A duality theorem for the matrix paradigms in queueing theory

V. Ramaswami

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Advanced Queuing Theory Analysis Probability and Statistical Research Probability and Risk Models

Abstract

For Markov renewal processes of GI/M/1 type, we define a double transform R(z,s), a generalization of Neuts’ rate matrix R, and establish an interesting duality theorem. That result expresses R(z,s) explicitly in terms of the double transform G(z,s) of Neuts for a suitably defined dual process of M/G/1 type. For the GI/PH/1 queue, the G(z,s) matrix arising in the duality theorem is identified as the matrix of the dual PH/G/1 queue with a time reversed representation of the PH-distribution. The duality theorem unifies the results on the two paradigms and enables one to obtain many results concerning one from those of the other by simple arguments.

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References

1963Introduction to the Theory of Queues.1980The N/G/1 queue and its detailed analysis 1978Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector 1989Markov processes whose steady state distribution is matrix-exponential with an application to theGI/PH/1 queue 1976Moment formulas for the Markov renewal branching process 1985Algorithms for the multi-server queue with phase type service

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2003Fluid Flow Models and Queues—A Connection by Stochastic Coupling 1996Some properties of the rate perators in level dependent uasi-birth-and-death processes with countable number of phases 1990Probabilistic interpretations of some duality results for the matrix paradigms in queueing theory 2003Phase-type distributions and representations: Some results and open problems for system theory 1999The quasi-birth-death type markov chain with a tree structure 1990From the matrix-geometric to the matrix-exponential 2006Sojourn time distributions in the queue defined by a general QBD process

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