On Cox processes and gamma renewal processes

1988 ◽  
Vol 25 (2) ◽  
pp. 423-427 ◽  
Author(s):  
Nikos Yannaros
Keyword(s):  
Gamma Distribution ◽  
Renewal Process ◽  
Shape Parameter ◽  
Sufficient Conditions ◽  
Short Description ◽  
Necessary And Sufficient Conditions ◽  
Renewal Processes ◽  
Cox Process ◽  
Cox Processes ◽  
Necessary And Sufficient

It is shown that the gamma distribution with shape parameter α can be obtained through a p-thinning for every 0 < p < 1, when 0 < α ≦ 1. In the case α > 1, the gamma distribution cannot be obtained through thinning. The class of renewal processes with gamma-distributed times between events is considered. It is shown that an ordinary gamma renewal process is a Cox process if and only if 0 < α ≦ 1. Necessary and sufficient conditions for delayed gamma renewal processes to be Cox are also given. Finally, a short description of the gamma renewal process as a Cox process is given.

On Cox processes and gamma renewal processes

1988 ◽  
Vol 25 (02) ◽  
pp. 423-427 ◽  
Author(s):  
Nikos Yannaros
Keyword(s):  
Gamma Distribution ◽  
Renewal Process ◽  
Shape Parameter ◽  
Sufficient Conditions ◽  
Short Description ◽  
Necessary And Sufficient Conditions ◽  
Renewal Processes ◽  
Cox Process ◽  
Cox Processes ◽  
Necessary And Sufficient

It is shown that the gamma distribution with shape parameter α can be obtained through a p-thinning for every 0 &lt; p &lt; 1, when 0 &lt; α ≦ 1. In the case α &gt; 1, the gamma distribution cannot be obtained through thinning. The class of renewal processes with gamma-distributed times between events is considered. It is shown that an ordinary gamma renewal process is a Cox process if and only if 0 &lt; α ≦ 1. Necessary and sufficient conditions for delayed gamma renewal processes to be Cox are also given. Finally, a short description of the gamma renewal process as a Cox process is given.

Characterizing pure loss GI/G/1 queues with renewal output

1974 ◽  
Vol 75 (1) ◽  
pp. 103-107 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Poisson Process ◽  
Service Time ◽  
Renewal Process ◽  
Queueing System ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Arrival Process ◽  
Renewal Function ◽  
Service Distribution ◽  
Necessary And Sufficient

SummaryIn a pure loss GI/G/1 queueing system, necessary and sufficient conditions are given for the output to be a renewal process. These conditions involve dependence between the service distribution and the renewal function of the arrival process: for example, if pr {service time < ξ} = 0 for some ξ > 0, then it is sufficient for the renewal function to be that of a quasi-Poisson process with index ξ.

The inverses of thinned renewal processes

1988 ◽  
Vol 25 (4) ◽  
pp. 822-828 ◽  
Author(s):  
Nikos Yannaros
Keyword(s):  
Gamma Distribution ◽  
Renewal Process ◽  
Shape Parameter ◽  
Renewal Processes ◽  
The Given

The class of inverses of a p-thinned renewal process is considered. It is shown that this class consists of renewal processes. It consists of Cox and renewal processes if and only if the given thinned process is Cox and renewal. In the non-Cox case, there exists a unique top renewal process, which by thinning generates all the possible inverses. Conditions for a renewal process to be a top process are given. Finally, a gamma renewal process is shown to be a top process when α > 1, where α is the shape parameter of the gamma distribution.

A branching process with disasters

1975 ◽  
Vol 12 (1) ◽  
pp. 47-59 ◽  
Author(s):  
Norman Kaplan ◽  
Aidan Sudbury ◽  
Trygve S. Nilsen
Keyword(s):  
Asymptotic Behavior ◽  
Renewal Process ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Population Process ◽  
Age Dependent ◽  
The Mean ◽  
Necessary And Sufficient

A population process is considered where particles reproduce according to an age-dependent branching process, and are subjected to disasters which occur at the epochs of an independent renewal process. Each particle alive at the time of a disaster, survives it with probability p and the survival of any particle is assumed independent of the survival of any other particle. The asymptotic behavior of the mean of the process is determined and as a consequence, necessary and sufficient conditions are given for extinction.

A branching process with disasters

1975 ◽  
Vol 12 (01) ◽  
pp. 47-59 ◽  
Author(s):  
Norman Kaplan ◽  
Aidan Sudbury ◽  
Trygve S. Nilsen
Keyword(s):  
Asymptotic Behavior ◽  
Renewal Process ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Population Process ◽  
Age Dependent ◽  
The Mean ◽  
Necessary And Sufficient

A population process is considered where particles reproduce according to an age-dependent branching process, and are subjected to disasters which occur at the epochs of an independent renewal process. Each particle alive at the time of a disaster, survives it with probability p and the survival of any particle is assumed independent of the survival of any other particle. The asymptotic behavior of the mean of the process is determined and as a consequence, necessary and sufficient conditions are given for extinction.

The inverses of thinned renewal processes

1988 ◽  
Vol 25 (04) ◽  
pp. 822-828 ◽  
Author(s):  
Nikos Yannaros
Keyword(s):  
Gamma Distribution ◽  
Renewal Process ◽  
Shape Parameter ◽  
Renewal Processes ◽  
The Given

The class of inverses of a p-thinned renewal process is considered. It is shown that this class consists of renewal processes. It consists of Cox and renewal processes if and only if the given thinned process is Cox and renewal. In the non-Cox case, there exists a unique top renewal process, which by thinning generates all the possible inverses. Conditions for a renewal process to be a top process are given. Finally, a gamma renewal process is shown to be a top process when α &gt; 1, where α is the shape parameter of the gamma distribution.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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