Bounds and error bounds for queueing networks

Queueing networks are an important means to model and evaluate a variety of practical systems. Unfortunately, analytic results are often not available. Numerical computation may then have to be employed. Or, system modifications might be suggested to obtain simple bounds or computationally easy approximations. Formal analytic support for the accuracy or nature of such modifications or approximations then becomes of interest. To this end, the Markov reward approach is surveyed and illustrated as a technique to conclude a priori error bounds as well as to formally prove bounds when comparing two related systems. More precisely, the technique can be applied to: perturbations; finite truncations; infinite approximations; system modifications, or system simplifications (bounds). A general comparison and error bound theorem are provided. The conditions and technical steps are illustrated in detail for a non-product form queueing network subject to breakdowns. This illustration highlights the technical difference with and extension of the stochastic comparison approach. In addition, some practical applications are given which illustrate the various types of applications.