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We consider an irreducible positive-recurrent discrete-time Markov process on the state space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>=</mo><msubsup><mo>ℤ</mo><mo>+</mo><mi>M</mi></msubsup><mo>×</mo><mi>J</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>ℤ</mo><mo>+</mo></msub></mrow></semantics></math></inline-formula> is the set of non-negative integers and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></semantics></math></inline-formula>. The number of states in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>J</mi></semantics></math></inline-formula> may be either finite or infinite. We assume that the process is a homogeneous quasi-birth-and-death process (QBD). It means that the one-step transition probability between non-boundary states <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> may depend on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle></mrow></semantics></math></inline-formula> but not on the specific values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle></semantics></math></inline-formula>. It is shown that the stationary probability vector of the process is expressed through square matrices of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>, which are the minimal non-negative solutions to nonlinear matrix equations.