Note on the reversible counters system of Lampard

1974 ◽  
Vol 11 (3) ◽  
pp. 624-628 ◽  
Author(s):  
R. M. Phatarfod
Keyword(s):  
Poisson Process ◽  
Negative Binomial ◽  
Branching Processes ◽  
Poisson Processes ◽  
Time Interval ◽  
Input Process ◽  
Markov Sequence ◽  
The Times ◽  
Bivariate Poisson Process

The paper considers an extension to the reversible counters system of Lampard [1]. In Lampard's model the input processes are two independent Poisson processes; this results in a gamma Markov sequence for the time-interval between successive output pulses and a negative binomial Markov sequence for the counts at the times of out-put pulses. We consider the input process to be a bivariate Poisson process and show that the first out-put process given above is not affected, while the second out-put-process becomes of a type studied in the theory of branching processes.

Note on the reversible counters system of Lampard

1974 ◽  
Vol 11 (03) ◽  
pp. 624-628 ◽  
Author(s):  
R. M. Phatarfod
Keyword(s):  
Poisson Process ◽  
Negative Binomial ◽  
Branching Processes ◽  
Poisson Processes ◽  
Time Interval ◽  
Input Process ◽  
Markov Sequence ◽  
The Times ◽  
Bivariate Poisson Process

The paper considers an extension to the reversible counters system of Lampard [1]. In Lampard's model the input processes are two independent Poisson processes; this results in a gamma Markov sequence for the time-interval between successive output pulses and a negative binomial Markov sequence for the counts at the times of out-put pulses. We consider the input process to be a bivariate Poisson process and show that the first out-put process given above is not affected, while the second out-put-process becomes of a type studied in the theory of branching processes.

scholarly journals Poisson superposition processes

2015 ◽  
Vol 52 (04) ◽  
pp. 1013-1027
Author(s):  
Harry Crane ◽  
Peter Mccullagh
Keyword(s):  
Power Series ◽  
Poisson Process ◽  
Explicit Expression ◽  
Negative Binomial ◽  
Poisson Processes ◽  
Domain Restriction ◽  
Point Configuration ◽  
Point Configurations ◽  
Algebraic Properties ◽  
Formal Power

Superposition is a mapping on point configurations that sends the n-tuple into the n-point configuration , counted with multiplicity. It is an additive set operation such that the superposition of a k-point configuration in is a kn-point configuration in . A Poisson superposition process is the superposition in of a Poisson process in the space of finite-length -valued sequences. From properties of Poisson processes as well as some algebraic properties of formal power series, we obtain an explicit expression for the Janossy measure of Poisson superposition processes, and we study their law under domain restriction. Examples of well-known Poisson superposition processes include compound Poisson, negative binomial, and permanental (boson) processes.

scholarly journals Counting processes with Bernštein intertimes and random jumps

2015 ◽  
Vol 52 (04) ◽  
pp. 1028-1044 ◽  
Author(s):  
Enzo Orsingher ◽  
Bruno Toaldo
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Negative Binomial ◽  
Counting Processes ◽  
Lévy Measure ◽  
Fractional Poisson Process ◽  
Independent Increments ◽  
Random Jumps ◽  
The Times ◽  
Passage Times

In this paper we consider point processes Nf (t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernštein functions f with Lévy measure v. We obtain the general expression of the probability generating functions Gf of Nf , the equations governing the state probabilities pk f of Nf , and their corresponding explicit forms. We also give the distribution of the first-passage times Tk f of Nf , and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times of jumps with height lj () under the condition N(t) = k for all these special processes is investigated in detail.

Some remarks on a counter process

1976 ◽  
Vol 13 (03) ◽  
pp. 623-627
Author(s):  
Lajos Takács
Keyword(s):  
Poisson Process ◽  
Poisson Processes ◽  
Limit Distributions ◽  
Input And Output ◽  
Counter Process ◽  
Bivariate Poisson Process

In 1968 Lampard determined some limit distributions for a counter process in which the input and output are independent Poisson processes. In 1974 Phatarfod dealt with the generalizations of Lampard's formulas for the case when the input and output form a bivariate Poisson process; however, his reasoning is erroneous. The object of this paper is to determine the correct limit distributions for the generalized process.

