On two dimensional Markov processes with branching property

Several queueing processes may be modeled as random walks on a multidimensional grid. In this paper the equilibrium distribution for the case of a two-dimensional grid is considered. In previous research it has been shown that for some two-dimensional random walks the equilibrium distribution has the form of an infinite series of products of powers which can be constructed with a compensation procedure. The object of the present paper is to investigate under which conditions such an elegant solution exists and may be found with a compensation approach. The conditions can be easily formulated in terms of the random behaviour in the inner area and the drift on the boundaries.

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Extended continued fractions, recurrence relations and two-dimensional Markov processes

Advances in Applied Probability ◽

10.2307/1427164 ◽

1989 ◽

Vol 21 (2) ◽

pp. 357-375 ◽

Cited By ~ 21

Author(s):

C. E. M. Pearce

Keyword(s):

Markov Processes ◽

Continued Fractions ◽

Recurrence Relations ◽

Equilibrium Distribution ◽

Linear Recurrence ◽

Natural State ◽

Two Dimensional ◽

State Spaces ◽

General Order ◽

Dimension 2

Connections between Markov processes and continued fractions have long been known (see, for example, Good [8]). However the usefulness of extended continued fractions in such a context appears not to have been explored. In this paper a convergence theorem is established for a class of extended continued fractions and used to provide well-behaved solutions for some general order linear recurrence relations such as arise in connection with the equilibrium distribution of state for some Markov processes whose natural state spaces are of dimension 2. Specific application is made to a multiserver version of a queueing problem studied by Neuts and Ramalhoto [13] and to a model proposed by Cohen [5] for repeated call attempts in teletraffic.

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Extended continued fractions, recurrence relations and two-dimensional Markov processes

Advances in Applied Probability ◽

10.1017/s0001867800018589 ◽

1989 ◽

Vol 21 (02) ◽

pp. 357-375 ◽

Cited By ~ 5

Author(s):

C. E. M. Pearce

Keyword(s):

Markov Processes ◽

Continued Fractions ◽

Recurrence Relations ◽

Equilibrium Distribution ◽

Linear Recurrence ◽

Natural State ◽

Two Dimensional ◽

State Spaces ◽

General Order ◽

Dimension 2

Connections between Markov processes and continued fractions have long been known (see, for example, Good [8]). However the usefulness of extended continued fractions in such a context appears not to have been explored. In this paper a convergence theorem is established for a class of extended continued fractions and used to provide well-behaved solutions for some general order linear recurrence relations such as arise in connection with the equilibrium distribution of state for some Markov processes whose natural state spaces are of dimension 2. Specific application is made to a multiserver version of a queueing problem studied by Neuts and Ramalhoto [13] and to a model proposed by Cohen [5] for repeated call attempts in teletraffic.

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Learning the dynamics of two-dimensional stochastic Markov processes

Open Systems & Information Dynamics ◽

10.1007/bf02228842 ◽

1992 ◽

Vol 1 (3) ◽

pp. 311-326 ◽

Cited By ~ 5

Author(s):

Lisa Borland ◽

Hermann Haken

Keyword(s):

Markov Processes ◽

Two Dimensional

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On multitype processes based on progeny length of particles of a supercritical Galton-Watson process

Journal of Applied Probability ◽

10.2307/3214055 ◽

1987 ◽

Vol 24 (1) ◽

pp. 14-24 ◽

Cited By ~ 3

Author(s):

V. G. Gadag ◽

M. B. Rajarshi

Keyword(s):

Weak Convergence ◽

Branching Process ◽

Two Dimensional ◽

One Dimensional ◽

Convergence Results ◽

Finite Line ◽

Probability Spaces ◽

Watson Process ◽

Branching Property

Based on infinite and finite line of descent of particles of a one-dimensional supercritical Galton-Watson branching process (GWBP), we construct an associated bivariate process. We show that this bivariate process is a two-dimensional, supercritical GWBP. We also show that this process retains its branching property on appropriate probability spaces, when conditioned on set of non-extinction and set of extinction. Some asymptotic and weak convergence results for this process have been established. A generalisation of these results to a multitype p-dimensional GWBP has also been carried out.

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On multitype processes based on progeny length of particles of a supercritical Galton-Watson process

Journal of Applied Probability ◽

10.1017/s0021900200030576 ◽

1987 ◽

Vol 24 (01) ◽

pp. 14-24 ◽

Cited By ~ 2

Author(s):

V. G. Gadag ◽

M. B. Rajarshi

Keyword(s):

Weak Convergence ◽

Branching Process ◽

Two Dimensional ◽

One Dimensional ◽

Convergence Results ◽

Finite Line ◽

Probability Spaces ◽

Watson Process ◽

Branching Property

Based on infinite and finite line of descent of particles of a one-dimensional supercritical Galton-Watson branching process (GWBP), we construct an associated bivariate process. We show that this bivariate process is a two-dimensional, supercritical GWBP. We also show that this process retains its branching property on appropriate probability spaces, when conditioned on set of non-extinction and set of extinction. Some asymptotic and weak convergence results for this process have been established. A generalisation of these results to a multitype p-dimensional GWBP has also been carried out.

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