inq the major stream are arbitrary but identically and independently distributed random variables. [ABSTRACT FROM AUTHOR]

This paper studies some of the theoretical questions of large opemngs or gaps in a single stream of traffic A gap in the traffic stream is defined as a headway between vehicles greater than or equal to some minimum size -- say x Several authors have studied the probability distribution of the wait that a randomly located observer must endure before he finds a gap This paper, while briefly reviewing the solutions of this well-known problem, is primarily concerned with expressions for (i) the distribution of gap sizes, (ii) the distribution of spacings between vehicles and gaps, (iii) the mean and variance of intervehicle and intergap spacings, (iv) the stationary flow rates of gaps, and (v) the distribution of blocked and unblocked periods It is assumed that the origin of measurements may be located (i) with the passing of a vehicle, (ii) at the beginning of a gap, or (iii) at random It is also assumed that the distribution of intervehicle spacings are independently, but identically, distributed random variables. [ABSTRACT FROM AUTHOR]

The approach to the traffic-flow problem based on an integral differential equation of the Boltzmann type which has been considered by one of us in a recent paper is further developed. The possibility of passing is explicitly introduced into the equation for the velocity distribution function. As in the previous paper, it is shown that at sufficiently high concentration a collective flow process must take place. In order to study more specifically the effects of one car on another, we define reduced n-car distribution functions giving the probability of finding a cluster of n cars all having the same velocity. We derive an equation for the evolution of this distribution function. Study of it yields some information as to the way traffic changes from relatively free flow to completely hindered, 'condensed' flow. [ABSTRACT FROM AUTHOR]

A commuter living outside a circular city arrives at a point on the circumference and is equally likely to wish to travel to every point of the interior. In the present paper we give the probability distribution for trip length, and the probability distribution for traffic weight at each point, under the assumption of six different routing systems. [ABSTRACT FROM AUTHOR]

A vehicle waiting at an intersection with a major road makes a merging maneuver into the mainstream traffic, thus possibly requiring oncoming traffic to slow down. This paper examines the resulting disturbance that this forced entry may create in the main stream, assumed to be a renewal process. After showing that the disturbance propagation is formally equivalent to the busy period of a related queuing model, explicit results on the length of disturbance period and the number of vehicles affected are obtained for the case of Poisson traffic. It is shown that there is some minimal main-stream headway which should be forced in order to maximize the rate at which entries can be made from the secondary road. Finally, two measures of accident potential for the merging maneuver are discussed. [ABSTRACT FROM AUTHOR]

In a recent paper a traffic assignment model has been constructed in which the cost on a link may depend not only on its load, but also on the loads on other links of the network. In this paper it is shown that this model is also capable of handling the case of several classes of users in the same transportation network each of which has an individual cost function and, at the same time, contributes to its own and other classes' cost functions in an individual way. Typical applications arise not only in street networks where vehicles of different types share the same roads (e.g., trucks and passenger cars) but also in other types of transportation networks (e.g., telephone networks). An algorithm is constructed for finding the system-optimizing flow pattern for such a multiclass-user transportation network. [ABSTRACT FROM AUTHOR]

The article focuses on the report of the fourth international symposium on the theory of traffic flow, that was held in Karlsruhe, Germany, in June, 1968. The symposium was attended by 154 participants from all over the world. Here, authors present brief comment on each paper discussed in the symposium. The paper entitled "A Study of Driver-Aided Car Following," by R. E. Fenton and W. B. Montano described recent experiments with a car controlled by a control, stick with a finger presenting a built-in kinesthetic tactile aid to the driver. The paper entitled "On Some Theoretical Traffic Problems" by J. Gustavsson, dealt with two topics, the asymptotic average speed of a vehicle on a long journey, and numerical examples illustrating the author's model for overtaking on a two-lane road. "An Analysis of Delays to Vehicle Platoons at Traffic Signals," by R.E. Allops examined the problem of the delay to a traffic platoon at an unsaturated traffic signal, the theoretical result being expressed in terms of the arrival pattern and duration of the platoon and the arrival time of the leading vehicle.

