Multi-Dimensional Markov Chains of M/G/1 Type
We consider an irreducible discrete-time Markov process with states represented as (<b><i>k</i></b><i>, i</i>) where <b><i>k</i></b> is an <i>M</i>-dimensional vector with non-negative integer entries, and <i>i</i> i...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/2/209 |
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| Summary: | We consider an irreducible discrete-time Markov process with states represented as (<b><i>k</i></b><i>, i</i>) where <b><i>k</i></b> is an <i>M</i>-dimensional vector with non-negative integer entries, and <i>i</i> indicates the state (phase) of the external environment. The number <i>n</i> of phases may be either finite or infinite. One-step transitions of the process from a state (<b><i>k</i></b><i>, i</i>) are limited to states (<b><i>n</i></b>, <i>j</i>) such that <b>n</b> ≥ <b>k</b><i>−</i><b>1</b>, where <b>1</b> represents the vector of all 1s. We assume that for a vector <b>k</b> ≥ <b>1</b>, the one-step transition probability from a state (<b><i>k</i></b><i>, i</i>) to a state (<b><i>n</i></b>, <i>j</i>) may depend on <i>i, j</i>, and <b><i>n</i></b> − <i><b>k</b></i>, but not on the specific values of <b><i>k</i></b> and <b><i>n</i></b>. This process can be classified as a Markov chain of M/G/1 type, where the minimum entry of the vector <b><i>n</i></b> defines the level of a state (<b><i>n</i></b>, <i>j</i>). It is shown that the first passage distribution matrix of such a process, also known as the matrix <b>G</b>, can be expressed through a family of nonnegative square matrices of order <i>n</i>, which is a solution to a system of nonlinear matrix equations. |
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| ISSN: | 2227-7390 |