On branching models with alarm triggerings

Two main methods are used to solve continuous-time quasi birth-and-death processes: matrix geometric (MG) and probability generating functions (PGFs). MG requires a numerical solution (via successive substitutions) of a matrix quadratic equation A0 + RA1 + R2A2 = 0. PGFs involve a row vector $\vec{G}(z)$ of unknown generating functions satisfying $H(z)\vec{G}{(z)^\textrm{T}} = \vec{b}{(z)^\textrm{T}},$ where the row vector $\vec{b}(z)$ contains unknown “boundary” probabilities calculated as functions of roots of the matrix H(z). We show that: (a) H(z) and $\vec{b}(z)$ can be explicitly expressed in terms of the triple A0, A1, and A2; (b) when each matrix of the triple is lower (or upper) triangular, then (i) R can be explicitly expressed in terms of roots of $\det [H(z)]$ ; and (ii) the stability condition is readily extracted.

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Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

Fractal and Fractional ◽

10.3390/fractalfract4040051 ◽

2020 ◽

Vol 4 (4) ◽

pp. 51 ◽

Cited By ~ 1

Author(s):

Thomas M. Michelitsch ◽

Federico Polito ◽

Alejandro P. Riascos

Keyword(s):

Cauchy Problem ◽

Random Walks ◽

Poisson Process ◽

Continuous Time ◽

Laplacian Matrix ◽

Space Time ◽

The State ◽

Diffusion Limit ◽

Matrix Functions ◽

Continuous Time Random Walks

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

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Maximizing the Expected Duration of Owning a Relatively Best Object in a Poisson Process with Rankable Observations

Journal of Applied Probability ◽

10.1017/s0021900200005544 ◽

2009 ◽

Vol 46 (02) ◽

pp. 402-414

Author(s):

Aiko Kurushima ◽

Katsunori Ano

Keyword(s):

Poisson Process ◽

Stopping Time ◽

Fixed Time ◽

Time Interval ◽

Prior Density ◽

Deterministic Equation ◽

Fixed Time Interval ◽

Unknown Number ◽

Random Intensity ◽

Time Selection

Suppose that an unknown number of objects arrive sequentially according to a Poisson process with random intensity λ on some fixed time interval [0,T]. We assume a gamma prior density G λ(r, 1/a) for λ. Furthermore, we suppose that all arriving objects can be ranked uniquely among all preceding arrivals. Exactly one object can be selected. Our aim is to find a stopping time (selection time) which maximizes the time during which the selected object will stay relatively best. Our main result is the following. It is optimal to select the ith object that is relatively best and arrives at some time s i (r) onwards. The value of s i (r) can be obtained for each r and i as the unique root of a deterministic equation.

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Maximizing the Expected Duration of Owning a Relatively Best Object in a Poisson Process with Rankable Observations

Journal of Applied Probability ◽

10.1239/jap/1245676096 ◽

2009 ◽

Vol 46 (2) ◽

pp. 402-414 ◽

Cited By ~ 3

Author(s):

Aiko Kurushima ◽

Katsunori Ano

Keyword(s):

Poisson Process ◽

Stopping Time ◽

Fixed Time ◽

Time Interval ◽

Prior Density ◽

Deterministic Equation ◽

Fixed Time Interval ◽

Unknown Number ◽

Random Intensity ◽

Time Selection

Suppose that an unknown number of objects arrive sequentially according to a Poisson process with random intensity λ on some fixed time interval [0,T]. We assume a gamma prior density Gλ(r, 1/a) for λ. Furthermore, we suppose that all arriving objects can be ranked uniquely among all preceding arrivals. Exactly one object can be selected. Our aim is to find a stopping time (selection time) which maximizes the time during which the selected object will stay relatively best. Our main result is the following. It is optimal to select the ith object that is relatively best and arrives at some time si(r) onwards. The value of si(r) can be obtained for each r and i as the unique root of a deterministic equation.

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On a class of continuous-time Markov processes

Journal of Applied Probability ◽

10.1017/s0021900200108046 ◽

1985 ◽

Vol 22 (04) ◽

pp. 804-815 ◽

Cited By ~ 2

Author(s):

J. Gani ◽

Pyke Tin

Keyword(s):

Markov Process ◽

Markov Processes ◽

Generating Functions ◽

Continuous Time ◽

Time Dependent ◽

Stationary Case ◽

Stationary Distributions ◽

General Process ◽

Probability Generating Functions ◽

Queueing Processes

This paper considers a certain class of continuous-time Markov processes, whose time-dependent and stationary distributions are studied. In the stationary case, the analogy with Whittle's relaxed Markov process is pointed out. The derivation of the probability generating functions of the general process provides useful results for the analysis of some population and queueing processes.

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THE CONTINUOUS-TIME TIME BERNULLI PROCESS

Science Evolution ◽

10.21603/2500-1418-2017-2-2-79-87 ◽

2017 ◽

pp. 79-87

Author(s):

Valery Pavsky ◽

Kirill Pavsky ◽

Svetlana Ivanova ◽

...

