A Highway Traffic Model

1975 ◽  
Vol 12 (S1) ◽  
pp. 303-309
Author(s):  
Herbert Solomon
Keyword(s):  
Poisson Process ◽  
Poisson Distribution ◽  
Random Measure ◽  
Traffic Model ◽  
Highway Traffic ◽  
Intensity Parameter ◽  
Time Space ◽  
Random Line ◽  
Straight Line ◽  
Poisson Fields

The trajectory of a car traveling at a constant speed on an idealized infinite highway can be viewed as a straight line in the time-space plane. Entry times are governed by a Poisson process with intensity parameter A leading to all trajectories as random lines in a plane. The Poisson distribution of number of encounters of cars on the highway is developed through random line models and non-homogeneous Poisson fields, and its parameter, which depends on the specific random measure employed, is obtained explicitly.

Some recent developments concerning asymptotic distributions of pontograms

1990 ◽  
Vol 108 (3) ◽  
pp. 559-567 ◽  
Author(s):  
Vera R. Eastwood
Keyword(s):  
Stochastic Process ◽  
Poisson Process ◽  
Asymptotic Distributions ◽  
Testing Device ◽  
Analytical Problem ◽  
Intensity Parameter ◽  
Straight Line ◽  
Data Problem ◽  
Recent Developments ◽  
Image Position

The data analytical problem of testing whether an empirical set of n given points in the plane could be considered to contain too many straight line configurations in a situation where the generating mechanism of the n points is unknown, was recently reintroduced to the literature by D. G. Kendall and W. S. Kendall[7]. Taking the Land's end data problem (cf. also Broadbent [1] and further references given there) as the anchor point and motivation for their discussion, D. G. and W. S. Kendall developed a new testing device, called the pontogram. The pontogram is a one- parameter stochastic process defined pointwise bywhere N denotes a Poisson process in [0, 1] with N(0) = 0 and unknown intensity parameter μ > 0 and where R = N(1) gives the number of Poisson events in the entire [0, 1] section.

Convergence to equilibrium in a traffic model with restricted passing

1979 ◽  
Vol 16 (4) ◽  
pp. 881-889 ◽  
Author(s):  
Hans Dieter Unkelbach
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Point Processes ◽  
Road Traffic ◽  
Poisson Processes ◽  
Traffic Model ◽  
Cluster Point ◽  
Convergence To Equilibrium ◽  
Constant Intensity

A road traffic model with restricted passing, formulated by Newell (1966), is described by conditional cluster point processes and analytically handled by generating functionals of point processes.The traffic distributions in either space or time are in equilibrium, if the fast cars form a Poisson process with constant intensity combined with Poisson-distributed queues behind the slow cars (Brill (1971)). It is shown that this state of equilibrium is stable, which means that this state will be reached asymptotically for general initial traffic distributions. Furthermore the queues behind the slow cars dissolve asymptotically like independent Poisson processes with diminishing rate, also independent of the process of non-queuing cars. To get these results limit theorems for conditional cluster point processes are formulated.

Rarefactions of compound point processes

1984 ◽  
Vol 21 (04) ◽  
pp. 710-719
Author(s):  
Richard F. Serfozo
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Classical Result ◽  
Point Processes ◽  
Rare Events ◽  
Random Measure ◽  
Bernoulli Trials ◽  
Random Measures ◽  
Infinitely Divisible ◽  
Independent Increments

The Poisson process is regarded as a point process of rare events because of the classical result that the number of successes in a sequence of Bernoulli trials is asymptotically Poisson as the probability of a success tends to 0. It is shown that this rareness property of the Poisson process is characteristic of any infinitely divisible point process or random measure with independent increments. These processes and measures arise as limits of certain rarefactions of compound point processes: purely atomic random measures with uniformly null atom sizes. Examples include thinnings and partitions of point processes.

scholarly journals Lidar Data Analysis for Time to Headway Determination in the DriveSafe Project Field Tests

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
İlker Altay ◽  
Bilin Aksun Güvenç ◽  
Levent Güvenç
Keyword(s):  
Field Tests ◽  
Simple Algorithm ◽  
Forward Direction ◽  
Tracking Algorithm ◽  
Lidar Data ◽  
Highway Traffic ◽  
Huge Amount ◽  
Detection And Tracking ◽  
Straight Line ◽  
Point Distance

The DriveSafe project was carried out by a consortium of university research centers and automotive OEMs in Turkey to reduce accidents caused by driver behavior. A huge amount of driving data was collected from 108 drivers who drove the instrumented DriveSafe vehicle in the same route of 25 km of urban and highway traffic in Istanbul. One of the sensors used in the DriveSafe vehicle was a forward-looking LIDAR. The data from the LIDAR is used here to determine and record the headway time characteristics of different drivers. This paper concentrates on the analysis of LIDAR data from the DriveSafe vehicle. A simple algorithm that only looks at the forward direction along a straight line is used first. Headway times based on this simple approach are presented for an example driver. A more accurate detection and tracking algorithm taken from the literature are presented later in the paper. Grid-based and point distance-based methods are presented first. Then, a detection and tracking algorithm based on the Kalman filter is presented. The results are demonstrated using experimental data.