Some remarks on a counter process

1976 ◽  
Vol 13 (3) ◽  
pp. 623-627 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Poisson Process ◽  
Poisson Processes ◽  
Limit Distributions ◽  
Input And Output ◽  
Counter Process ◽  
Bivariate Poisson Process

In 1968 Lampard determined some limit distributions for a counter process in which the input and output are independent Poisson processes. In 1974 Phatarfod dealt with the generalizations of Lampard's formulas for the case when the input and output form a bivariate Poisson process; however, his reasoning is erroneous. The object of this paper is to determine the correct limit distributions for the generalized process.

scholarly journals Poisson superposition processes

2015 ◽  
Vol 52 (4) ◽  
pp. 1013-1027
Author(s):  
Harry Crane ◽  
Peter Mccullagh
Keyword(s):  
Power Series ◽  
Poisson Process ◽  
Explicit Expression ◽  
Negative Binomial ◽  
Poisson Processes ◽  
Domain Restriction ◽  
Point Configuration ◽  
Point Configurations ◽  
Algebraic Properties ◽  
Formal Power

Superposition is a mapping on point configurations that sends the n-tuple into the n-point configuration , counted with multiplicity. It is an additive set operation such that the superposition of a k-point configuration in is a kn-point configuration in . A Poisson superposition process is the superposition in of a Poisson process in the space of finite-length -valued sequences. From properties of Poisson processes as well as some algebraic properties of formal power series, we obtain an explicit expression for the Janossy measure of Poisson superposition processes, and we study their law under domain restriction. Examples of well-known Poisson superposition processes include compound Poisson, negative binomial, and permanental (boson) processes.

A bivariate Poisson queueing process that is not infinitely divisible

1972 ◽  
Vol 72 (3) ◽  
pp. 449-450 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Queueing System ◽  
Poisson Processes ◽  
Single Server ◽  
Infinitely Divisible ◽  
Queueing Process ◽  
Input And Output ◽  
Poisson Arrivals ◽  
Bivariate Poisson Process

We are using the term ‘bivariate Poisson process’ to describe a bivariate point process (N1(.), N2(.)) whose components (or, marginal processes) are Poisson processes. In this we are following Milne (2) who amongst his examples cites the case where N1(.) and N2(.) refer to the input and output processes respectively of the M/G/∈ queueing system. Such a bivariate point process is infinitely divisible. We shall now show that in a stationary M/M/1 queueing system (i.e. Poisson arrivals at rate λ, exponential service at rate µ > λ, single-server) a similar identification of (N1(.), N2(.)) yields a bivariate Poisson process that is not infinitely divisible.

scholarly journals Correlated fractional counting processes on a finite-time interval

2015 ◽  
Vol 52 (04) ◽  
pp. 1045-1061 ◽  
Author(s):  
Luisa Beghin ◽  
Roberto Garra ◽  
Claudio Macci
Keyword(s):  
Poisson Process ◽  
Finite Time ◽  
Negative Binomial ◽  
Space Time ◽  
Time Interval ◽  
Counting Processes ◽  
Finite Time Interval ◽  
Fractional Counting ◽  
Fractional Poisson Process ◽  
Slight Generalization

We present some correlated fractional counting processes on a finite-time interval. This will be done by considering a slight generalization of the processes in Borges et al. (2012). The main case concerns a class of space-time fractional Poisson processes and, when the correlation parameter is equal to 0, the univariate distributions coincide with those of the space-time fractional Poisson process in Orsingher and Polito (2012). On the one hand, when we consider the time fractional Poisson process, the multivariate finite-dimensional distributions are different from those presented for the renewal process in Politi et al. (2011). We also consider a case concerning a class of fractional negative binomial processes.

scholarly journals Counting processes with Bernštein intertimes and random jumps

2015 ◽  
Vol 52 (4) ◽  
pp. 1028-1044 ◽  
Author(s):  
Enzo Orsingher ◽  
Bruno Toaldo
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Negative Binomial ◽  
Counting Processes ◽  
Lévy Measure ◽  
Fractional Poisson Process ◽  
Independent Increments ◽  
Random Jumps ◽  
The Times ◽  
Passage Times

In this paper we consider point processes Nf (t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernštein functions f with Lévy measure v. We obtain the general expression of the probability generating functions Gf of Nf, the equations governing the state probabilities pkf of Nf, and their corresponding explicit forms. We also give the distribution of the first-passage times Tkf of Nf, and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times of jumps with height lj () under the condition N(t) = k for all these special processes is investigated in detail.

scholarly journals Correlated fractional counting processes on a finite-time interval

2015 ◽  
Vol 52 (4) ◽  
pp. 1045-1061 ◽  
Author(s):  
Luisa Beghin ◽  
Roberto Garra ◽  
Claudio Macci
Keyword(s):  
Poisson Process ◽  
Finite Time ◽  
Negative Binomial ◽  
Space Time ◽  
Time Interval ◽  
Counting Processes ◽  
Finite Time Interval ◽  
Fractional Counting ◽  
Fractional Poisson Process ◽  
Slight Generalization

We present some correlated fractional counting processes on a finite-time interval. This will be done by considering a slight generalization of the processes in Borges et al. (2012). The main case concerns a class of space-time fractional Poisson processes and, when the correlation parameter is equal to 0, the univariate distributions coincide with those of the space-time fractional Poisson process in Orsingher and Polito (2012). On the one hand, when we consider the time fractional Poisson process, the multivariate finite-dimensional distributions are different from those presented for the renewal process in Politi et al. (2011). We also consider a case concerning a class of fractional negative binomial processes.

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