Keyword(s):

Poisson Process ◽

Poisson Distribution ◽

Continuous Time ◽

Queuing Theory ◽

Probability Space ◽

Fixed Time ◽

Theory Model ◽

Bernoulli Process ◽

Queuing Model ◽

First Case

A random Bernoulli process with continuous time and a finite number of states (random events) is proposed. The process is obtained by two mutually complementary methods - directly from the Poisson process with an intensity parameter that depends on time and methods of queuing theory, from a queuing system with two parameters. In the first case, the process was formalized on a probability space with measure, as a measurable function of time. The intensity of the Poisson process was considered as a measure. The Bernoulli process for each fixed time was obtained as a conditional distribution from a suitable Poisson distribution. The parameter of the Poisson distribution was determined from the differential equation, in the formulation of which the approximation of the Bernoulli formula by the Poisson formula was essentially used. In the second method, standard methods of queuing theory were used. A two- parameter queuing model was formulated in which for all customer flows the time between occurrence of neighboring customers was a random value satisfying the exponential law. The model was formalized by a system of differential equations, whose analytical solution represented the continuous-time Bernoulli process. In finding solutions, the method of generating functions was used. It is of interest to derive the Bernoulli process both from the probability space constructed for the Poisson process and from the queuing theory model. The authors believe that the proposed process can be generalized to a wider class of functions than that used in the work, down to measurable ones. The possibilities for the practical application of the continuous-time Bernoulli process will undoubtedly be expanded, since its discrete analog is well known in many fields of science and technology.

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On a class of continuous-time Markov processes

Journal of Applied Probability ◽

10.2307/3213948 ◽

1985 ◽

Vol 22 (4) ◽

pp. 804-815 ◽

Cited By ~ 3

Author(s):

J. Gani ◽

Pyke Tin

Keyword(s):

Markov Process ◽

Markov Processes ◽

Generating Functions ◽

Continuous Time ◽

Time Dependent ◽

Stationary Case ◽

Stationary Distributions ◽

General Process ◽

Probability Generating Functions ◽

Queueing Processes

This paper considers a certain class of continuous-time Markov processes, whose time-dependent and stationary distributions are studied. In the stationary case, the analogy with Whittle's relaxed Markov process is pointed out. The derivation of the probability generating functions of the general process provides useful results for the analysis of some population and queueing processes.

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Generalized fractional Poisson process and related stochastic dynamics

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0034 ◽

2020 ◽

Vol 23 (3) ◽

pp. 656-693 ◽

Cited By ~ 2

Author(s):

Thomas M. Michelitsch ◽

Alejandro P. Riascos

Keyword(s):

Poisson Process ◽

Continuous Time ◽

Counting Process ◽

Stochastic Dynamics ◽

Waiting Times ◽

Fractional Diffusion ◽

Diffusion Limit ◽

Fractional Operators ◽

Fractional Poisson Process ◽

Erlang Process

AbstractWe survey the ‘generalized fractional Poisson process’ (GFPP). The GFPP is a renewal process generalizing Laskin’s fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the ‘well-scaled’ diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.

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Design of regular positive and stable descriptor systems via state-feedbacks for descriptor continuous-time linear systems with singular pencils

Archives of Control Sciences ◽

10.2478/acsc-2014-0017 ◽

2014 ◽

Vol 24 (3) ◽

pp. 289-297

Author(s):

Tadeusz Kaczorek

Keyword(s):

Linear Systems ◽

Continuous Time ◽

Special Form ◽

The State ◽

New Method ◽

Descriptor Systems ◽

Descriptor System ◽

Asymptotically Stable ◽

Numerical Example ◽

Time Linear

Abstract A new method is proposed of design of regular positive and asymptotically stable descriptor systems by the use of state-feedbacks for descriptor continuous-time linear systems with singular pencils. The method is based on the reduction of the descriptor system by elementary row and column operations to special form. A procedure for the design of the state-feedbacks gain matrix is presented and illustrated by a numerical example

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On mutations in the branching model for multitype populations

Advances in Applied Probability ◽

10.1017/apr.2018.24 ◽

2018 ◽

Vol 50 (2) ◽

pp. 543-564 ◽

Cited By ~ 1

Author(s):

Loïc Chaumont ◽

Thi Ngoc Anh Nguyen

Keyword(s):

Infectious Diseases ◽

Asymptotic Behaviour ◽

Continuous Time ◽

Mutation Rates ◽

Genetic Evolution ◽

Connected Component ◽

Branching Model ◽

Limiting Behaviour ◽

Number Of Individuals

AbstractThe forest of mutations associated to a multitype branching forest is obtained by merging together all vertices in each of its clusters and by preserving connections between them. (Here, by cluster, we mean a maximal connected component of the forest in which all vertices have the same type.) We first show that the forest of mutations of any multitype branching forest is itself a branching forest. Then we give its progeny distribution and we describe some of its crucial properties in terms of the initial progeny distribution. We also obtain the limiting behaviour of the number of mutations both when the total number of individuals tends to ∞ and when the number of roots tends to ∞. The continuous-time case is then investigated by considering multitype branching forests with edge lengths. When mutations are nonreversible, we give a representation of their emergence times which allows us to describe the asymptotic behaviour of the latter, under certain conditions on the mutation rates. These results have potential relevance for emergence of mutations in population cells, particularly for genetic evolution of cancer or development of infectious diseases.

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