Some results on a traffic model of Rényi

1969 ◽  
Vol 6 (02) ◽  
pp. 293-300
Author(s):  
Mark Brown
Keyword(s):  
Poisson Process ◽  
Constant Velocity ◽  
Random Variables ◽  
Traffic Model ◽  
Homogeneous Poisson Process ◽  
Image Position

In [5] Renyi considers the following traffic model: Vehicles enter a highway at times 〈Ti , i = 1, 2, … 〉, forming a homogeneous Poisson process of intensity λ. The vehicle entering at time Ti will choose a velocity Vi and will travel at that constant velocity. The random variables 〈Vi , i = 1, 2, …〉 are independently and identically distributed (i.i.d.) and independent of 〈Ti 〉 with c.d.f. F satisfying All vehicles travel in the same direction.

Convergence to equilibrium in a traffic model with restricted passing

1979 ◽  
Vol 16 (04) ◽  
pp. 881-889 ◽  
Author(s):  
Hans Dieter Unkelbach
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Point Processes ◽  
Road Traffic ◽  
Poisson Processes ◽  
Traffic Model ◽  
Cluster Point ◽  
Convergence To Equilibrium ◽  
Constant Intensity

A road traffic model with restricted passing, formulated by Newell (1966), is described by conditional cluster point processes and analytically handled by generating functionals of point processes. The traffic distributions in either space or time are in equilibrium, if the fast cars form a Poisson process with constant intensity combined with Poisson-distributed queues behind the slow cars (Brill (1971)). It is shown that this state of equilibrium is stable, which means that this state will be reached asymptotically for general initial traffic distributions. Furthermore the queues behind the slow cars dissolve asymptotically like independent Poisson processes with diminishing rate, also independent of the process of non-queuing cars. To get these results limit theorems for conditional cluster point processes are formulated.

scholarly journals Modeling Stochastic Variability in the Numbers of Surviving Salmonella enterica, Enterohemorrhagic Escherichia coli, and Listeria monocytogenes Cells at the Single-Cell Level in a Desiccated Environment

2016 ◽  
Vol 83 (4) ◽  
Author(s):  
Kento Koyama ◽  
Hidekazu Hokunan ◽  
Mayumi Hasegawa ◽  
Shuso Kawamura ◽  
Shigenobu Koseki
Keyword(s):  
Escherichia Coli ◽  
Listeria Monocytogenes ◽  
Probability Distribution ◽  
Poisson Process ◽  
Poisson Distribution ◽  
Salmonella Enterica ◽  
Bacterial Cells ◽  
Content Type ◽  
Cell Numbers ◽  
Inactivation Process

ABSTRACT Despite effective inactivation procedures, small numbers of bacterial cells may still remain in food samples. The risk that bacteria will survive these procedures has not been estimated precisely because deterministic models cannot be used to describe the uncertain behavior of bacterial populations. We used the Poisson distribution as a representative probability distribution to estimate the variability in bacterial numbers during the inactivation process. Strains of four serotypes of Salmonella enterica, three serotypes of enterohemorrhagic Escherichia coli, and one serotype of Listeria monocytogenes were evaluated for survival. We prepared bacterial cell numbers following a Poisson distribution (indicated by the parameter λ, which was equal to 2) and plated the cells in 96-well microplates, which were stored in a desiccated environment at 10% to 20% relative humidity and at 5, 15, and 25°C. The survival or death of the bacterial cells in each well was confirmed by adding tryptic soy broth as an enrichment culture. Changes in the Poisson distribution parameter during the inactivation process, which represent the variability in the numbers of surviving bacteria, were described by nonlinear regression with an exponential function based on a Weibull distribution. We also examined random changes in the number of surviving bacteria using a random number generator and computer simulations to determine whether the number of surviving bacteria followed a Poisson distribution during the bacterial death process by use of the Poisson process. For small initial cell numbers, more than 80% of the simulated distributions (λ = 2 or 10) followed a Poisson distribution. The results demonstrate that variability in the number of surviving bacteria can be described as a Poisson distribution by use of the model developed by use of the Poisson process. IMPORTANCE We developed a model to enable the quantitative assessment of bacterial survivors of inactivation procedures because the presence of even one bacterium can cause foodborne disease. The results demonstrate that the variability in the numbers of surviving bacteria was described as a Poisson distribution by use of the model developed by use of the Poisson process. Description of the number of surviving bacteria as a probability distribution rather than as the point estimates used in a deterministic approach can provide a more realistic estimation of risk. The probability model should be useful for estimating the quantitative risk of bacterial survival during inactivation.

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