Asymptotics for the stationary distribution in a discrete‐time two‐dimensional quasi‐birth‐and‐death process

Asymptotics for the stationary distribution in a discrete‐time two‐dimensional quasi‐birth‐and‐death process

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Article ID: iaor20132897
Volume: 74
Issue: 2
Start Page Number: 109
End Page Number: 149
Publication Date: Jun 2013
Journal: Queueing Systems
Authors:
Keywords: markov processes
Abstract:

We consider a discrete‐time two‐dimensional process { ( L n ( 1 ) , L n ( 2 ) ) } equ1 on + 2 equ2 with a background process {J n } on a finite set, where individual processes { L n ( 1 ) } equ3 and { L n ( 2 ) } equ4 are both skip free. We assume that the joint process { Y n } = { ( L n ( 1 ) , L n ( 2 ) , J n ) } equ5 is Markovian and that the transition probabilities of the two‐dimensional process { ( L n ( 1 ) , L n ( 2 ) ) } equ6 are modulated depending on the state of the background process {J n }. This modulation is space homogeneous, but the transition probabilities in the inside of + 2 equ7 and those around the boundary faces may be different. We call this process a discrete‐time two‐dimensional quasi‐birth‐and‐death (2D‐QBD) process, and obtain the decay rates of the stationary distribution in the coordinate directions. We also distinguish the case where the stationary distribution asymptotically decays in the exact geometric form, in the coordinate directions.

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