The use of non-linear programming in matrix analytic methods

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Abstract
In this paper, we present new methods for finding the R and G matrices which play a crucial role in determining the steady-state distribution of Markov chains of the GI/M/1 and M/G/l type respectively. The methods involve finding solutions to some non-linear programming problems. We obtain the Karush-Kuhn-Tucker (KKT) conditions for two of the non-linear programming problems and, for the M/G/l case, show that there is a simple relationship between the dual variables and the G matrix. We present two algorithms for solving the non-linear programming problems. We have also carried out numerical investigations for the M/G/l problem and found that one of the algorithms often performs much better than what we call the “standard method” of solution

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