An eclectic survey of bounding methods for Markov chain models

Markov models are often used for performance modeling. However most models do not have closed form solutions, and numerical solutions are often not feasible due to the large (or even infinite) state space of models of practical interest. One can sometimes take advantage of special structures such as nearly completely decomposable models or models with matrix analytic solutions. This paper presents a brief survey of some work over the past few years on finding bounds on performance measures for models that are otherwise intractable. The bounds are found by modifying the original model such that: (a) the modified model is efficiently solvable and (b) the modified model is proven to provide an upper (lower) bound on the value of the performance measure in the original model. We have successfully applied this methodology in a number of studies which are briefly surveyed here. This is not meant to be a broad, inclusive survey but rather we concentrate on an approach that the authors have been exploring