Limit theorems for thinning of renewal point processes

10.1017/s0021900200036214 ◽

1972 ◽

Vol 9 (04) ◽

pp. 847-851 ◽

Cited By ~ 1

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.

Convergence to equilibrium in a traffic model with restricted passing

10.2307/3213153 ◽

1979 ◽

Vol 16 (4) ◽

pp. 881-889 ◽

Cited By ~ 2

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

10.1017/s0021900200033568 ◽

1979 ◽

Vol 16 (04) ◽

pp. 881-889 ◽

Cited By ~ 1

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.

The coincidence approach to stochastic point processes

10.2307/1425855 ◽

1975 ◽

Vol 7 (1) ◽

pp. 83-122 ◽

Cited By ~ 129

Author(s):

Odile Macchi

Keyword(s):

Poisson Process ◽

Probability Space ◽

Point Processes ◽

Counting Process ◽

Regular Point ◽

General Point ◽

Completely Regular ◽

Photon Process ◽

Stochastic Point Processes

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

Densities for stationary random sets and point processes

10.1017/s0001867800022552 ◽

1984 ◽

Vol 16 (02) ◽

pp. 324-346 ◽

Cited By ~ 19

Author(s):

Wolfgang Weil ◽

John A. Wieacker

Keyword(s):

Poisson Process ◽

Point Processes ◽

Convex Set ◽

Compact Convex ◽

Random Sets ◽

Intensity Measure ◽

Additive Functionals ◽

Compact Convex Set ◽

Classical Integral

For certain stationary random setsX, densitiesDφ(X) of additive functionalsφare defined and formulas forare derived whenKis a compact convex set in. In particular, for the quermassintegrals and motioninvariantX, these formulas are in analogy with classical integral geometric formulas. The case whereXis the union set of a Poisson processYof convex particles is considered separately. Here, formulas involving the intensity measure ofYare obtained.

Densities for stationary random sets and point processes

10.2307/1427072 ◽

1984 ◽

Vol 16 (2) ◽

pp. 324-346 ◽

Cited By ~ 37

Author(s):

Wolfgang Weil ◽

John A. Wieacker

Keyword(s):

Poisson Process ◽

Point Processes ◽

Convex Set ◽

Compact Convex ◽

Random Sets ◽

Intensity Measure ◽

Additive Functionals ◽

Compact Convex Set ◽

Classical Integral

For certain stationary random sets X, densities Dφ (X) of additive functionals φ are defined and formulas for are derived when K is a compact convex set in . In particular, for the quermassintegrals and motioninvariant X, these formulas are in analogy with classical integral geometric formulas. The case where X is the union set of a Poisson process Y of convex particles is considered separately. Here, formulas involving the intensity measure of Y are obtained.

A model for interaction of a Poisson and a renewal process and its relation with queuing theory

10.2307/3212816 ◽

1972 ◽

Vol 9 (2) ◽

pp. 451-456 ◽

Cited By ~ 9

Author(s):

Lennart Råde

Keyword(s):

Poisson Process ◽

Renewal Process ◽

Queuing Theory ◽

Random Number ◽

Sufficient Conditions ◽

Stieltjes Transform ◽

Time Intervals ◽

Response Process ◽

Necessary And Sufficient ◽

Special Case

This paper discusses the response process when a Poisson process interacts with a renewal process in such a way that one or more points of the Poisson process eliminate a random number of consecutive points of the renewal process. A queuing situation is devised such that the c.d.f. of the length of the busy period is the same as the c.d.f. of the length of time intervals of the renewal response process. The Laplace-Stieltjes transform is obtained and from this the expectation of the time intervals of the response process is derived. For a special case necessary and sufficient conditions for the response process to be a Poisson process are found.

Limit theorems for the minimal position in a branching random walk with independent logconcave displacements

10.1017/s0001867800009824 ◽

2000 ◽

Vol 32 (01) ◽

pp. 159-176 ◽

Cited By ~ 5

Author(s):

Markus Bachmann

Keyword(s):

Random Walk ◽

Limit Theorems ◽

Random Number ◽

Time Lag ◽

Extreme Value Distribution ◽

Extreme Value ◽

Limiting Distribution ◽

Branching Random Walk ◽

Minimal Position ◽

Conditional Limit

Consider a branching random walk in which each particle has a random number (one or more) of offspring particles that are displaced independently of each other according to a logconcave density. Under mild additional assumptions, we obtain the following results: the minimal position in the nth generation, adjusted by its α-quantile, converges weakly to a non-degenerate limiting distribution. There also exists a ‘conditional limit’ of the adjusted minimal position, which has a (Gumbel) extreme value distribution delayed by a random time-lag. Consequently, the unconditional limiting distribution is a mixture of extreme value distributions.