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There has been a revival of interest in traffic flow theory in the late 1990s, mostly because vehicle detectors have been installed at many highway locations throughout the world to [...]

A model is described for the motion of two elevators serving traffic only from a single lobby floor to multiple higher floors. Over a certain range of passenger arrival rates, the average round trip time of an elevator can be approximated by a linear function of the number of passengers carried which, in turn, is proportional to the time since the last elevator departure from the lobby (the headway). A pair of deterministic equations for the means of two state variables describes the approximate evolution of the system behavior. These equations lead to a very strange chaotic type of motion in the two-dimensional state space. For moderately light traffic the two elevators tend to travel close together, but for heavy traffic the headways tend to be nearly uniformly distributed over the mean round trip time of a single elevator.

A diffusion model is used to find heavy traffic approximate autocorrelation functions for several variables in queueing systems (i.e. for waiting time, system time, number in system and unfinished work). A table is given by which correlations can easily be found for each variable in any GI/G/1 queue. Further, the infinite sum or integral of the autocorrelations is also found, and the spectral density function. The sum has applications in statistical analysis of queues.Extensive comparisons of approximate and exact correlations and their sums are reported, particularly for waiting times and system times, but also including number in system in M/M/1 queues. In general the correlations have similar accuracy to the probability distributions found by diffusion approximations. The percentage error is less for number in system than for waiting time.

Suppose that the arrival rate ?(t) of customers to a service facility increases with time at a nearly constant rate, d?(t)/dt = a, so as to pass through the saturation condition, ?(t) = µ =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 Tproportional to a–2/3such that if , then the queue distribution stays close to the prevailing equilibrium distribution associated with the ?(t) and µ, evaluated at time t.For |t| = O(T), however, the mean queue length is much less than the equilibrium mean, and is measured in units of some characteristic length Lwhich is proportional to a–1/3. For , the queue is approximately normally distributed with a mean of the order Llarger 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.

During a rush hour, the arrival rate ?(t) of customers to a service facility is assumed to increase to a maximum value exceeding the service rate µ, and then decrease again. In Part I it was shown that, after ?(t) has passed µ, the expected queue E{X(t)} exceeds that given by the deterministic theory by a fixed amount, (0.95)L, which is proportional to the (–1/3) power of a(t) = d?(t)/dtevaluated at time t= 0 when ?(t) = µ.The maximum of E{X(t)}, therefore, occurs when ?(t) again is equal to µat time t1as predicted by deterministic queueing theory, but is larger than given by the deterministic theory by this same constant (0.95)L(provided t1is sufficiently large). It is shown here that the maximum queue, suptX(t) is, approximately normally distributed with a mean (0.95) (L+ L1) larger than predicted by deterministic theory where L, is proportional to the (–1/3) power of a(t1). We also investigate the distribution of X(t) at the end of the rush hour when the queue distribution returns to equilibrium. During the transition, the queue distribution is approximately a mixture of a truncated normal and the equilibrium distributions. These results are applied to a case where ?(t) is quadratic in t.

The arrival rate of customers to a service facility is assumed to have the form ?(t) = ?(0) — ßt2for some constant ß.Diffusion approximations show that for ?(0) sufficiently close to the service rate µ, the mean queue length at time 0 is proportional to ß–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for all ?(0) and ß.Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

If on a long homogeneous highway there is no interaction between cars, then, under a wide range of conditions, an initial distribution of cars will in the course of time tend toward that of a Poisson process with statistically independent velocities for the cars in any finite interval of highway. Here we will generalize this known property to obtain the following. Suppose cars do interact in such a way as to delay a car when it passes another, but the density of cars is so low that we can neglect simultaneous interactions between three or more cars. There will again be equilibrium distributions of cars to which general classes of initial distributions will converge. These equilibrium distributions are superpositions of two statistically independent processes, one a Poisson process of single free cars with statistically independent velocities, and the other a Poisson process of interacting pairs of cars with various velocities. In the limit of zero interaction, the density of pairs vanishes leaving only the Poisson process of single cars as a special case. To the same order of approximation, including the first order effects of interactions, the headway distribution between consecutive cars will still have exponential tail outside the range of interaction.