We model a long queue of humans by a continuous-space model in which, when a customer moves forward, they stop a random distance behind the previous customer, but do not move at all if their distance behind the previous customer is below a threshold. The latter assumption leads to " of motion in which only some random number W of customers move. We prove that P(W > k) decreases as order k 1=2 ; in other words, for large k the k’th customer moves on average only once every order k 1=2 service times. A more rened analysis relies on a non-obvious asymptotic relation to the coalescing Brownian motion process; we give a careful outline of such an analysis without attending to all the technical details.
@cite_0: A new class of exclusion type processes acting in continuum with synchronous updating is introduced and studied. Ergodic averages of particle velocities are obtained and their connections to other statistical quantities, in particular to the particle density (the so called Fundamental Diagram) is analyzed rigorously. The main technical tool is a “dynamical” coupling applied in a nonstandard fashion: we do not prove the existence of the successful coupling (which even might not hold) but instead use its presence absence as an important diagnostic tool. Despite that this approach cannot be applied to lattice systems directly, it allows to obtain new results for the lattice systems embedding them to the systems in continuum. Applications to the traffic flows modelling are discussed as well.
@cite_9: We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and determine the kernel in the scaling limit. This result has been announced first in a letter by one of us (Sasamoto in J. Phys. A 38:L549–L556, 2005) and here we provide a self-contained derivation. Connections to last passage directed percolation and random matrices are also briefly discussed.
@cite_7: An anonymous Bluetooth tracking system was deployed at the new Indianapolis International Airport in Indiana from May 8 to June 2, 2009, to measure the time for passengers (a) to move from the nonsterile side of the airport (presecurity), (b) to clear the security screening checkpoint, and (c) to enter the walkway to Concourse B on the sterile side. The maximum passenger transit time between these checkpoints was observed on Monday mornings at approximately 0600, when it could take passengers up to 20 min to transit the security queue and screening and to walk to Concourse B. Depending on the day of the week, this approach was demonstrated to sample between 5 and 6.8 of passengers. This modest sample size provides a more robust measurement of screening times than the current system of manually distributing time-stamped cards as passengers enter the queue and collecting them where passengers pass through the magnetometer. Furthermore, because the final passenger reference point used in this study is on the sterile concourse, it captures the time associated with passengers repackaging their belongings and redonning their shoes. The data from this pilot study suggest the feasibility of using an automated system to provide quantitative information to managers for more effective allocation of scarce resources, as well as providing the traveling public with necessary information about the amount of time they should allocate for transiting the security screening process. The paper concludes by suggesting that additional pilot studies should be performed at several airports with alternative checkpoint configurations to develop a consensus on best practices for locating sensors to measure passage times at airport security screening.
@cite_3: We introduce and analyze a simple probabilistic cellular automaton which emulates the flow of cars along a highway. Our Traffic CA captures the essential features of several more complicated algorithms, studied numerically by K. Nagel and others over the past decade as prototypes for the emergence of traffic jams. By simplifying the dynamics, we are able to identify and precisely formulate the self-organized critical evolution of our system. We focus here on the Cruise Control case, in which well-spaced cars move deterministically at maximal speed, and we obtain rigorous results for several special cases. Then we introduce a symmetry assumption that leads to a two-parameter model, described in terms of acceleration (α) and braking (β) probabilities. Based on the results of simulations, we map out the (α, β) phase diagram, identifying three qualitatively distinct varieties of traffic which arise, and we derive rigorous bounds to establish the existence of a phase transition from free flow to jams. Many other results and conjectures are presented. From a mathematical perspective, Traffic CA provides local, particle-conserving, one-dimensional dynamics which cluster, and converge to a mixture of two distinct equilibria.
We do not know any previous work on closely related models. In one way, our model could be regarded as a continuum, syncronous-moves analog of the (discrete-space, asynchronous moves) TASEP which has been intensively studied @cite_9 owing to its connections with other statistical physics models. The small literature on such analogs of TASEP (see @cite_0 and citations therein) has focused on ergodic properties for processes on the doubly-infinite line. In another way, our phenomenon could be regarded as analogous to stop-and-go motion in traffic jams, for which realistic models would need to consider both positions and velocities of vehicles, although discrete-space discrete-time synchronous models have also been studied @cite_3 . Academic literature such as @cite_7 studying real world security lines does not address the specific wave" feature studied in this paper.