G. F. Newell

A simplified theory of kinematic waves in highway traffic, part II: Queueing at freeway bottlenecks

For a freeway having various entrance and exit ramps, the methods described in Part I are used to relate the cumulative flow curve at any junction to the net cumulative entrance flow at this junction, and the cumulative flow curves for the freeway at the next upstream junction and/ or the next downstream junction. If the type of flow-density relations typical of freeway traffic are idealized by a triangular shaped curve with only two wave speeds, one for free-flowing traffic (positive) and the other for congested traffic (negative), then the relationship is easy to evaluate. The cumulative flow curve at the junction is simply the lower envelope of a translation of the cumulative curve from upstream and a different translation of the cumulative curve from downstream. This relationship is the basic building block for a freeway flow prediction model described in Part III.

A simplified car-following theory: a lower order model

Abstract A very simple “car-following” rule is proposed wherein, if an nth vehicle is following an (n−1)th vehicle on a homogeneous highway, the time-space trajectory of the nth vehicle is essentially the same as the (n−1)th vehicle except for a translation in space and in time. It seems that such a rule is at least as accurate as any of the more elaborate rules of car-following that have been proposed over the last 50 years or so. Actually, the proposed model could be interpreted as a special case of existing models but with fewer parameters and a different logic. At least this should form a reasonable starting point for investigating other phenomena.

The Morning Commute for Nonidentical Travelers

In previous theories of the morning rush hour, travelers with different work starting times are assumed to attach certain values to queueing delay and schedule delay (deviations from their work starting times), the same values for all travelers. The goal was to establish and evaluate a stable assignment of trip starting times such that no individual traveler can find a more desirable time than the one assigned. This theory is extended here to situations in which different travelers may attach different values to these delays. For a certain class of cost models it is shown that there is still a stable assignment but it is dictated by only a certain fraction of travelers who are most willing to be early or late for work. Other travelers will choose to be at work on time. The queueing pattern is significantly different from that of the previous theories.

A moving bottleneck

Suppose that a vehicle or convoy enters a two-lane unidirectional roadway and travels at a velocity v* less than the prevailing traffic. This moving bottleneck may cause a queue to form as vehicles try to pass the obstruction. It is shown that by going to a moving coordinate system traveling at velocity v* the analysis of this can be transformed into a corresponding analysis of flow past a stationary bottleneck. The theory is then extended to investigate possible consequences of trucks on grades.

A simplified theory of kinematic waves in highway traffic, part III: Multi-destination flows

Suppose that one specifies the times at which vehicles enter each freeway junction and where (but not when) they will exit. The problem is to determine the cumulative flows past various points of the freeway, trip times, etc. To solve this problem, one must combine the equations of traffic dynamics for the total flow (for all origins and destinations) as described in Part II, with equations specifying that the trip time of a vehicle between two successive junctions is independent of its origin or destination. One must evaluate not only the total flow past each point but the component flows past each point for each destination. A solution of this problem based on the theory in Part II is fairly straightforward.

Configuration of physical distribution networks

Distribution problems, including vehicle routing and warehouse location problems, are usually formulated by considering a finite number of possible locations for the customers, the warehouses, and vehicle stops. The question of selecting which of these points are actually used (and how) is a mixed-integer programming problem which is difficult to solve. Thus, such a discrete formulation results in a problem that has to be solved heuristically; it also entails a large data preparation effort each time a solution has to be developed in response to changing world conditions. The continuous approach used in this paper attempts to circumvent some of these drawbacks. We consider one source and its customers in a service area; customer locations are modeled by a density surface over the service area. With this information, and data about the cost of inventory and transportation, we can determine the number of transhipment points, and the frequency and routing of all the distribution vehicles. An example is given. The continuous approach does not yield a solution. It gives design guidelines, which ensure near minimum total cost. These design guidelines are based on general properties of optimal solutions (discussed at the beginning of the paper) and on the specific characteristics of the case at hand. Implementation of the guidelines to obtain a feasible configuration requires human intervention. While the continuous method involves approximations (the real world is discrete and considerably more complicated than in our model), it yields insight into the structure of logistic systems. This insight should not only help in the design process; it may well also lead to improved heuristic solution methods for discrete formulations. Hybrid methods may eventually emerge.

