Advances in Applied Probability

Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators

Allan J. Motyer and Peter G. Taylor

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We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the `level' process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples.

Article information

Adv. in Appl. Probab. Volume 38, Number 2 (2006), 522-544.

First available: 26 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) [See also 90Bxx] 90B22: Queues and service [See also 60K25, 68M20]

Decay rate QBD process countable phase tridiagonal block generator stationary distribution two-node Jackson network


Motyer, Allan J.; Taylor, Peter G. Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators. Advances in Applied Probability 38 (2006), no. 2, 522--544. doi:10.1239/aap/1151337083.

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