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.

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.

Limit theorems for thinning of renewal point processes

1972 ◽  
Vol 9 (4) ◽  
pp. 847-851 ◽  
Author(s):  
Råde L.
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Point Processes ◽  
Random Number

Limit theorems for the thinning of renewal point processes according to two different schemes are studied. In the first scheme when a point is retained a random number of succeeding points are deleted. According to the second scheme a random number of points are deleted by an inhibitory Poisson process.

An invariance property of Poisson processes

1969 ◽  
Vol 6 (02) ◽  
pp. 453-458 ◽  
Author(s):  
Mark Brown
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Random Variables ◽  
Poisson Processes ◽  
Invariance Property ◽  
Homogeneous Poisson Process ◽  
Arrival Times ◽  
Random Functions

In this paper we shall investigate point processes generated by random variables of the form 〈gi (Ti ]), i=± 1, ± 2, … 〉, where 〈Ti, i= ± 1, … 〉 is the set of arrival times from a (not necessarily homogeneous) Poisson process or mixture of Poisson processes, and 〈gi, i = ± 1, … 〉 is an independently and identically distributed (i.i.d.) or interchangeable sequence of random functions, independent of 〈Ti 〉.

scholarly journals The Berry-Esseen bound for the Poisson shot-noise

1987 ◽  
Vol 19 (2) ◽  
pp. 512-514 ◽  
Author(s):  
John A. Lane
Keyword(s):  
Statistical Analysis ◽  
Shot Noise ◽  
Point Processes ◽  
Normal Approximation ◽  
Poisson Processes ◽  
Cluster Point ◽  
Special Cases ◽  
Poisson Shot Noise

This note provides a useful extension of the Berry–Esseen bound on the error in the normal approximation for shot-noise. The special cases treated are of particular interest in the statistical analysis of Poisson processes and cluster point processes.

scholarly journals Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

2014 ◽  
Vol 46 (2) ◽  
pp. 348-364 ◽  
Author(s):  
Günter Last ◽  
Mathew D. Penrose ◽  
Matthias Schulte ◽  
Christoph Thäle
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Malliavin Calculus ◽  
Limit Theorems ◽  
Measurable Space ◽  
Central Limit Theorems ◽  
Poisson Processes ◽  
Chaos Expansion

This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Itô integrals with respect to the compensated Poisson process. Also, we present a multivariate central limit theorem for a vector whose components admit a finite chaos expansion of the type of a Poisson U-statistic. The approach is based on recent results of Peccati et al. (2010), combining Malliavin calculus and Stein's method; it also yields Berry-Esseen-type bounds. As applications, we discuss moment formulae and central limit theorems for general geometric functionals of intersection processes associated with a stationary Poisson process of k-dimensional flats in .

scholarly journals Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

2014 ◽  
Vol 46 (02) ◽  
pp. 348-364 ◽  
Author(s):  
Günter Last ◽  
Mathew D. Penrose ◽  
Matthias Schulte ◽  
Christoph Thäle
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Malliavin Calculus ◽  
Limit Theorems ◽  
Measurable Space ◽  
Central Limit Theorems ◽  
Poisson Processes ◽  
Chaos Expansion

This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Itô integrals with respect to the compensated Poisson process. Also, we present a multivariate central limit theorem for a vector whose components admit a finite chaos expansion of the type of a PoissonU-statistic. The approach is based on recent results of Peccatiet al.(2010), combining Malliavin calculus and Stein's method; it also yields Berry-Esseen-type bounds. As applications, we discuss moment formulae and central limit theorems for general geometric functionals of intersection processes associated with a stationary Poisson process ofk-dimensional flats in.

scholarly journals The Berry-Esseen bound for the Poisson shot-noise

1987 ◽  
Vol 19 (02) ◽  
pp. 512-514 ◽  
Author(s):  
John A. Lane
Keyword(s):  
Statistical Analysis ◽  
Shot Noise ◽  
Point Processes ◽  
Normal Approximation ◽  
Poisson Processes ◽  
Cluster Point ◽  
Special Cases ◽  
Poisson Shot Noise

This note provides a useful extension of the Berry–Esseen bound on the error in the normal approximation for shot-noise. The special cases treated are of particular interest in the statistical analysis of Poisson processes and cluster point processes.

Some Results on Interarrival Times of Nonhomogeneous Poisson Processes

1996 ◽  
Vol 10 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Subhash C. Kochar
Keyword(s):  
Poisson Process ◽  
Stochastic Order ◽  
Stochastic Ordering ◽  
Intensity Function ◽  
Poisson Processes ◽  
Constant Intensity ◽  
Nonhomogeneous Poisson Processes ◽  
Interarrival Times ◽  
Hazard Rate Ordering ◽  
Intensity Functions

It is well known that in the case of a Poisson process with constant intensity function the interarrival times are independent and identically distributed, each having exponential distribution. We study this problem when the intensity function is monotone. In particular, we show that in the case of a nonhomogeneous Poisson process with decreasing (increasing) intensity the interarrival times are increasing (decreasing) in the hazard rate ordering sense and they are also jointly likelihood ratio ordered (cf. Shanthikumar and Yao, 1991, Bivariate characterization of some stochastic order relations, Advances in Applied Probability 23: 642–659). This result is stronger than the usual stochastic ordering between the successive interarrival times. Also in this case, the interarrival times are conditionally increasing in sequence and, as a consequence, they are associated. We also consider the problem of comparing two nonhomogeneous Poisson processes in terms of the ratio of their intensity functions and establish some results on the successive number of events from one process occurring between two consecutive occurrences from the second process.

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