Reliability Governed by the Relative Locations of Random Variables Following a Homogeneous Poisson Process in a Finite Domain

2004 ◽  
Vol 53 (2) ◽  
pp. 226-237 ◽  
Author(s):  
M.T. Todinov
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Finite Domain ◽  
Homogeneous Poisson Process

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.

On the probability that a random point is the jth nearest neighbour to its own kth nearest neighbour

1986 ◽  
Vol 23 (01) ◽  
pp. 221-226 ◽  
Author(s):  
Norbert Henze
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Arbitrary Point ◽  
Limiting Distribution ◽  
Poisson Processes ◽  
Random Point ◽  
Independent Random Variables ◽  
Nearest Neighbour ◽  
Homogeneous Poisson Process

In a homogeneous Poisson process in R d , consider an arbitrary point X and let Y be its kth nearest neighbour. Denote by Rk the rank of X in the proximity order defined by Y, i.e., Rk = j if X is the jth nearest neighbour to Y. A representation for Rk in terms of a sum of independent random variables is obtained, and the limiting distribution of Rk, as k →∞, is shown to be normal. This result generalizes to mixtures of Poisson processes.

Time spent below a random threshold by a Poisson driven sequence of observations

2003 ◽  
Vol 40 (03) ◽  
pp. 807-814 ◽  
Author(s):  
S. N. U. A. Kirmani ◽  
Jacek Wesołowski
Keyword(s):  
Poisson Process ◽  
Asymptotic Distribution ◽  
Random Variables ◽  
Nonhomogeneous Poisson Process ◽  
System Level ◽  
Homogeneous Poisson Process ◽  
Random Threshold ◽  
The Mean ◽  
Current Value ◽  
Independent And Identically Distributed

The mean and the variance of the time S(t) spent by a system below a random threshold until t are obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/t as t → ∞ is derived.

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 〉.

Non-homogeneously paced random records and associated extremal processes

1978 ◽  
Vol 15 (3) ◽  
pp. 552-559 ◽  
Author(s):  
Donald P. Gaver ◽  
Patricia A. Jacobs
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rate Function ◽  
Extreme Value ◽  
Value Distribution ◽  
Homogeneous Poisson Process ◽  
Extremal Process ◽  
Extremal Processes ◽  
Non Homogeneous Poisson Process

A study is made of the extremal process generated by i.i.d. random variables appearing at the events of a non-homogeneous Poisson process, 𝒫. If 𝒫 has an exponentially increasing rate function, then records eventually occur in a homogeneous Poisson process. The size of the latest record has a classical extreme value distribution.

Non-homogeneously paced random records and associated extremal processes

1978 ◽  
Vol 15 (03) ◽  
pp. 552-559 ◽  
Author(s):  
Donald P. Gaver ◽  
Patricia A. Jacobs
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rate Function ◽  
Extreme Value ◽  
Value Distribution ◽  
Homogeneous Poisson Process ◽  
Extremal Process ◽  
Extremal Processes ◽  
Non Homogeneous Poisson Process

A study is made of the extremal process generated by i.i.d. random variables appearing at the events of a non-homogeneous Poisson process, 𝒫. If 𝒫 has an exponentially increasing rate function, then records eventually occur in a homogeneous Poisson process. The size of the latest record has a classical extreme value distribution.

Some results on a traffic model of Rényi

1969 ◽  
Vol 6 (2) ◽  
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.

On the probability that a random point is the jth nearest neighbour to its own kth nearest neighbour

1986 ◽  
Vol 23 (1) ◽  
pp. 221-226 ◽  
Author(s):  
Norbert Henze
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Arbitrary Point ◽  
Limiting Distribution ◽  
Poisson Processes ◽  
Random Point ◽  
Independent Random Variables ◽  
Nearest Neighbour ◽  
Homogeneous Poisson Process

In a homogeneous Poisson process in Rd, consider an arbitrary point X and let Y be its kth nearest neighbour. Denote by Rk the rank of X in the proximity order defined by Y, i.e., Rk = j if X is the jth nearest neighbour to Y. A representation for Rk in terms of a sum of independent random variables is obtained, and the limiting distribution of Rk, as k →∞, is shown to be normal. This result generalizes to mixtures of Poisson processes.

Time spent below a random threshold by a Poisson driven sequence of observations

2003 ◽  
Vol 40 (3) ◽  
pp. 807-814 ◽  
Author(s):  
S. N. U. A. Kirmani ◽  
Jacek Wesołowski
Keyword(s):  
Poisson Process ◽  
Asymptotic Distribution ◽  
Random Variables ◽  
Nonhomogeneous Poisson Process ◽  
System Level ◽  
Homogeneous Poisson Process ◽  
Random Threshold ◽  
The Mean ◽  
Current Value ◽  
Independent And Identically Distributed

The mean and the variance of the time S(t) spent by a system below a random threshold until t are obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/t as t → ∞ is derived.

An invariance property of Poisson processes

1969 ◽  
Vol 6 (2) ◽  
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〉.

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