Computable Bounds for Conditional Steady-State Probabilities in Large Markov Chains and Queueing Models

A method is presented to compute, for the steady-state conditional probabilities of a given Markov subchain, the best lower and upper bounds derivable from the submatrix of transition probabilities between the states of that subchain only. The bounds can be improved when additional information, even fragmentary, on the entire chain is available. When the submatrix has a special structure, analytical expressions of the bounds can be obtained. The method is shown to be useful and economical to bound performance measures in large nonproduct-form queueing network models of computer communication systems.