LINEAR INDEPENDENCE OF ROOT EQUATIONS FOR M G/1 TYPE MARKOV-CHAINS/

There is a classical technique for determining the equilibrium probabi lities of M/G/1 type Markov chains. After transforming the equilibrium balance equations of the chain, one obtains an equivalent system of e quations in analytic functions to be solved. This method requires find ing all singularities of a given matrix function in the unit disk and then using them to obtain a set of linear equations in the finite numb er of unknown boundary probabilities. The remaining probabilities and other measures of interest are then computed from the boundary probabi lities. Under certain technical assumptions, the linear independence o f the resulting equations is established by a direct argument involvin g only elementary results from matrix theory and complex analysis. Sim ple conditions for the ergodicity and nonergodicity of the chain are a lso given.