Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process

Toshihisa Ozawa () and Masahiro Kobayashi ()
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Toshihisa Ozawa: Komazawa University
Masahiro Kobayashi: Tokai University

Queueing Systems: Theory and Applications, 2018, vol. 90, issue 3, No 6, 403 pages

Abstract: Abstract We consider a discrete-time two-dimensional process $$\{(X_{1,n},X_{2,n})\}$$ { ( X 1 , n , X 2 , n ) } on $$\mathbb {Z}_+^2$$ Z + 2 with a supplemental process $$\{J_n\}$$ { J n } on a finite set, where the individual processes $$\{X_{1,n}\}$$ { X 1 , n } and $$\{X_{2,n}\}$$ { X 2 , n } are both skip-free. We assume that the joint process $$\{\varvec{Y}_n\}=\{(X_{1,n},X_{2,n},J_n)\}$$ { Y n } = { ( X 1 , n , X 2 , n , J n ) } is Markovian and that the transition probabilities of the two-dimensional process $$\{(X_{1,n},X_{2,n})\}$$ { ( X 1 , n , X 2 , n ) } are modulated depending on the state of the supplemental process $$\{J_n\}$$ { J n } . This modulation is space homogeneous except for the boundaries of $$\mathbb {Z}_+^2$$ Z + 2 . We call this process a discrete-time two-dimensional quasi-birth-and-death process. Under several conditions, we obtain the exact asymptotic formulae of the stationary distribution in the coordinate directions.

Keywords: Quasi-birth-and-death process; Stationary distribution; Asymptotic property; Matrix analytic method; Two-dimensional reflecting random walk; 60J10; 60K25 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (6)

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DOI: 10.1007/s11134-018-9586-x

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