Advances in Applied Probability

On the dual relationship between Markov chains of GI/M/1 and M/G/1 type

P. G. Taylor and B. Van Houdt

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In 1990, Ramaswami proved that, given a Markov renewal process of M/G/1 type, it is possible to construct a Markov renewal process of GI/M/1 type such that the matrix transforms G(z, s) for the M/G/1-type process and R(z, s) for the GI/M/1-type process satisfy a duality relationship. In his 1996 PhD thesis, Bright used time reversal arguments to show that it is possible to define a different dual for positive-recurrent and transient processes of M/G/1 type and GI/M/1 type. Here we compare the properties of the Ramaswami and Bright dual processes and show that the Bright dual has desirable properties that can be exploited in the design of algorithms for the analysis of Markov chains of GI/M/1 type and M/G/1 type.

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Adv. in Appl. Probab. Volume 42, Number 1 (2010), 210-225.

First available in Project Euclid: 26 March 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 90B22: Queues and service [See also 60K25, 68M20]

Matrix-analytic models quasi-birth-and-death process processes of GI/M/1 type processes of M/G/1 type


Taylor, P. G.; Van Houdt, B. On the dual relationship between Markov chains of GI/M/1 and M/G/1 type. Adv. in Appl. Probab. 42 (2010), no. 1, 210--225. doi:10.1239/aap/1269611150.

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