Let G be a directed graph such that ever5 edge v of G is associated with a positive integer, called the index of v. Then G is called a network graph if, at every vertex t of G, the sum of the indices of the edges terminating at v is equal to that of the edges incident from v. This paper obtains several theorems giving necessary and sufficient conditions for a directed graph to be a network graph, and applied the results to solving some problems of smoothing traffic flow. [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]

The problem of optimizing the control of two oversaturated traffic intersections is solved by using the semi-graphical methods employed in a previous paper for an isolated intersection. As in the case of a single intersection the optimum control involves values of the control variables that lie along edges of the control region, which in this case is defined by the permissible ranges of the green phase splits. An analytical formulation of the method using PONTRYAGIN's control theory is also given. [ABSTRACT FROM AUTHOR]

The various theoretical analyses that have been made of delay to traffic at a fixed time traffic signal are critically reviewed. The more practicably applicable expressions for the average delay per vehicle, especially those derived by Webster, Miller, and Newell, are examined in some detail, and this paper provides an introduction to numerical comparisons to be reported by Hutchinson in the sequel. [ABSTRACT FROM AUTHOR]

This paper concerns itself with the kinematic analysis of traffic flow on a signalized one-way artery. Using as mathematical model the continuum flow theory of LIGHTHILL -WHITHAM-RICHAJWS, the flow pattern is carefully investigated under the assumptions of parabolic volume-density diagram and constant flow on the artery. The interesting symmetry existing between fluid and congested regimes is noted and used in proving various results. The delay caused by an interruption of arbitrary regimes is analyzed as a function of space and time. It is shown that an interruption of an arbitrary regime is equivalent to a particular interruption of a corresponding uniform regime: this provides a very simple and general method for delay computation. The delay caused by a periodic interruption (traffic signal) is bounded by and is asymptotically equal to the duration of the red signal; more- over, both the maximum and the average delay at fixed distance from the signalized intersection are increasing functions of the cycle time. It is further shown that the results concerning the delay function apply to all convex volume-density diagrams. [ABSTRACT FROM AUTHOR]

The general problem is that of locating a central facility in a network so as to minimize the sum of its distances from the sources of flow to it, each distance being appropriately weighted to reflect the associated flow volume and/or cost. In this paper, simple one-pass solution algorithms are given for two classes of topologically simple networks, namely those which are either acyclic or contain exactly one cycle. The first algorithm is based on a reduction procedure that may also yield useful simplification of problems involving general networks. [ABSTRACT FROM AUTHOR]

Before we can construct models for multilane traffic flow, we must be able to describe single-lane traffic. This paper considers some of these methods of description including the use of headway and counting distributions. Another method of describing point processes is by means of product densities. These are essentially joint probabilities of two or more vehicles passing a point at different specified times. Product densities have been used in many fields including particle physics, ecology, and road traffic. The idea of using product densities is simple and attractive, but they do have two major disadvantages: (i) except in simple cases, headway distributions are very difficult to derive from product densities, and (ii) product density estimates for different time lags are cor- related, so that tests of hypotheses are not easy to perform. An alternative to product densities is proposed. For want of a better name these are called termination rates. The termination rate after lag τ is the probability of a vehicle passing a point in a small interval of time, divided by the length of the small interval and conditional upon the last vehicle having passed a time τ previously. These are very similar to mortality rates for humans or scrap page rates for commodities with limited life-times. The interrelations between these statistics are quoted but not derived. Various sets of data are used to illustrate the less common statistics. Some analysis is given of a sample of data from the Congress (now Eisenhower) Expressway in Chicago. Variance to mean ratios are used to show that there is a strong correlation between lanes even though the autocorrelations of headways within lanes are very small. Bivariate termination rates have been estimated for each lane. Given a vehicle in one lane, it was found that the expected number of vehicles in a neighboring lane within 2 or 3 sec was enhanced by about 10-15 percent. [ABSTRACT FROM AUTHOR]

Investigations of motorist delays at the intersection of a major and minor road have involved variations of one or the other of two merging strategies: (i) MILLER'S[8] model that assumes that a minor-road motorist makes consecutive decisions consistently, and (ii) that of WEISS AND MARADUDIN[7] that assumptions that a different, independent decision is made for each headway. In this paper we show that results for both models may be obtained easily, using the results of queuing theory, if certain reasonable assumptions are made. Simple expressions for the Laplace-Stieltjes transforms and means of the delays are derived in the case of independent major-road headways with a common, general distribution. For exponential headways, a numerical study shows the mean Miller delay to increase with the variation in the merge times, while the mean Weiss-Maradudin delay decreases. To illustrate the application of the expressions obtained, we investigate the reduction in the delay achieved by inserting a traffic island between two streams of major-road traffic. [ABSTRACT FROM AUTHOR]

In this paper the authors find the optimal allocation of servicing periods to a facility servicing N incoming traffic streams. In our model it is assumed that stream flows are deterministic, that there is a fixed amount of lime lost, τi , when the server switches attention from the ith to the i + 1st stream and that the objective is to minimize time average delay. [ABSTRACT FROM AUTHOR]

The article presents an alternative method for determining the state probabilities in the multiple-channel case with fixed holding times and random arrivals. The author of this article claims that this method is less mathematically sophisticated and possesses practical advantages from the point of view of simplicity and convenience. This method is used to calculate the equilibrium-state probabilities in congestion problems in railroads. The author of this article outlines how the equations for this method is solved, as well as its variables used.

