SPECTRAL-ANALYSIS OF M G/1 AND G/M/1 TYPE MARKOV-CHAINS/

When analyzing the equilibrium behavior of MIG/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid tec hnical problems that arise in applying results from complex analysis a nd linear algebra. It is shown that such restrictive assumptions are u nnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or sec ond moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounde d solutions of the system of homogeneous vector-valued Wiener-Hopf equ ations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are s hown to correspond to a spectral basis of the shift operator on this s ubspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of propertie s of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary i s identified and shown to be associated with certain eigenvalues and v ectors of one of these generating functions. An equilibrium analysis o f the class of G/M/1 type Markov chains by similar methods is also pre sented.