Queues with time-dependent arrival rates I—the transition through saturation

Suppose that the arrival rate A(t) of customers to a service facility increases with time at a nearly constant rate, dA(t)/dt = a, so as to pass through the saturation condition, ,A(t)= =t = service capacity, at some time which we label as t = 0. The stochastic properties of the queue are investigated here through use of the diffusion approximation (Fokker-Planck equation). It is shown that there is a characteristic time T proportional to a-2/3 such that if t 0, It 1> T, the queue is approximately normally distributed with a mean of the order L larger than that predicted by deterministic queueing models. Numerical estimates are given for the mean and variance of the distribution for all t. The queue distributions are also evaluated in non-dimensional units.

Theory of oscillation type viscometers V: Disk oscillating between fixed plates

A disk oscillating between two fixed plates is considered as an instrument for absolute measurements of viscosity. The existing theories relating the viscosity to the decrement of oscillation are improved by calculating the effects of the edge. This is done by assuming that the separation between the plates is small compared with both the radius of the disk and the boundary layer thickness. A comparison is made with the experimental data ofKestin andPilarczyk for which the present theory is estimated to be correct to 0·1%.ZusammenfassungEs wird untersucht, ob eine zwischen zwei feststehenden Platten schwingende Scheibe als Gerät für die Absolutmessung der Zähigkeit von Flüssigkeiten benutzt werden kann. Die dafür bestehenden Theorien, bei denen die Viskosität aus dem logarithmischen Dekrement der Schwingung berechnet wird, sind hier durch die Berücksichtigung der Randeffekte erweitert worden, unter der Voraussetzung, dass der Abstand zwischen den beiden feststehenden Platten klein ist im Verhältnis zum Radius der Scheibe und zur Grenzschichtdicke. Die Theorie stimmt mit den vonKestin undPilarczyk experimentell gefundenen Werten auf 0,1% genau überein.

Approximate stochastic behavior of n-server service systems with large n

I. General Formulation.- 1. Introduction.- 2. Graphical Representations.- 3. Stochastic Properties.- II. Approximation Methods.- 1. Introduction.- 2. Approximations - No Customer Queue.- 3. Approximations with Queueing and Large Sk = s.- 4. Queueing with Random S, CS ?1.- 5. Queueing with Random S, CS ~ 1.- III. Approximations for Short Service Times.- 1. Introduction.- 2. Deterministic Approximations.- 3. Small Queues.- 4. Transition Behavior.- 5. The Final Transition.- IV. Equilibrium Distributions.- 1. Introduction.- 2. Approximation Equilibrium Distributions.- 3. Equilibrium Distributions for M/M/n.- 4. Equilibrium Distributions for G/M/n.- 5. Equilibrium Distributions for M/D/n or G/D/n.- 6. Concluding Comments.- References.

Vibration Spectrum and Heat Capacity of a Chain Polymer Crystal

The Born‐von Karman model for a chain polymer crystal previously studied by Stockmayer and Hecht is re‐examined using only analytic approximations rather than numerical methods. The analytic approach brings out many peculiar properties of the chain polymer model that did not appear in the numerical treatment.One branch of the frequency distribution g(v) is shown to be proportional to v2 for the smallest values of v, approximately proportional to v32 for slightly larger v, to v½ for still larger v and to v—½ in still a fourth range of small values of v. An accurate graph of g(v) is constructed for the entire range of frequencies using values of the force constants suggested by Stockmayer and Hecht. g(v) is shown to have approximate singularities of a type not anticipated by van Hove in his broad treatment of singularities for general systems. This anomalous behavior results from having strong valence forces resisting the bending of bond angles. A classification and description of various kinds of singulari...