Some traffic service systems are characterized by service times that increase with the number of patrons undergoing service simultaneously. Two examples, the operation of a road traffic circle and the placement of toll telephone calls, are modelled to explore this property. The traffic circle model may yield information to determine whether a flow control system is justified. The telephone model leads to a doctrine for the rejection of calls during congested operation. Generally, the traffic capacity of such systems can be maximized only if some of the customers are thus rejected or retarded. [ABSTRACT FROM AUTHOR]

The results of a series of experiments carried out to determine the characteristics of single-lane bus flow are reported. Using pairs of buses theoretical 'car-following' models of single-lane traffic flow, which had been shown by previous work to be valid for automobiles, are examined for this type of heavy vehicle and found to provide a good representation of the detailed manner in which one bus follows another. Flow characteristics of buses were also determined by using platoons of up to ten buses. Both of these approaches, which can be termed microscopic and macroscopic, respectively, indicated that a single lane of buses form a stable stream of traffic and have an optimal flow of about 1450 buses/hour at a speed of 33 miles/hour for an expressway type facility. [ABSTRACT FROM AUTHOR]

Traffic signals can be synchronized so that a car, starting at one end of a street and traveling at preassigned speeds, can go to the other end without stopping for a red light. The portion of a signal cycle for which this is possible is called the bandwidth for that direction. Ordinarily the band-width in each direction is single, i.e., is not split into two or more intervals within a cycle. We solve two problems for this case: (1) Given an arbitrary number of signals along a street, a common signal period, the green and red times for each signal, and specified vehicle speeds in each direction between adjacent signals, synchronize the signals to produce bandwidths that are equal in each direction and as large as possible. (2) Adjust the synchronization to increase one bandwidth to some specified, feasible value and maintain the other as large as is then possible. The method of calculation has been programmed for a digital computer and results have been used to synchronize signals on a street in Cleveland. [ABSTRACT FROM AUTHOR]

A linear control algorithm is proposed for the computer operation of a signalized traffic intersection. A theoretical analysis of the algorithm under the assumption of constant vehicular arrival and departure rates shows that the control is stable under unsaturated conditions, and with proper choice of control constants usually gives a rapid convergence to a limit cycle that minimizes the average delay. [ABSTRACT FROM AUTHOR]

The homogeneous time independent solution for a one-car speed distribution function is derived for a Boltzmann-like approach to the statistical theory of traffic flow Numerical results are discussed for a number of different desired speed-distribution functions Among these distribution functions, chosen for mathematical sunplicity are, for example, the exponential, linear, and parabolic Flow versus concentration curves and distribution function versus speed curves are given for several of the models. [ABSTRACT FROM AUTHOR]

The authors have tried to improve the traffic dynamic theory of Professor Pipes and have found the fundamental equation of traffic dynamics for a case m which the velocity of the following vehicle is determined not only by the space between following and lead cars but also by the velocity of the lead car. The safety of the following car with respect to rear end collision when the lead car is in sinusoidal motion is quantified by introducing a safety index. At the same time, the authors describe the stability of the indicial response of the following car, and the stability of the propagation of a sinusoidal disturbance down a line of cars. [ABSTRACT FROM AUTHOR]

The steady-state flow is examined for a car-following model in which the acceleration at time t of a car attempting to follow a lead car is proportional to the relative velocity at a time t-Δ and in which the sensitivity &lamda; is no longer taken constant as in previous work but is inversely proportional to the car spacing. The characteristics of the steady-state flow for this model are described and compared with experimental data. [ABSTRACT FROM AUTHOR]

The manner in which vehicles follow each other on a highway (without passing) and the propagation disturbances down a line of vehicles has been investigated Experimental data is presented which indicates that the acceleration at time t of a car which is attempting to follow a leader is proportional to the difference in velocity of the two cars at a time (t-Δ) ,Δ being about 1 5 sec and the proportionality constant being about 037 sec-1 It is shown theoretically that the motion of a long line of vehicles becomes unstable when the product of the lag time and the proportionality constant exceeds one-half The experimental data implies that drying is done on the verge of instability A variety of other laws of following is analyzed theoretically [ABSTRACT FROM AUTHOR]

Recently developed techniques for traffic operational changes involving traffic signal control equipment are discussed Present practice is very largely based on intuition and qualitative analysis Quantitative analysis is limited because a general theory of traffic flow has not yet been developed Studies aimed at the eventual formulation of a theory of traffic flow are being conducted at the Road Research Laboratory (British), Brown University, the University of Michigan, and the University of California In combination with the flow theory the greatest promise appears to be in the simulation approach, some aspects of which are discussed [ABSTRACT FROM AUTHOR]